Proof. The inequality

$$\begin{align*}\left\|m\right\|_{MS_p(L_2(Y',g_*\nu),L_2(X',f_*\mu))} \leq \left\|m\right\|_{MS_p(L_2(Y',\nu'),L_2(X',\mu'))}\end{align*}$$

follows directly from [Reference Lafforgue and de la Salle28, Lemma 1.9] and the absolute continuity assumption. So our goal will be to prove the following equality:

$$\begin{align*}\left\|m \circ (f \times g)\right\|_{MS_p(L_2(Y,\nu),L_2(X,\mu))} = \left\|m\right\|_{MS_p(L_2(Y',g_*\nu),L_2(X',f_*\mu))}.\end{align*}$$

To lighten the notation, let us assume that $(X,\mu )=(Y,\nu )$ , $(X',\mu ')=(Y',\nu ')$ and $f=g$ . Let $\mathcal B, \mathcal B'$ be the underlying $\sigma $ -algebras, and consider $\mathcal A:=f^{-1}(\mathcal B')$ . Then f allows to identify $L_2(X',\mathcal B',f_*\mu )$ with $L_2(X,\mathcal A,\mu )$ . In particular,

$$\begin{align*}\left\|m\circ(f \times f)\right\|_{MS_p(L_2(X,\mathcal A,\mu))} = \left\|m\right\|_{MS_p(L_2(X',\mathcal B',f_*\mu))},\end{align*}$$

and similarly for the cb-norm. However, [Reference Lafforgue and de la Salle28, Lemma 1.13] implies that the cb norms of $m \circ (f\times f)$ on $S_p(L_2(\mathcal A,\mu ))$ and $S_p(L_2(\mathcal B,\mu ))$ coincide, so we deduce

$$\begin{align*}\left\|m\circ(f \times f)\right\|_{\mathrm{cb}MS_p(L_2(X,\mathcal B,\mu))} = \left\|m\right\|_{\mathrm{cb}MS_p(L_2(X',\mathcal B',f_*\mu))}.\end{align*}$$

Then [Reference Lafforgue and de la Salle28, Theorem 1.18] allows us to conclude. Indeed, our assumptions that $\mu $ and $\mu '$ have no atoms imply that both cb-norms are equal to their norms.

Proof. Given $u \in \mathbf {R}^n \setminus \{0\}$ , define

$$\begin{align*}m_u(\xi,\eta) = \chi_{\langle \xi-\eta,u\rangle>0} = \begin{cases} 1 & \textrm{if }\langle \xi-\eta,u\rangle >0\\ 0 & \textrm{otherwise.} \end{cases} \end{align*}$$

Then, the proof relies on the following two claims:

1. (A) Let $(x,y)$ be a transverse point in the boundary of $\partial \Sigma $ and let $T \in \mathrm {GL}_n(\mathbf {R})$ be such that $T^* \mathbf {n}_1(x,y) = - \mathbf {n}_2(x,y)$ . Then, the following identity holds for almost every $\xi ,\eta \in \mathbf {R}^n$ :

   $$\begin{align*}\lim_{\varepsilon\to 0^+} \chi_\Sigma \big( x+\varepsilon T\xi, y+\varepsilon \eta \big) = m_{\mathbf{n}_2(x,y)}(\xi,\eta).\end{align*}$$

2. (B) Let $u_j$ be as in the statement. Then the Schur multiplier

   $$\begin{align*}M:\big( (\xi,j),\eta \big) \in \big( \mathbf{R}^n \times\{1,\dots,N\} \big) \times \mathbf{R}^n \mapsto m_{u_j}(\xi,\eta)\end{align*}$$
   is bounded on $S_p(L_2(\mathbf {R}^n),L_2(\mathbf {R}^n \times \{1,2,\ldots ,N\}))$ with norm $\le C$ .

Assuming the validity of the above claims, we may now conclude the proof using standard transference ideas that go back at least to the work of Bożejko and Fendler [Reference Bożejko and Fendler2]. Consider $f_1,f_2,\dots ,f_N \in L_p(\mathbf {R}^n)$ . If $e_{j,1}$ denotes the standard elementary matrices, if $C=\sum _{j=1}^N c_j e_{j,1}$ , we have $|C|^p=(C^*C)^{\frac p 2} = (\sum _j |c_j|^2)^{\frac p 2} e_{1,1}$ . Taking $c_j=f_j(x)$ and integrating with respect to x, we get

$$\begin{align*}\Big\| \Big( \sum_{j=1}^N |f_j|^2 \Big)^{\frac{1}{2}} \Big\|_{L_p(\mathbf{R}^n)} = \Big\| \sum_{j=1}^N f_j \otimes e_{j,1} \Big\|_{L_p(\mathbf{R}^n;S_p)}.\end{align*}$$

