1.1 Admissible manifolds: discussion

Let us now briefly discuss and compare the different assumptions we will have to make about the function $\varphi $ in the definition of the metric in equation (1.5) in order to prove our estimates.

Assumptions (A0). Let $(\mathcal {M},g)$ be a Lorentzian manifold of dimension $n+1\geq 4$ defined by $\mathcal {M}=\mathbb {R}_{t}\times \Sigma $ , with $(\Sigma ,h)$ a warped product: that is, a Riemannian manifold in the form $\Sigma =\mathbb {R}_{r}^{+}\times \mathbb {K}^{n-1}$ , where $\mathbb {K}^{n-1}$ is an $(n-1)$ -dimensional compact spin manifold and $\Sigma $ is equipped with the Riemannian metric

(1.10) $$ \begin{align} dr^{2}+\varphi(r)^{2}d\omega_{\mathbb{K}^{n-1}}^{2}, \end{align} $$

where $d\omega _{\mathbb {K}^{n-1}}^{2}$ the Riemannian metric on $\mathbb {K}^{n-1}$ and where the function $\varphi :\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ is $C^{\infty }(\mathbb {R}^{+})$ , is strictly positive on $(0,+\infty )$ and is such that

(1.11) $$ \begin{align} \varphi(0)=\varphi^{(2n)}(0)=0,\qquad \varphi^{\prime}(0)=1,\qquad \frac{\varphi^{\prime}(r)}{\varphi(r)}\in L^{\infty}. \end{align} $$

Notice that Assumptions (A0), in the case $n=3$ and $\mathbb {K}=\mathbb {S}$ , are essentially the ones we retained in [Reference Cacciafesta and de Suzzoni11] in order to prove local-in-time Strichartz estimates.

In order to prove global-in-time Strichartz estimates for the dynamics restricted to an eigenspace of $\mathcal {D}_{\mathbb {K}^{n-1}}$ , we need to complement (A0) with the following:

Assumptions (A1). Let $(\mathcal {M},g)$ satisfy assumptions (A0). Assume that the operator $\mathcal {D}_{\mathbb {K}^{n-1}}$ has no eigenvalue $\mu $ with $|\mu |<\frac 12$ , and let $\mu $ be in the spectrum of $\mathcal {D}_{\mathbb {K}^{n-1}}$ . Let $V_{\mu } = \frac {\mu (\mu + \varphi ^{\prime })}{\varphi ^{2}}$ and

(1.12) $$ \begin{align} \delta_{\varphi}(\mu) = \min(1,\inf (4r^{2}{V}_{\mu} + 1),\inf (-4r^{2} {V}_{\mu}- 4r^{3}{V}_{\mu}^{\prime}+1)). \end{align} $$

Assume that the function $\varphi $ in the metric in equation (1.10) satisfies

(1.13) $$ \begin{align} \displaystyle\delta_{\varphi}(-\mu),\delta_{\varphi}(\mu)> 0,\quad r^{2}{V}_{\mu} \in L^{\infty}. \end{align} $$

In order to prove global-in-time Strichartz estimates for the complete flow, we need to strengthen our assumptions some more, having in mind as a main example asymptotically flat manifolds. We thus set the following:

Assumptions (A2). Let $(\mathcal {M},g)$ be defined by $\mathcal {M}=\mathbb {R}_{t}\times \Sigma $ , with $(\Sigma ,h)$ a spherically symmetric manifold of dimension $n\geq 3$ with the metric given by equation (1.5) with $\mathbb {K}=\mathbb {S}$ . Let $\varphi \in C^{\infty }(\mathbb {R^{+}})$ be such that $\varphi (0) = 0$ , $\varphi ^{\prime }(0) = 1$ , and for all $k\in \mathbb {N}$ , $\varphi ^{(2k)}(0) = 0$ . We assume that there exists $\varphi _{1} \in \mathcal C^{\infty } (\mathbb {R}^{+})$ such that

$$ \begin{align*} \varphi : r\mapsto r(1 + \varphi_{1}(r)) \end{align*} $$

with the following assumptions on $\varphi _{1}$ :

• • $\varphi _{1}$ is non-negative;

• • $\sup _{r\geq 0}(|\varphi _{1}(r)|+|r\varphi _{1}(r)^{\prime }|+ |r^{2}\varphi _{1}^{\prime \prime }(r)|)\ll 1 $ .

