2 The paraboloid: a quantum mechanical viewpoint

In the particular case when S is the paraboloid, the phase-space representation (1.7) has a well-known quantum mechanical derivation going back to the original work of Wigner [Reference Wigner53]. As may be expected, this involves the classical Wigner transform, and as we shall see in this section, leads to some additional insights and simplifications in our arguments. Moreover, parametrised formulations of the Stein and Mizohata–Takeuchi inequalities (1.2) and (1.4) will emerge rather naturally from these classical considerations, permitting them some physical (or probabilistic) interpretations.

The Wigner transform is defined (see, e.g., [Reference Folland28]) for $g_1, g_2\in L^2(\mathbb {R}^d)$ by

(2.1) $$ \begin{align} W(g_1,g_2)(x,v)=\int_{\mathbb{R}^d}g_1\left(x+\frac{y}{2}\right)\overline{g_2\left(x-\frac{y}{2}\right)}e^{-2\pi iv\cdot y}\mathrm{d}y. \end{align} $$

For a solution $u:\mathbb {R}^d\times \mathbb {R}\rightarrow \mathbb {C}$ of the Schrödinger equation

$$ \begin{align*}2\pi i\frac{\partial u}{\partial t}=\Delta_{x} u\end{align*} $$

with initial data $u_0\in L^2(\mathbb {R}^d)$ , it is a classical observation dating back to Wigner [Reference Wigner53] that

(2.2) $$ \begin{align} f(x,v,t):=W(u(\cdot, t), u(\cdot, t))(x,v) \end{align} $$

satisfies the kinetic transport equation

$$ \begin{align*}\frac{\partial f}{\partial t}=2v\cdot\nabla_x f \end{align*} $$

from classical mechanics. Consequently

$$ \begin{align*}f(x,v,t)=f_0(x+2tv,v),\end{align*} $$

where $f_0=W(u_0,u_0):\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}$ is the Wigner distribution of the initial data $u_0$ . We note that the function f may be reconciled with the corresponding f in the forthcoming Sections 3 and 4 using the Fourier invariance property (see 1.94 of [Reference Folland28])

(2.3) $$ \begin{align} W(g_1,g_2)(x,v)=W(\widehat{g}_1,\widehat{g}_2)(-v,x). \end{align} $$

By the classical marginal property

(2.4) $$ \begin{align} \int_{\mathbb{R}^d}W(g,g)(x,v)\mathrm{d}v=|g(x)|^2 \end{align} $$

of the Wigner distribution we obtain the phase-space representation

(2.5) $$ \begin{align} |u(x,t)|^2=\int_{\mathbb{R}^d}f(x,v,t)\mathrm{d}v=\int_{\mathbb{R}^d}f_0(x+2tv,v)\mathrm{d}v=:\rho(f_0)(x,t). \end{align} $$

The operator $\rho $ , which is referred to as a velocity averaging operator in kinetic theory, is easily seen to be a certain (parametrised) adjoint space-time X-ray transform, indeed

$$ \begin{align*}\rho^*(g)(x,v)=\int_{\mathbb{R}}g(x-2tv,t)\mathrm{d}t, \end{align*} $$

which is of course an integral of the space-time function g along the line through the point $(x,0)$ with direction $(-2v,1)$ . We caution that the parameter v, being a velocity, describes the direction of this line. This differs from elsewhere in this paper where v is used as a translation (or position) parameter.

As we have indicated in the introduction, the above phase-space representation is particularly natural if one is interested in weighted $L^2$ norms of u, since by duality

(2.6) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t=\int_{\mathbb{R}^d\times\mathbb{R}^d}W(u_0,u_0)(x,v)\rho^*w(x,v)\mathrm{d}x\mathrm{d}v. \end{align} $$

We refer to [Reference Dendrinos, Mustata and Vitturi24] where this identity was recently derived directly. If the initial data $u_0$ is a Gaussian then $W(u_0,u_0)$ is also a (real) Gaussian, and being nonnegative it follows that

$$ \begin{align*} \begin{aligned} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t&\leq\int_{\mathbb{R}^d}\left(\int_{\mathbb{R}^d}W(u_0,u_0)(x,v)\mathrm{d}x\right)\sup_x\rho^*w(x,v)\mathrm{d}v\\&= \int_{\mathbb{R}^d}|\widehat{u}_0(v)|^2\sup_x\rho^*w(x,v)\mathrm{d}v, \end{aligned} \end{align*} $$

which in turn implies that

$$ \begin{align*}\int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\leq\sup_{\substack{x\in\mathbb{R}^d\\ v\in\mathrm{supp\,}(\widehat{u}_0)}}\rho^* w(x,v)\;\|u_0\|_2^2. \end{align*} $$

Here we have used the further marginal property

(2.7) $$ \begin{align} \int_{\mathbb{R}^d}W(g,g)(x,v)\mathrm{d}x=|\widehat{g}(v)|^2 \end{align} $$

of the Wigner distribution, followed by Plancherel’s theorem. It is therefore reasonably natural to ask whether

(2.8) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\lesssim \int_{\mathbb{R}^d}|\widehat{u}_0(v)|^2\sup_x\rho^*w(x,v)\mathrm{d}v, \end{align} $$

and thus

(2.9) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\lesssim\sup_{\substack{x\in\mathbb{R}^d\\ v\in\mathrm{ supp\,}(\widehat{u}_0)}}\rho^* w(x,v)\;\|u_0\|_2^2 \end{align} $$

might hold for general $u_0$ . As we clarify shortly in Remark 2.3, the inequalities (2.8) and (2.9) are parabolic forms of the Stein (1.2) and Mizohata–Takeuchi (1.4) inequalities and as such also fail (see Remark 1.2).

Remark 2.1. As indicated in Remark 1.2, the recent counterexamples in [Reference Cairo18] leave open the possibility that for $\widehat {u}_0$ supported in the unit ball (say),

(2.10) $$ \begin{align} \int_{|(x,t)|\leq R}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\leq C_\varepsilon R^\varepsilon \int_{\mathbb{R}^d}|\widehat{u}_0(v)|^2\sup_x\rho^*w(x,v)\mathrm{d}v \end{align} $$

and

(2.11) $$ \begin{align} \int_{|(x,t)|\leq R}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\leq C_\varepsilon R^\varepsilon \sup_{\substack{x\in\mathbb{R}^d\\ v\in\mathrm{supp\,}(\widehat{u}_0)}}\rho^* w(x,v)\;\|u_0\|_2^2 \end{align} $$

might hold for each $\varepsilon>0$ and all $R\gg 1$ ; in other words, (2.8) and (2.9) under the assumption that w is supported in the space-time ball $B(0,R)$ , accepting a factor of $R^\varepsilon $ in the implicit constant on the right-hand sides for each $\varepsilon>0$ . The requirement that $\widehat {u}_0$ is supported in some fixed compact set (the unit ball here) prevents scale-invariance considerations reducing (2.10) and (2.11) to (2.8) and (2.9) respectively. We remark that these inequalities are naturally referred to as Strichartz estimates, being bounds on space-time norms.

Remark 2.2 (A quasi-probabilistic interpretation).

In the phase-space formulation of quantum mechanics the Wigner distribution $W(u_0,u_0)$ is interpreted as a (quasi-) probability distribution on position-momentum space for a quantum particle, and so the inequalities (2.8) and (2.9) for any given weight w are the assertions that

(2.12) $$ \begin{align}\mathbb{E}_{x,v}(\rho^*w)\lesssim\mathbb{E}_{x,v}(\|\rho^*w\|_{L^\infty_x}) \end{align} $$

and

(2.13) $$ \begin{align}\mathbb{E}_{x,v}(\rho^*w)\lesssim\|\rho^*w\|_\infty \end{align} $$

respectively; we recall that these inequalities are known to fail for general w unless we make some additional localisations (see Remark 2.1). Here the expectation is taken with respect to the quasi-probability density $W(u_0,u_0)$ , where of course $\|u_0\|_2=1$ . Note that $\mathbb {E}_{x,v}(\|\rho ^*w\|_{L^\infty _x})=\mathbb {E}_{v}(\|\rho ^*w\|_{L^\infty _x})$ by the marginal property (2.7), where $\mathbb {E}_{v}$ is taken with respect to the probability density $|\widehat {u}_{0}(v)|^{2}$ . The forthcoming Theorems 2.4–2.8 may be interpreted similarly. Evidently the subtleties in (2.12), (2.13) and all of these inequalities arise from the fact that the Wigner distribution typically takes both positive and negative values.

Remark 2.3. Although (2.8) is false in general (see Remark 2.1), for any given weight w it may be seen as an instance of (1.2) where $d=n-1$ and

(2.14) $$ \begin{align} S=\mathbb{P}^d:=\{u=(u',u_{d+1})\in\mathbb{R}^d\times\mathbb{R}:u_{d+1}=|u'|^2\} \end{align} $$

is the paraboloid. This is a consequence of a certain change-of-measure invariance property enjoyed by the general inequality (1.2): specifically, if $\mathrm {d}\tilde {\sigma }(u)=a(u)\mathrm {d}\sigma (u)$ for some density a on S, then (1.5) quickly implies that

(2.15) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\tilde{\sigma}}(x)|^2w(x)\mathrm{d}x\leq C\int_S|g(u)|^2\sup_{v\in T_uS}a(u)Xw(N(u),v)\mathrm{d}\tilde{\sigma}(u). \end{align} $$

Next we define the (affine surface) measure $\mathrm {d}\tilde {\sigma }$ on $\mathbb {P}^d$ by

(2.16) $$ \begin{align} \int_S\Phi \mathrm{d}\tilde{\sigma}=\int_{\mathbb{R}^d}\Phi(u',|u'|^2)\mathrm{d}u', \end{align} $$

so that $a(u)=(1+4|u|^2)^{-1/2}$ . With these choices, a scalar change of variables reveals that

$$ \begin{align*}\sup_{v\in T_u S}a(u)Xw(N(u),v)=\sup_x\rho^*w(x,u'). \end{align*} $$

Finally, defining $g: S\rightarrow \mathbb {C}$ by $g(\cdot ,|\cdot |^2)=\widehat {u}_0$ , we have that $u(x,t)=\widehat {g\mathrm {d}\tilde {\sigma }}(x,t)$ , from which (2.8) follows. The change-of-measure invariance property (2.15) enjoyed by (1.2) is not inherited by the corresponding Mizohata–Takeuchi inequality (1.4), meaning that there is in principle a different Mizohata–Takeuchi inequality for each density a – namely

(2.17) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\tilde{\sigma}}(x)|^2w(x)\mathrm{d}x\leq C\sup_{(u,v)\in TS}a(u)Xw(N(u),v)\|g\|^2_{L^2(\mathrm{d}\tilde{\sigma})}, \end{align} $$

where again, the supremum is restricted to $u\in \mathrm {supp\,}(g)$ . It is straightforward to verify that (2.9) coincides with (2.17) with the above choice of density a on the paraboloid. Similar change-of-measure arguments relate the paraboloid-carried Wigner distribution referred to in (1.7) to the classical Wigner distribution (2.1), reconciling (2.5) with (1.7). We clarify this in Remark 4.7 in Section 4.

Perhaps the most obvious difficulty in going beyond Gaussian initial data is that $W(u_0,u_0)$ is everywhere nonnegative if and only if $u_0$ is a Gaussian (this is known as Hudson’s theorem, see [Reference Folland28] for a treatment of this and other fundamental properties of the Wigner transform), and the inequality $\|W(u_0,u_0)\|_1\lesssim \|u_0\|_2^2$ fails for general $u_0$ (see [Reference Lerner37]). Of course the $L^p$ estimates that do hold for the Wigner distribution (see [Reference Lieb38]) yield variants of (2.9) via Hölder’s inequality, such as

(2.18) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\lesssim\|\rho^* w\|_{L^2(\mathbb{R}^d\times [-1,1]^d)}\|u_0\|_2^2, \end{align} $$

as was observed in [Reference Dendrinos, Mustata and Vitturi24] whenever $\widehat {u}_0$ is supported in the cube $[-1,1]^d$ . Here we observe that further variants arise from certain Sobolev estimates on the Wigner transform. For example, we have the following:

Theorem 2.4. For $s>d/2$ ,

(2.19) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\leq\int_{\mathbb{R}^d}\widetilde{I}_{2s}(|\widehat{u}_0|^2,|\widehat{u}_0|^2)(v)^{1/2}\|\rho^* w(\cdot,v)\|_{H_x^s}\mathrm{d}v, \end{align} $$

where

$$ \begin{align*}\widetilde{I}_s(g_1,g_2)(v):=\int_{\mathbb{R}^d}\frac{g_1\left(v+\frac{\xi}{2}\right)g_2\left(v-\frac{\xi}{2}\right)}{(1+|\xi|^2)^{s/2}}\mathrm{d}\xi \end{align*} $$

and $H_x^s$ denotes the usual inhomogeneous $L^2$ Sobolev space in the variable x.

Theorem 2.5. For $s>d/2$ ,

(2.20) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\lesssim\sup_{v\in\frac{1}{2}(\mathrm{supp\,}(\widehat{u}_0)+\mathrm{ supp\,}(\widehat{u}_0))}\|\rho^* w(\cdot,v)\|_{H_x^s}\|u_0\|_2^2, \end{align} $$

where the implicit constant depends on at most d and s.

Remark 2.6. As our arguments quickly reveal, Theorems 2.4 and 2.5 require no positivity hypothesis on the weight w. This point aside, Theorems 2.4 and 2.5 may be viewed as substitutes for (2.8) and (2.9), which are false for general weights; see Remark 2.1. This is a consequence of the elementary Sobolev embedding $H^s(\mathbb {R}^d)\subset L^\infty (\mathbb {R}^d)$ , which holds whenever $s>d/2$ . It is natural to ask whether the stronger

(2.21) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\lesssim\sup_{v\in\mathrm{supp\,}(\widehat{u}_0)}\|\rho^* w(\cdot,v)\|_{H_x^s}\|u_0\|_2^2 \end{align} $$

holds, as suggested by (2.9) for arbitrary positive weights.

Proof of Theorem 2.4.

By (2.6) and an application of the duality of $H^s$ and $H^{-s}$ we have

$$ \begin{align*}\int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\leq\int_{\mathbb{R}^d}\|W(u_0,u_0)(\cdot, v)\|_{H_x^{-s}}\|\rho^* w(\cdot,v)\|_{H_x^s}\mathrm{d}v, \end{align*} $$

and so it remains to show that

(2.22) $$ \begin{align} \|W(u_0,u_0)(\cdot, v)\|_{H_x^{-s}}^2=\widetilde{I}_{2s}(|\widehat{u}_0|^2,|\widehat{u}_0|^2)(v). \end{align} $$

This follows from the classical Fourier invariance property (2.3), which implies

(2.23) $$ \begin{align} \mathcal{F}_x^{-1}W(g_1,g_2)(\xi, v)=\widehat{g}_1\left(-v+\frac{\xi}{2}\right)\overline{\widehat{g}_2\left(-v-\frac{\xi}{2}\right)}, \end{align} $$

where $\mathcal {F}_x$ denotes the Fourier transform in x. The identity (2.22) now follows by Plancherel’s theorem and the definition of the inhomogeneous Sobolev norm.

Proof of Theorem 2.5.

Observe first that $\widetilde {I}_{2s}(|\widehat {u}_0|^2,|\widehat {u}_0|^2)(v)=0$ whenever

$$ \begin{align*}v\not\in\frac{1}{2}(\mathrm{ supp\,}(\widehat{u}_0)+\mathrm{supp\,}(\widehat{u}_0)).\end{align*} $$

Hence, by Theorem 2.4, it suffices to show that

(2.24) $$ \begin{align} \|\widetilde{I}_s(g_1,g_2)\|_{L^{1/2}(\mathbb{R}^d)}\lesssim\|g_1\|_1\|g_2\|_1 \end{align} $$

whenever $s>d$ . The operator $\widetilde {I}_s$ is a variant (with singularity only at infinity) of the bilinear fractional integral operator

(2.25) $$ \begin{align} \displaystyle I_s(g_1,g_2)(v):=\int_{\mathbb{R}^d}\frac{g_1\left(v+\frac{\xi}{2}\right)g_2\left(v-\frac{\xi}{2}\right)}{|\xi|^{s}}\mathrm{d}\xi \end{align} $$

treated by Kenig and Stein in [Reference Kenig and Stein33] and Grafakos and Kalton in [Reference Grafakos and Kalton31] (see also [Reference Grafakos30] for estimates above $L^1$ ), and the bound (2.24) follows a brief inspection of their arguments. For similar arguments, see also Section 7.

Theorems 2.4 and 2.5 cease to be natural if the initial datum $u_0$ has compact Fourier support, as they involve inhomogeneous Sobolev spaces, which respond to high frequencies of $u_0$ only. The appropriate substitutes are the following, which align with our main Theorems 1.7 and 1.8:

Theorem 2.7 (Parabolic Sobolev–Stein).

For $s<d/2$ ,

(2.26) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\leq\int_{\mathbb{R}^d}I_{2s}(|\widehat{u}_0|^2,|\widehat{u}_0|^2)(v)^{1/2}\|\rho^* w(\cdot,v)\|_{\dot{H}_x^s}\mathrm{d}v, \end{align} $$

where $I_s(g_1,g_2)$ is given by (2.25) and $\dot {H}_x^s$ denotes the usual homogeneous $L^2$ Sobolev space in the variable x.

Theorem 2.8 (Parabolic Sobolev–Mizohata–Takeuchi).

For $s<d/2$ ,

(2.27) $$ \begin{align} \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\lesssim\sup_{v\in\frac{1}{2}(\mathrm{supp\,}(\widehat{u}_0)+\mathrm{ supp\,}(\widehat{u}_0))}\|\rho^* w(\cdot,v)\|_{\dot{H}_x^s}\|u_0\|_2^2 \end{align} $$

whenever $\mathrm {supp\,}(\widehat {u}_0)\subseteq B(0;1)$ . The implicit constant depends on at most d and s.

Remark 2.9. Theorems 2.7 and 2.8 also permit signed weights. Restricting to positive weights, Theorems 2.7 and 2.8 are also easily seen to be respectively weaker than (2.8) and (2.9) via a Sobolev embedding. Specifically, by the support hypothesis on $\widehat {u}_0$ we may find a spatial bump function $\Phi $ such that

$$ \begin{align*}\int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2w(x,t)\mathrm{d}x\mathrm{d}t\leq \int_{\mathbb{R}^d\times\mathbb{R}}|u(x,t)|^2\Phi*w(x,t)\mathrm{d}x\mathrm{d}t, \end{align*} $$

and so it suffices to observe that for any $v\in \mathbb {R}^d$ ,

$$ \begin{align*} \begin{aligned} \|\rho^*(\Phi*w)(\cdot,v)\|_{\infty}\lesssim \|\rho^* w(\cdot,v)\|_{\dot{H}_x^s} \end{aligned} \end{align*} $$

whenever $s<d/2$ . This follows by Plancherel’s identity and the Cauchy–Schwarz inequality.

The proofs of Theorems 2.7 and 2.8 are very similar to those of Theorems 2.4 and 2.5 above, the essential difference being the use of homogeneous rather than inhomogeneous Sobolev norms, and matters are reduced to an $L^1\times L^1\rightarrow L^{1/2}$ bound on the bilinear operator

$$ \begin{align*}T(g_1,g_2)(v):=\int_{B(0;1)}\frac{g_1\left(v+\frac{\xi}{2}\right)g_2\left(v-\frac{\xi}{2}\right)}{|\xi|^{s}}\mathrm{d}\xi. \end{align*} $$

This is a local form of the bilinear fractional integral operator $I_s$ defined in (2.25), and again the required bound follows a brief inspection of the arguments in [Reference Kenig and Stein33].