We know from [Reference Caspers and de la Salle7, Theorem 5.2] that there is an ultrafilter $\mathcal {U}$ on $\mathbf {N}$ and a completely isometric map

$$\begin{align*}j_p\colon L_p(\mathbf{R}^n) \to \prod_{\mathcal{U}} S_p(L_2(\mathbf{R}^n))\end{align*}$$

that intertwines Fourier and Schur multipliers. The notation $\prod _{\mathcal {U}} S_p(L_2(\mathbf {R}^n))$ stands for the Banach space ultraproduct, that is the quotient of

$$\begin{align*}\prod_{\alpha \in \mathbf{N}} S_p(L_2(\mathbf{R}^n)):=\Big\{(A_\alpha)_{\alpha \in \mathbf{N}}\mid A_\alpha \in S_p(L_2(\mathbf{R}^n)), \sup_\alpha \|A_\alpha\|_p<\infty\Big\}\end{align*}$$

by its closed subspace $\{(A_\alpha )_\alpha \mid \lim _{\alpha \to \mathcal {U}} \|A_\alpha \|_p=0\}$ . Pick a representative $(A_{j,\alpha })_{\alpha \in \mathbf {N}}$ of $j_p(f_j)$ . This gives that $(S_{m_u}(A_{j,\alpha }))_{\alpha \in \mathbf {N}}$ is a representative of $j_p(H_u(f_j))$ for every $u \in \mathbf {R}^n$ . That $j_p$ is a complete isometry gives

(2.2) $$ \begin{align} \Big\| \Big( \sum_{j=1}^N |f_j|^2 \Big)^{\frac{1}{2}} \Big\|_{L_p(\mathbf{R}^n)} = \lim_{\alpha \to \mathcal{U}} \Big\| \sum_{j=1}^N A_{j,\alpha} \otimes e_{j,1} \Big\|_{S_p}, \end{align} $$

and applying it to $H_{u_j}(f_j)$ instead of $f_j$ , we obtain

$$\begin{align*}\Big\| \Big( \sum_{j=1}^N |H_{u_j}(f_j)|^2 \Big)^{\frac{1}{2}} \Big\|_{L_p(\mathbf{R}^n)} = \lim_{\alpha \to \mathcal{U}} \Big\| \sum_{j=1}^N S_{m_{u_j}} (A_{j,\alpha}) \otimes e_{j,1} \Big\|_{S_p}.\end{align*}$$

In the preceding equality and in what follows, if $A_1,\dots ,A_N \in S_p(L_2(\mathbf {R}^n))$ , we see the sum $\sum _{j=1}^N A_j \otimes e_{j,1}$ as the element of $S_p(L_2(\mathbf {R}^n),L_2(\mathbf {R}^n \times \{1,2,\ldots ,N\}))$ mapping $g \in L_2(\mathbf {R}^n)$ to the function $(\xi ,j) \in \mathbf {R}^n \times \{1,2,\ldots ,N\} \mapsto (A_jg)(\xi )$ . In the particular case when $A_j \in S_p \cap S_2$ has kernel $K_j \in L_2(\mathbf {R}^n \times \mathbf {R}^n)$ , $\sum _{j=1}^N A_j \otimes e_{j,1}$ maps g to the function $(\xi ,j)\mapsto \int K_j(\xi ,\eta ) g(\eta ) d\eta $ , so it is the Hilbert-Schmidt operator with kernel $((\xi ,j),\eta )\mapsto K_j(\xi ,\eta )$ . Similarly, $\sum _{j=1}^N S_{m_j}(A_j)\otimes e_{j,1}$ is the Hilbert-Schmidt operators with kernel $((\xi ,j),\eta )\mapsto m_j(\xi ,\eta ) K_j(\xi ,\eta )$ . Therefore, we have

(2.3) $$ \begin{align} S_M\Big(\sum_{j=1}^N A_j \otimes e_{j,1}\Big) = \sum_{j=1}^N S_{m_j}(A_j)\otimes e_{j,1}, \end{align} $$

where $S_M$ is the Schur multiplier appearing in claim (B). Claim (B) therefore implies that (2.3) holds for every $A_1,\dots ,A_N \in S_p(L_2(\mathbf {R}^n))$ and that the $S_p$ -norm of (2.3) is $\leq C\|\sum _{j=1}^N A_j \otimes e_{j,1}\|_p$ .

Let us apply this with $A_j=A_{j,\alpha }$ . According to claim (2.2), we get

$$ \begin{align*}\Big\| \Big( \sum_{j=1}^N |H_{u_j}(f_j)|^2 \Big)^{\frac{1}{2}} \Big\|_{L_p(\mathbf{R}^n)} & \leq C \lim_{\alpha \to \mathcal{U}} \Big\| \sum_{j=1}^N A_{j,\alpha} \otimes e_{j,1} \Big\|_{S_p}\\ &= C \Big\| \Big( \sum_{j=1}^N |f_j|^2 \Big)^{\frac{1}{2}} \Big\|_{L_p(\mathbf{R}^n)}. \end{align*} $$

Thus, the assertion is a consequence of claim (B), for which we need to justify claim (A) first. To do so, we can assume that $\Sigma = f^{-1}(-\infty ,0)$ for a $\mathcal {C}^1$ -submersion $f \colon U \times V \to \mathbf {R}$ . Then $\nabla f(x,y)= (\nabla _x f(x,y),\nabla _y f(x,y))$ is a normal vector to the boundary $\partial \Sigma = f^{-1}(0)$ at every $(x,y) \in \partial \Sigma $ , pointing away from $\Sigma $ . Thus, replacing f by a positive multiple, we can assume that its gradient is $(\mathbf {n}_1(x,y),\mathbf {n}_2(x,y))$ . Then, the Taylor expansion of f gives

$$ \begin{align*} f \big( x+\varepsilon T \xi, y+\varepsilon \eta \big) & = \varepsilon \big\langle \mathbf{n}_1(x,y), T\xi \big\rangle + \varepsilon \big\langle \mathbf{n}_2(x,y), \eta \big\rangle + o(\varepsilon) \\ & = \varepsilon \big\langle \mathbf{n}_2(x,y), \eta - \xi \big\rangle + o(\varepsilon). \end{align*} $$