Remark 1.1. The map

$$ \begin{align*} \varphi_{1} = \varepsilon\frac{r^{\alpha}}{\langle r \rangle^{\beta}} \end{align*} $$

with $\beta \geq \alpha> 0$ , with $\alpha ,\beta \in \mathbb {N}$ , $\alpha $ even, satisfies these assumptions.

Remark 1.2. Our aim is to apply this type of result to well-known Lorentzian manifolds such as black holes. It is known that there are spherically symmetric back holes, such as the Schwarzschild or Reissner-Nordström one; in these cases, the metric (outside the black hole) writes

$$ \begin{align*} ds^{2} = F(r) dt^{2} - F^{-1}(r) dr^{2} -r^{2} d\omega_{\mathbb S^{2}}, \end{align*} $$

where $d\omega _{\mathbb S^{2}}$ is the metric on the sphere $\mathbb S^{2}$ and F is defined as

$$ \begin{align*} F(r) = 1 - \frac{A}{r} + \frac{B}{r^{2}}, \end{align*} $$

with $A,B\geq 0$ and $B = 0$ in the case of the Schwarzschild metric.

The metric couples time and space, but the Dirac equation does not, and in fact it is written (we refer to [Reference Nicolas31, Eq. (12)]) as follows

$$ \begin{align*} \Big[\gamma^{0} \partial_{t} + F\gamma^{1} \Big(\partial_{r} + \frac1{r} + \frac{F^{\prime}}{4F}\Big) + \frac{F^{1/2}}{r} \mathcal D_{\mathbb S^{2}} + i F^{1/2} m\Big] u = 0, \end{align*} $$

where $\mathcal D_{\mathbb S^{2}}$ is an operator acting only on the angular variable and can be diagonalized in the same way as we did in this paper.

By changing variables, we get an equation of the type

$$ \begin{align*} \Big[\gamma^{0} \partial_{t} + \gamma^{1} \Big(\partial_{r} + \psi_{1}(r)\Big) + \psi_{2}(r) \mathcal D_{\mathbb S^{2}} + i \psi_{3}(r) m\Big] u = 0, \end{align*} $$

where $\psi _{j}$ , $j=1,2,3$ are functions of the radial variable such that

$$ \begin{align*} \psi_{1}(r), \psi_{2}(r) = \frac1{r} + \mathcal O(1/r^{2}), \quad\psi_{3}(r) = 1 + \mathcal O(1/r). \end{align*} $$

The behavior as $r\rightarrow \infty $ is the same as in our current case, especially if $m=0$ . The difficulty arises when one looks at the region close to the black hole, as there, indeed, the functions $\psi _{2} $ and $\psi _{3}$ are not differentiable.

Therefore, this paper has to be thought of as a first step toward the study of the dispersion of the Dirac operator in a spherically symmetric black hole metric, as we here tackle a similar case for the behavior at $\infty $ , but we do not tackle the difficult task of looking at what happens close to the black hole.

1.2 Main results

We are now ready to state the main results. For a definition of functional spaces, we refer to Subsection 1.3.

Definition 1.3. We say that the triple $(p,q,m)$ is admissible, either if $m=0$

$$ \begin{align*} \frac2{p} + \frac{n-1}{q} = \frac{n-1}2, \quad p> 2, \quad q\in [2,\infty) , \end{align*} $$

or if $m\neq 0$

$$ \begin{align*} \frac2{p} + \frac{n}{q} = \frac{n}2, \quad p> 2, \quad q\in [2,\infty). \end{align*} $$

The first result we prove is a global-in-time Strichartz estimate for the Dirac flow restricted to eigenspaces of the operator $\mathcal {D}_{\mathbb {K}^{n-1}}$ .