3 The sphere: an optical viewpoint

The extension operator for the sphere

$$ \begin{align*}\widehat{g\mathrm{d}\sigma}(x):=\int_{\mathbb{S}^{n-1}}e^{-2\pi ix\cdot\omega}g(\omega)\mathrm{d}\sigma(\omega)\end{align*} $$

is of central importance in optics, providing a description of a unit-wavelength (or monochromatic) optical wave field as a superposition of plane waves; note that $\widehat {g\mathrm {d}\sigma }$ solves the Helmholtz equation $ \Delta u +u=0$ on $\mathbb {R}^n$ . Of particular physical significance is $|\widehat {g\mathrm {d}\sigma }|^2$ , sometimes referred to as the local intensity of the field; see, for example, [Reference Alonso3]. The Stein and Mizohata–Takeuchi inequalities (1.2) and (1.4), when specialised to the sphere $S=\mathbb {S}^{n-1}$ , become statements about this intensity, namely

(3.1) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\leq C\int_{\mathbb{S}^{n-1}}|g(\omega)|^2\sup_{v\in\langle\omega\rangle^\perp}Xw(\omega,v)\mathrm{d}\sigma(\omega), \end{align} $$

and

(3.2) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\leq C\sup_{\omega\in\mathrm{ supp\,}(g)}\|Xw(\omega,\cdot)\|_{L^\infty(\langle\omega\rangle^\perp)}\|g\|_{L^2(\mathbb{S}^{n-1})}^2 \end{align} $$

respectively. These conjectural inequalities are well-known for radial weights, as discussed in the introduction, although we recall that for general weights they should carry a further localisation hypothesis following the counterexamples of Cairo [Reference Cairo18]; see Remark 1.2 for clarification. Both (3.1) and (3.2) capture the expectation that the intensity $|\widehat {g\mathrm {d}\sigma }|^2$ concentrates on rays (lines), and as such connect physical optics to geometric optics. A good illustration of this is found in the high-frequency limiting identity

(3.3) $$ \begin{align} \limsup_{R\rightarrow\infty}R^{n-1}\int_{\mathbb{R}^{n}}|\widehat{g\mathrm{d}\sigma}(Rx)|^2w(x)\mathrm{d}x=\frac{1}{(2\pi)^{n+1}}\int_{\mathbb{S}^{n-1}}|g(\xi)|^2\left(\int_{\mathbb{R}}w(t\xi)\mathrm{d}t\right)\mathrm{d}\sigma(\xi), \end{align} $$

established (for compactly supported w) by Agmon and Hörmander in [Reference Agmon and Hörmander1]; see [Reference Barceló, Ruiz and Vega5]. Accordingly (3.1) and (3.2) call for an optical (or spherical) analogue of the quantum-mechanical (or parabolic) phase-space perspective from Section 2. Fortunately such a perspective is well-known in modern optics (see [Reference Alonso3]) and involves the spherical Wigner transform that we define next. For $g_1,g_2\in L^2(\mathbb {S}^{n-1})$ let

(3.4) $$ \begin{align} W_{\mathbb{S}^{n-1}}(g_1,g_2)(\omega, v)=\int_{\mathbb{S}^{n-1}}g_1(\omega')\overline{g_2(R_\omega \omega')}e^{-2\pi iv\cdot(\omega'-R_\omega\omega')}J(\omega,\omega')\mathrm{d}\sigma(\omega'). \end{align} $$

Here $\omega \in \mathbb {S}^{n-1}$ , $v\in \langle \omega \rangle ^\perp $ , and for a point $\omega '\in \mathbb {S}^{n-1}$ , the point $R_\omega \omega '$ is defined to be the unique $\omega "\in \mathbb {S}^{n-1}$ for which $\omega $ is the geodesic midpoint of $\omega '$ and $\omega "$ ; that is,

(3.5) $$ \begin{align} R_\omega \omega'=2(\omega\cdot\omega')\omega-\omega'. \end{align} $$

The function $J(\omega ,\omega '):=2^{n-2}|\omega \cdot \omega '|^{n-2}$ (see the forthcoming Remark 4.7) is chosen so that

$$ \begin{align*}\int_{\mathbb{S}^{n-1}}\Phi(R_\omega\omega')J(\omega,\omega')\mathrm{d}\sigma(\omega)=\int_{\mathbb{S}^{n-1}}\Phi \mathrm{d}\sigma \end{align*} $$

for each $\omega '$ . This expression for J may be obtained by direct computation, noting that the map ${\omega \mapsto \omega ":=R_\omega \omega '}$ is not a bijection; it maps each component of $\mathbb {S}^{n-1}\backslash \langle \omega '\rangle ^\perp $ bijectively to $\mathbb {S}^{n-1}\backslash \{-\omega '\}$ with

(3.6) $$ \begin{align} \mathrm{d}\sigma(\omega")=2^{n-1}|\omega\cdot\omega'|^{n-2}\mathrm{d}\sigma(\omega). \end{align} $$

The essential features of this construction are those described in [Reference Alonso3]; see also [Reference Kowalski and Ławniczak34].

Motivated by the role of the transport equation in Section 2, for $g\in L^2(\mathbb {S}^{n-1})$ we define the auxiliary function $f:\mathbb {S} ^{n-1}\times \mathbb {R}^n\rightarrow \mathbb {R}$ (not to be confused with (2.2)) by

$$ \begin{align*}f(\omega,x)=\int_{\mathbb{S}^{n-1}}g(\omega')\overline{g(R_\omega \omega')}e^{-2\pi ix\cdot(\omega'-R_\omega\omega')}J(\omega,\omega')\mathrm{d}\sigma(\omega'), \end{align*} $$

so that $W_{\mathbb {S}^{n-1}}(g,g)$ is the restriction of f to the tangent bundle $T\mathbb {S}^{n-1}:=\{(\omega ,v):\omega \in \mathbb {S}^{n-1}, v\in \langle \omega \rangle ^\perp \}$ . That f is real-valued follows from the fact that $R_\omega \circ R_\omega =I$ for each $\omega $ . Evidently f satisfies the transport equation

(3.7) $$ \begin{align} \omega\cdot\nabla_xf=0, \end{align} $$

meaning that $f(\omega ,x)=f(\omega ,x_{\langle \omega \rangle ^\perp })=W_{\mathbb {S}^{n-1}}(g,g)(\omega ,x_{\langle \omega \rangle ^\perp })$ , where $x_{\langle \omega \rangle ^\perp }$ is the orthogonal projection of x onto $\langle \omega \rangle ^\perp $ . The functions f and $W_{\mathbb {S}^{n-1}}$ have some nice features; for example, we have the marginal identity

(3.8) $$ \begin{align} \int_{\mathbb{S}^{n-1}}f(\omega,x)\mathrm{d}\sigma(\omega)=|\widehat{g\mathrm{d}\sigma}(x)|^2, \end{align} $$

by Fubini’s theorem and the definition of J. We note in passing that we have the additional marginal property

$$ \begin{align*}\int_{\langle\omega\rangle^\perp}W_{\mathbb{S}^{n-1}}(g,g)(\omega,v)\mathrm{d}v=\frac{1}{2}\left(|g(\omega)|^2+|g(-\omega)|^2\right), \end{align*} $$

very much as in the setting of the classical Wigner distribution. This may be obtained by fixing $\omega $ and considering the contributions to $W_{\mathbb {S}^{n-1}}(g,g)$ coming from the integrals over the hemispheres $\mathbb {S}^{n-1}_{\pm }:=\{\omega ':\pm \omega \cdot \omega '>0\}$ and using the fact that the mapping $\omega '\mapsto \omega '-R_\omega \omega '$ is a bijection from each of $\mathbb {S}^{n-1}_{\pm }$ to the unit ball of $\langle \omega \rangle ^\perp $ ; see the forthcoming proof of Theorem 3.2 for a similar argument.

These observations lead to the desired spherical analogue of (2.5):

Proposition 3.1 (Spherical phase-space representation).

$$ \begin{align*}|\widehat{g\mathrm{d}\sigma}|^2=X^*W_{\mathbb{S}^{n-1}}(g,g).\end{align*} $$

Proof. By (3.8), (3.7) and Fubini’s theorem,

$$ \begin{align*} \begin{aligned} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x&=\int_{\mathbb{R}^n}\int_{\mathbb{S}^{n-1}}f(\omega,x)\mathrm{d}\sigma(\omega)w(x)\mathrm{d}x\\ &=\int_{\mathbb{S}^{n-1}}\int_{\langle\omega\rangle^\perp}f(\omega,x_{\langle\omega\rangle^\perp})\left(\int_{\langle\omega\rangle}w(x_{\langle\omega\rangle}+x_{\langle\omega\rangle^\perp})\mathrm{d}x_{\langle\omega\rangle}\right)\mathrm{d}x_{\langle\omega\rangle^\perp}\mathrm{d}\sigma(\omega)\\ &=\int_{\mathbb{S}^{n-1}}\int_{\langle\omega\rangle^\perp}W_{\mathbb{S}^{n-1}}(g,g)(\omega,v)Xw(\omega,v)\mathrm{d}v\mathrm{d}\sigma(\omega)\\ &=\int_{\mathbb{R}^n}X^*W_{\mathbb{S}^{n-1}}(g,g)(x)w(x)\mathrm{d}x \end{aligned} \end{align*} $$

for all test functions w.

As we have already indicated, Proposition 3.1 is well-known in some form in optics (at least in low dimensions) where it provides a representation of the local intensity of an optical field as a linear superposition of light rays – a useful and explicit connection between physical and geometric optics; see Alonso [Reference Alonso3]. Proposition 3.1 may be used to prove the following spherical versions of Theorems 1.7 and 1.8:

Theorem 3.2 (Spherical Sobolev–Stein).

For $s<\frac {n-1}{2}$ , there exists a dimensional constant c such that

(3.9) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\leq c\int_{\mathbb{S}^{n-1}}I_{\mathbb{S}^{n-1},2s}(|g|^2,|g|^2)(\omega)^{1/2}\|Xw(\omega,\cdot)\|_{\dot{H}^s(\langle\omega\rangle^\perp)}\mathrm{d}\sigma(\omega), \end{align} $$

where

$$ \begin{align*}I_{\mathbb{S}^{n-1},s}(g_1,g_2)(\omega):=\int_{\mathbb{S}^{n-1}}\frac{g_1(\omega')g_2(R_\omega\omega')}{|\omega'-R_\omega\omega'|^{s}}|\omega\cdot\omega'|^{n-2}\mathrm{d}\sigma(\omega'). \end{align*} $$

Theorem 3.4 (Spherical Sobolev–Mizohata–Takeuchi).

For $s<\frac {n-1}{2}$ ,

(3.10) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\lesssim\sup_{\omega\in\mathrm{ supp\,}^*(g)}\|Xw(\omega,\cdot)\|_{\dot{H}^s(\langle\omega\rangle^\perp)}\|g\|_{L^2(\mathbb{S}^{n-1})}^2, \end{align} $$

where $\mathrm {supp\,}^*(g)$ is the set of all geodesic midpoints of pairs of points from $\mathrm {supp\,}(g)$ . The implicit constant depends on at most n and s.

Proof of Theorem 3.2.

By partitioning $\mathbb {S}^{n-1}$ into boundedly many (depending only on n) geodesically convex subsets (caps), it suffices to show (3.9) under the assumption that g is supported in a cap S satisfying $\omega \cdot \omega '\geq \tfrac {1}{2}$ for all points $\omega ,\omega '\in S$ (in line with (1.12)). By Proposition 3.1 and the Cauchy–Schwarz inequality it suffices to show that

(3.11) $$ \begin{align} \|W_{\mathbb{S}^{n-1}}(g,g)(\omega,\cdot)\|_{\dot{H}^{-s}(\langle\omega\rangle^\perp)}^2\lesssim I_{\mathbb{S}^{n-1},2s}(|g|^2,|g|^2)(\omega), \end{align} $$

for some implicit constant depending only on n. Next, for fixed $\omega \in S$ we make the change of variables $\xi =\omega '-R_\omega \omega '$ , which maps S bijectively to a subset U of $\langle \omega \rangle ^\perp $ . Defining $\omega ':U\rightarrow S$ by $\xi =\omega '(\xi )-R_\omega \omega '(\xi )$ we have

$$ \begin{align*} \begin{aligned} W_{\mathbb{S}^{n-1}}(g,g)(\omega,v)&=\int_{S}g(\omega')\overline{g(R_\omega \omega')}e^{iv\cdot(\omega'-R_\omega\omega')}J(\omega,\omega')\mathrm{d}\sigma(\omega')\\ &=\int_{U}g(\omega'(\xi))\overline{g(R_\omega \omega'(\xi))}e^{iv\cdot\xi}\frac{J(\omega,\omega'(\xi))}{\widetilde{J}(\omega,\omega'(\xi))}\mathrm{d}\xi, \end{aligned} \end{align*} $$

where $\widetilde {J}(\omega ,\omega ')=2^{n-1}\omega \cdot \omega '\sim 1$ is the Jacobian of the change of variables. Hence

(3.12)

and so by Plancherel’s theorem on $\langle \omega \rangle ^\perp $ ,

(3.13) $$ \begin{align} \begin{aligned} \|W_{\mathbb{S}^{n-1}}(g,g)(\omega,\cdot)\|_{\dot{H}^{-s}(\langle\omega\rangle^\perp)}^2&=\int_{U}\left||\xi|^{-s}g(\omega'(\xi))\overline{g(R_\omega \omega'(\xi))}\frac{J(\omega,\omega'(\xi))}{\widetilde{J}(\omega,\omega'(\xi))}\right|^2\mathrm{d}\xi\\ &=\int_{S}|\omega'-R_\omega\omega'|^{-2s}|g(\omega')|^2|g(R_\omega \omega')|^2\frac{J(\omega,\omega')^2}{\widetilde{J}(\omega,\omega')}\mathrm{d}\sigma(\omega')\\ &\lesssim\int_{S}|\omega'-R_\omega\omega'|^{-2s}|g(\omega')|^2|g(R_\omega \omega')|^2|\omega\cdot\omega'|^{n-2}\mathrm{d}\sigma(\omega')\\ &=I_{\mathbb{S}^{n-1},2s}(|g|^2,|g|^2)(\omega), \end{aligned} \end{align} $$

The inequality (3.11) follows.

Proof of Theorem 3.4.

Arguing as in the proof of Theorem 3.2, it suffices to establish (3.10) for g supported in a single cap S. Since $I_{\mathbb {S}^{n-1},2s}(|g|^2,|g|^2)(\omega )=0$ if $\omega \not \in \mathrm {supp\,}^*(g)$ ,

$$ \begin{align*}\int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\lesssim\sup_{\omega\in\mathrm{supp\,}^*(g)}\|Xw(\omega,\cdot)\|_{\dot{H}^s(\langle\omega\rangle^\perp)}\|I_{\mathbb{S}^{n-1},2s}(|g|^2,|g|^2)\|_{L^{1/2}(S)}^{1/2}, \end{align*} $$

by Theorem 3.2. It therefore suffices to show that

is bounded from $L^1\times L^1$ into $L^{1/2}$ whenever $s<n-1$ . This will be established in Section 7, where more general surface-carried bilinear fractional integral operators are estimated.

4 General submanifolds: a geometric viewpoint

As we shall see, identifying a phase-space representation of $|\widehat {g\mathrm {d}\sigma }|^2$ that is explicit enough to establish Theorems 1.7 and 1.8 requires some careful geometric analysis, beginning with the identification of a suitable generalised Wigner distribution (or transform). We present this for general smooth submanifolds of $\mathbb {R}^n$ that are strictly convex in the sense that their shape operators $\mathrm {d}N_u$ are positive definite at all points $u\in S$ .

4.1 Surface-carried Wigner transforms

The general procedure for constructing a suitable Wigner transform on a submanifold of Euclidean space is again well-known in optics [Reference Alonso3], [Reference Petruccelli and Alonso45]; see, for example, [Reference Gneiting, Fischer and Hornberger29] for related intrinsic constructions in quantum physics. As is pointed out in [Reference Alonso3], for $n\geq 3$ matters are considerably more involved as there is some choice to be exercised.

For compactly supported function $g_1,g_2\in L^2(S)$ let

(4.1) $$ \begin{align} W_S(g_1,g_2)(u, v)=\int_{S}g_1(u')\overline{g_2(R_u u')}e^{-2\pi iv\cdot(u'-R_uu')}J(u,u')\mathrm{d}\sigma(u'). \end{align} $$

Here $u\in S$ , $v\in T_uS$ , and we define, for $u'\neq u$ , $R_u u'$ to be the unique point $u"\in S$ with $u"\not =u'$ such that

(4.2) $$ \begin{align} (u'-u")\cdot N(u)=0 \end{align} $$

and

(4.3) $$ \begin{align} N(u)\wedge N(u')\wedge N(u")=0. \end{align} $$

Define $R_{u}u:=u$ for all $u\in S$ . Condition (4.2) stipulates that $u'-u"\in T_u S$ , which as we shall see, is necessary for the phase-space representation (1.7); see Figure 1. Condition (4.3), which stipulates that $N(u), N(u'), N(u")$ lie on a great circle, is where we have exercised some choice. This appears to be physically significant and is at least implicitly referred to in the optics literature; see, for example, [Reference Alonso3] (p. 346) in the context of the sphere. Moreover, the appropriateness of (4.3) is particularly apparent when S is the paraboloid, as we clarify in the forthcoming Remark 4.7. In (4.1) the function $J(u,u')$ is the reciprocal of the Jacobian of the mapping $u\mapsto R_uu'$ , so that

(4.4) $$ \begin{align} \int_S\Phi(R_uu')J(u,u')\mathrm{d}\sigma(u)=\int_{S}\Phi \mathrm{d}\sigma \end{align} $$

for each $u'\in S$ . The required bijectivity here follows from the assumed geodesic convexity of $N(S)$ referred to in Section 1. We refer to $W_S(g_1,g_2)$ as the Wigner transform on S, and $W_S(g,g)$ as the Wigner distribution on S. As we shall see shortly, the Jacobian J is a bounded function on compact subsets of $S\times S$ , allowing $W_S(g_1,g_2)$ to be defined as a Lebesgue integral.

Figure 1 A depiction of the choice of $u"$ via the conditions (4.2) and (4.3).

The point $u"$ may seem rather difficult to identify at first sight, although it has a simple alternative description that is constructive. This is shown in Figure 2, and will play an important role in our analysis.

Figure 2 The construction of $u"$ via parallel supporting hyperplanes in $T_uS+\{u'\}$ .

Remark 4.1 (Existence of $u"$ ).

There is a technical point that we have glossed over in the above definition of $W_S$ and Figures 1 and 2. For given $u,u'\in S$ our hypotheses do not guarantee the existence of such a point $u":=R_u u'$ , unless S is closed (the boundary of a convex body in $\mathbb {R}^n$ ). One way to remedy this might be to continue S to a closed submanifold, upon which $R_u u'$ may always be defined, and observe that the resulting function $W_S(g_1,g_2)$ is independent of the choice of extension since $g_2$ is supported on S. In any event, the integral in (4.1) should be interpreted as taken over

$$ \begin{align*}\{u'\in S: (u'-u")\cdot N(u)=0 \text{ and } N(u)\wedge N(u')\wedge N(u")=0\text{ for some }u"\not=u'\}.\end{align*} $$

Naturally such domain restrictions will be apparent in our analysis of the Jacobian J in Section 6.

Remark 4.2 (Differentiability of $u"$ ).

We expect that the maps $u\mapsto R_u u'$ and $u'\mapsto R_u u'$ are differentiable away from $u=u'$ and that this should follow from (4.2) and (4.3) by a suitable application of the implicit function theorem; see Figure 2. This smoothness is of course clear when S is the sphere thanks to the explicit formula (3.5) and is assumed to be true of the submanifolds S considered here.

Remark 4.3 (Rationale for the choice of third point $u"$ ).

As is pointed out in [Reference Alonso3] and [Reference Petruccelli and Alonso45], for $n\geq 3$ there are many possible ways of defining the third point $u"$ in terms of $u'$ and u, although for the purposes of proving Theorems 1.7 and 1.8 there are a number of natural requirements that significantly constrain this choice. First of all, the choice should be ‘nondegenerate’ in the sense that the distances $|u'-u|$ and $|u'-u"|$ should be comparable (suitably uniformly in terms of the geometry of S); it should be symmetric so that the resulting Wigner distribution is real-valued (and the Wigner transform is conjugate symmetric), and it should be geometrically/physically natural, so that the Jacobian J may be expressed in terms of the Gauss map N and its derivative $\mathrm {d}N$ (the shape operator). The forthcoming Propositions 4.4 and 4.5 show that our choice of $u"$ has these features. As we shall see, the coplanarity condition (4.3) is natural as it allows the mapping $u\mapsto R_u u'$ to be transformed to a relatively simple ‘outward vector field’ on the tangent space $T_{u'}S$ . This involves parametrising S using the Gauss map followed by stereographic projection (a composition that may also be found in the theory of minimal surfaces).

It will be important for us to understand how the distances between the three points $u, u', u"$ relate to each other. This is provided by the following proposition, whose proof is deferred to Section 5. In particular it tells us that the function $\rho (u,u'):=|u'-R_uu'|$ on $S\times S$ is a quasi-distance, as we clarify in Section 8.

Proposition 4.4 (Distance estimates).

For all $u,u', u"\in S$ with $u"=R_uu'$ ,

(4.5) $$ \begin{align} |u'-u"|\lesssim Q(S)^{1/2}|u-u'| \end{align} $$

and

(4.6) $$ \begin{align} |u'-u"|\gtrsim \frac{1}{Q(S)}|u-u'|. \end{align} $$

We now turn from the metric properties to the measure-theoretic properties of the map $R_u$ , and a host of explicit identities satisfied by the Wigner transform $W_S$ .

To see that $W_S$ is conjugate-symmetric, which in particular implies that the Wigner distribution $W_S(g,g)$ is real-valued, already appears to require some work. For fixed $u\in S$ observe first that if $u"=R_u u'$ then $u'=R_u u"$ , and so by a change of variables,

$$ \begin{align*} \begin{aligned} W_S(g_1,g_2)(u,v)=\int_S g_1(R_u u")\overline{g_2(u")}e^{-2\pi iv\cdot(R_u u"-u")}J(u,R_u u")\Delta(u,u")\mathrm{d}\sigma(u"), \end{aligned} \end{align*} $$

where $\Delta (u,u")$ is the Jacobian of the change of variables $u'=R_u u"$ . It therefore remains to show that

$$ \begin{align*} J(u,u')\Delta(u,u")=J(u,u"), \end{align*} $$

recalling that J was defined in (4.4). Fortunately we have explicit formulae for the Jacobians J and $\Delta $ from which this quickly follows. In the following proposition we denote by $K(u)$ the Gaussian curvature of S at the point u, recalling that $K(u)$ is the determinant of the shape operator $\mathrm {d}N_u$ . Further, we denote by $P_{W}v$ the orthogonal projection of a vector $v\in \mathbb {R}^n$ onto a subspace W of $\mathbb {R}^n$ .