Therefore, if $\eta - \xi $ is not orthogonal to $\mathbf {n}_2(x,y)$ (a condition that holds for almost every $\xi $ and $\eta $ ), we have $\chi _\Sigma (x+\varepsilon T\xi , y+\varepsilon \eta ) =m_{\mathbf {n}_2(x,y)}(\xi ,\eta )$ for every $\varepsilon>0$ small enough. This proves claim (A).

To prove claim (B) we apply (A). More precisely, let $T_j \in \mathrm {GL}_n(\mathbf {R})$ be such that $T_j^* \mathbf {n}_1(x_j,y) = - \mathbf {n}_2(x_j,y) = - u_j$ for every $j = 1,2,\ldots ,N$ . The existence of these maps is clear, because by the transversality assumption, both $\mathbf {n}_1(x_j,y)$ and $\mathbf {n}_2(x_j,y)$ are nonzero vectors in $\mathbf {R}^n$ and $\mathrm {GL}_n(\mathbf {R})$ acts transitively on them. By Lemma 2.1, the Schur multiplier with symbol

$$\begin{align*}m_\varepsilon \big( (\xi,j),\eta \big) = \chi_\Sigma \big( x_j + \varepsilon T_j \xi, y + \varepsilon \eta \big) \end{align*}$$

is bounded with norm $\leq C$ for every $\varepsilon>0$ . Taking $\varepsilon \to 0^+$ , we obtain that the almost everywhere limit of $m_\varepsilon $ is $S_p$ -bounded with norm $\leq C$ . However, this limit is $m_{u_j}(\xi ,\eta )$ from claim (A). This proves claim (B).

Proof of (1) $\Rightarrow $ (2) in Theorem A.

Let us assume that (1) in Theorem A holds. By taking charts, we can and will assume that M and N are open subsets of $\mathbf {R}^{m}$ and $\mathbf {R}^n$ , respectively. We shall further assume that $m=n$ . For instance, if $m<n$ , we can replace M by $M \times \mathbf {R}^{n-m}$ and $\Sigma $ by

$$\begin{align*}\Big\{ \big( (x,x'),y \big) \mid (x,y) \in \Sigma, x' \in \mathbf{R}^{n-m} \Big\}.\end{align*}$$

By the transversality assumption, the map $z \mapsto \mathbf {n}_2(z)/\|\mathbf {n}_2(z)\|$ is continuous on a neighbourhood of the transverse point $(x_0,y_0)$ in Theorem A. Moreover, for y close to $y_0$ , we have that $\partial \Sigma ^y$ is locally a manifold, so is connected. Thus, if (2) was not true, there would exist y close to $y_0$ such that the subset of the sphere $X = \{\mathbf {n}_2(x',y)/\|\mathbf {n}_2(x',y)\|: x' \in \partial \Sigma ^y \cap U\}$ contains a connected subset not reduced to a point. According to (1) and Lemma 2.3, this would imply that the square function inequality holds uniformly in $L_p(\mathbf {R}^n)$ for any finite set in X. However, Fefferman’s main result in his proof of the ball multiplier theorem [Reference Fefferman15] claims that such a uniform inequality cannot hold. In fact, Fefferman stated it for $n=2$ , but the result in arbitrary dimension follows from the $2$ -dimensional case by K. de Leeuw’s restriction theorem [Reference de Leeuw29]. Hence, the zero-curvature condition (2) must hold.

Proof. By the transversality assumption that $T_{z_0}\Pi $ surjects onto $T_{y_0} M$ , we see that $T_{z_0} \Pi \cap (T_{x_0}M \oplus 0) \neq T_{x_0} M \oplus 0$ . Thus, by applying a local diffeomorphism $\phi :M \to \mathbf {R}^m$ , we can assume that $M=\mathbf {R}^m$ , $x_0=0$ and $(1,0, \ldots ,0)\notin T_{z_0} \Pi $ . Then by the implicit function theorem, there is a $\mathcal {C}^1$ function $h \colon \mathbf {R}^{m-1} \times N \to \mathbf {R}$ such that, on a neighbourhood of $(0,y_0)$ ,

$$ \begin{align*}\Pi = \big\{(x,y) \mid x_1 = h(x_2,\dots,x_m,y)\big\}.\end{align*} $$

The function $h(0,\cdot )$ vanishes at $y_0$ and, by the second half of the transversality assumption, has nonzero differential at $y_0$ . By the implicit function theorem (or the surjection theorem) again, there is a diffeomorphism $\psi $ from a neighbourhood of $y_0$ into $\mathbf {R}^n$ vanishing at $y_0$ and such that $h(0,y) = \psi (y)_1$ for every y close enough to $y_0$ . This proves the lemma with $g(0,y)=h(0,\psi ^{-1}(y))$ .

Proof of Theorem 2.5.