Theorem 1.4. Let $(\mathcal {M},g)$ satisfy assumptions (A0) and (A1). Then, for any admissible triple $(p,q,m)$ in Definition 1.3 and any $\varepsilon>0$ , there exists a constant C depending only on $m,p,q, \varphi , \varepsilon $ (but not on $\mu $ ) such that for all $v_{0} \in H^{1/2}_{\varphi }$ ,

(1.14) $$ \begin{align} \begin{aligned} \big\| \Big( \frac{\varphi(r)}{r}\Big)^{\frac{(n-1)}2\left(1-\frac2q\right)} &e^{-ith_{\mu}}v_{0} \big\|_{L^{p}(\mathbb{R},W^{1/q-1/p,q}_{\varphi} )} \\ &\leq C |\mu|^{5/p+\varepsilon}(\delta_{\varphi}(\mu)^{1/p+\varepsilon}+\delta_{\varphi}(-\mu)^{1/p+\varepsilon}) \|v_{0}\|_{H^{1/2}_{\varphi}}, \end{aligned} \end{align} $$

where $W^{1/q-1/p,q}_{\varphi } $ and $H^{1/2}_{\varphi }$ are Sobolev spaces on the manifold $\Sigma $ for radial functions defined in Subsection 1.3.

Remark 1.6. Note that assuming that the compact manifold $\mathbb {K}^{n-1}$ satisfies that the (discrete) spectrum of the Dirac operator on $\mathbb {K}^{n-1}$ is included in $(-\infty ,- \frac 12]\cup [\frac 12, \infty )$ ensures that the Dirac operator on $\Sigma $ with $\varphi =r$ is self-adjoint (see [Reference Chou15, Theorem 3.2] and references therein). The operator $h_{\mu }$ being isomorphic to an $L^{\infty }$ perturbation of

$$ \begin{align*} \tilde h_{\mu} = \begin{pmatrix} m & -\Big( \partial_{r} + \frac1{r}\Big) + \frac\mu{r} \\ \Big( \partial_{r} + \frac1{r}\Big) + \frac\mu{r} & -m \end{pmatrix}, \end{align*} $$

we get that $h_{\mu }$ is self-adjoint. This will be further commented upon in Remark 2.2.

Remark 1.7. When $\mathbb {K}^{n-1}$ is the $(n-1)$ -dimensional sphere, then the manifold $\Sigma $ is smooth and in fact geometrically complete, which ensures the self-adjointness of the Dirac operator. Also, in this case, by relying on the endpoint Strichartz estimate proved in [Reference Machihara, Nakamura, Nakanishi and Ozawa29] and on mixed Strichartz-local smoothing estimates ([Reference Cacciafesta and D’Ancona9]), it is possible to recover the endpoint as well but with extra (global) derivatives, namely the estimates

(1.15) $$ \begin{align} \left\|u\left(\frac{\varphi(r)}r\right)^{\frac{(n-1)}2}\right\|_{L^{2}_{t}(I, L^{\infty}(\mathbb{R}_{+},L^{p}(\mathbb S^{n-1})))}\leq \sqrt p |\mu|^{5/2}(\delta_{\varphi}(\mu)^{1/2}+\delta_{\varphi}(-\mu)^{1/2}) \|u_{0}\|_{H^{2}(\Sigma)} \end{align} $$

when $m=0,n=3$ and

$$ \begin{align*} \big\| \Big( \frac{\varphi(r)}{r}\Big)^{\frac{(n-1)}2\left(1-\frac2q\right)} &e^{-ith_{\mu}}v_{0} \big\|_{L^{2}(\mathbb{R},W^{1/q-1/2,q}_{\varphi} )} \\ &\leq C |\mu|^{5/2}(\delta_{\varphi}(\mu)^{1/2}+\delta_{\varphi}(-\mu)^{1/2}) \|v_{0}\|_{H^{3/2}_{\varphi}}, \end{align*} $$

where $q = \frac {2(n-1)}{n-3}$ if $m=0$ and $q = \frac {2n}{n-2}$ otherwise.