Proposition 4.5 (Jacobian identities).

For all $u,u', u"\in S$ with $u"=R_uu'$ ,

(4.7) $$ \begin{align} J(u,u')=\left(\frac{|N(u')\wedge N(u")|}{|N(u)\wedge N(u')|}\right)^{n-2}\left|\frac{\langle u"-u',N(u")\rangle}{\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle}\right|\frac{K(u)}{K(u")}, \end{align} $$

(4.8) $$ \begin{align} \kern-6pt \Delta(u,u')=\left(\frac{|N(u)\wedge N(u")|}{|N(u)\wedge N(u')|}\right)^{n-1}\frac{|\langle P_{T_{u'}S}N(u),(\mathrm{d}N_{u'})^{-1}(P_{T_{u'}S}N(u))\rangle|}{|\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle|}\frac{K(u')}{K(u")} \end{align} $$

and

(4.9) $$ \begin{align} J(u,u')\Delta(u,u")=J(u,u"). \end{align} $$

We defer the proof of Proposition 4.5 to Section 6.

Remark 4.6 (Interpreting J).

The expression for J in Proposition (4.5), while seemingly rather complicated, may be understood in somewhat simple geometric terms. In particular:

1. (i) Matters are much simpler when $n=2$ , where we may write

   $$ \begin{align*} \begin{aligned} \displaystyle J(u,u')&\displaystyle =\left|\frac{\langle u"-u',P_{T_uS}N(u")\rangle}{\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle}\right|\frac{K(u)}{K(u")} \\ &\displaystyle = \frac{|u"-u'|\cdot |N(u)\wedge N(u")|}{|P_{T_{u"}S}N(u)|^{2}}K(u) \\ &\displaystyle = \frac{|u"-u'|}{|N(u)\wedge N(u")|}K(u). \end{aligned} \end{align*} $$
   Here we have used the (two-dimensional) formula
   $$ \begin{align*}\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle=\frac{1}{K(u")}|P_{T_{u"}S}N(u)|^{2},\end{align*} $$
   along with the elementary identities $|P_{T_{u"}S}N(u)|=|P_{T_{u}S}N(u")|=|N(u)\wedge N(u")|$ .

2. (ii) The factor

   (4.10) $$ \begin{align}\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle^{-1} \end{align} $$
   is bounded above by $\langle P_{T_{u"}S}N(u),\mathrm {d}N_{u"}(P_{T_{u"}S}N(u))\rangle $ by the harmonic-arithmetic mean inequality. This bound is (up to a suitable normalisation factor) the directional curvature of S at the point $u"$ in the direction $P_{T_{u"}S}N(u)$ . One might therefore interpret the factor (4.10) as a certain ‘harmonic directional curvature’.

3. (iii) The factor

   $$ \begin{align*}\frac{|N(u')\wedge N(u")|}{|N(u)\wedge N(u')|}\end{align*} $$
   quantifies (in relative terms) the transversality of the tangent spaces to S at the points $u,u', u"$ , and is therefore also a manifestation of the curvature profile of S; see Figure 1.

4. (iv) The factor $\langle u"-u',N(u")\rangle $ is different in nature as it explicitly relates to the positions of the points $u', u"$ . It is instructive to use the fact that $u'-u"\in T_uS$ to write this as

   $$ \begin{align*}\langle u"-u',P_{T_uS}N(u")\rangle=|N(u)\wedge N(u")||u"-u'|\left\langle \tfrac{\displaystyle u"-u'}{\displaystyle |u"-u'|},\tfrac{\displaystyle P_{T_uS}N(u")}{\displaystyle |P_{T_uS}N(u")|}\right\rangle.\end{align*} $$
   We observe that the inner product in the final expression above quantifies the extent to which $u"$ is displaced from the line through $u'$ in the direction $P_{T_uS}N(u")$ ; see Figure 2.

5. (v) The Jacobian J is scale-invariant in the sense that an isotropic scaling of S leaves J unchanged. This is apparent from the definition of J but is also manifest in the formula (4.7).

Remark 4.7 (Examples).

Proposition 4.5 is easily applied to examples.

1. (i) If $S=\mathbb {P}^{n-1}$ , the paraboloid (2.14), then a careful calculation using Proposition 4.5 reveals that

   $$ \begin{align*}J(u,u')=2^{n-1}\left(\frac{1+4|x"|^2}{1+4|x|^2}\right)^{1/2}, \end{align*} $$
   where we are writing $u=(x,|x|^2)$ , $u'=(x',|x'|^2)$ , $u":=R_uu'=(x",|x"|^2)$ . As should be expected from our analysis in Section 2, the parabolic Wigner distribution $W_{\mathbb {P}^{n-1}}$ may be pulled back to the classical Wigner distribution via a suitable map $\Phi :\mathbb {R}^d\times \mathbb {R}^d\rightarrow T\mathbb {P}^d$ ; in this case $\Phi (x,v)=((x,|x|^2),P_{T_{(x,|x|^2)}\mathbb {P}^d}(v,0))$ . This uses the simple geometric fact that the coplanarity condition (4.3) transforms to a colinearity condition in parameter space. More specifically, if for a function $g:\mathbb {R}^d\rightarrow \mathbb {C}$ we let $Lg(x,|x|^2)=(1+4|x|^2)^{-\frac {1}{2}}g(x)$ , and for a function $h:T\mathbb {P}^d\rightarrow \mathbb {C}$ we let $Uh(x,v)=(1+4|x|^2)^{1/2}h(\Phi (x,v))$ , then
   $$ \begin{align*}UW_{\mathbb{P}^d}(Lg,Lg)=W(g,g).\end{align*} $$
   Moreover, $X_{\mathbb {P}^d}^*h=\rho (Uh)$ , allowing one to deduce the quantum-mechanical phase-space representation (2.5) from the forthcoming Proposition 4.8. We refer to [Reference Alonso3] (p. 353) for a similar remark.

2. (ii) If $S=\mathbb {S}^{n-1}$ , evidently $K\equiv 1$ and $N(\omega )=\omega $ , and to be consistent with Section 3 we use $\omega $ rather than u to represent a point. We may use the explicit formula (3.5) to write

   $$ \begin{align*} \frac{|N(\omega')\wedge N(\omega")|}{|N(\omega)\wedge N(\omega')|}=\frac{(1-(\omega'\cdot\omega")^{2})^{\frac{1}{2}}}{(1-(\omega\cdot\omega')^{2})^{\frac{1}{2}}}=\frac{|P_{\langle\omega'\rangle^{\perp}}\omega"|}{|P_{\langle\omega\rangle^{\perp}}\omega'|}=2|\omega\cdot\omega'|. \end{align*} $$
   On the other hand, since $\langle \omega "-\omega ',N(\omega ")\rangle =\langle \omega "-\omega ',P_{\langle \omega \rangle ^{\perp }}\omega "\rangle $ , projecting both sides of (3.5) to $\langle \omega \rangle ^{\perp }$ yields
   $$ \begin{align*} \left|\frac{\langle \omega"-\omega',N(\omega")\rangle}{\langle P_{T_{\omega"}S}N(\omega),(\mathrm{d}N_{\omega"})^{-1}(P_{T_{\omega"}S}N(\omega))\rangle}\right|=\frac{|\langle \omega"-\omega',\omega'\rangle|}{|1-(\omega\cdot\omega")^{2}|}=\frac{|1-(\omega'\cdot\omega")|}{|1-(\omega\cdot\omega')^{2}|}=2, \end{align*} $$
   since $\omega \cdot \omega " = \omega \cdot \omega '$ and $\omega '\cdot \omega "=2(\omega \cdot \omega ')^{2}-1$ . Altogether we conclude that
   $$ \begin{align*}J(\omega,\omega')=2^{n-1}|\omega\cdot\omega'|^{n-2},\end{align*} $$
   as appears in (3.6).

We now come to the phase-space representation of $|\widehat {g\mathrm {d}\sigma }|^2$ , and we begin by defining an auxiliary function $f:S\times \mathbb {R}^n\rightarrow \mathbb {R}$ by

$$ \begin{align*}f(u,x)=\int_{S}g(u')\overline{g(R_uu')}e^{-2\pi ix\cdot(u'-R_uu')}J(u,u')\mathrm{d}\sigma(u'), \end{align*} $$

so that $W_S(g,g)$ is the restriction of f to the tangent bundle $TS:=\{(u,v):u\in S, v\in T_uS\}$ . As in the spherical case, we continue to have the marginal identity

(4.11) $$ \begin{align} \int_{S}f(u,x)\mathrm{d}\sigma(u)=|\widehat{g\mathrm{d}\sigma}(x)|^2 \end{align} $$

by Fubini’s theorem and the definition of J. While we shall not need to use it, it is pertinent to also note the second marginal property

(4.12) $$ \begin{align} \int_{T_uS}W_S(g,g)(u,v)\mathrm{d}v=|g(u)|^2 \end{align} $$

here (possibly subject to an additional regularity assumption on S) referred to in the introduction; we refer to Section 8 for clarification of this, along with the sense in which it holds as a pointwise identity. Another key property is that f satisfies the transport equation

(4.13) $$ \begin{align} N(u)\cdot\nabla_xf=0, \end{align} $$

meaning that $f(u,x)=W_S(g,g)(u,P_{T_uS}x)$ , where $P_{T_uS}:\mathbb {R}^n\rightarrow T_uS$ is the orthogonal projection onto $T_uS$ .

Proposition 4.8 (General phase-space representation).

(4.14) $$ \begin{align} |\widehat{g\mathrm{d}\sigma}|^2=X_S^*W_S(g,g) \end{align} $$

where $X_Sw(u,v):=Xw(N(u),v)$ , the pullback of $Xw$ under the Gauss map

$$ \begin{align*}TS\ni (u,v)\mapsto (N(u),v)\in T\mathbb{S}^{d-1}. \end{align*} $$

We note that for a phase-space function $h:TS\rightarrow \mathbb {C}$ we have the explicit expression

$$ \begin{align*}X_S^*h(x)=\int_{S}h(u,P_{T_uS}x)\mathrm{d}\sigma(u).\end{align*} $$

Proof of Proposition 4.8.

By (4.11), (4.13) and Fubini’s theorem,

$$ \begin{align*} \begin{aligned} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x&=\int_{\mathbb{R}^n}\int_{S}f(u,x)\mathrm{d}\sigma(u)w(x)\mathrm{d}x\\ &=\int_{S}\int_{T_uS}f(u,v)\left(\int_{(T_uS)^\perp}w(v+z)\mathrm{d}z\right)\mathrm{d}v\mathrm{d}\sigma(u)\\ &=\int_S\int_{T_uS}W_S(g,g)(u,v)Xw(N(u),v)\mathrm{d}v\mathrm{d}\sigma(u)\\ &=\int_{\mathbb{R}^n}X_S^*W(g,g)(x)w(x)\mathrm{d}x \end{aligned} \end{align*} $$

for all test functions w.

Remark 4.9 (A polarised form).

The polarised form

$$ \begin{align*}\widehat{g_1\mathrm{d}\sigma}\:\overline{\widehat{g_2\mathrm{d}\sigma}}=X_S^*W_S(g_1,g_2) \end{align*} $$

of (4.14) may be established similarly, and indeed may be deduced directly from (4.14).

Remark 4.10. There is a point of contact here with [Reference Bennett, Nakamura and Shiraki15], where among other things it is shown that the classical Radon transform fails to distinguish $|\widehat {g\mathrm {d}\sigma }|^2$ from $X_S^*\nu $ for a large class of distributions $\nu $ on $TS$ , provided a suitable transversality condition is satisfied. Perhaps unsurprisingly, $W_S(g,g)$ is easily seen to be an example of such a distribution.

We are now ready to state or main theorems (Theorems 1.7 and 1.8) in full.

Theorem 4.11 (L ² Sobolev–Stein inequality).

Suppose that S is a smooth strictly convex surface with curvature quotient $Q(S)$ , and $s<\frac {n-1}{2}$ . Then there is a dimensional constant c such that

(4.15) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\leq c Q(S)^{\frac{5n-8}{4}}\int_S I_{S,2s}(|g|^2,|g|^2)(u)^{1/2}\|X_Sw(u,\cdot)\|_{\dot{H}^s(T_u S)}\mathrm{d}\sigma(u), \end{align} $$

where

(4.16) $$ \begin{align} I_{S,s}(g_1,g_2)(u):=\int_S\frac{g_1(u')g_2(R_u u')}{|u'-R_u u'|^s}J(u,u')\mathrm{d}\sigma(u'). \end{align} $$

Remark 4.12. The S-carried fractional integral $I_{S,s}$ is natural for a number of reasons relating to the presence of the Jacobian factor J. In particular, it is symmetric thanks to (4.9) (a property that is analogous to the conjugate symmetry of the Wigner transform $W_S$ ), and as we shall see in Section 7, its Lebesgue space bounds do not depend on any lower bound on the curvature of S. The restriction $s<\frac {n-1}{2}$ ensures that the kernel of $I_{S,s}$ is locally integrable.

Theorem 4.13 (L ² Sobolev–Mizohata–Takeuchi inequality).

Suppose that S is a smooth strictly convex surface with curvature quotient $Q(S)$ , and $s<\frac {n-1}{2}$ . Then there exists a constant c, depending on at most n, s and the diameter of S, such that

(4.17) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\leq cQ(S)^{\frac{9n-12}{4}}\sup_{u\in\mathrm{ supp\,}^*(g)}\|X_Sw(u,\cdot)\|_{\dot{H}^s(T_u S)}\|g\|_{L^2(S)}^2, \end{align} $$

where $ \mathrm {supp\,}^*(g):=\{u\in S: R_u u'\in \mathrm {supp\,}(g)\;\text { for some }\; u'\in \mathrm {supp\,}(g)\}$ .

Remark 4.14. We remark that $\mathrm {supp\,}(g)\subseteq \mathrm {supp\,}^*(g)$ , and often this containment is strict. When S is the sphere, $\mathrm { supp\,}^*(g)$ is the ‘support midpoint set’, consisting of all geodesic midpoints of pairs of points from the support of g. Hence $\mathrm {supp\,}^*(g)\subseteq \mathrm {cvx\,}\mathrm {supp\,}(g)$ in this case, where $\mathrm {cvx\,}$ forms the geodesic convex hull. More generally, $\mathrm { supp\,}^*(g)\subseteq N^{-1}\mathrm {cvx\,}(N(\mathrm {supp\,}(g)))$ , so that

$$ \begin{align*}\int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\leq cQ(S)^{\frac{9n-12}{4}}\sup_{\omega\in\mathrm{cvx\,}(N(\mathrm{supp\,}(g)))}\|Xw(\omega,\cdot)\|_{\dot{H}^s(T_u S)}\|g\|_{L^2(S)}^2. \end{align*} $$

Remark 4.15. While we expect that the power of $Q(S)$ in the statement of Theorem 4.11 is sharp when $n=2$ , it seems unlikely that it is in higher dimensions. The power of $Q(S)$ in the statement of Theorem 4.13 is of course larger still, incurring extra factors from the bounds on the bilinear fractional integrals $I_{S,s}$ in Section 7.

4.2 Proof of the Sobolev–Stein inequality (Theorem 1.7)

In this section we prove Theorem 1.7, or more specifically, Theorem 4.11. We begin with an application of Proposition 4.8 and the Cauchy–Schwarz inequality to write

(4.18) $$ \begin{align} \int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}|^2w\leq \int_S \|W_S(g,g)(u,\cdot)\|_{\dot{H}^{-s}(T_u S)}\|X_Sw(u,\cdot)\|_{\dot{H}^s(T_u S)}\mathrm{d}\sigma(u) \end{align} $$

for any $s\in \mathbb {R}$ . In order to estimate the Sobolev norm of the Wigner distribution above we fix $u\in S$ and make the change of variables

(4.19) $$ \begin{align} \xi=u'-R_u u'. \end{align} $$

Since S is the graph of a strictly convex function, the map $u'\mapsto \xi $ is a bijection from S to a subset U of $T_uS$ . To see this it suffices to establish injectivity, and hence we look to show that $u'-R_uu'\not =\tilde {u}'-R_u\tilde {u}'$ for $u'\neq \tilde {u}'$ . We may assume that $u'-R_uu'$ and $\tilde {u}'-R_u\tilde {u}'$ are parallel, as otherwise the desired conclusion is immediate. Observe that $u'-\tilde {u}'\not \in T_uS$ , as otherwise strict convexity of the level sets of S (sections of S by translates of $T_uS$ ) would force $u'=\tilde {u}'$ or $R_uu'=\tilde {u}'$ ; see Figure 2. Since $u'-\tilde {u}'\not \in T_uS$ and S is the graph of a function, the level sets of S through $u'$ and $\tilde {u}'$ respectively, when projected onto $T_uS$ , are both enclosed by the supporting hyperplanes depicted in Figure 2; this may require interchanging the roles of $u'$ and $\tilde {u}'$ , as we may. Since $u'-R_uu'$ and $\tilde {u}'-R_u\tilde {u}'$ are parallel, it follows that $|\tilde {u}'-R_u\tilde {u}'|<|u'-R_uu'|$ , and thus $u'-R_uu'\not =\tilde {u}'-R_u\tilde {u}'$ . As a result of this bijectivity,

$$ \begin{align*}\|W_{S}(g,g)(u,\cdot)\|_{H^{-s}(T_u S)}^2=\int_{T_u S}\left|\int_U g(u'(\xi))\overline{g(R_u u'(\xi))}|\xi|^{-s}e^{-2\pi iv\cdot\xi}J(u,u'(\xi))\frac{\mathrm{d}\xi}{\widetilde{J}(u,u'(\xi))}\right|^2\mathrm{d}v, \end{align*} $$

where $\widetilde {J}(u,u')$ is the Jacobian of the map $u'\mapsto \xi $ . Hence by Plancherel’s theorem on $T_u S$ ,

$$ \begin{align*} \begin{aligned} \|W_{S}(g,g)(u,\cdot)\|_{H^{-s}(T_u S)}^2&=\int_{U}\left|g(u'(\xi))\overline{g(R_u u'(\xi))}|\xi|^{-s}\frac{J(u,u'(\xi))}{\widetilde{J}(u,u'(\xi))}\right|^2\mathrm{d}\xi\\ &=\int_S\frac{|g(u')|^2|g(R_u u')|^2}{|u'-R_u u'|^{2s}}\frac{J(u,u')^2}{\widetilde{J}(u,u')}\mathrm{d}\sigma(u'). \end{aligned} \end{align*} $$

In order to complete the proof of Theorem 4.11 it therefore suffices to prove that

(4.20) $$ \begin{align} \frac{J(u,u')}{\widetilde{J}(u,u')}\lesssim Q(S)^{\frac{5n-8}{2}} \end{align} $$

with implicit constant depending only on the dimension. We do this in two steps.

Step 1: Bounding $\widetilde {J}(u,u')$

The goal here is to obtain a suitable lower bound for $\widetilde {J}(u,u')$ .

Proposition 4.16. We have that

(4.21) $$ \begin{align} \widetilde{J}(u,u')\geq (1+\Delta(u,u')^{2})^{\frac{1}{2}} \end{align} $$

for all $u,u'\in S$ .

Proof. Let $u\in S$ be fixed. The Jacobian $\widetilde {J}$ of the change of variables

$$ \begin{align*} \xi(u')=u'-R_{u}u', \end{align*} $$

may be expressed as

(4.22) $$ \begin{align} \widetilde{J}(u,u')=\frac{|(\mathrm{d}\xi)_{u'}(v_{1})\wedge\cdots\wedge (\mathrm{d}\xi)_{u'}(v_{n-1})|}{|v_{1}\wedge\cdots\wedge v_{n-1}|}, \end{align} $$

where $v_{1},\ldots ,v_{n-1}$ is a basis for $T_{u'}S$ . We remark that

$$ \begin{align*}(\mathrm{d}\xi)_{u'}(v_{1})\wedge\cdots\wedge (\mathrm{d}\xi)_{u'}(v_{n-1})\in\Lambda^{n-1}(T_{u}S)\quad\mathrm{and}\quad v_{1}\wedge\cdots\wedge v_{n-1}\in\Lambda^{n-1}(T_{u'}S),\end{align*} $$

and we identify the exterior algebras $\Lambda ^{n-1}(T_{u'}S)$ and $\Lambda ^{n-1}(T_{u}S)$ with subspaces of $\Lambda ^{n-1}(\mathbb {R}^{n})$ via the natural embedding induced by the inclusions $T_{u'}S\subset \mathbb {R}^{n}$ and $T_{u}S\subset \mathbb {R}^{n}$ , respectively.