By symmetry of the two variables, it is enough to prove (a) $\Leftrightarrow $ (c). The implication (c) $\Rightarrow $ (a) is clear with the same U and V because in that case, $T_y \Pi _x$ is the kernel of $d_y g$ , which is independent of x. It remains to prove the implication (a) $\Rightarrow $ (c). Observe that both conditions are unchanged if we replace $(x_0,y_0,\Pi )$ by $(\phi (x_0),\psi (y_0),\phi \times \psi (\Pi ))$ for local diffeomorphisms. Therefore, by the normal form Lemma 2.6, we may assume that $M \times N=\mathbf {R}^m \times \mathbf {R}^n$ , $(x_0,y_0)=(0,0)$ and

$$ \begin{align*}\Pi \cap (U\times V) = \Big\{ (x,y)\in U \times V \mid x_1 = g(x_2,\dots,x_m,y)\Big\}\end{align*} $$

for some $\mathcal {C}^1$ function $g\colon \mathbf {R}^{m-1}\times \mathbf {R}^n \to \mathbf {R}$ satisfying $g(0,y)=y_1$ . Then, for every $(x,y) \in \Pi $ with $x=(x_1,\tilde {x})$ , we have $T_y \Pi _x = \ker ( d_y g(\tilde x,y))$ . Let $\tilde U \subset \mathbf {R}^{m-1},\tilde V \subset V$ be square neighbourhoods of $0$ such that $(g(\tilde x,y),\tilde x) \in U$ for $(\tilde x,y) \in \tilde U \times \tilde V$ . Then for every such $\tilde x,y$ , condition (a) applied to $x=(g(\tilde x,y),\tilde x)$ and $x' = (g(0,y),0)$ yields $\ker d_y g( \tilde x,y) = \ker d_y g(0,y) = \operatorname {span}(e_2,\dots ,e_n)$ . In particular, $\partial _{y_j} g(\tilde x,y)=0$ for every $j \geq 2$ , so (since $\tilde V$ is a square) $g(\tilde x,y) = w(\tilde x,y_1)$ for certain $\mathcal {C}^1$ function $w:\mathbf {R}^{m-1}\times \mathbf {R} \to \mathbf {R}$ satisfying $w(0,s)=s$ for all s. By the implicit function theorem, we get

$$ \begin{align*}\big\{(x_1,\tilde x,s) \mid x_1= w(\tilde x,s) \big\} = \big\{(x_1,\tilde x,s) \mid s = u(x_1,\tilde x) \big\}\end{align*} $$

locally for a $\mathcal {C}^1$ function $u \colon \mathbf {R}^m \to \mathbf {R}$ . This completes the proof of Theorem 2.5.

Proof of (2) $\Rightarrow $ (3) in Theorem A.

This is (a) $\Rightarrow $ (c) in Theorem 2.5 for $\Pi =\partial \Sigma $ .

Proof of (3) $\Rightarrow $ (1) in Theorem A.

This is immediate from the classical boundedness of the triangular projection on Schatten p-classes for $1 < p < \infty $ (due to Macaev [Reference Macaev48]; see also [Reference Gohberg and Krein17, Chap III, §6]) and the transference Lemma 2.1 above.

Proof. Spherical Hilbert transforms $H_{\mathbf {S},\delta }$ arise from relatively compact domains $\Sigma _\delta $ whose boundary is fully transverse. In particular, we may apply Theorem A. In dimension $1$ , the assertion follows immediately since the zero-curvature condition (2) is trivially satisfied. Alternatively, the symbol can be expressed as a triangular truncation in terms of the polar coordinates of x and y. When $n \ge 2$ , it is easily checked that the tangent spaces at $\partial \Sigma _{x_1}$ and $\partial \Sigma _{x_2}$ differ at their intersection points. This was illustrated for $n=2$ in Figure 1. Theorem A implies the assertion.

Proof. The main difficulty is to prove the equivalence (1) $\Leftrightarrow $ (4), which we leave to the end of the proof. The implication (2) $\Rightarrow $ (4) is clear, with $\mathrm {H} = \exp (\mathfrak {h})$ the stabilizer of $0$ . Under the assumption that $\mathrm {G}$ is simply connected, the converse (4) $\Rightarrow $ (2) holds by a Theorem of Mostow [Reference Mostow33], which implies that $\mathrm {H}$ is a closed subgroup. Therefore, $\mathrm {G}/\mathrm {H}$ is a $1$ -dimensional manifold that is simply connected, and so is diffeomorphic to $\mathbf {R}$ . The implication (3) $\Rightarrow $ (2) is also clear because $\mathrm {G}_j$ is given as a group of diffeomorphisms of $\mathbf {R}$ with $\Omega _j=\{g \in \mathrm {G}_j\mid g \cdot 0>0\}$ . The converse (2) $\Rightarrow $ (3) follows from Lie’s classification of (local) actions by diffeomorphism on the real line [Reference Lie30]; see also [Reference Tits47] for modern presentations and [Reference Ghys16] for the global aspect. More precisely, the fact that $g_0$ belongs to the boundary of $\Omega $ implies that $0$ is not fixed by $\mathrm {G}$ and – here we use that $\mathrm {G}$ is connected – the $\mathrm {G}$ -orbit of $0$ is an open interval, so by identifying it with $\mathbf {R}$ , we can assume that the $\mathrm {G}$ -action is transitive. In that case, the image of $\mathrm {G}$ in $\mathrm {Diff}(\mathbf {R})$ is one of the three groups in condition (3) of Corollary B1, see [Reference Ghys16, Section 4.1] for the details.