The fact that the constant on the right-hand side of the estimate in equation (1.14) is a function of $\mu $ suggests that it might be possible to prove Strichartz estimates with loss of angular derivatives: such estimates are quite classical in the context of dispersive PDEs, and the local-in-time case (in dimension $3$ ) has been already discussed in the predecessor of this paper ([Reference Cacciafesta and de Suzzoni11]). For the next theorem, we shall indeed restrict to the case $\mathbb {K}^{n-1}=\mathbb {S}^{n-1}$ in order to be able to resort to the well-established Littlewood-Paley theory on the sphere. It is in fact possible to ‘sum’ the Strichartz estimates in equation (1.14) in order to obtain Strichartz estimates for general initial data by requiring additional regularity in the angular variable (we postpone to Subsection 1.3 the precise definitions of the spaces $H^{a,b}(\Sigma )$ ).

The result is the following:

Theorem 1.10. Let $(\mathcal {M},g)$ satisfy assumptions (A2). Let $p,q\in [2,\infty ]$ and $a,b \geq 0$ . Assume that $(p,q,m)$ is admissible and $\frac 5{pb} + \frac {1}{2a} < 1$ . Then the solutions u to equation (1.2) with initial data $ u_{0}\in H^{a,b}(\Sigma )$ satisfy the estimates

(1.16) $$ \begin{align} \left\|\left(\frac{\varphi(r)}r\right)^{\frac{(n-1)}2\left(1-\frac2q\right)}u \right\|_{L^{p}_{t}(\mathbb{R},W^{1/q-1/p,q}(\Sigma))}\leq C\| u_{0}\|_{H^{a,b}(\Sigma)}. \end{align} $$

As a matter of fact, the starting point in the proof of Theorem 1.10 is showing that within the assumptions (A2), the crucial condition given by (A1) is satisfied. In other words, Assumptions (A2) (asymptotically flat, spherically symmetric manifolds) provide an explicit example of ‘admissible manifolds’ for the validity of Theorem 1.4. In Subsection 4.2, we will thus prove the following:

The plan of the paper is the following. In Section 2, we review the separation of variables procedure for the Dirac equation in the warped products setting and show how to reduce to the Klein-Gordon dynamics. In Section 3, we discuss the classical Kato argument to obtain the Strichartz estimates for the Klein-Gordon dynamics with potentials of critical decay. Finally, in Section 4, we show that asymptotically flat manifolds are admissible, and we prove Strichartz estimates for general initial data in the spherically symmetric setting.

1.3 Notations

We will use the standard notation $L^{p}$ , $\dot {H}^{s}$ , $H^{s}$ , $W^{p,q}$ to denote, respectively, the Lebesgue and homogeneous/non homogeneous Sobolev spaces of functions from $\mathbb {R}^{n}$ to $\mathbb {C}^{M}$ . We will use the same notation to denote these functional spaces on the (spatial) manifold $(\Sigma ,h)$ , which is in our structure in equation (1.1), that is, with time and space already decoupled, by adding the dependence $L^{p}(\Sigma )$ , $\dot {H}^{s}(\Sigma )$ , $H^{s}(\Sigma )$ , $W^{p,q}(\Sigma )$ : for example, the norm $L^{p}(\Sigma )$ will be given by

$$ \begin{align*}\|f\|_{L^{p}(\Sigma)}^{p}:=\int |f(x)|^{p} \sqrt{\det(h(x))}dx \end{align*} $$

and so on.

The space $L^{2}(\Sigma )$ is thus endowed with the usual Hilbertian structure.

The space $\dot H^{1}(\Sigma )$ , is induced by the norm

$$ \begin{align*} \|f\|_{\dot H^{1}(\Sigma)}^{2} : = \|\sqrt{h^{ij}\langle D_{i} f,D_{j} f \rangle_{\mathbb{C}^{M}}}\|_{L^{2}(\Sigma)}, \end{align*} $$

where the $D_{j}$ are covariant derivatives for Dirac bispinors.

The space $W^{1,p}$ , $p\in [1,\infty ]$ is induced by

$$ \begin{align*} \|f\|_{W^{1,p}(\Sigma)} = \|\sqrt{h^{ij}\langle D_{i} f,D_{j} f \rangle_{\mathbb{C}^{M}}}\|_{L^{p}(\Sigma)} + \|f\|_{L^{p}(\Sigma)}. \end{align*} $$

The spaces $\dot H^{s}(\Sigma )$ and $W^{s,p}(\Sigma )$ with $s\in [-1,1]$ are defined by interpolation and duality.