It will be convenient to fix $u'$ and express (4.22) in terms of unit velocities of trajectories along smooth curves in S emanating from $u'$ . In what follows $c:I\rightarrow S$ will denote the arc-length parametrisation of such a curve, where I is an open interval containing $0$ such that $c(0)=u'$ . If $\mathcal {C}$ denotes the set of all such mappings c, then evidently

$$ \begin{align*} T_{u'}S = \langle\left\{\dot{c}(0): c\in\mathcal{C}\right\}\rangle. \end{align*} $$

By the strict convexity of S, the $(n-1)$ -dimensional spaces $T_{u'}S$ and $T_{u}S$ intersect in an $(n-2)$ -dimensional subspace $\mathcal {H}$ . We then pick curves $c_{1},\ldots ,c_{n-2}\in \mathcal {C}$ such that

$$ \begin{align*}\mathcal{H}=\langle \dot{c}_{1}(0),\ldots,\dot{c}_{n-2}(0)\rangle,\end{align*} $$

and the set $\{\dot {c}_{i}(0)\}_{1\leq i\leq n-2}$ is orthonormal. To obtain an orthonormal basis for $T_{u'}S$ , we simply take any other curve $c_{n-1}\in \mathcal {C}$ such that $\dot {c}_{n-1}(0)\in \mathcal {H}^{\perp }\cap T_{u'}S$ . There is one more degree of freedom in choosing $c_{n-1}$ , and we assume without loss of generality that $\dot {c}_{n-1}(0)\cdot N(u)\geq 0$ . This gives

$$ \begin{align*}T_{u'}S=\langle \dot{c}_{1}(0),\ldots,\dot{c}_{n-2}(0),\dot{c}_{n-1}(0)\rangle.\end{align*} $$

Since

$$ \begin{align*} (\mathrm{d}\xi)_{u'}(\dot{c}_{i}(0))= (\xi\circ c_{i})'(0) = \dot{c}_{i}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{i}(0)),\quad 1\leq i\leq n-1, \end{align*} $$

then, since $|\dot {c}_{1}(0)\wedge \cdots \wedge \dot {c}_{n-1}(0)|=1$ by orthonormality of the chosen basis of $T_{u'}S$ ,

(4.23) $$ \begin{align} \begin{aligned} \displaystyle\widetilde{J}(u,u')&\displaystyle =|(\mathrm{d}\xi)_{u'}(\dot{c}_{1}(0))\wedge\cdots\wedge (\mathrm{d}\xi)_{u'}(\dot{c}_{n-1}(0))| \\ &\displaystyle= |(\dot{c}_{1}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{1}(0)))\wedge\cdots\wedge (\dot{c}_{n-1}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)))| \\ &\displaystyle= |W_{1}-W_{2}|, \end{aligned} \end{align} $$

where

$$ \begin{align*} \begin{aligned} \displaystyle W_{1}&\displaystyle := (\dot{c}_{1}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{1}(0)))\wedge\cdots\wedge (\dot{c}_{n-2}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-2}(0)))\wedge \dot{c}_{n-1}(0), \\ \displaystyle W_{2}&\displaystyle := (\dot{c}_{1}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{1}(0)))\wedge\cdots\wedge (\dot{c}_{n-2}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-2}(0)))\wedge (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)). \end{aligned} \end{align*} $$

The next claim collects a few useful facts about the action of $(\mathrm {d}R_{u})_{u'}$ on $\mathcal {H}$ .

Claim 4.17. The following hold:

1. 1. The subspace $\mathcal {H}=T_{u'}S\cap T_{u}S$ generated by the set of vectors $\{\dot {c}_{1}(0),\ldots ,\dot {c}_{n-2}(0)\}$ is invariant under the map $(\mathrm {d}R_{u})_{u'}$ . Moreover, $\left .(\mathrm {d}R_{u})_{u'}\right |{}_{\mathcal {H}}:\mathcal {H}\longrightarrow \mathcal {H}$ is an isomorphism. Equivalently,

   (4.24) $$ \begin{align} \mathcal{H}=\langle \dot{c}_{1}(0),\ldots,\dot{c}_{n-2}(0)\rangle=\langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{1}(0)),\ldots,(\mathrm{d}R_{u})_{u'}(\dot{c}_{n-2}(0))\rangle. \end{align} $$

2. 2. Let $M_{u,u'}:=\left .(\mathrm {d}R_{u})_{u'}\right |{}_{\mathcal {H}}: \mathcal {H}\longrightarrow \mathcal {H}$ denote the restriction of $(\mathrm {d}R_{u})_{u'}$ to the invariant subspace $\mathcal {H}$ . Then $I-M_{u,u'}:\mathcal {H}\rightarrow \mathcal {H}$ satisfies

   (4.25) $$ \begin{align} \det{(I-M_{u,u'})} \geq 1. \end{align} $$

Proof. Let $\omega := N(u)$ . Notice that the coplanarity condition (4.3) implies that

(4.26) $$ \begin{align} v_{1}:=\frac{P_{\langle\omega\rangle^{\perp}}N(u')}{|P_{\langle\omega\rangle^{\perp}}N(u')|}=-\frac{P_{\langle\omega\rangle^{\perp}}N(u")}{|P_{\langle\omega\rangle^{\perp}}N(u")|}=:-v_{2}. \end{align} $$

On the other hand, $v_{1}$ and $v_{2}$ are the outward normal vectors (in $T_{u}S$ ) of the convex submanifold

(4.27) $$ \begin{align} \mathcal{S}_{u,u'}:=S\cap (T_{u}S + u') \end{align} $$

at $u'$ and $u"$ respectively, hence

$$ \begin{align*} T_{u'}\mathcal{S}_{u,u'} = T_{u"}\mathcal{S}_{u,u'}, \end{align*} $$

from which (4.24) follows; see Figure 2. Observe also that on $\mathcal {S}_{u,u'}$ we have

(4.28) $$ \begin{align} R_{u}u' = \widetilde{N}^{-1}(-\widetilde{N}(u')), \end{align} $$

where $\widetilde {N}:\mathcal {S}_{u,u'}\rightarrow \mathbb {S}^{n-2}$ is the Gauss map of $\mathcal {S}_{u,u'}\subset u'+T_{u}S$ . Computing derivatives, $\left .(\mathrm {d}R_{u})_{u'}\right |{}_{\mathcal {H}}: \mathcal {H}\longrightarrow \mathcal {H}$ satisfies

$$ \begin{align*} (\mathrm{d}R_{u})_{u'} = -\mathrm{d}\widetilde{N}^{-1}_{-\widetilde{N}(u')}\circ\mathrm{d}\widetilde{N}_{u'}. \end{align*} $$

Finally, since $\mathrm {d}\widetilde {N}^{-1}_{-\widetilde {N}(u')}$ and $\mathrm {d}\widetilde {N}_{u'}$ are positive definite (recall that our assumptions on S imply positive definiteness of $\mathrm {d}N_{u}$ for all $u\in S$ , hence the same holds for $\mathrm {d}\widetilde {N}_{u}$ ) the product $\mathrm {d}\widetilde {N}^{-1}_{-\widetilde {N}(u')}\circ \mathrm {d}\widetilde {N}_{u'}$ has positive eigenvalues, therefore

$$ \begin{align*} \det{(I-M_{u,u'})}=\det{(I+\mathrm{d}\widetilde{N}^{-1}_{-\widetilde{N}(u')}\circ\mathrm{d}\widetilde{N}_{u'})}\geq 1.\\[-41pt] \end{align*} $$

The next claim contains three key identities involving $W_{1}$ and $W_{2}$ .

Claim 4.18. The following identities hold:

$$ \begin{align*} \begin{aligned} \displaystyle \langle W_{1},W_{1}\rangle &\displaystyle = \det{(I-M_{u,u'})}^{2}, \\ \displaystyle\langle W_{1},W_{2}\rangle &=\displaystyle \det{(I-M_{u,u'})}^{2}\langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)),\dot{c}_{n-1}(0)\rangle \\ \displaystyle \langle W_{2},W_{2}\rangle&\displaystyle = \Delta(u,u')^{2}\det{(M_{u,u'}^{-1}-I)}^{2}. \end{aligned} \end{align*} $$

Proof. Let $\textbf {0}_{1\times (n-2)}$ be the $1\times (n-2)$ zero row and let $\textbf {X}_{u,u'}$ be the $(n-2)\times (n-2)$ matrix whose $(i,j)$ entry is given by

$$ \begin{align*} (\textbf{X}_{u,u'})_{i,j}:= \langle \dot{c}_{i}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{i}(0)),\dot{c}_{j}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{j}(0))\rangle. \end{align*} $$

Observe that

$$ \begin{align*} \begin{aligned} \displaystyle\langle W_{1},W_{1}\rangle&\displaystyle=\det{\begin{pmatrix} \textbf{X}_{u,u'} & \textbf{0}_{1\times (n-2)}^{\top} \\ \textbf{0}_{1\times (n-2)} & 1 \end{pmatrix}} =\displaystyle\det{(\textbf{X}_{u,u'})}=\displaystyle \det{(I-M_{u,u'})}^{2}, \end{aligned} \end{align*} $$

where we used the facts that $\mathcal {H}$ is invariant under $(\mathrm {d}R_{u})_{u'}$ (as verified in Claim 4.17) and that $\dot {c}_{n-1}(0)$ is orthogonal to $\mathcal {H}$ . Now let $\textbf {Y}_{u,u'}$ be the $(n-2)\times (n-2)$ matrix whose $(i,j)$ entry is given by

$$ \begin{align*} (\textbf{Y}_{u,u'})_{i,j}:= \langle (\mathrm{d}R_{u})_{u'}^{-1}(\dot{c}_{i}(0))-\dot{c}_{i}(0),(\mathrm{d}R_{u})_{u'}^{-1}(\dot{c}_{j}(0))-\dot{c}_{j}(0)\rangle. \end{align*} $$

Analogously,

$$ \begin{align*} \begin{aligned} \displaystyle\langle W_{2},W_{2}\rangle &\displaystyle = |(\dot{c}_{1}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{1}(0)))\wedge\cdots\wedge (\dot{c}_{n-2}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-2}(0)))\wedge (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0))|^{2} \\ &\displaystyle = \Delta(u,u')^{2}\left|\left(\bigwedge_{j=1}^{n-2}(\mathrm{d}R_{u})_{u'}^{-1}[\dot{c}_{j}(0)- (\mathrm{d}R_{u})_{u'}(\dot{c}_{j}(0))]\right)\wedge \dot{c}_{n-1}(0)\right|^{2} \\ &\displaystyle = \Delta(u,u')^{2}\left|\left(\bigwedge_{j=1}^{n-2}[(\mathrm{d}R_{u})_{u'}^{-1}-I](\dot{c}_{j}(0))\right)\wedge \dot{c}_{n-1}(0)\right|^{2} \\ &=\Delta(u,u')^{2}\det(\textbf{Y}_{u,u'}) \\ &= \Delta(u,u')^{2}\det{(M_{u,u'}^{-1}-I)}^{2}. \end{aligned} \end{align*} $$

Finally,

$$ \begin{align*} \begin{aligned} \displaystyle\langle W_{1},W_{2}\rangle &\displaystyle= \det{\begin{pmatrix} \textbf{X}_{u,u'} & A_{(n-2)\times 1} \\ \textbf{0}_{1\times (n-2)} & \langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)),\dot{c}_{n-1}(0)\rangle \end{pmatrix}} \\ &=\displaystyle\det{(I-M_{u,u'})}^{2}\langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)),\dot{c}_{n-1}(0)\rangle, \end{aligned} \end{align*} $$

where $A_{(n-2)\times 1}$ is a $(n-2)\times 1$ column that does not feature in the final expression.

Expanding $|W_{1}-W_{2}|^{2}$ using the standard scalar product on the exterior algebra $\Lambda ^{n-1}(\mathbb {R}^{n})$ ,

(4.29) $$ \begin{align} \begin{aligned} |W_{1}-W_{2}|^{2} &= \langle W_{1}, W_{1}\rangle - 2\langle W_{1}, W_{2}\rangle + \langle W_{2}, W_{2}\rangle \\&= \det{(I-M_{u,u'})}^{2} - 2\det{(I-M_{u,u'})}^{2}\langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)),\dot{c}_{n-1}(0)\rangle\\& \quad + \Delta(u,u')^{2}\det{(M_{u,u'}^{-1}-I)}^{2}, \end{aligned} \end{align} $$

thanks to Claim 4.18. We continue with the following key observation:

Claim 4.19. Under (1.12), it holds that

$$ \begin{align*} \langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)),\dot{c}_{n-1}(0)\rangle < 0. \end{align*} $$

Proof. Recall that $\dot {c}_{n-1}(0)\cdot N(u)> 0$ by assumption, therefore differentiating at $t=0$ the identity

$$ \begin{align*}\langle R_{u}(c_{n-1}(t)) - c_{n-1}(t), N(u)\rangle =0\end{align*} $$

gives $\langle (\mathrm {d}R_{u})_{u'}(\dot {c}_{n-1}(0)), N(u)\rangle>0$ . Next, observe that $N(u'), N(u), N(u")$ and $\dot {c}_{n-1}(0)$ are in $\mathcal {H}^{\perp }$ , the (two-dimensional) orthogonal complement of $\mathcal {H}$ in $\mathbb {R}^{n}$ . Since $N(p)\cdot N(q)\geq \frac {1}{2}$ for all $p,q\in S$ by assumption, the angles $\alpha _{1}$ (between $N(u')$ and $N(u)$ ) and $\alpha _{2}$ (between $N(u)$ and $N(u")$ ) are such that $0<\alpha _{1}+\alpha _{2}<\frac {\pi }{2}$ . Since $N(u)\in \mathcal {H}^{\perp }$ , we have by the self-adjointness of the projection operator $P_{\mathcal {H}^{\perp }}$ ,

$$ \begin{align*}0<\langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)), N(u) \rangle = \langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)), P_{\mathcal{H}^{\perp}}N(u)\rangle = \langle P_{\mathcal{H}^{\perp}}[(\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0))], N(u)\rangle ,\end{align*} $$

which implies that $P_{\mathcal {H}^{\perp }}[(\mathrm {d}R_{u})_{u'}(\dot {c}_{n-1}(0))]$ is in the upper-half space of $\mathcal {H}^{\perp }$ (here we are assuming without loss of generality that $N(u)=e_{2}$ , the second canonical vector of $\mathcal {H}^{\perp }\cong \mathbb {R}^{2}$ ). On the other hand, $(\mathrm {d}R_{u})_{u'}(\dot {c}_{n-1}(0))\in T_{u"}S$ , hence $\langle P_{\mathcal {H}^{\perp }}[(\mathrm {d}R_{u})_{u'}(\dot {c}_{n-1}(0))], N(u") \rangle =0$ , that is, the angle between $N(u")$ and $P_{\mathcal {H}^{\perp }}[(\mathrm {d}R_{u})_{u'}(\dot {c}_{n-1}(0))]$ is $\frac {\pi }{2}$ . Since $\theta :=\frac {\pi }{2}-(\alpha _{1}+\alpha _{2})$ is strictly positive, the angle $\gamma :=\frac {\pi }{2}+\theta $ between $P_{\mathcal {H}^{\perp }}[(\mathrm {d}R_{u})_{u'}(\dot {c}_{n-1}(0))]$ and $\dot {c}_{n-1}(0)$ is strictly larger than $\frac {\pi }{2}$ (see Figure 3), which implies that

$$ \begin{align*}\langle P_{\mathcal{H}^{\perp}}[(\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0))], \dot{c}_{n-1}(0)\rangle < 0.\end{align*} $$

Figure 3 A graphical representation of the proof of Claim 4.19.

Finally, again by the self-adjointness of $P_{\mathcal {H}^{\perp }}$ ,

$$ \begin{align*} \begin{aligned} \displaystyle \langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)), \dot{c}_{n-1}(0)\rangle &\displaystyle= \langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)), P_{\mathcal{H}^{\perp}}[\dot{c}_{n-1}(0)]\rangle \\ &\displaystyle= \langle P_{\mathcal{H}^{\perp}}[(\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0))], \dot{c}_{n-1}(0)\rangle \\ &\displaystyle < 0, \end{aligned} \end{align*} $$

which concludes the proof of the claim.

Returning to (4.29),

(4.30) $$ \begin{align} \begin{aligned} |W_{1}-W_{2}|^{2} &\displaystyle= \det{(I-M_{u,u'})}^{2} - 2\det{(I-M_{u,u'})}^{2}\langle (\mathrm{d}R_{u})_{u'}(\dot{c}_{n-1}(0)),\dot{c}_{n-1}(0)\rangle\\& \quad + \Delta(u,u')^{2}\det{(M_{u,u'}^{-1}-I)}^{2}\\&\geq 1+\Delta(u,u')^{2}, \end{aligned} \end{align} $$

by (4.25) and Claim 4.19, which concludes the proof of Proposition 4.16.

Step 2: Bounding $J/\widetilde {J}$

Let $\lambda _{1}(p)\leq \lambda _{2}(p)\leq \cdots \leq \lambda _{n-1}(p)$ be the eigenvalues of the shape operator $\mathrm {d}N$ at p. Since $u"-u'\in \langle N(u)\rangle ^{\perp }$ ,

$$ \begin{align*} \begin{aligned} \displaystyle\left|\frac{\langle u"-u',N(u")\rangle}{\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle}\right|&\displaystyle = \left|\frac{\langle u"-u',N(u")-N(u)\rangle}{\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle}\right| \\ &\leq\displaystyle \lambda_{n-1}(u") \frac{|\langle u"-u',N(u")-N(u)\rangle|}{|P_{T_{u"}S}N(u)|^{2}}. \end{aligned} \end{align*} $$

Using the fact that $|P_{T_{u"}S}N(u)|=|N(u")\wedge N(u)|\approx |N(u")-N(u)|$ , which follows from (1.12), we have

$$ \begin{align*} \begin{aligned} \displaystyle J(u,u')&\displaystyle =\left(\frac{|N(u')\wedge N(u")|}{|N(u)\wedge N(u')|}\right)^{n-2}\left|\frac{\langle u"-u',N(u")\rangle}{\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle}\right|\frac{K(u)}{K(u")} \\ &\displaystyle \lesssim\left(\frac{|N(u')-N(u")|}{|N(u)- N(u')|}\right)^{n-2}\frac{|\langle u"-u',N(u")-N(u)\rangle|}{|P_{T_{u"}S}N(u)|^{2}}\frac{\prod_{j=1}^{n-1}\lambda_{j}(u)}{\prod_{j=1}^{n-1}\lambda_{j}(u")} \lambda_{n-1}(u") \\ &\lesssim\displaystyle\left(\frac{|u'-u"|}{|u-u'|}\right)^{n-2}\left(\frac{\sup_{p}\lambda_{n-1}(p)}{\inf_{p}\lambda_{1}(p)}\right)^{n-2} \frac{|u"-u'|\cdot|N(u")-N(u)|}{|N(u")-N(u)|^{2}}\frac{\prod_{j=1}^{n-2}\lambda_{j}(u)}{\prod_{j=1}^{n-2}\lambda_{j}(u")}\lambda_{n-1}(u) \\ &\lesssim\displaystyle\left(\frac{|u'-u"|}{|u-u'|}\right)^{n-2}\frac{|u"-u'|}{|N(u")-N(u)|} Q(S)^{2(n-2)} \sup_{p}\lambda_{n-1}(p). \end{aligned} \end{align*} $$

Hence by (4.5) and the fact that $\widetilde {J}(u,u')\geq 1$ (see Proposition 4.16),

(4.31) $$ \begin{align} \displaystyle\frac{J(u,u')}{\widetilde{J}(u,u')}\lesssim Q(S)^{\frac{5(n-2)}{2}} \frac{|u"-u'|}{|N(u")-N(u)|} \sup_{p}\lambda_{n-1}(p). \end{align} $$

On the other hand, using the fact that $\widetilde {J}(u,u')\geq \Delta (u,u')$ , which also follows from Proposition 4.16,

$$ \begin{align*} \begin{aligned} \displaystyle \frac{J(u,u')}{\widetilde{J}(u,u')}\leq\displaystyle \frac{J(u,u')}{\Delta(u,u')}=J(u,u")&\lesssim\displaystyle\left(\frac{|u'-u"|}{|u-u"|}\right)^{n-2}\frac{|u"-u'|}{|N(u')-N(u)|} Q(S)^{2(n-2)} \sup_{p}\lambda_{n-1}(p)\\ &\lesssim\displaystyle Q(S)^{\frac{5(n-2)}{2}}\frac{|u"-u'|}{|N(u')-N(u)|} \sup_{p}\lambda_{n-1}(p), \end{aligned} \end{align*} $$

by the distance estimate (4.5) and by (4.9). Consequently,

(4.32) $$ \begin{align} \begin{aligned} \displaystyle\frac{J(u,u')}{\widetilde{J}(u,u')}&\displaystyle\lesssim |u"-u'| Q(S)^{\frac{5(n-2)}{2}}\frac{1}{\max\{|N(u")-N(u)|,|N(u')-N(u)|\}}\sup_{p}\lambda_{n-1}(p) \\ &\displaystyle\lesssim |u"-u'| Q(S)^{\frac{5(n-2)}{2}}\frac{1}{|N(u")-N(u)|+|N(u')-N(u)|} \sup_{p}\lambda_{n-1}(p) \\ &\displaystyle\lesssim \frac{|u"-u'|}{|N(u")-N(u')|} Q(S)^{\frac{5(n-2)}{2}} \sup_{p}\lambda_{n-1}(p) \\ &\displaystyle\lesssim \frac{1}{\inf_{p}\lambda_{1}(p)} Q(S)^{\frac{5(n-2)}{2}} \sup_{p}\lambda_{n-1}(p) \\ &\displaystyle\lesssim Q(S)^{\frac{5n-8}{2}}, \end{aligned} \end{align} $$

by the mean-value inequality applied to the Gauss map N. This implies (4.20), completing the proof of Theorem 1.7 (Theorem 4.11).