Next, let us focus on the equivalence (1) $\Leftrightarrow $ (4) for general Lie groups. If we translate $\Omega $ by $g_0^{-1}$ , we may assume that $g_0=e$ and the tangent space of $\mathrm {G}$ at $g_0$ identifies with its Lie algebra $\mathfrak {g}$ . Also, the tangent space of $\partial \Omega $ at $g_0$ identifies with a codimension $1$ subspace $\mathfrak h$ of $\mathfrak g$ . Define the $\mathcal {C}^1$ -manifold

$$\begin{align*}\widetilde{\Omega} = \Big\{(g,h) \in \mathrm{G}\times \mathrm{G} \mid gh \in \Omega\Big\}.\end{align*}$$

Its sections $\widetilde {\Omega }_g$ and $\widetilde {\Omega }^h$ are left and right translates of the $\mathcal {C}^1$ -domain $\Omega $

(3.1) $$ \begin{align} \widetilde{\Omega}_g:= \big\{h: (g,h) \in \widetilde{\Omega}\big\} = g^{-1}\Omega \quad \text{and} \quad \widetilde{\Omega}^h := \big\{g : (g,h) \in \widetilde{\Omega}\big\}=\Omega h^{-1}. \end{align} $$

In particular, $\widetilde {\Omega }$ is transverse at every point of its boundary. By Lemma 2.1 and Theorem 3.1, we know that (1) is equivalent to the existence of a neighbourhood of the identity $U \subset \mathrm {G}$ such that $\chi _{\widetilde {\Omega }}$ defines a Schur multiplier on $S_p(L_2(U))$ . By Theorem A, this is equivalent to the existence of a neighbourhood of the identity $V \subset \mathrm {G}$ such that both conditions below hold:

(3.2) $$ \begin{align} T_{h} \partial\widetilde{\Omega}_{g_1} = T_h \partial\widetilde{\Omega}_{g_2} &\textrm{ for every }g_1,g_2,h \in V \textrm{ such that }g_1h,g_2h \in \partial \Omega. \end{align} $$

(3.3) $$ \begin{align} T_{g} \partial\widetilde{\Omega}^{h_1} = T_g \partial\widetilde{\Omega}^{h_2} &\textrm{ for every }g, h_1,h_2 \in V \textrm{ such that }gh_1,gh_2 \in \partial \Omega. \end{align} $$

By the above idenfications (3.1), if we denote by $L_x,R_x\colon \mathrm {G}\to \mathrm {G}$ the left and right multiplication by x, these conditions are equivalent to the existence of a neighbourhood of the identity W such that

(3.4) $$ \begin{align} d_{x_1} L_{x_2x_1^{-1}} (T_{x_1} \partial \Omega) = T_{x_2} \partial \Omega &\textrm{ for every }x_1,x_2 \in \partial \Omega \cap W. \end{align} $$

(3.5) $$ \begin{align} d_{x_1} R_{x_1^{-1}x_2} (T_{x_1} \partial \Omega) = T_{x_2} \partial \Omega &\textrm{ for every }x_1,x_2 \in \partial \Omega \cap W. \end{align} $$

Indeed, taking $x_j = g_j h$ for $j=1,2$ we have

$$\begin{align*}T_h (g_j^{-1} \partial \Omega) = d_{x_j} L_{g_j^{-1}}(T_{x_j} \partial \Omega).\end{align*}$$

Composing by $(d_{x_2} L_{g_2^{-1}})^{-1} = d_h L_{g_2}$ , and using $(d_h L_{g_2}) \circ (d_{x_1} L_{g_1^{-1}})= d_{x_1} L_{x_2x_1^{-1}}$ by the chain rule, we see that (3.2) is equivalent to (3.4). The equivalence for right multiplication maps is entirely similar. Next, recalling that $T_e \partial \Omega = \mathfrak {h}$ , the above conditions can be written in the equivalent forms:

(3.6) $$ \begin{align} d_{e} L_x(\mathfrak{h}) =T_x \partial \Omega&\textrm{ for every }x \in \partial \Omega \cap W. \end{align} $$

(3.7) $$ \begin{align} d_{e} R_{x} (\mathfrak{h}) = T_{x} \partial \Omega &\textrm{ for every }x \in \partial \Omega \cap W. \end{align} $$