Due to the warped product structure of the metric in equation (1.5), for a radial function $f_{rad}(|x|)$ we define

$$ \begin{align*}\|f_{rad}\|_{L^{p}_{\varphi}}^{p}:=\int_{0}^{+\infty} |f_{rad}(r)|^{p} \varphi(r)^{n-1}dr \sim \|f_{rad}\|_{L^{p}(\Sigma)}^{p}. \end{align*} $$

For the Sobolev spaces, we use the compatible notations

$$ \begin{align*} \|f_{rad}\|_{\dot H^{1}_{\varphi}} := \|\partial_{r} f_{rad}\|_{L^{2}(\Sigma)} \end{align*} $$

and

$$ \begin{align*} \|f_{rad}\|_{W^{1,p}_{\varphi}} := \|\partial_{r} f_{rad}\|_{L^{p}(\Sigma)} + \|f_{rad}\|_{L^{p}(\Sigma)}. \end{align*} $$

We define $\dot H^{s}_{\varphi }$ and $W^{s,p}_{\varphi }$ , $s\in (0,1)$ , by interpolation, and $\dot H^{s}_{\varphi }, W^{s,p}_{\varphi }$ , $s\in [-1,1], p\in (1,\infty )$ , by duality.

Note that since we are dealing with vectors in $\mathbb {C}^{M}$ , $|f(x)|$ should be understood as

$$ \begin{align*}|f(x)| = \sqrt{\langle f(x),f(x) \rangle_{\mathbb{C}^{M}}}. \end{align*} $$

The norms in time will be denoted by $L^{p}_{t}$ . The mixed Strichartz spaces will be standardly denoted by $ L^{p}_{t}L^{q}(\Sigma )=L^{p}(I;L^{q}(\Sigma ,\mathbb {C}^{M}))$ .

We finally introduce the spaces $H^{a,b}$ for $a \in [-1,1], b\in \mathbb {R}$ by defining the norms

$$ \begin{align*}\|f\|_{H^{a,b}(\Sigma)} = \Big( \|f\|_{H^{a}(\Sigma)}^{2} + \|(-\Delta_{\mathbb S^{n-1}})^{b/2}f\|_{L^{2}(\Sigma)} \Big)^{1/2}. \end{align*} $$

2.1 The separation of variables

We start by recalling that the Dirac operator on a Lorentzian manifold $(\mathcal {M},g)$ of dimension $n+1$ admitting a spin structureFootnote ¹ and with decoupled space and time writes

$$ \begin{align*}\mathcal{D} = m\gamma^{0} - i\gamma^{0}\underline \gamma^{j} D_{j}, \end{align*} $$

where the implicit summation on j is taken from $1$ to n. In the above, $m\in \mathbb {R}$ is the mass of the electron, $\gamma ^{0}$ is a self-adjoint matrix of size $M\times M$ with $M=2^{\lfloor (n+1)/2 \rfloor }$ with values in $\mathbb {C}$ whose square is the identity, $\underline \gamma ^{j}$ are anti-hermitian matrix bundles that satisfy

and $D_{j}$ are covariant derivatives for spinor bundles.

Writing $(\mathcal M, g)$ as $\mathcal M = \mathbb {R}_{t} \times \Sigma $ and

$$ \begin{align*} g = \begin{pmatrix} 1 & (0) \\ (0) & -h \end{pmatrix}, \end{align*} $$

where h is the (Riemannian) metric of $\Sigma $ , we endow the spinorial Riemannian manifold (i.e., a Riemannian manifold with a spin structure) $(\Sigma , h)$ with a vierbein $e^{j}_{\; a}$ (chosen such that for all $j,k$ , $e^{j}_{\; a}\delta ^{ab} e^{k}_{\; b} = h^{jk}$ ), and we fix

$$ \begin{align*}\underline \gamma^{j} = e^{j}_{\; a} \gamma^{a}. \end{align*} $$