4.3 Proof of the Sobolev–Mizohata–Takeuchi inequality (Theorem 1.8)

In this section we prove Theorem 1.8, or more specifically, Theorem 4.13. We begin by observing that if $u\not \in \mathrm {supp\,}^*(g)$ and $u'\in S$ , then either $u'\not \in \mathrm {supp\,}(g)$ or $R_u u'\not \in \mathrm {supp\,}(g)$ , meaning that ${I_{S,s}(|g|^2, |g|^2)(u)=0}$ . Consequently, by Theorem 4.11,

$$ \begin{align*}\int_{\mathbb{R}^n}|\widehat{g\mathrm{d}\sigma}|^2w\leq c Q(S)^{\frac{5n-8}{4}} \sup_{u\in\mathrm{supp\,}^*(g)}\|X_Sw(u,\cdot)\|_{\dot{H}^s(T_u S)}\int_S I_{S,s}(|g|^2,|g|^2)(u)^{1/2}\mathrm{d}\sigma(u), \end{align*} $$

and so we are reduced to proving a suitable $L^1(S)\times L^1(S)\rightarrow L^{1/2}(S)$ estimate on the bilinear operator

(4.33) $$ \begin{align} I_{S,s}(g_1,g_2)(u):=\int_{S}\frac{g_1(u')g_2(R_u u')}{|u'-R_u u'|^{s}}J(u,u')\mathrm{d}\sigma(u') \end{align} $$

whenever $s<n-1$ . This follows by a direct application of the forthcoming Theorem 7.2.

4.4 Improved Sobolev–Stein constants in the plane

Our proof of Theorem 1.7 identifies $\|J/\widetilde {J}\|_\infty ^{1/2}$ as the naturally occurring dilation-invariant functional on the surface S, rather than the power of the curvature quotient $Q(S)$ that we use to bound it. In two dimensions our expression for J, being relatively simple, permits the bound $\|J/\widetilde {J}\|_\infty ^{1/2}\lesssim \Lambda (S)$ , where $\Lambda (S)$ is defined in (1.13). To see this we argue as in (4.32), using Propositions 4.5 and 4.16 to write

$$ \begin{align*} \begin{aligned} \frac{J(u,u')}{\widetilde{J}(u,u')}\leq \min\{J(u,u'),J(u,u")\}=|u'-u"|K(u)\min\left\{\frac{1}{|N(u)\wedge N(u")|}, \frac{1}{|N(u)\wedge N(u')|}\right\} \lesssim \Lambda(S). \end{aligned} \end{align*} $$

The two-dimensional case of Theorem 1.7 may then be strengthened to the following:

Theorem 4.20 (Improved Sobolev–Stein in the plane).

Suppose that $s<\frac {1}{2}$ . There is an absolute constant c such that

$$ \begin{align*}\int_{\mathbb{R}^2}|\widehat{g\mathrm{d}\sigma}(x)|^2w(x)\mathrm{d}x\leq c\Lambda(S)\int_S I_{S,2s}(|g|^2,|g|^2)(u)^{1/2}\|X_Sw(u,\cdot)\|_{\dot{H}^s(T_u S)}\mathrm{d}\sigma(u). \end{align*} $$

A similar, although potentially rather more complicated statement is possible in higher dimensions, and is left to the interested reader.

5 Estimating distances: the proof of Proposition 4.4

We begin with (4.5), and the elementary observation that if $\pi $ is 2-plane that is normal to S at a point u, then by (1.12), it must be close to normal at all points of intersection with S. More specifically, for $\widetilde {u}\in S$ we have

$$ \begin{align*}|P_\pi N(\widetilde{u})|\geq|P_{(T_uS)^\perp}N(\widetilde{u})|=N(u)\cdot N(\widetilde{u})\geq 1/2.\end{align*} $$

It follows by Meusnier’s theorem that for such a $\pi $ , the curvature of the curve $S\cap \pi $ at a point is comparable to a normal curvature of S at that same point. This allows us to transfer the curvature quotient of S to such curves, and we shall appeal to this momentarily.

Now let $\pi '$ and $\pi "$ be the normal 2-planes at the point u that pass through the points $u'$ and $u"$ respectively. Let x be the orthogonal projection of u onto the plane $T_uS+\{u'\}$ , and note that $\{u,u',x\}$ and $\{u,u",x\}$ are the vertices of right-angled triangles in the 2-planes $\pi '$ and $\pi "$ respectively. Next observe that by the triangle inequality and Pythagoras’ theorem, it is enough to show that

(5.1) $$ \begin{align} |x-u"|\lesssim Q(S)^{1/2}|x-u'|. \end{align} $$

To see this we write S as a graph over $T_uS+\{u'\}$ as follows: let $\phi _u:T_uS+\{u'\}\rightarrow \mathbb {R}$ be such that $x'\mapsto x'+\phi _u(x')N(u)$ is a bijective map from a subset $U\subset T_uS$ into S; see Figure 1. That this is possible, and indeed that $\phi _u$ is uniquely defined, follows from (1.12) (a point that is elaborated in [Reference Bennett, Nakamura and Shiraki15]). Notice that

(5.2) $$ \begin{align} \phi_u(u')=0, \phi_u(x)=|x-u|\;\;\text{ and }\;\;\nabla\phi_u(x)=0, \end{align} $$

by construction. Assuming that $N(u)=e_n$ , as we may, the graph condition (1.12) implies that the normal vector $(\nabla \phi _u, -1)$ lies in some fixed (proper) vertical cone, and so in particular we also have

(5.3) $$ \begin{align} |\nabla\phi_u|\lesssim 1. \end{align} $$

We now apply Taylor’s theorem on the line segment $[x,u']$ , along with (5.2), to obtain

$$ \begin{align*}|x-u|=\phi_u(x)-\phi_u(u')=\frac{1}{2}k'(u,u')|x-u'|^2, \end{align*} $$

where $k'(u,u')$ is a quantity comparable to some normal curvature of S at some point. Here we have used (5.3) along with our initial observation via Meusnier’s theorem. By symmetry a similar statement may be made with $u"$ in place of $u'$ , from which we deduce that

$$ \begin{align*}k'(u,u')|x-u'|^2=k"(u,u")|x-u"|^2.\end{align*} $$

The inequality (5.1) now follows from the definition of $Q(S)$ and taking square roots.

Turning to (4.6), we fix u and exploit the properties of the map $H:=H_{\omega }= N^{-1}\circ \Phi _{\omega }$ from Section 6. By the mean value theorem and Claim 6.4,

$$ \begin{align*} \begin{aligned} \displaystyle |u-u'| {\kern-1pt}={\kern-1pt} |H(0)-H(x')| &\displaystyle{\kern-1pt}\leq{\kern-1pt} \sup_{\theta}\|\mathrm{d}H_{\theta}\|\cdot |x'| \displaystyle{\kern-1pt}\leq{\kern-1pt} \sup_{\theta}\|\mathrm{d}H_{\theta}\|\cdot \frac{|(1-\widetilde{\eta}(x'))x'|}{|1-\widetilde{\eta}(x')|} \displaystyle=\sup_{\theta}\|\mathrm{d}H_{\theta}\|\cdot \frac{|x'-x"|}{|1-\widetilde{\eta}(x')|}, \end{aligned} \end{align*} $$

where $x"$ is such that $H(x")=u"$ . Consequently,

$$ \begin{align*} \begin{aligned} \displaystyle |u-u'| &\displaystyle\leq\sup_{\theta}\|\mathrm{d}H_{\theta}\|\cdot \frac{|H^{-1}(H(x'))-H^{-1}(H(x"))|}{|1-\widetilde{\eta}(x')|} \\ &\displaystyle\leq\sup_{\theta}\|\mathrm{d}H_{\theta}\|\cdot \sup_{\widetilde{\theta}}\|\mathrm{d}H^{-1}_{\widetilde{\theta}}\|\cdot \frac{|H(x')-H(x")|}{|1-\widetilde{\eta}(x')|} \\ &\displaystyle=\sup_{\theta}\|\mathrm{d}H_{\theta}\|\cdot \sup_{\widetilde{\theta}}\|\mathrm{d}H^{-1}_{\widetilde{\theta}}\|\cdot \frac{|u'-u"|}{|1-\widetilde{\eta}(x')|}, \end{aligned} \end{align*} $$

and therefore

$$ \begin{align*} |u'-u"|\geq \frac{|1-\widetilde{\eta}(x')|}{\sup_{\theta}\|\mathrm{d}H_{\theta}\|\cdot \sup_{\widetilde{\theta}}\|\mathrm{d}H^{-1}_{\widetilde{\theta}}\|}\cdot |u-u'|. \end{align*} $$

We also have, for a fixed $\theta $ ,

$$ \begin{align*} \|\mathrm{d}H_{\theta}\|\leq \|\mathrm{d}N^{-1}_{\Phi(\theta)}\|\cdot \|\mathrm{d}\Phi_{\theta}\|\leq\frac{1}{\inf_{p\in S}\lambda_1(p)}\cdot \|\mathrm{d}\Phi_{\theta}\|_{L^{\infty}_{\theta}}, \end{align*} $$

where $\inf _{p\in S}\lambda _1(p)$ is the infimum over $p\in S$ of the smallest eigenvalue $\lambda _1(p)$ of the shape operator $\mathrm {d}N_{p}$ . Similarly,

$$ \begin{align*} \|\mathrm{d}H^{-1}_{\widetilde{\theta}}\|\leq \|\mathrm{d}\Phi^{-1}_{\widetilde{\theta}}\|\cdot \|\mathrm{d}N_{\Phi(\widetilde{\theta})}\|\leq\|\mathrm{d}\Phi^{-1}_{\widetilde{\theta}}\|_{L^{\infty}_{\widetilde{\theta}}}\cdot\sup_{p}\lambda_{n-1}(p), \end{align*} $$

where $\sup _{p\in S}\lambda _{n-1}(p)$ is the supremum over $p\in S$ of the largest eigenvalue $\lambda _{n-1}(p)$ . Consequently,

(5.4) $$ \begin{align} |u'-u"|\gtrsim |1-\widetilde{\eta}(x')|\cdot \frac{\inf_{p\in S}\lambda_1(p)}{\sup_{p\in S}\lambda_{n-1}(p)}\cdot |u-u'|\gtrsim \frac{1}{Q(S)}\cdot |u-u'|, \end{align} $$

since $\widetilde {\eta }<0$ by the strict convexity of S.

6 Computing Jacobians: the proof of Proposition 4.5

In this section we provide detailed proofs of (4.7), (4.8) and (4.9). The key idea is that the maps $u\mapsto R_u u'$ and $u'\mapsto R_uu'$ may be transformed into outward vector fields on Euclidean spaces (specifically $T_{u'}S$ and $T_uS$ respectively) by conjugating them with a composition of the Gauss map and a suitable stereographic projection. The derivatives of such vector fields have only two eigenspaces, allowing the computation of their Jacobians to be reduced to the identification of just two eigenvalues, one of which has multiplicity $n-2$ (see the forthcoming Lemma 6.2). This is manifested in the factor raised to the power $n-2$ in the formula (4.7) for J. We begin by recalling and introducing the notation and geometric objects that will feature in our computations of J and $\Delta $ .

• • $N:S\rightarrow \mathbb {S}^{n-1}$ is the Gauss map, $\mathrm {d}N_{u}:T_{u}S\rightarrow T_{N(u)}\mathbb {S}^{n-1}$ is the shape operator (recall that $T_{u}S= T_{N(u)}\mathbb {S}^{n-1}$ ), and $K(u)=\det (\mathrm {d}N_{u})$ is the Gaussian curvature at $u\in S$ .

• • The formulas of this section will be written in terms of the parameters u, $u'$ and $u"=R_{u}u'$ , which are points on S. We will denote their images via the Gauss map by $\omega $ , $\omega ^{\prime }$ and $\omega ^{\prime \prime }$ , respectively.

• • For a fixed $\omega ^{\prime }\in \mathbb {S}^{n-1}$ , $\Phi _{\omega ^{\prime }}:\langle \omega ^{\prime }\rangle ^{\perp }\rightarrow \mathbb {S}^{n-1}$ denotes the inverse of the stereographic projection map with respect to $-\omega ^{\prime }$ . Explicitly

  (6.1) $$ \begin{align} \Phi_{\omega^{\prime}}(x)=\left(\frac{2x}{1+|x|^{2}},\frac{1-|x|^{2}}{1+|x|^{2}}\right) \end{align} $$
  via the identification $\mathbb {R}^n=\langle \omega '\rangle ^\perp \times \langle \omega '\rangle $ . If $\omega =\Phi _{\omega ^{\prime }}(x)$ , it follows that
  (6.2) $$ \begin{align} x=\frac{\omega-\langle\omega,\omega^{\prime}\rangle\omega^{\prime}}{1+\langle\omega,\omega^{\prime}\rangle}. \end{align} $$
  The differential $(\mathrm {d}\Phi _{\omega ^{\prime }})_{x}:\langle \omega ^{\prime }\rangle ^{\perp }\rightarrow \langle \omega \rangle ^{\perp }$ satisfies
  (6.3) $$ \begin{align} (\mathrm{d}\Phi_{\omega^{\prime}})_{x}(x)=\langle\omega,\omega^{\prime}\rangle\omega-\omega^{\prime}. \end{align} $$
  The determinants of $(\mathrm {d}\Phi _{\omega ^{\prime }})_{x}$ and its inverse are, respectively,
  (6.4) $$ \begin{align} \det((\mathrm{d}\Phi_{\omega^{\prime}})_{x})=\left(\frac{2}{1+|x|^{2}}\right)^{n-1}=(1+\langle\omega,\omega^{\prime}\rangle)^{n-1} \end{align} $$
  and
  (6.5) $$ \begin{align} \det((\mathrm{d}\Phi_{\omega^{\prime}}^{-1})_{\omega})=\left(\frac{1}{1+\langle\omega,\omega^{\prime}\rangle}\right)^{n-1}. \end{align} $$
  We refer the reader to Chapter 4 of [Reference Lieb and Loss39] for further discussion on the properties of these maps.

• • For $\omega $ fixed, set

  $$ \begin{align*} H_{\omega}=N^{-1}\circ\Phi_{\omega}. \end{align*} $$
  $H_{\omega }$ will play a crucial role in this section. As we shall see, it allows us to reduce the computations of J and $\Delta $ to certain Euclidean analogues with simple spectral structure (outward vector fields, as discussed above and alluded to in Remark 4.3).

We are now ready to prove (4.7), (4.8) and (4.9).

6.1 Computing J

For fixed $\omega '$ we define the map $\Psi _{\omega ^{\prime }}:N(S)\rightarrow \mathbb {S}^{n-1}$ by

(6.6) $$ \begin{align} \Psi_{\omega^{\prime}}(\omega)=N(R_{N^{-1}(\omega)}N^{-1}(\omega^{\prime})). \end{align} $$

Strictly speaking the domain of $\Psi _{\omega ^{\prime }}$ depends on $\omega '$ , as we allude to in Remark 4.1. The parameter $\omega \in \mathbb {S}^{n-1}$ will be a variable and we will use $x\in \langle \omega ^{\prime }\rangle ^{\perp }$ to represent its preimage by the map $\Phi _{\omega ^{\prime }}$ . Explicitly,

By (6.6) and the definition of $J(u,u')$ , along with the fact that the Gaussian curvature $K(u)$ is the determinant of the shape operator $\mathrm {d}N_u$ , we have

(6.7) $$ \begin{align} J(u,u')=\left|\det{\left(\mathrm{d}\Psi_{N(u')}(N(u))\right)}\right|\frac{K(u)}{K(u")}. \end{align} $$

The next step is to reduce the computation of the Jacobian determinant $\det {\left (\mathrm {d}\Psi _{N(u')}(N(u))\right )}$ to one of a much simpler outward vector field $\varphi $ on the tangent space at $u'$ (see Lemma 6.2 below). This will be achieved by combining properties of the inverse stereographic projection map $\Phi _{\omega ^{\prime }}$ with the geometric condition (4.3). To this end we define the map $\varphi :\langle \omega ^{\prime }\rangle ^{\perp }\rightarrow \langle \omega ^{\prime }\rangle ^{\perp }$ by

$$ \begin{align*}\varphi(x):=\Phi_{\omega^{\prime}}^{-1}\circ\Psi_{\omega^{\prime}}\circ\Phi_{\omega^{\prime}}(x).\end{align*} $$

Claim 6.1. The vector field $\varphi :\langle \omega '\rangle ^\perp \rightarrow \langle \omega '\rangle ^\perp $ is given by

(6.8) $$ \begin{align} \varphi(x) = \eta(x)x, \end{align} $$

where

(6.9) $$ \begin{align} \eta(x) = \frac{\langle x,H_{\omega^{\prime}}^{-1}(R_{H_{\omega^{\prime}}(x)}H_{\omega^{\prime}}(0))\rangle}{|x|^{2}}=\frac{\langle x,\Phi_{\omega^{\prime}}^{-1}(\omega^{\prime\prime})\rangle}{|x|^{2}}. \end{align} $$

Proof of Claim 6.1.

By definition of the map $R_{(\cdot )}u'$ , the normals $\omega $ , $\omega ^{\prime }$ and $\omega ^{\prime \prime }$ are coplanar; therefore, they lie on a great circle. This implies that

$$ \begin{align*}\varphi(x)=\mu(x)x\end{align*} $$

for some $\mu (x)$ , which we conclude to be equal to $\eta (x)$ by taking scalar products with x on both sides of the equation above.

By the chain rule,

$$ \begin{align*}\det(\mathrm{d}\varphi(x))=\det((\mathrm{d}\Phi_{\omega^{\prime}}^{-1})(\omega^{\prime\prime}))\det((\mathrm{d}\Psi_{\omega^{\prime}})(\omega))\det((\mathrm{d}\Phi_{\omega^{\prime}})(x)),\end{align*} $$

hence

$$ \begin{align*} \det((\mathrm{d}\Psi_{\omega^{\prime}})(\omega))=\frac{\det(\mathrm{d}\varphi(x))}{\det((\mathrm{d}\Phi_{\omega^{\prime}}^{-1})(\omega^{\prime\prime}))\det((\mathrm{d}\Phi_{\omega^{\prime}})(x))}. \end{align*} $$

This implies, by (6.7),

(6.10) $$ \begin{align} J(u,u')=\frac{|\det(\mathrm{d}\varphi(x))|}{|\det((\mathrm{d}\Phi_{\omega^{\prime}}^{-1})(\omega^{\prime\prime}))||\det((\mathrm{d}\Phi_{\omega^{\prime}})(x))|}\frac{K(u)}{K(u")}. \end{align} $$

We are now in a position to invoke the following elementary lemma, whose proof is left to the reader:

Lemma 6.2 (Differential structure of an outward vector field).