If we remember that $\operatorname {\mathrm {Ad}}_x =d_e (R_{x^{-1}} L_x)$ , we obtain that this system is equivalent to $d_{e} L_x(\mathfrak {h}) =T_x \partial \Omega $ and $\operatorname {\mathrm {Ad}}_x \mathfrak {h} = \mathfrak {h}$ for every $x \in \partial \Omega \cap W$ . Therefore, we have proved that (1) at $g_0=e$ is equivalent to the existence of a neighbourhood of the identity W such that $T_x \partial \Omega = d_e L_x( \mathfrak {h})$ and $\operatorname {\mathrm {Ad}}_x \mathfrak {h} = \mathfrak {h}$ for every $x \in \partial \Omega \cap W$ . These conditions clearly hold if $\mathfrak {h}$ is a Lie algebra and $\partial \Omega $ locally coincides with the exponential of a neighbourhood of $0$ in $\mathfrak {h}$ . Conversely, assume $T_x \partial \Omega = d_e L_x(\mathfrak {h})$ and $\operatorname {\mathrm {Ad}}_x(\mathfrak {h}) = \mathfrak {h}$ for every $x \in \partial \Omega \cap W$ . Making x go to the identity element e in the second condition, we deduce that $\operatorname {\mathrm {ad}}_{\mathrm {X}}(\mathfrak h) \subset \mathfrak {h}$ for every $\mathrm {X} \in \mathfrak {h}$ . That is, $\mathfrak {h}$ is a Lie subalgebra. By the local uniqueness of a manifold in $\mathrm {G}$ containing e and whose tangent space at x is $d_e L_x(\mathfrak {h})$ (Frobenius’ theorem), we deduce that $\partial \Omega $ is locally the exponential of a neighbourhood of $0$ in $\mathfrak {h}$ . This completes the proof.

Proof of Corollary B2.

Assertion i) follows since the quotient of a nilpotent Lie algebra remains nilpotent, so the nonnilpotent examples in Theorem 3.3 (3) cannot happen when $\mathrm {G}$ is nilpotent. Assertions ii) and iii) follow immediately from Lie’s classification [Reference Lie30]: up to isomorphism, there is a unique pair $(\mathfrak {h},\mathfrak {g})$ where $\mathfrak {g}$ is a simple Lie algebra and $\mathfrak {h}$ is a codimension $1$ subalgebra. It is given by $\mathfrak {g}=\mathfrak {sl}_2$ and $\mathfrak {h}$ the subalgebra of upper-triangular matrices. This completes the proof.

Proof of Lemma 3.5 for $\mathrm {G}$ unimodular.

Translating V and W, we can assume that the identity belongs to V and W and $g_0=e$ . Let U be a neighbourhood of e such that $U \subset V$ and $U^{-1} U \subset W$ . Let $\phi = \frac {1}{|U|} \chi _U$ and $\psi = \chi _{U^{-1} U}$ , so that

$$ \begin{align*}\int \phi(g h) \psi(h) dh = 1 \quad \text{for every} g \in U.\end{align*} $$

Consider the map

$$\begin{align*}J_p\colon L_p(\mathcal{L} \mathrm{G}) \ni x \mapsto \phi^{\frac 1 p} x \psi^{\frac 1 p} \in S_p(L_2(V),L_2(W)),\end{align*}$$

where we identify $\phi ,\psi $ with the operators of multiplication by $\phi ,\psi $ . The convention is that $0^{\frac 1 \infty } = 0$ . We claim that the maps $J_p$ are completely bounded with cb-norm

$$\begin{align*}\|J_p\|_{\mathrm{cb}(L_p,S_p)} \leq \|\phi\|_{L_2(\mathrm{G})}^{\frac 1 p} \|\psi\|_{L_2(\mathrm{G})}^{\frac 1 p}\end{align*}$$

whenever $1 \le p \le \infty $ . By interpolation, it is enough to justify the extreme cases $p=1$ and $p=\infty $ . The case $p = \infty $ is clear. For the case $p=1$ , we factorize $x = x_1 x_2$ so that $J_1(x) = \phi x_1 \cdot x_2 \psi $ . Take them so that $\|x\|_1 = \|x_1\|_2 \|x_2\|_2$ , and it suffices to show that both factors are bounded in $S_2(L_2(\mathrm {G})) \simeq L_2(\mathrm {G} \times \mathrm {G})$ by $\|\phi \|_{L_2(\mathrm {G})} \|x_1\|_{L_2(\mathcal {L} \mathrm {G})}$ and $\|\psi \|_{L_2(\mathrm {G})} \|x_2\|_{L_2(\mathcal {L} \mathrm {G})}$ , respectively. Using that

$$\begin{align*}J_\infty: x \mapsto \Big( \widehat{x}(gh^{-1}) \Big), \end{align*}$$

the expected bounds follow from Plancherel theorem $L_2 (\mathcal {L} \mathrm {G}) \simeq L_2(\mathrm {G})$ . Moreover when $x= \lambda (f)$ , the operator $J_p(x)$ has kernel $(\phi (g)^{1/p} f(gh^{-1}) \psi (h)^{1/p})$ . Thus, it is clear that the map $J_p$ intertwines the Fourier multiplier with symbol $g \mapsto m(g)$ and the Schur multiplier with symbol $(g,h) \mapsto m(gh^{-1})$ .