The implicit summation for a is taken from $1$ to n. The family $(\gamma ^{a})_{0\leq a\leq n}$ satisfies the anticommutation relations

$$ \begin{align*}\{\gamma^{a},\gamma^{b}\} = 2\eta^{ab} \end{align*} $$

where

$$ \begin{align*}\eta = \begin{pmatrix} 1 & & & \\ & -1 & (0) & \\ & (0) & \ddots & \\ & & & -1 \end{pmatrix} \end{align*} $$

is the Minkowski metric in $\mathbb {R}^{1+n}$ . Writing $\alpha ^{0} = \gamma ^{0}$ and $\alpha ^{a} = \gamma ^{0} \gamma ^{a}$ , we have that the family $(\alpha ^{a})_{0\leq a \leq n}$ satisfy the canonical anticommutation relations

$$ \begin{align*}\{\alpha^{a},\alpha^{b}\} = 2\delta^{ab} \end{align*} $$

and are self-adjoint matrices. What is more, the Dirac operator now writes

$$ \begin{align*}\mathcal{D} = m\alpha^{0} -i e^{j}_{\; a} \alpha^{a} D_{j}. \end{align*} $$

Details on this construction (as well as the definition and the main properties of a vierbein) can be found in [Reference Parker and Toms32, Section 3.9 page, 144]. The minimal dimension for such a family of matrices is $2^{\lfloor (n+1)/2 \rfloor }$ . An easy way to see that this dimension is big enough is to consider for $n=2$ , the Pauli matrices

$$ \begin{align*}\alpha^{0} = \sigma_{1} = \begin{pmatrix} 1 &0 \\ 0 & -1\end{pmatrix},\quad \alpha^{1} = \sigma_{2} = \begin{pmatrix} 0 & i \\-i & 0 \end{pmatrix} , \quad \alpha^{2} = \sigma_{3} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \end{align*} $$

and for $n=2k+2$ even, given a family $(\tilde \alpha ^{a})_{0\leq a \leq 2k}$ of self-adjoint matrices of size $K\times K$ satisfying canonical anticommutation relations, the matrices written by block

(2.1)

Therefore, a natural way to pass from dimension $n = 2k$ even to $n+1 = 2k+1$ odd is to pass from the family of matrices $(\tilde \alpha ^{a})_{0\leq a \leq n}$ to

and to pass from dimension $n+1 = 2k+1$ odd to dimension $n+2$ even is simply to add the matrix

$$ \begin{align*}\alpha^{n} = \begin{pmatrix} (0) & i \textrm{Id}_{K} \\ -i \textrm{Id}_{K} & (0) \end{pmatrix}. \end{align*} $$

However, it is also natural to pass from an odd to an even dimension in the same way as to pass from an even to an odd. The reason is that, because of the theory of Clifford algebras, the algebra generated by the family $(\alpha ^{a})_{0\leq a\leq n+2}$ defined as in equation (2.1) is canonically isomorphic to the one generated by

(see [Reference Parker and Toms32] Section 5.6.2, page 229, for further details). We now consider the following setting: $(\Sigma ,\sigma )$ is a warped product – that is, a Riemannian manifold in the form $\Sigma =\mathbb {R}^{+}_{r}\times \mathbb {K}^{n-1}_{\phi }$ – where $\mathbb {K}^{n-1}$ is a $(n-1)$ -dimensional compact spin manifold, and $\Sigma $ is equipped with the Riemannian metric

$$ \begin{align*}dh^{2}=dr^{2}+\varphi(r)^{2}d\phi^{2}, \end{align*} $$

where $\varphi :\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ and $d\phi ^{2}$ is the Riemannian metric over $\mathbb {K}^{n-1}$ . In other words,

$$ \begin{align*} h = \begin{pmatrix} 1 & (0) \\ (0) & \varphi^{2} \kappa \end{pmatrix}, \end{align*} $$

where $\kappa $ is the Riemannian metric of $\mathbb {K}^{n-1}$ .