Let $\eta :\mathbb {R}^{d}\rightarrow \mathbb {R}$ be a $C^{1}$ function and let $\varphi :\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ be given by

(6.11) $$ \begin{align} \varphi(x)=\eta(x)x. \end{align} $$

The linear map

$$ \begin{align*}\mathrm{d}\varphi(x)=x(\nabla\eta(x))^{\top}+\eta(x) I_{d}\end{align*} $$

has eigenvalues $\lambda _{1}(x)=\eta (x)$ and $\lambda _{2}(x)=\langle \nabla \eta (x),x\rangle + \eta (x)$ of multiplicity $(d-1)$ and $1$ , respectively. The eigenspaces associated to these eigenvalues are

$$ \begin{align*} \begin{aligned} \displaystyle E_{\lambda_{1}(x)}&\displaystyle :=\langle\nabla\eta(x)\rangle^{\perp}, \\ \displaystyle E_{\lambda_{2}(x)}&\displaystyle :=\langle x\rangle. \end{aligned} \end{align*} $$

In particular,

(6.12) $$ \begin{align} \det(\mathrm{d}\varphi(x))=[\eta(x)]^{d-1}(\langle\nabla\eta(x),x\rangle + \eta(x)). \end{align} $$

The parameter $u'\in S$ is fixed in this subsection; therefore, $\omega ^{\prime }$ will also be fixed, and we write $H_{\omega ^{\prime }}=H$ to simplify notation. Let us use (6.12) to compute $\det (\mathrm {d}\varphi (x))$ . The eigenvalue $\lambda _{1}(x)$ of $\mathrm {d}\varphi (x)$ is

$$ \begin{align*} \lambda_{1}(x)=\eta(x)=\frac{\langle x,H^{-1}(R_{H(x)}H(0))\rangle}{|x|^{2}}, \end{align*} $$

hence, by (6.12), all there is left to do is to compute the eigenvalue $\lambda _{2}(x)$ of $\mathrm {d}\varphi (x)$ . By definition of the map $R_{(\cdot )}u'$ , the vector $R_{u}u'-u'$ is in the tangent space of S at u. In short,

$$ \begin{align*} \langle R_{u}u'-u', N(u)\rangle = 0. \end{align*} $$

Equivalently,

(6.13) $$ \begin{align} \langle H(\eta(x)x)-H(0), N(H(x))\rangle = 0. \end{align} $$

Differentiating both sides of (6.13) with respect to x,

$$ \begin{align*} \displaystyle 0 = \mathrm{d}(N\circ H)_{x}^{\top}\left(H(\eta(x)x)-H(0)\right) + \left(x\cdot\nabla\eta(x)^{\top}+\eta(x)I_{n-1}\right)^{\top}\mathrm{d}H_{\eta(x) x}^{\top}\left(N\circ H(x)\right). \end{align*} $$

Taking scalar products on both sides with x and using that $N\circ H=\Phi _{\omega '}$ , we have

$$ \begin{align*} \displaystyle 0 = \langle H(\eta(x)x)-H(0),(\mathrm{d}\Phi_{\omega^{\prime}})_{x}(x)\rangle + \langle\mathrm{d}H_{\eta(x) x}^{\top}\left(\Phi_{\omega^{\prime}}(x)\right),\left(x\cdot\nabla\eta(x)^{\top}+\eta(x)I_{n-1}\right)(x)\rangle. \end{align*} $$

By Lemma 6.2,

$$ \begin{align*}\left(x\cdot\nabla\eta(x)^{\top}+\eta(x)I_{n-1}\right)(x)=(\langle\nabla\eta(x),x\rangle+\eta(x)) x,\end{align*} $$

and hence

$$ \begin{align*} \lambda_2(x)=\langle\nabla\eta(x),x\rangle+\eta(x) = -\frac{\langle H(\eta(x)x)-H(0),(\mathrm{d}\Phi_{\omega^{\prime}})_{x}(x)\rangle}{\langle\mathrm{d}H_{\eta(x) x}^{\top}\left(\Phi_{\omega^{\prime}}(x)\right),x\rangle}. \end{align*} $$

By Lemma 6.2 again,

$$ \begin{align*} \det(\mathrm{d}\varphi(x))=-[\eta(x)]^{n-2}\frac{\langle H(\eta(x)x)-H(0),(\mathrm{d}\Phi_{\omega^{\prime}})_{x}(x)\rangle}{\langle\mathrm{d}H_{\eta(x) x}^{\top}\left(\Phi_{\omega^{\prime}}(x)\right),x\rangle}. \end{align*} $$

By (6.10),

(6.14) $$ \begin{align} J(u,u')=|\eta(x)|^{n-2}\frac{1}{|\langle\mathrm{d}H_{\eta(x) x}^{\top}\left(\Phi_{\omega^{\prime}}(x)\right),x\rangle|}\frac{|\langle H(\eta(x)x)-H(0),(\mathrm{d}\Phi_{\omega^{\prime}})_{x}(x)\rangle|}{|\det((\mathrm{d}\Phi_{\omega^{\prime}}^{-1})(\omega^{\prime\prime}))||\det((\mathrm{d}\Phi_{\omega^{\prime}})(x))|}\frac{K(u)}{K(u")}. \end{align} $$

To proceed, we need to understand each factor in the formula above, which is the content of the next claim.

Claim 6.3. The following identities hold:

(6.15) $$ \begin{align} |\eta(x)|=\left|\frac{\langle\omega,\omega^{\prime\prime}\rangle-\langle\omega,\omega^{\prime}\rangle\langle\omega^{\prime},\omega^{\prime\prime}\rangle}{(1+\langle\omega^{\prime\prime},\omega^{\prime}\rangle)(1-\langle\omega,\omega^{\prime}\rangle)}\right|; \end{align} $$

(6.16) $$ \begin{align} \langle H(\eta(x)x)-H(0),(\mathrm{d}\Phi_{\omega^{\prime}})_{x}(x)\rangle = -\langle u"-u',\omega'\rangle; \end{align} $$

(6.17) $$ \begin{align} \langle\mathrm{d}H_{\eta(x) x}^{\top}\left(\Phi_{\omega^{\prime}}(x)\right),x\rangle = \frac{1}{\eta(x)}\langle\omega,\mathrm{d}N^{-1}_{\omega^{\prime\prime}}(\langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime\prime}-\omega^{\prime})\rangle. \end{align} $$

Let us assume Claim 6.3 for the moment and complete the proof of the proposition. By the claim, (6.4) and (6.5),

(6.18) $$ \begin{align} \displaystyle J(u,u')&=\displaystyle\left|\frac{\langle\omega,\omega^{\prime\prime}\rangle-\langle\omega,\omega^{\prime}\rangle\langle\omega^{\prime},\omega^{\prime\prime}\rangle}{(1+\langle\omega^{\prime\prime},\omega^{\prime}\rangle)(1-\langle\omega,\omega^{\prime}\rangle)}\right|{}^{n-1}\frac{|\langle u"-u',\omega'\rangle|}{|\langle\omega,\mathrm{d}N^{-1}_{\omega^{\prime\prime}}(\langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime\prime}-\omega^{\prime})\rangle|}\frac{|1+\langle\omega^{\prime\prime},\omega^{\prime}\rangle|^{n-1}}{|1+\langle\omega,\omega^{\prime}\rangle|^{n-1}}\frac{K(u)}{K(u")} \nonumber\\&=\displaystyle\left|\frac{\langle\omega,\omega^{\prime\prime}\rangle-\langle\omega,\omega^{\prime}\rangle\langle\omega^{\prime},\omega^{\prime\prime}\rangle}{1-\langle\omega,\omega^{\prime}\rangle^{2}}\right|{}^{n-1}\frac{|\langle u"-u',\omega'\rangle|}{|\langle\omega,\mathrm{d}N^{-1}_{\omega^{\prime\prime}}(\langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime\prime}-\omega^{\prime})\rangle|}\frac{K(u)}{K(u")}. \nonumber\\&=\displaystyle\left(\frac{(1-\langle\omega',\omega"\rangle^{2})^{\frac{1}{2}}}{(1-\langle\omega,\omega^{\prime}\rangle^{2})^{\frac{1}{2}}}\right)^{n-1}\frac{|\langle u"-u',\omega'\rangle|}{|\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(\langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime\prime}-\omega^{\prime})\rangle|}\frac{K(u)}{K(u")}, \end{align} $$

where we used the facts that $\langle \omega ,v\rangle =\langle P_{T_{u"}S}N(u),v\rangle $ for every $v\in T_{u"}S$ , and that three coplanar vectors $\omega ,\omega '$ and $\omega "$ on the sphere satisfy

$$ \begin{align*} \langle\omega,\omega^{\prime\prime}\rangle-\langle\omega,\omega^{\prime}\rangle\langle\omega^{\prime},\omega^{\prime\prime}\rangle = (1-\langle\omega',\omega"\rangle^{2})^{\frac{1}{2}}(1-\langle\omega,\omega'\rangle^{2})^{\frac{1}{2}}. \end{align*} $$

We exploit the coplanarity of $\omega ,\omega '$ and $\omega "$ twice more. First, it implies the existence of $a,b\in \mathbb {R}$ such that

(6.19) $$ \begin{align} \omega"=a\omega+b\omega'. \end{align} $$

Consequently,

$$ \begin{align*} \frac{|\langle u"-u',\omega^{\prime\prime}\rangle|}{|\langle u"-u',\omega^{\prime}\rangle|}=\frac{|\langle u"-u',a\omega+b\omega'\rangle|}{|\langle u"-u',\omega^{\prime}\rangle|}=|b|, \end{align*} $$

since $u"-u'$ is perpendicular to $N(u)=\omega $ . On the other hand, projecting both sides of (6.19) to $\langle \omega \rangle ^{\perp }$ gives

$$ \begin{align*} P_{\langle\omega\rangle^{\perp}}\omega" = b P_{\langle\omega\rangle^{\perp}}\omega'\Longrightarrow |b|=\frac{|P_{\langle\omega\rangle^{\perp}}\omega"|}{|P_{\langle\omega\rangle^{\perp}}\omega'|}, \end{align*} $$

which in turn implies

(6.20) $$ \begin{align} \frac{|\langle u"-u',\omega^{\prime\prime}\rangle|}{|\langle u"-u',\omega^{\prime}\rangle|}=\frac{|P_{\langle\omega\rangle^{\perp}}\omega"|}{|P_{\langle\omega\rangle^{\perp}}\omega'|} \Longrightarrow |\langle u"-u',\omega^{\prime}\rangle|=\frac{(1-\langle\omega,\omega^{\prime}\rangle^{2})^{\frac{1}{2}}}{(1-\langle\omega,\omega"\rangle^{2})^{\frac{1}{2}}}|\langle u"-u',N(u")\rangle|. \end{align} $$

Second, the fact that $\omega ,\omega '$ and $\omega "$ are coplanar also gives us that $P_{\langle \omega "\rangle ^{\perp }}\omega '$ and $P_{\langle \omega "\rangle ^{\perp }}\omega $ are parallel, therefore

(6.21) $$ \begin{align} \langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime\prime}-\omega^{\prime}=P_{\langle\omega"\rangle^{\perp}}\omega'=\frac{|P_{\langle\omega"\rangle^{\perp}}\omega'|}{|P_{\langle\omega"\rangle^{\perp}}\omega|}P_{\langle\omega"\rangle^{\perp}}\omega =\frac{(1-\langle\omega',\omega"\rangle^{2})^{\frac{1}{2}}}{(1-\langle\omega,\omega"\rangle^{2})^{\frac{1}{2}}}P_{T_{u"}S}N(u). \end{align} $$

Likewise, or by symmetry,

(6.22) $$ \begin{align} \langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime}-\omega^{\prime\prime}=P_{\langle\omega'\rangle^{\perp}}\omega"=\frac{|P_{\langle\omega'\rangle^{\perp}}\omega"|}{|P_{\langle\omega'\rangle^{\perp}}\omega|}P_{\langle\omega'\rangle^{\perp}}\omega=\frac{(1-\langle\omega',\omega"\rangle^{2})^{\frac{1}{2}}}{(1-\langle\omega,\omega'\rangle^{2})^{\frac{1}{2}}}P_{T_{u'}S}N(u). \end{align} $$

Using (6.20) and (6.21) in (6.18) gives (4.7). We now move to the final part of the argument.

Proof of Claim 6.3.

By (6.9) and (6.2),

$$ \begin{align*} \begin{aligned} \displaystyle|\eta(x)|&\displaystyle=\left|\left\langle \frac{\omega-\langle\omega,\omega^{\prime}\rangle\omega^{\prime}}{1+\langle\omega,\omega^{\prime}\rangle},\frac{\omega^{\prime\prime}-\langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime}}{1+\langle\omega^{\prime\prime},\omega^{\prime}\rangle}\right\rangle\right|\frac{|1+\langle\omega,\omega^{\prime}\rangle|^{2}}{|\omega-\langle\omega,\omega^{\prime}\rangle\omega^{\prime}|^{2}} \\ &\displaystyle= \left|\frac{\langle\omega,\omega^{\prime\prime}\rangle-\langle\omega,\omega^{\prime}\rangle\langle\omega^{\prime},\omega^{\prime\prime}\rangle}{(1+\langle\omega^{\prime\prime},\omega^{\prime}\rangle)(1-\langle\omega,\omega^{\prime}\rangle)}\right|, \end{aligned} \end{align*} $$

which verifies (6.15). To establish (6.16), we simply observe that $H(\eta (x)x)-H(0)=u"-u'$ , and this together with (6.3) implies that

$$ \begin{align*} \langle H(\eta(x)x)-H(0),(\mathrm{d}\Phi_{\omega^{\prime}})_{x}(x)\rangle = \langle u"-u',\langle\omega,\omega^{\prime}\rangle\omega-\omega^{\prime}\rangle = -\langle u"-u',\omega'\rangle, \end{align*} $$

since $u"-u'$ is perpendicular to $\omega $ by definition of $u"$ . Finally, notice that $\Phi _{\omega ^{\prime }}(\eta (x)x)=\omega ^{\prime \prime }$ and that a direct computation gives

(6.23) $$ \begin{align} (\mathrm{d}\Phi_{\omega^{\prime}})_{\eta(x)x}(x)=\frac{1}{\eta(x)}\left(\langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime\prime}-\omega^{\prime}\right). \end{align} $$

Therefore by definition of H, the chain rule and (6.23), we have

$$ \begin{align*} \begin{aligned} \displaystyle \langle\mathrm{d}H_{\eta(x) x}^{\top}\left(\Phi_{\omega^{\prime}}(x)\right),x\rangle &\displaystyle = \langle\omega,\mathrm{d}H_{\eta(x) x}(x)\rangle \\ &\displaystyle= \langle\omega,\mathrm{d}N^{-1}_{\Phi_{\omega^{\prime}}(\eta(x)x)}\circ(\mathrm{d}\Phi_{\omega^{\prime}})_{\eta(x)x}(x)\rangle\\ &\displaystyle= \frac{1}{\eta(x)}\langle\omega,\mathrm{d}N^{-1}_{\omega^{\prime\prime}}(\langle\omega^{\prime\prime},\omega^{\prime}\rangle\omega^{\prime\prime}-\omega^{\prime})\rangle, \end{aligned} \end{align*} $$

which concludes the proof of Claim 6.3.

6.2 Computing $\Delta $

Arguing as in Section 6.1, for fixed $\omega $ we define the map $\widetilde {\Psi }_{\omega }:N(S)\rightarrow \mathbb {S}^{n-1}$ by

(6.24) $$ \begin{align} \widetilde{\Psi}_{\omega}(\omega')=N(R_{N^{-1}(\omega)}N^{-1}(\omega^{\prime})). \end{align} $$

Recalling from Section 4 that $\Delta (u,u")$ is the Jacobian of the change of variables $u'=R_u u"$ , it follows that

(6.25) $$ \begin{align} \Delta(u,u')=\left|\det{\left(\mathrm{d}\widetilde{\Psi}_{N(u)}(N(u'))\right)}\right|\frac{K(u')}{K(u")}. \end{align} $$

Recall that $\omega ^{\prime }\in \mathbb {S}^{n-1}$ is a variable now. We will use $x'\in \langle \omega \rangle ^{\perp }$ to represent its preimage by the map $\Phi _{\omega }$ :

Once more we reduce the computation of $\det {\left (\mathrm {d}\widetilde {\Psi }_{N(u)}(N(u'))\right )}$ to an application of Lemma 6.2. Define $\widetilde {\varphi }:\langle \omega \rangle ^{\perp }\rightarrow \langle \omega \rangle ^{\perp }$ by

$$ \begin{align*}\widetilde{\varphi}(x'):=\Phi_{\omega}^{-1}\circ\widetilde{\Psi}_{\omega}\circ\Phi_{\omega}(x').\end{align*} $$

Claim 6.4. $\widetilde {\varphi }$ is given by

(6.26) $$ \begin{align} \widetilde{\varphi}(x') = \widetilde{\eta}(x')x', \end{align} $$

where

(6.27) $$ \begin{align} \widetilde{\eta}(x') = \frac{\langle x',H_{\omega}^{-1}(R_{H_{\omega}(0)}H_{\omega}(x'))\rangle}{|x'|^{2}}=\frac{\langle x',\Phi_{\omega}^{-1}(\omega^{\prime\prime})\rangle}{|x'|^{2}}. \end{align} $$

The proof of Claim 6.4 is similar to the one of Claim 6.1. By the chain rule,

$$ \begin{align*}\det(\mathrm{d}\widetilde{\varphi}(x'))=\det((\mathrm{d}\Phi_{\omega}^{-1})(\omega^{\prime\prime}))\det((\mathrm{d}\Psi_{\omega})(\omega^{\prime}))\det((\mathrm{d}\Phi_{\omega})(x')),\end{align*} $$

hence

$$ \begin{align*} \det((\mathrm{d}\widetilde{\Psi}_{\omega})(\omega^{\prime}))=\frac{\det(\mathrm{d}\widetilde{\varphi}(x'))}{\det((\mathrm{d}\Phi_{\omega}^{-1})(\omega^{\prime\prime}))\det((\mathrm{d}\Phi_{\omega})(x'))}. \end{align*} $$

This implies, by (6.25), that

(6.28) $$ \begin{align} \Delta(u,u')=\frac{|\det(\mathrm{d}\widetilde{\varphi}(x'))|}{|\det((\mathrm{d}\Phi_{\omega}^{-1})(\omega^{\prime\prime}))||\det((\mathrm{d}\Phi_{\omega})(x'))|}\frac{K(u')}{K(u")}. \end{align} $$

The parameter $u\in S$ is fixed in this subsection (and therefore so is $\omega \in \mathbb {S}^{n-1}$ ), so we lighten notation by writing $H_{\omega }=H$ . We may again compute $\det (\mathrm {d}\widetilde {\varphi }(x'))$ using Lemma 6.2. The eigenvalue $\widetilde {\lambda }_{1}(x')$ of $\mathrm {d}\widetilde {\varphi }(x')$ is

$$ \begin{align*} \widetilde{\lambda}_{1}(x')=\widetilde{\eta}(x')=\frac{\langle x',H^{-1}(R_{H(0)}H(x'))\rangle}{|x'|^{2}}, \end{align*} $$

hence we just have to compute the eigenvalue $\widetilde {\lambda }_{2}(x')$ of $\mathrm {d}\widetilde {\varphi }(x')$ and use (6.12). Recall that

$$ \begin{align*} \langle R_{u}u'-u', N(u)\rangle = 0. \end{align*} $$

Equivalently,

(6.29) $$ \begin{align} \langle H(\widetilde{\eta}(x')x')-H(x'), N(H(0))\rangle = 0. \end{align} $$

Differentiating both sides of (6.29) with respect to $x'$ and taking scalar products with $x'$ as well gives

$$ \begin{align*} \langle (\mathrm{d}H_{\widetilde{\varphi}(x')}\circ\mathrm{d}\varphi_{x'})^{\top}(\Phi_{\omega}(0)),x'\rangle = \langle\omega,\mathrm{d}H_{x'}(x')\rangle, \end{align*} $$

which in turn implies that

$$ \begin{align*} \langle (\mathrm{d}H_{\widetilde{\varphi}(x')})^{\top}(\omega),(\mathrm{d}\widetilde{\varphi}_{x'})(x')\rangle =\langle(\mathrm{d}H_{x'})^{\top}\omega,(x')\rangle= \langle\omega,\mathrm{d}H_{x'}(x')\rangle. \end{align*} $$

Using the fact that $x'$ is an eigenvector of $\mathrm {d}\widetilde {\varphi }(x')$ with eigenvalue $\widetilde {\lambda }_{2}(x')$ and that $H=N^{-1}\circ \Phi _{\omega }$ yields

$$ \begin{align*} \begin{aligned} \displaystyle\widetilde{\lambda}_{2}(x')&\displaystyle =\frac{\langle\omega,\mathrm{d}N^{-1}_{\omega'}\circ(\mathrm{d}\Phi_{\omega})_{x'}(x')\rangle}{\langle\omega,\mathrm{d}N^{-1}_{\omega"}\circ(\mathrm{d}\Phi_{\omega})_{\widetilde{\eta}(x')x'}(x')\rangle} =\displaystyle \widetilde{\eta}(x')\frac{\langle\omega,\mathrm{d}N^{-1}_{\omega'}(\langle\omega,\omega'\rangle\omega'-\omega)\rangle}{\langle\omega,\mathrm{d}N^{-1}_{\omega"}(\langle\omega",\omega\rangle\omega"-\omega)\rangle}. \end{aligned} \end{align*} $$

By Lemma 6.2 once more,

$$ \begin{align*} \det(\mathrm{d}\widetilde{\varphi}(x'))=[\widetilde{\eta}(x')]^{n-1}\frac{\langle\omega,\mathrm{d}N^{-1}_{\omega'}(\langle\omega,\omega'\rangle\omega'-\omega)\rangle}{\langle\omega,\mathrm{d}N^{-1}_{\omega"}(\langle\omega",\omega\rangle\omega"-\omega)\rangle}. \end{align*} $$

By (6.28),

$$ \begin{align*} \Delta(u,u')=\frac{1}{|\det((\mathrm{d}\Phi_{\omega}^{-1})(\omega^{\prime\prime}))||\det((\mathrm{d}\Phi_{\omega})(x'))|}|\widetilde{\eta}(x')|^{n-1}\frac{|\langle\omega,\mathrm{d}N^{-1}_{\omega'}(\langle\omega,\omega'\rangle\omega'-\omega)\rangle|}{|\langle\omega,\mathrm{d}N^{-1}_{\omega"}(\langle\omega",\omega\rangle\omega"-\omega)\rangle|}\frac{K(u')}{K(u")}. \end{align*} $$

By (6.27),

(6.30) $$ \begin{align} \begin{aligned} \displaystyle|\widetilde{\eta}(x')|&\displaystyle=\left|\left\langle \frac{\omega'-\langle\omega,\omega^{\prime}\rangle\omega}{1+\langle\omega,\omega^{\prime}\rangle},\frac{\omega^{\prime\prime}-\langle\omega^{\prime\prime},\omega\rangle\omega}{1+\langle\omega^{\prime\prime},\omega\rangle}\right\rangle\right|\frac{|1+\langle\omega,\omega^{\prime}\rangle|^{2}}{|\omega'-\langle\omega,\omega^{\prime}\rangle\omega|^{2}} \\ &\displaystyle= \left|\frac{\langle\omega',\omega^{\prime\prime}\rangle-\langle\omega,\omega^{\prime}\rangle\langle\omega,\omega^{\prime\prime}\rangle}{(1+\langle\omega^{\prime\prime},\omega\rangle)(1-\langle\omega,\omega^{\prime}\rangle)}\right|. \end{aligned} \end{align} $$

By (6.4), (6.5), and (6.30),

$$ \begin{align*} \begin{aligned} \displaystyle\Delta(u,u')&=\displaystyle \left|\frac{\langle\omega',\omega^{\prime\prime}\rangle-\langle\omega,\omega^{\prime}\rangle\langle\omega,\omega^{\prime\prime}\rangle}{(1+\langle\omega^{\prime\prime},\omega\rangle)\cdot(1-\langle\omega,\omega^{\prime}\rangle)}\right|{}^{n-1}\left|\frac{1+\langle\omega",\omega\rangle}{1+\langle\omega',\omega\rangle}\right|{}^{n-1}\frac{|\langle\omega,\mathrm{d}N^{-1}_{\omega'}(\langle\omega,\omega'\rangle\omega'-\omega)\rangle|}{|\langle\omega,\mathrm{d}N^{-1}_{\omega"}(\langle\omega",\omega\rangle\omega"-\omega)\rangle|}\frac{K(u')}{K(u")} \\ &\displaystyle= \left(\frac{|N(u)\wedge N(u")|}{|N(u)\wedge N(u')|}\right)^{n-1}\frac{|\langle P_{T_{u'}S}N(u),(\mathrm{d}N_{u'})^{-1}(P_{T_{u'}S}N(u))\rangle|}{|\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle|}\frac{K(u')}{K(u")}, \end{aligned} \end{align*} $$

by (6.21), (6.22) and by similar geometric considerations to those in Section 6.1. This establishes (4.8).