The inequality $C^{-1}\|x\|_p \leq \|J_p(x)\|_p$ is a bit more involved. Let $f \in M_n \otimes \mathcal {C}_c(U)$ with $\lambda (f) \in L_p(M_n \otimes \mathcal {L} \mathrm {G})$ and assume that $x = \lambda (f)$ by density. Let q be the conjugate exponent of p and $\gamma \in M_n \otimes \mathcal {C}_c(\mathrm {G})$ with $\lambda (\gamma ) \in L_q(M_n \otimes \mathcal {L} \mathrm {G})$ . Then we have

$$ \begin{align*} \operatorname{\mathrm{Tr}} \otimes \operatorname{\mathrm{Tr}}_n \big( J_p(\lambda(f)) J_{q} (\lambda(\gamma))^* \big) & = \operatorname{\mathrm{Tr}}_n \int_{\mathrm{G} \times \mathrm{G}} \phi(g) f \gamma^*(gh^{-1}) \psi(h) ds dt\\ & = \int_{\mathrm{G}} \operatorname{\mathrm{Tr}}_n \big( f(g) \gamma(g)^* \big) \Big[ \int_{\mathrm{G}} \phi(gh) \psi(h) \, dh \Big] \, dg\\ & = \int_{\mathrm{G}} \operatorname{\mathrm{Tr}}_n \big( f(g) \gamma(g)^* \big) \, dg = \tau\otimes \operatorname{\mathrm{Tr}}_n \big( \lambda(f) \lambda(\gamma)^* \big). \end{align*} $$

In the last line, we used that $\int _{\mathrm {G}} \phi (gh) \psi (h) \, dh=1$ on $\mathrm {supp} f \subset U$ . By Hölder’s inequality, we get

$$ \begin{align*} \big| \tau\otimes \operatorname{\mathrm{Tr}}_n( \lambda(f) \lambda(\gamma)^*) \big| & \leq \|J_p(\lambda(f))\|_p \|J_q(\lambda(\gamma)\|_q\\ & \leq \|J_q\|_{\mathrm{cb}} \|J_p(\lambda(f))\|_p \|\lambda(\gamma)\|_q\\ & \leq \|J_q\|_{\mathrm{cb}} \|\lambda(\gamma)\|_q \|J_p(\lambda(f))\|_p. \end{align*} $$

Taking the sup over $\gamma $ , we get $C^{-1}\|x\|_p \leq \|J_p(x)\|_p$ for $C = \|J_q\|_{\mathrm {cb}}< \infty $ .

Proof of Theorem 3.1.

The implication (a) $\Rightarrow $ (b) is easy. Indeed, if (a) holds and $\varphi \in A(\mathrm {G})$ is supported in U and equal to $1$ on a neighbourhood of $g_0$ (for the construction of $\varphi $ , see the proof of Lemma 3.5), then $T_\varphi $ is completely bounded on $L_p(\mathcal {L} \mathrm {G})$ for every $1 \leq p \leq \infty $ , and takes values in the space of elements Fourier supported in U. In particular, $T_{m\varphi } = T_{m,U}\circ T_\varphi $ is also completely bounded. The implication (b) $\Rightarrow $ (c) follows from [Reference Caspers and de la Salle7, Theorem 4.2], which implies that

$$ \begin{align*}(g,h) \in \mathrm{G} \times \mathrm{G} \mapsto m(gh^{-1})\varphi(gh^{-1})\end{align*} $$

defines a completely bounded Fourier $L_p$ -multiplier. Thus, we get (c) if $V,W$ are chosen so that $\varphi =1$ on $V W^{-1}$ . Finally, (c) $\Rightarrow $ (a) follows from Lemma 3.5.

Proof. Let m be the indicator function of $\{g \in \mathrm {G} \mid g \cdot 0>0\}$ . It suffices to prove the following implication for every $2 \leq p<\infty $ : if m defines a completely bounded Fourier $L_p$ -multiplier with norm $\leq C_p$ , then it defines a completely bounded Fourier $L_{2p}$ -multiplier with norm $\leq 2 C_p$ . Indeed, using that $C_2=\|m\|_\infty =1$ , we deduce $C_{2^N} \leq 2^N$ for every integer N, so by interpolation, $C_p \leq 2p$ for all $p\geq 2$ . By duality, the conclusion also holds for $p\leq 2$ .

Let r be the dual exponent of $2p$ . Let $f \in \mathcal {C}_c(\mathrm {G})$ and consider $X =\mathcal {F}_r(f)$ and $Y=\mathcal F_r(mf)$ ; these are well-defined elements of $L_{2p}(\mathcal {L} G)$ by [Reference Terp46]. Then, we claim that the equality below holds:

(3.9) $$ \begin{align} Y^* Y = T_m(Y^*X) + T_m(Y^*X)^*. \end{align} $$

Indeed, this inequality is equivalent to the almost everywhere equality

$$\begin{align*}(mf)^* \ast (mf) = m \big( (mf)^* \ast f \big) + \big( m \big( (mf)^* \ast f \big)\big)^*,\end{align*}$$

which follows for the fact that $m(g^{-1}) m(g^{-1}h) = m(h) m(g^{-1}) + m(h^{-1})m(g^{-1}h)$ for almost every $g,h \in \mathrm {G}$ . If the whole group $\mathrm {G}$ fixes $0$ , this is obvious because m is identically $0$ . Otherwise, the stabilizer of $0$ is a closed subgroup, so it has measure $0$ and it is enough to justify the equality for $h \cdot 0 \neq 0$ . Set $(\alpha ,\beta ) = (g \cdot 0 ,h \cdot 0)$ and observe that $m(g^{-1}) m(g^{-1}h)=1$ if and only if $\alpha < \min \{0, \beta \}$ . Similarly, we have $m(h) m(g^{-1}) = 1$ iff $\alpha < 0 < \beta $ and $m(h^{-1})m(g^{-1}h)=1$ iff $\alpha < \beta < 0$ . Therefore, the expected identity reduces to the trivial one $\chi _{\alpha < 0 \wedge \beta } = \chi _{\alpha < 0 < \beta } + \chi _{\alpha < \beta < 0}$ . This justifies (3.9), both sides of which are in $L_{p}(\mathcal {L} \mathrm {G})$ . Thus, taking the norm and applying the triangle inequality, the hypothesis and Hölder’s inequality leads to