6.3 Relating J and $\Delta $

Here we establish (4.9), the ‘switching property’ of $\Delta $ . By (4.7) we have

$$ \begin{align*} \begin{aligned} \displaystyle\frac{J(u,u")}{J(u,u')} &\displaystyle=\left(\frac{|N(u)\wedge N(u')|}{|N(u)\wedge N(u")|}\right)^{n-2}\left|\frac{\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle}{\langle P_{T_{u'}S}N(u),(\mathrm{d}N_{u'})^{-1}(P_{T_{u'}S}N(u))\rangle}\right|\frac{|\langle u"-u',N(u')\rangle|}{|\langle u"-u',N(u")\rangle|}\frac{K(u")}{K(u')}. \end{aligned} \end{align*} $$

Using the coplanarity condition (4.3), an elementary argument similar to that leading to (6.20) reveals that

$$ \begin{align*} \frac{|\langle u"-u',N(u')\rangle|}{|\langle u"-u',N(u")\rangle|}=\frac{|P_{\langle\omega\rangle^{\perp}}N(u')|}{|P_{\langle\omega\rangle^{\perp}}N(u")|} =\frac{(1-\langle\omega,\omega^{\prime}\rangle^{2})^{\frac{1}{2}}}{(1-\langle\omega,\omega"\rangle^{2})^{\frac{1}{2}}}=\frac{|N(u)\wedge N(u')|}{|N(u)\wedge N(u")|}, \end{align*} $$

from which (4.9) follows.

7 Surface-carried fractional integrals

In this section we establish Lebesgue space bounds on the bilinear fractional integrals

$$ \begin{align*}I_{S,s} (g_1,g_2)(u) := \int_S \frac{g_1(u') g_2( R_uu' )}{ |u' - R_uu'|^{s} } J(u,u') \mathrm{d}\sigma(u') \end{align*} $$

arising in Section 4.

Remark 7.1 (Relation to classical fractional integral operators).

This is a surface-carried variant of the bilinear fractional integral operator

$$ \begin{align*}I_s(f_1,f_2)(x):=\int_{\mathbb{R}^d}\frac{f_1\left(x+\frac{y}{2}\right)f_2\left(x-\frac{y}{2}\right)}{|y|^s}dy \end{align*} $$

that naturally arises when S is the paraboloid (see Section 2), and has been studied by several authors; we refer to [Reference Grafakos30] and [Reference Kenig and Stein33].

As indicated in Section 4, the presence of the factor J in the kernel implies that this operator is symmetric – that is, $I_{S,s}(g_1,g_2)=I_{S,s}(g_2,g_1)$ . It is also natural for geometric reasons, allowing for bounds that are independent of any lower bounds on the curvature of S. For example, we have

(7.1) $$ \begin{align} \begin{aligned} \|I_{S,s}(f_1,f_2)\|_1&=\int_S\int_S \frac{f_1(u')f_2(R_u u')}{|u'-R_u u'|^s}J(u,u')\mathrm{d}\sigma(u)\mathrm{d}\sigma(u')\\&=\int_S\int_S\frac{f_1(u')f_2(u")}{|u'-u"|^s}\mathrm{d}\sigma(u')\mathrm{d}\sigma(u")\\ &\leq C_s\|f_1\|_2\|f_2\|_2, \end{aligned} \end{align} $$

where

$$ \begin{align*}C_s:=\sup_{u\in S}\int_S\frac{\mathrm{d}\sigma(u')}{|u-u'|^s}.\end{align*} $$

Evidently $C_s$ does not depend on any lower bound on the curvature of S. More generally we have the following:

Theorem 7.2. Let $0<s<n-1$ , $q\in [\frac 12,1]$ , and S be as above. Then

$$ \begin{align*}\|I_{S,s}(g_1,g_2)\|_{L^q(S)} \lesssim Q(S)^{2(n-1)}\|g_1\|_{L^{2q}(S)} \|g_2\|_{L^{2q}(S)}, \end{align*} $$

where the implicit constant depends on $n,s,q$ and the diameter of S.

In order to prove Theorem 7.2 we adapt the argument of Kenig and Stein [Reference Kenig and Stein33] from the Euclidean setting.

Proof of Theorem 7.2.

For each dyadic scale $\lambda \lesssim \mathrm {diam\,}(S)$ we decompose S into a collection $\Theta _\lambda $ of $\lambda $ -caps $\theta $ , noting that $|\theta | \sim \lambda ^{n-1}$ for such a cap. Performing a dyadic decomposition and using the embedding $\ell ^q \subset \ell ^1$ , for $q\le 1$ , we have that (recall that $u"=R_{u}u'$ )

$$ \begin{align*} \int_S I_{S,s}(g_1,g_2)^q \mathrm{d}\sigma(u) &\lesssim \sum_{0<\lambda \lesssim\mathrm{diam\,}(S)} \lambda^{-qs} \int_S \Bigg( \int_{u'\in S: |u'-u"|\sim \lambda}g_1(u') g_2(u" ) J(u,u') \mathrm{d}\sigma(u') \Bigg)^{q} \mathrm{d}\sigma(u). \end{align*} $$

Next, we fix an arbitrary dyadic scale $\lambda $ and decompose

$$ \begin{align*} \int_S \Bigg( \int_{u'\in S: |u'-u"|\sim \lambda}&g_1(u') g_2( u" ) J(u,u') \mathrm{d}\sigma(u') \Bigg)^q \mathrm{d}\sigma(u)\\ &= \sum_{\theta\in\Theta_\lambda} \int_\theta \Bigg( \int_{u'\in S: |u'-u"|\sim \lambda}g_1(u') g_2( u" ) J(u,u') \mathrm{d}\sigma(u') \Bigg)^q \mathrm{d}\sigma(u)\\ &\lesssim \lambda^{(n-1)(1-q)} \sum_{\theta\in\Theta_\lambda}\Bigg( \int_\theta \int_{u'\in S: |u'-u"|\sim \lambda}g_1(u') g_2( u" ) J(u,u') \mathrm{d}\sigma(u') \mathrm{d}\sigma(u)\Bigg)^q. \end{align*} $$

Here we used that $0<q\le 1$ once more. Recall that $|u-u'|\lesssim Q(S) |u' - u"|$ for all $u,u'\in S$ by Proposition 4.4. Thus if $u \in \theta \in \Theta _\lambda $ and $|u' - u"| \sim \lambda $ , then $|u-u'|\lesssim Q(S)\lambda $ which means that $u' \in \theta ^*$ , where $\theta ^*$ is an $O(Q(S))$ dilate of $\theta $ . Similarly, $u" \in \theta ^*$ . Consequently,

for $p\ge 1$ , where we have used that

$$ \begin{align*}\int_S\int_S f(u)g(u") J(u,u') \mathrm{d}\sigma(u)\mathrm{d}\sigma(u') = \|f\|_{L^1(S)}\|g\|_{L^1(S)}.\end{align*} $$

Since $q\ge \frac 12$ and $p=2q$ , we obtain that

$$ \begin{align*} \begin{aligned} \displaystyle\int_S \Bigg( \int_{u'\in S: |u'-u"|\sim \lambda}&g_1(u') g_2( u" ) J(u,u') \mathrm{d}\sigma(u') \Bigg)^q \mathrm{d}\sigma(u) \\ &\displaystyle\lesssim \lambda^{(n-1)(1-q)} (Q(S)\lambda)^{(n-1)\frac{2q}{p'}} Q(S)^{n-1}\|g_1 \|_{L^p(S)}^{q} \|g_2 \|_{L^p(S)}^{q} \\ &\displaystyle=\lambda^{(n-1)(1-q)+(n-1)\frac{2q}{p'}} Q(S)^{2q(n-1)}\|g_1 \|_{L^p(S)}^{q} \|g_2 \|_{L^p(S)}^{q}, \end{aligned} \end{align*} $$

since the set of dilated caps $\{\theta ^{\ast }:\theta \in \Theta _{\lambda }\}$ covers S with a $Q(S)^{n-1}$ overlap factor. The geometric series converges as long as $-qs + (n-1)( 1- q + \frac {2q}{p'} )>0$ . Since $p=2q$ , this is equivalent to $s<n-1$ .

8 Surface-carried maximal operators

Recall from Section 4 that the geometric Wigner distribution $W_S(g,g)$ possesses the marginal properties (4.11) and (4.12). In the (superficially) more general polarised form these are the identities

(8.1) $$ \begin{align} \int_{S}W_S(g_1,g_2)(u,P_{T_uS}x)\mathrm{d}\sigma(u)=\widehat{g_1\mathrm{d}\sigma}(x)\overline{\widehat{g_2\mathrm{d}\sigma}(x)} \end{align} $$

and

(8.2) $$ \begin{align} \int_{T_uS}W_S(g_1,g_2)(u,v)\mathrm{d}v=g_1(u)\overline{g_2(u)} \end{align} $$

respectively. While (8.1) is an elementary consequence of Fubini’s theorem and the definition of the Jacobian J, the property (8.2) appears to be a little more delicate in general. In particular, for $g_1,g_2$ merely in $L^2$ , the integral in identity (8.2) should be interpreted as a suitable pointwise limit – see the forthcoming Proposition 8.4. As may be expected, a maximal analogue of the bilinear fractional integral operator $I_{S,s}$ of Section 7 naturally arises in our analysis. For locally integrable functions $f_1,f_2:S\rightarrow \mathbb {R}_+$ and $0<\delta <1$ we define the ‘averaging’ operator

$$ \begin{align*}A_{S,\delta}(f_1,f_2)(u)=\delta^{-(n-1)}\int_{|u'-R_uu'|<\delta}f_1(u')f_2(R_uu')J(u,u')\mathrm{d}\sigma(u'), \end{align*} $$

and maximal operator

$$ \begin{align*}M_S(f_1,f_2)(u)=\sup_{0<\delta<1}A_{S,\delta}(f_1,f_2)(u). \end{align*} $$

Remark 8.1 (Relation to classical maximal operators).

The operator $M_S$ is a surface-carried variant of the classical bi(sub)-linear Hardy–Littlewood maximal operator

$$ \begin{align*}M(f_1,f_2)(x)=\sup_{\delta>0}\frac{1}{|B(0,\delta)|}\int_{B(0,\delta)}f_1\left(x+\frac{y}{2}\right)f_2\left(x-\frac{y}{2}\right)\mathrm{d}y \end{align*} $$

on a Euclidean space.

We shall need the following estimate:

Theorem 8.2. If S is smooth, strictly convex and has finite curvature quotient $Q(S)$ , then

(8.3) $$ \begin{align} M_{S}:L^2(S)\times L^2(S)\rightarrow L^{1,\infty}(S). \end{align} $$

Proof. We begin by using the Cauchy–Schwarz inequality to write

$$ \begin{align*} \begin{aligned} A_{S,\delta}(f_1,f_2)(u)&\leq\left(\delta^{-(n-1)}\int_{|u'-R_uu'|<\delta}f_1(u')^2J(u,u')\mathrm{d}\sigma(u')\right)^{1/2}\\&\;\;\;\;\;\;\times\left(\delta^{-(n-1)}\int_{|u'-R_uu'|<\delta}f_2(R_uu')^2J(u,u')\mathrm{d}\sigma(u')\right)^{1/2}. \end{aligned} \end{align*} $$

Making the change of variables $R_uu'=u"$ in the second factor above, using Proposition 4.5, and the fact that $R_uu"=u'$ , we see that

$$ \begin{align*}\begin{aligned} \int_{|u'-R_uu'|<\delta}f_2(R_uu')^2J(u,u')\mathrm{d}\sigma(u')&=\int_{|u"-R_uu"|<\delta}f_2(u")^2J(u,u')\Delta(u,u")\mathrm{d}\sigma(u")\\&=\int_{|u"-R_uu"|<\delta}f_2(u")^2J(u,u")\mathrm{d}\sigma(u")\\&=\int_{|u'-R_uu'|<\delta}f_2(u')^2J(u,u')\mathrm{d}\sigma(u'). \end{aligned} \end{align*} $$

Thus,

$$ \begin{align*}M_S(f_1,f_2)(u)\leq M_S^1(f_1^2)(u)^{1/2}M_S^1(f_2^2)(u)^{1/2},\end{align*} $$

where

$$ \begin{align*}M_S^1(f)(u):=\sup_{0<\delta<1}\delta^{-(n-1)}\int_{|u'-R_uu'|<\delta}f(u')J(u,u')\mathrm{d}\sigma(u').\end{align*} $$

Hence

$$ \begin{align*} \begin{aligned} \lambda\left|\left\{u\in S:M_S(f_1,f_2)(u)>\lambda\right\}\right|&\leq\lambda\left|\left\{u\in S:M_S^1(f_1^2)(u)M_S^1(f_2^2)(u)>\lambda^2\right\}\right|\\&\leq \lambda\left|\left\{u\in S:M_S^1(f_1^2)(u)>\varepsilon\lambda\right\}\right|\\ &\quad +\lambda\left|\left\{u\in S:M_S^1(f_2^2)(u)>\varepsilon^{-1}\lambda\right\}\right| \end{aligned} \end{align*} $$

for all $\varepsilon>0$ . We claim that the sublinear operator $M_S^1$ is of weak-type (1,1), and assuming this momentarily we have

$$ \begin{align*} \begin{aligned} \lambda\left|\left\{u\in S:M_S(f_1,f_2)(u)>\lambda\right\}\right|&\lesssim \varepsilon^{-1}\|f_1\|_2^2+\varepsilon\|f_2\|_2^2 \end{aligned} \end{align*} $$

uniformly in $\varepsilon $ . Optimising in $\varepsilon $ now yields the claimed weak-type bound on the bi-sublinear operator $M_S$ . A similar argument in a Euclidean context may be found in [Reference Grafakos30].

It remains to establish that $M_S^1:L^1(S)\rightarrow L^{1,\infty }$ , and we do this by applying the well-known abstract form of the classical Hardy–Littlewood maximal theorem presented in [Reference Stein48]. To this end we let $B_\delta (u)=\{u'\in S: \rho (u,u')<\delta \}$ , the ball in S centred at u with respect to the function $\rho (u,u'):=|u'-R_uu'|$ . By Proposition 4.4 it follows that $\rho $ is a quasi-distance, as defined in [Reference Stein48] (p. 10). Specifically, we may quickly verify that (i) $\rho (x,y)=0\iff x=y$ , (ii) $\rho (x,y)\leq c\rho (y,x)$ , and (iii) $\rho (x,y)\leq c(\rho (x,z)+\rho (y,z))$ , for some positive constant c depending on $Q(S)$ . By the change of variables (4.19) and an application of Proposition 4.16,

(8.4) $$ \begin{align} |B_\delta(u)|=\int_{|\xi|\leq \delta}\widetilde{J}(u,u'(\xi))^{-1}\mathrm{d}\xi\leq \delta^{n-1}, \end{align} $$

so that

$$ \begin{align*}M_S^1f(u)\leq\sup_{0<\delta<1}\frac{1}{|B_\delta(u)|}\int_{B_\delta(u)}f(u')J(u,u')\mathrm{d}\sigma(u').\end{align*} $$

Arguing as in the proof of (4.31), we have

$$ \begin{align*} \displaystyle J(u,u')\lesssim Q(S)^{\frac{5(n-2)}{2}} \frac{|u"-u'|}{|N(u")-N(u)|} \sup_{p}\lambda_{n-1}(p), \end{align*} $$

which by a further use of Proposition 4.4 and the mean value theorem applied to the Gauss map shows that $J(u,u')$ is, up to a dimensional constant, bounded from above by a power of $Q(S)$ . Consequently,

$$ \begin{align*}M_S^1f(u)\lesssim \sup_{0<\delta<1}\frac{1}{|B_\delta(u)|}\int_{B_\delta(u)}f(u')\mathrm{d}\sigma(u'),\end{align*} $$

where the implicit constant is permitted to depend on $Q(S)$ . It remains to show that the surface measure on S is doubling with respect to the family of balls $B_\delta (u)$ , as we may then apply the abstract Hardy–Littlewood maximal theorem of [Reference Stein48] (see p. 37). By (8.4) it suffices to show that $|B_\delta (u)|\geq cQ(S)^{n-1}\delta ^{n-1}$ , for some dimensional constant c. However, this follows from Proposition 4.4 since

$$ \begin{align*}B_\delta(u)\supseteq\{u'\in S: |u'-u|\lesssim Q(S)\delta\}.\\[-37pt] \end{align*} $$

Equipped with the above maximal theorem, we may now clarify the marginal property (8.2). While we expect that (8.2) (suitably interpreted) holds for all of the submanifolds S that we consider in this paper, our approach seems to require the additional assumption that

(8.5) $$ \begin{align} \lim_{u'\rightarrow u}(\mathrm{d}R_u)_{u'}\;\;\text{ exists.} \end{align} $$

We note that (8.5) requires some interpretation since for each $u'\not =u$ , the map $(\mathrm {d}R_u)_{u'}:T_{u'}S\rightarrow T_{u"}S$ , and the limit should be interpreted as a linear transformation of $T_u S$ . One way to do this is to parametrise S by $T_uS$ , upon which the map $R_u$ may be parametrised by a map $y_u$ on the fixed domain $T_uS$ . We clarify this technical point in the arguments that follow. The local statement (8.5) appears to be an extremely mild assumption. It is straightforward to verify for parabolic S, and since a smooth strictly convex surface is locally parabolic (by Taylor’s theorem), one might reasonably expect it to be verifiable in general.

Proposition 8.4. Let S be smooth and strictly convex. Suppose $\chi $ is a Schwartz function on $T_uS$ with $\chi (0)=1$ , and $\chi _r(v)=\chi (v/r)$ for each $r>0$ . Then for compactly supported $g_1,g_2\in L^2(S)$ ,

$$ \begin{align*}\int_{T_uS}W_S(g_1,g_2)(u,v)\chi_r(v)\mathrm{d}v\rightarrow g_1(u)\overline{g_2(u)} \end{align*} $$

as $r\rightarrow \infty $ for almost every $u\in S$ . Moreover, if $g_1,g_2$ are continuous, then this convergence holds at all points u.

Before we turn to the proof of Proposition 8.4, we state a lemma whose (somewhat technical) proof we leave to the end of the section.

Lemma 8.5. If the limit (8.5) exists then for each $u\in S$ ,

$$ \begin{align*}\lim_{u'\rightarrow u}\widetilde{J}(u,u')=2^{n-1}\end{align*} $$

and

$$ \begin{align*}\lim_{\substack{u'\rightarrow u\\u'-u"\in\langle\omega\rangle}}J(u,u')=2^{n-1}\end{align*} $$

for each $\omega \in T_uS\backslash \{0\}$ .

Proof of Proposition 8.4.

We begin by writing

(8.6) $$ \begin{align} \int_{T_uS}W_S(g_1,g_2)(u,v)\chi_r(v)\mathrm{d}v&=\int_{T_uS}\int_S g_1(u')\overline{g_2(R_uu')}e^{-2\pi i v\cdot(u'-R_uu')}J(u,u')\mathrm{d}\sigma(u')\chi_r(v)\mathrm{d}v \nonumber\\&=\int_S g_1(u')\overline{g_2(R_uu')}\widehat{\chi}_r(u'-R_uu')J(u,u')\mathrm{d}\sigma(u') \nonumber\\&=:\mathcal{A}_{S,r}(g_1,g_2)(u). \end{align} $$

Since $\widehat {\chi }$ is a bump function, it follows that $\mathcal {M}_S(g_1,g_2)(u)\lesssim M_S(|g_1|,|g_2|)(u)$ where

$$ \begin{align*}\mathcal{M}_S(g_1,g_2)(u):=\sup_{r>1}|\mathcal{A}_{S,r}(g_1,g_2)(u)|. \end{align*} $$

Consequently

(8.7) $$ \begin{align} \mathcal{M}_S:L^2(S)\times L^2(S)\rightarrow L^{1,\infty}(S), \end{align} $$

by Theorem 8.2. Proposition 8.4 requires us to show that

(8.8) $$ \begin{align} \mathcal{A}_{S,r}(g_1,g_2)(u)\rightarrow g_1(u)\overline{g_2(u)}\;\text{ for almost every }\;u\in S. \end{align} $$

The first step, which uses a minor variant of a standard argument in the setting of sublinear maximal operators (see, e.g., [Reference Stein and Weiss49]), is to use the maximal estimate (8.7) to reduce to the case of continuous $g_1,g_2$ . We leave this classical exercise to the reader. Suppose now that $g_1,g_2$ are continuous functions. It suffices to show that

(8.9) $$ \begin{align} \mathcal{A}_{S,r}(1,1)(u):=\int_S \widehat{\chi}_r(u'-R_uu')J(u,u')\mathrm{d}\sigma(u')\rightarrow 1. \end{align} $$

Invoking the change of variables (4.19) and using polar coordinates in $T_uS$ we have

$$ \begin{align*} \begin{aligned} \mathcal{A}_{S,r}(1,1)(u)&=\int_{T_uS}\widehat{\chi}_r(\xi)\frac{J(u,u'(\xi))}{\widetilde{J}(u,u'(\xi))}\mathrm{d}\xi\\ &=\int_0^\infty \int_{\mathbb{S}^{n-2}(T_uS)}r^{n-1}\widehat{\chi}(rt\omega)\frac{J(u,u'(t\omega))}{\widetilde{J}(u,u'(t\omega))}\mathrm{d}\sigma(\omega)t^{n-2}\mathrm{d}t\\ &=\int_0^\infty \int_{\mathbb{S}^{n-2}(T_uS)}\widehat{\chi}(s\omega)\frac{J(u,u'(r^{-1}s\omega))}{\widetilde{J}(u,u'(r^{-1}s\omega))}\mathrm{d}\sigma(\omega)\mathrm{d}s, \end{aligned} \end{align*} $$

where $\mathbb {S}^{n-2}(T_uS)$ denotes the unit sphere in $T_uS$ . The limit (8.9) now follows by Lemma 8.5 since $u'(r^{-1}s\omega )\rightarrow u$ as $r\rightarrow \infty $ , while $u'(r^{-1}s\omega )-R_uu'(r^{-1}s\omega )=r^{-1}s\omega \in \langle \omega \rangle $ .

It remains to prove Lemma 8.5.

Proof of Lemma 8.5.

We begin by clarifying the hypothesis (8.5), and showing that this limit must actually equal $-I$ , where I denotes the identity on $T_uS$ . This reflects a crucial ‘limiting symmetry’ of the configuration of points $u,u',u"$ as $u'\rightarrow u$ . By translation and rotation invariance we may suppose that $u=0$ and $S=\{(x',\phi (x')):x'\in X\}$ , for some smooth real-valued function $\phi $ on a subset X of $T_uS$ satisfying $\nabla \phi (0)=0$ and $\mathrm {Hess\,}(\phi )(x')>_{pd}0$ for all $x'$ . The map $R:=R_u$ then takes the form $R(x',\phi (x'))=(x",\phi (x"))$ , for some unique $x"\in T_uS$ satisfying

(8.10) $$ \begin{align} \phi(x")=\phi(x') \end{align} $$

and

(8.11) $$ \begin{align} \frac{\nabla\phi(x")}{|\nabla\phi(x")|}=-\frac{\nabla\phi(x')}{|\nabla\phi(x')|}. \end{align} $$

Observe that (8.10) follows by (4.2) and (8.11) is a consequence of (4.28). Writing $x"=y(x')$ allows us to interpret (8.5) as the existence of the limit $\mathrm {d}y_0:=\lim _{x'\rightarrow 0}\mathrm {d}y_{x'}:T_{u}S\rightarrow T_{u}S$ . In order to show that $\mathrm {d}y_0=-I$ , we fix $v\in T_{u}S$ and let $x_{k}'\rightarrow 0$ be a sequence in $T_{u}S$ satisfying

$$ \begin{align*}\frac{\nabla\phi(y(x_{k}'))}{|\nabla\phi(y(x_{k}'))|}=v\end{align*} $$

for all k. This sequence exists as the Gauss maps $\widetilde {N}$ of the sections $\mathcal {S}_{u,u'}$ (see (4.27)) are bijections. Differentiating (8.10) at the points of this sequence, we have

$$ \begin{align*} \mathrm{d}y(x_{k}')^{\top}(\nabla\phi(y(x_{k}')))=\nabla\phi(x'). \end{align*} $$

Using (8.11),

$$ \begin{align*} \mathrm{d}y(x_{k}')^{\top}\left(\frac{\nabla\phi(x_{k}')}{|\nabla\phi(x_{k}')|}\right)=-\frac{|\nabla\phi(x_{k}')|}{|\nabla\phi(y(x_{k}'))|}\frac{\nabla\phi(x_{k}')}{|\nabla\phi(x_{k}')|}, \end{align*} $$

which implies

(8.12) $$ \begin{align} \mathrm{d}y(x_{k}')^{\top}(v)=-\frac{|\nabla\phi(x_{k}')|}{|\nabla\phi(y(x_{k}'))|}v \end{align} $$

for all $k\in \mathbb {N}$ . By the mean-value inequality,

$$ \begin{align*} \frac{|\nabla\phi(x_{k}')|}{|\nabla\phi(y(x_{k}'))|}\leq\frac{\sup\|\mathrm{Hess\,}\phi\|_{\infty}}{\inf\|\mathrm{ Hess\,}\phi\|_{\infty}}\frac{|x_{k}'|}{|y(x_{k}')|}\leq \frac{\sup\|\mathrm{Hess\,}\phi\|_{\infty}}{\inf\|\mathrm{ Hess\,}\phi\|_{\infty}}\frac{1}{\|\mathrm{d}y(c_{k})\|_{\infty}} \end{align*} $$

for some $c_{k}$ with $c_{k}\rightarrow 0$ . On the other hand, $y(y(x_{k}'))=x_{k}'$ (recall that $R_{u}(R_{u}u')=u'$ ), hence $\mathrm {d}y(y(x_{k}'))\circ \mathrm {d}y(x_{k}')=I$ , which gives $\mathrm {d}y_0^{2}=I$ , therefore $\|\mathrm {d}y(c_{k})\|_{\infty }$ does not approach $0$ and the sequence

$$ \begin{align*}\frac{|\nabla\phi(x_{k}')|}{|\nabla\phi(y(x_{k}'))|}\end{align*} $$

is bounded. By passing to a subsequence and by taking limits, we conclude from (8.12) that

$$ \begin{align*} \mathrm{d}y_0^{\top}(v)=-Lv \end{align*} $$

for some positive real number L and for all $v\in T_{u}S$ . On the other hand, since $\mathrm {d}y_0^{2}=I$ , the only possible eigenvalues of $\mathrm {d}y_0$ are $\pm 1$ , hence $\mathrm {d}y_0=-I$ . Finally, taking the limit as $u'\rightarrow u$ in the first identity of (4.30) gives

(8.13) $$ \begin{align} \lim_{u'\rightarrow u}\widetilde{J}(u,u')=2^{n-1}. \end{align} $$

Turning to the limiting identity for J, we first establish some bounds relating to the limiting arrangements of the points $u, u', u"$ and their normals $N(u), N(u'), N(u")$ , beginning with

(8.14) $$ \begin{align} u'+u"-2u=o(|u-u'|). \end{align} $$

To see this (recalling that we are supposing $u=0$ ) observe that $u'+u"=(x'+y(x'), 2\phi (x'))$ , and since $\phi (x')=O(|x'|^2)$ , it remains to show that $h(x'):=x'+y(x')=o(|x'|)$ . By the mean value theorem, it suffices to observe that $\mathrm {d}h_{x'}=I+\mathrm {d}y_{x'}=o(1)$ as $x'\rightarrow 0$ , since $\mathrm {d}y_{x'}\rightarrow -I$ . A similar, albeit lengthier argument reveals that

(8.15) $$ \begin{align} N(u')+N(u")-2N(u)=o(|u-u'|). \end{align} $$

Recalling the formula for $J(u,u')$ , we observe first that the factor

$$ \begin{align*}\frac{|N(u')\wedge N(u")|}{|N(u')\wedge N(u)|}=\frac{2|N(u')\wedge N(u)|+o(|u'-u|)}{|N(u')\wedge N(u)|}\rightarrow 2 \end{align*} $$

as $u'\rightarrow u$ . Here we are also using (1.12), which tells us that $|N(u')\wedge N(u)|\sim |u'-u|$ . It remains to show that for each unit vector $\omega \in T_uS$ ,

(8.16) $$ \begin{align} \left|\frac{\langle u"-u',N(u")\rangle}{\langle P_{T_{u"}S}N(u),(\mathrm{d}N_{u"})^{-1}(P_{T_{u"}S}N(u))\rangle}\right|\rightarrow 2 \end{align} $$

as $u'\rightarrow u$ with $u'-u"\in \langle \omega \rangle $ . Noting that $\langle u"-u', N(u")\rangle =\langle u"-u', P_{T_uS} N(u")\rangle $ , by (8.14) we are reduced to showing that

$$ \begin{align*}\lim_{\substack{u'\to u\\u'-u"\in\langle\omega\rangle}} \frac{ \langle u"-u, P_{T_uS} N(u") \rangle }{ \langle P_{T_{u"}S} N(u), (\mathrm{d}N_{u"})^{-1} P_{T_{u"}S} N(u) \rangle } =1. \end{align*} $$

By symmetry, we may replace $u"$ by $u'$ here, so that the objective is to show that

(8.17) $$ \begin{align} \lim_{\substack{u'\to u\\u'-u"\in\langle\omega\rangle}} \frac{ \langle u'-u, P_{T_uS} N(u') \rangle }{ \langle P_{T_{u'}S} N(u), (\mathrm{d}N_{u'})^{-1} P_{T_{u'}S} N(u) \rangle } =1. \end{align} $$

To this end we Taylor expand $N(u')$ about $0$ via the parametrisation $u'=(x',\phi (x')) =: \Phi (x')$ to obtain

$$ \begin{align*} N(u') &= N\circ \Phi(x') = N\circ \Phi (0) + \mathrm{d}(N\circ \Phi)_0 x' + O(|x'|^2)\\ &= N(u) + (\mathrm{d}N)_u\circ (\mathrm{d}\Phi)_0 x' + O(|x'|^2) \\ &= N(u) + (\mathrm{d}N)_u x' + O(|x'|^2), \end{align*} $$

where we have used that $(\mathrm {d}\Phi )_{x'} = \begin {pmatrix} \mathrm {id}_{\mathbb {R}^{n-1}} & \mathbf {0} \\ \nabla _{n-1} \phi (x') & 0\end {pmatrix}$ and $\nabla \phi (0)=0$ . Thus, in view of the fact that $|x'|= O(|u'-u|)$ we have

$$ \begin{align*}x' = (\mathrm{d}N)_u^{-1} \big( N(u') - N(u) + O(|u'-u|^2) \big). \end{align*} $$

The numerator of (8.17) now becomes

$$ \begin{align*} \langle u'-u, P_{T_uS} N(u') \rangle &= \langle P_{T_uS}(u'-u), P_{T_uS} N(u') \rangle \\ &= \langle x', P_{T_uS} N(u') \rangle \\ &= \big\langle (\mathrm{d}N)_u^{-1} \big( N(u') - N(u) + O(|u'-u|^2) \big), P_{T_uS} N(u') \big\rangle. \end{align*} $$

Note that

$$ \begin{align*}P_{T_uS} N(u') = N(u')-N(u) + O(|u'-u|^2), \end{align*} $$

and so

$$ \begin{align*}\langle u'-u, P_{T_uS} N(u') \rangle = \big\langle (\mathrm{d}N)_u^{-1} \big( N(u') - N(u) + O(|u'-u|^2) \big), \big( N(u') - N(u) + O(|u'-u|^2) \big) \big\rangle. \end{align*} $$

This is now similar to the denominator of (8.17). In fact,

$$ \begin{align*}P_{T_{u'}S} N(u) = - (N(u')-N(u)) + O(|u'-u|^2), \end{align*} $$

and so

$$ \begin{align*} &\frac{ \langle u'-u, P_{T_uS} N(u') \rangle }{ \langle P_{T_{u'}S} N(u), (\mathrm{d}N_{u'})^{-1} P_{T_{u'}S} N(u) \rangle }\\ &\;\;\;\;\;\;\;\;= \frac{ \big\langle (\mathrm{d}N)_u^{-1} \big( N(u') - N(u) + O(|u'-u|^2) \big), \big( N(u') - N(u) + O(|u'-u|^2) \big) \big\rangle }{ \big\langle (\mathrm{d}N)_{u'}^{-1} \big( N(u') - N(u) + O(|u'-u|^2) \big), \big( N(u') - N(u) + O(|u'-u|^2) \big) \big\rangle }. \end{align*} $$

Further, from (8.15) we have

$$ \begin{align*}N(u') - N(u) = \frac12( N(u') - N(u") ) + o(|u-u'|), \end{align*} $$

and hence,

$$ \begin{align*} &\frac{ \langle u'-u, P_{T_uS} N(u') \rangle }{ \langle P_{T_{u'}S} N(u), (\mathrm{d}N_{u'})^{-1} P_{T_{u'}S} N(u) \rangle }\\ &\;\;\;\;\;\;\;\;= \frac{ \big\langle (\mathrm{d}N)_u^{-1} \big( N(u') - N(u") + o(|u'-u|) \big), \big( N(u') - N(u") + o(|u'-u|) \big) \big\rangle }{ \big\langle (\mathrm{d}N)_{u'}^{-1} \big( N(u') - N(u") + o(|u'-u|) \big), \big( N(u') - N(u") + o(|u'-u|) \big) \big\rangle }. \end{align*} $$

Consequently, if

$$ \begin{align*}\lim_{\substack{u'\to u\\ u'-u"\in\langle\omega\rangle}} \frac{N(u') - N(u")}{|N(u') - N(u")|}\end{align*} $$

exists, then (8.17) follows. Here we have also appealed to the fact that

$$ \begin{align*}|N(u')-N(u")| = |(\mathrm{d}N)_u(u'-u") + O(|u-u'|^2)| \gtrsim |u-u'|. \end{align*} $$

Arguing similarly using Taylor’s theorem, we also have

$$ \begin{align*}N(u")-N(u) = (\mathrm{d}N)_u x" + O(|x"|^2), \end{align*} $$

from which it follows that

$$ \begin{align*}N(u') - N(u") = (\mathrm{d}N)_u x' -(\mathrm{d}N)_u x" + O(|x'|^2) + O(|x"|^2) = (\mathrm{d}N)_u ( u'-u" ) + O(|u-u'|^2), \end{align*} $$

and so

$$ \begin{align*}\frac{N(u') - N(u")}{|N(u') - N(u")|} = \frac{(\mathrm{d}N)_u ( u'-u" ) + O(|u-u'|^2)}{|(\mathrm{d}N)_u ( u'-u" )| + O(|u-u'|^2)} =\frac{(\mathrm{d}N)_u ( \omega ) + O(|u-u'|)}{|(\mathrm{d}N)_u ( \omega )| + O(|u-u'|)}, \end{align*} $$

which converges (to $(\mathrm {d}N)_u\omega /|(\mathrm {d}N)_u\omega |$ ) as $u'\rightarrow u$ with $u'-u"\in \langle \omega \rangle $ , as required.

9 Tomographic constructions

In this section we show that the explicit geometric Wigner distributions from Section 4 may be constructed tomographically from the corresponding extension operators, at least when $n=2$ . This is motivated by the tomographic approach to weighted extension inequalities developed in [Reference Bennett and Nakamura14, Reference Bennett, Nakamura and Shiraki15]. For the submanifolds S considered in Section 4, we saw that the natural tomographic transform is the S-parametrised X-ray transform $X_Sw(u,v):=Xw(N(u),v)$ . Here X denotes the standard X-ray transform and N the Gauss map of S. We remark that if the Gauss map is bijective, such as when S is strictly convex and closed, the operator $X_S$ is easily seen to inherit the inversion formula

$$ \begin{align*}c_nX_S^*K(-\Delta_v)^{1/2}X_S\psi=\psi \end{align*} $$

from the classical inversion formula for X; here $\psi $ is a suitably well-behaved function and $K(u)$ is the Gaussian curvature of S at a point u (acting multiplicatively). This suggests the following:

Proposition 9.1. Let $\Phi $ be a smooth bump function on $\mathbb {R}^2$ such that $\Phi (0)=1$ , and let $\Phi _\lambda (x)=\Phi (x/\lambda )$ for each $\lambda>0$ . If S is a strictly convex smooth curve in the plane then

$$ \begin{align*}\lim_{\lambda\rightarrow\infty}K(u)(-\Delta_v)^{1/2}X_S(\Phi_\lambda|\widehat{g\mathrm{d}\sigma}|^2)(u,v)=W_S(g,g)(u,v) \end{align*} $$

for all compactly supported smooth functions g on S.

Proof of Proposition 9.1.

A routine (distributional) argument, using the well-known fact that

$$ \begin{align*}\mathcal{F}_v(Xf)(\omega,\xi)=\widehat{f}(\xi),\;\;\xi\in\langle\omega\rangle^\perp,\end{align*} $$

reveals that

(9.1) $$ \begin{align} \begin{aligned} (-\Delta_v)^{1/2}X_S( \Phi_\lambda|\widehat{g\mathrm{d}\sigma}|^2)(u,v)& =\int_{T_u S}e^{2\pi i\xi\cdot v}|\xi|\widehat{\Phi}_\lambda\ast (g\mathrm{d}\sigma)*(\widetilde{g\mathrm{d}\sigma})(\xi)\mathrm{d}\xi. \end{aligned} \end{align} $$

In order to take the limit as $\lambda \to \infty $ it suffices, by the dominated convergence theorem, to show that

(9.2) $$ \begin{align} \sup_{\lambda\geq 1}|\xi| |\widehat{\Phi}_\lambda|\ast (g\mathrm{d}\sigma)*(\widetilde{g\mathrm{d}\sigma})(\xi)\lesssim(1+|\xi|)^{-N} \end{align} $$

for some sufficiently large $N\in \mathbb {N}$ . This may be seen by first appealing to the strict convexity of S, along with the assumed properties of g, to show that for some ball $B\subset \mathbb {R}^2$ ; see [Reference Silva43, Section 2] for the appropriate detailed computations. The estimate (9.2) then follows using the rapid decay of $\widehat {\Phi }$ . Taking this limit, it follows that

$$ \begin{align*} \lim_{\lambda\to\infty}(-\Delta_v)^{1/2}&X_S( \Phi_\lambda|\widehat{g\mathrm{d}\sigma}|^2)(u,v) \\&=\int_{T_u S}e^{2\pi i\xi\cdot v}|\xi| (g\mathrm{d}\sigma)*(\widetilde{g\mathrm{d}\sigma})(\xi)\mathrm{d}\xi \\&=\int_S\int_Sg(u')\overline{g(u")}e^{2\pi i(u'-u")\cdot v}|u'-u"|\delta((u'-u")\cdot N(u))\mathrm{d}\sigma(u")\mathrm{d}\sigma(u'). \end{align*} $$

Now, for fixed $u,u'$ the function $u"\mapsto (u'-u")\cdot N(u)$ vanishes if and only if either $u"=u'$ or $u"=R_u u'$ , as defined in Section 4, and so it remains to establish the formula

(9.3) $$ \begin{align} \int_S|u'-u"|\delta((u'-u")\cdot N(u))\mathrm{d}\sigma(u")=\frac{|u'-R_u u'|}{|N(u)\wedge N(R_u u')|} \end{align} $$

whenever $u'\not = u$ ; see Remark 4.6. Making the change of variables $u"'=u"-R_u u'$ (we stress that $u"$ is the variable of integration in (9.3) rather than a simplified notation for $R_{u}u'$ ), and using $\mathcal {H}^1$ to denote 1-dimensional Hausdorff measure in the plane, we have that

$$ \begin{align*}\begin{aligned} \int_S|u'-u"|\delta((u'-u")&\cdot N(u))\mathrm{d}\sigma(u")\\&=\int_{S-\{R_u u'\}}|u'-R_u u'-u"'|\delta(u"'\cdot N(u))\mathrm{d}\mathcal{H}^1(u"')\\ &=|u'-R_u u'|\lim_{\varepsilon\rightarrow 0}\frac{1}{2\varepsilon}\mathcal{H}^1\left(\{u"'\in (S-\{R_u u'\}):|u"'\cdot N(u)|<\varepsilon\}\right), \end{aligned} \end{align*} $$

from which (9.3) follows from the smoothness of S by elementary geometric considerations.