$$\begin{align*}\|Y\|_{2p}^2 \leq 2 C_{p} \|X\|_{2p} \|Y\|_{2p}.\end{align*}$$

We deduce $\|Y\|_{2p} \leq 2 C_{p} \|X\|_{2p}$ . Since $\mathcal {C}_c(\mathrm {G})$ is dense in $L_r(\mathrm {G})$ , we obtain that m defines a Fourier $L_{2p}$ -multiplier with norm $\leq 2C_p$ . A similar argument gives the same bound for the completely bounded norm, which concludes the proof.

Proof. When $\phi =\psi $ are indicator functions, the first two points were proved in [Reference Caspers and de la Salle7, Proposition 3.3, Theorem 5.2]. The same argument applies in our case. Let us justify identity (3.10). First, observe that $\phi \overline {\phi '}$ and $\psi \overline {\phi '}$ belong to $L_2(\mathrm {G})$ by Hölder’s inequality, so that m indeed belongs to $A(\mathrm {G})$ . In particular, m defines a completely bounded $L_1$ and $L_\infty $ Fourier multiplier. Thus, it also defines a Fourier $L_p$ -multiplier [Reference Caspers and de la Salle7, Definition-Proposition 3.5]. Therefore, by interpolation [Reference Caspers and de la Salle7, Section 6], it suffices to prove (3.10) for $p=1$ and $p=\infty $ . These two cases are formally equivalent, and we just consider $p=\infty $ . In that case, $y\in L_1(\mathcal {L} \mathrm {G})$ corresponds to an element $f \in A(\mathrm {G})$ and $\phi ' y \psi '$ is the trace class operator with kernel

$$\begin{align*}\Big( \phi'(g) f(hg^{-1}) \psi'(h) \Big)_{g,h \in \mathrm{G}};\end{align*}$$

see [Reference Caspers and de la Salle7, Lemma 3.4]. By a weak- $*$ density argument, it is enough to prove (3.10) for $x=\lambda (g_0)$ for some $g_0 \in \mathrm {G}$ . In that case, $T_m(x) = m(g_0) \lambda (g_0)$ and we obtain $\operatorname {\mathrm {tr}}( T_m(x) y^*) = m(g_0) \overline {f(g_0^{-1})}$ . We can compute

$$ \begin{align*} \operatorname{\mathrm{Tr}} \big( \phi x \psi (\phi' y \psi')^* \big) & = \operatorname{\mathrm{Tr}} \big( \lambda(g_0) \psi \overline{\psi'} y^* \phi \overline{\phi'} \big) \\ [5pt] & = \operatorname{\mathrm{Tr}} \Big[ \Big( \psi \overline{\psi'})(g_0^{-1}g) \overline{f(g_0^{-1} g h^{-1} )} (\phi \overline{\phi'})(h) \Big)_{g,h \in \mathrm{G}} \Big]\\ & = \int_{\mathrm{G}} (\psi \overline{\psi'})(g_0^{-1}g) \overline{f(g_0^{-1})} (\phi \overline{\phi'})(g) \, dg = m(g_0) \overline{f(g_0^{-1})}. \end{align*} $$

This justifies the identity (3.10) and completes the proof of the lemma.

Proof of Lemma 3.5, general case.

We take $\phi ,\psi \in \mathcal {C}_c(\mathrm {G})$ as in the proof in the unimodular case, and set $m(g) = \int \phi (gh) \psi (h) dh$ . By Lemma 3.10, we can define completely bounded maps

$$\begin{align*}J_p\colon L_p(\mathcal{L} \mathrm{G}) \ni x \mapsto \phi^{\frac 1 p} x \psi^{\frac 1 p} \in S_p(L_2(V),L_2(W)),\end{align*}$$

which intertwine Fourier and Schur multipliers. Now, if $x \in L_p(M_n \otimes \mathcal {L} \mathrm {G})$ is Fourier supported in U and $y \in L_q(M_n \otimes \mathcal {L} \mathrm {G})$ – for q being the dual exponent of p – we get

$$ \begin{align*} \operatorname{\mathrm{tr}}(x y^*) & = \operatorname{\mathrm{tr}}\big( T_{m}(x) y^*\big) = \operatorname{\mathrm{Tr}} \big( J_p(x) J_q (y)^* \big)\\ & \leq \|J_p(x)\|_{S_p} \|J_q(y)\|_{S_q} \leq \|J_q\|_{\mathrm{cb}} \|y\|_{L_q(\mathcal{L} \mathrm{G})} \|J_p(x)\|_{S_p}. \end{align*} $$

The first line is because $m=1$ on U and x is Fourier supported in U, and by (3.10). The last line is Hölder’s inequality. Taking suprema over y in the unit ball of $L_q(\mathcal {L} \mathrm {G})$ gives $\|x\|_{L_p(\mathcal {L} \mathrm {G})} \leq \|J_q\|_{\mathrm {cb}} \|J_p(x)\|_{S_p}$ .