In the case that interests us, we assume that a vierbein $\tilde e = (\tilde e_{j}^{\;a})$ has been set for $\mathbb {K}^{n-1}$ , which we assume admits a spin structure. As the equation is covariant, we may choose any convenient vierbein for $\Sigma $ : we use as a vierbein for $\Sigma $

$$ \begin{align*}e= \begin{pmatrix} 1 & (0) \\ (0) & \varphi(r)\tilde e \end{pmatrix}. \end{align*} $$

We set $(\tilde \alpha ^{a})_{0\leq a\leq n-1}$ a family of matrices satisfying canonical anticommutation relations and

We set also

and finally

We recall that the covariant derivatives for Dirac spinors are given by

$$ \begin{align*}D_{\mu} = \partial_{\mu} +i \omega_{\mu}^{\; ab} \Sigma_{a,b}, \end{align*} $$

where $\omega $ is the spin connection and

$$ \begin{align*}\Sigma_{a,b} = -\frac{i}{8} [\gamma_{a},\gamma_{b}]. \end{align*} $$

We have for all

$$ \begin{align*} [\gamma^{a},\gamma^{b}] = [\alpha^{0} \alpha^{a}, \alpha^{0}\alpha^{b}] = [\alpha^{b},\alpha^{a}] = \begin{pmatrix} [\tilde \alpha^{b-1},\tilde \alpha^{a-1}] & (0) \\ (0) & [\tilde \alpha^{b-1},\tilde \alpha^{a-1}] \end{pmatrix} & = \\ & \ \begin{pmatrix} [\tilde \gamma^{a-1}, \tilde \gamma^{b-1}] & (0) \\ (0) & [\tilde \gamma^{a-1}, \tilde \gamma^{b-1}] \end{pmatrix}. \end{align*} $$

Therefore, we have

$$ \begin{align*}\Sigma_{a,b} = \begin{pmatrix} \tilde \Sigma_{a-1,b-1} & (0) \\ (0) & \tilde \Sigma_{a-1,b-1} \end{pmatrix}, \end{align*} $$

where $\tilde \Sigma _{a,b} = -\frac {i}{8} [\tilde \gamma ^{a}, \tilde \gamma ^{b} ]$ .

We also have

$$ \begin{align*}de^{a} + \omega^{a}_{\; b} \wedge e^{b} = 0. \end{align*} $$

Since $e^{1} = dr$ , we have $de^{1} = 0$ and thus

$$ \begin{align*} \omega^{1}_{\; b} \wedge e^{b} = 0. \end{align*} $$

Therefore, we get $\omega ^{1}_{\; b} \sim e_{\; b}$ for all b and then $\omega _{1}^{\; 1b} = 0$ for all $b\geq 1$ .

Since for all $a>1$ , we have $e^{a} = \varphi (r)\tilde e^{a-1}$ , we get

$$ \begin{align*}de^{a} = \varphi^{\prime} e^{1} \wedge \tilde e^{a-1} + \varphi d\tilde e^{a-1} = \varphi^{\prime} e^{1} \wedge \tilde e^{a-1} - \varphi \tilde \omega^{a-1}_{\; b} \wedge \tilde e^{b} \end{align*} $$

and thus

$$ \begin{align*}\omega^{a}_{\; b} \wedge e^{b} = \varphi^{\prime} \tilde e^{a-1} \wedge e^{1} + \varphi \tilde\omega^{a-1}_{b}\wedge \tilde e^{b} =\varphi^{\prime} \tilde e^{a-1} \wedge e^{1} + \tilde\omega^{a-1}_{b}\wedge e^{b+1}. \end{align*} $$

Therefore,

$$ \begin{align*}\omega^{a}_{\; 1} = \varphi^{\prime} \tilde e^{a-1}\sim e^{a} \Rightarrow \omega_{1}^{\; a1} = 0 \textrm{ and for all }j>1, \; \omega_{j}^{\; a1} =\varphi^{\prime} \tilde e_{j-1}^{a-1}, \end{align*} $$

and for all $b>1$ ,

$$ \begin{align*}\omega^{a}_{\; b} = \tilde \omega^{a-1}_{\; (b-1)} \Rightarrow \omega_{1}^{\; ab} = 0 \textrm{ and for all }j>1 ,\; \omega_{j}^{\; ab} = \tilde \omega_{j-1}^{(a-1)(b-1)}. \end{align*} $$

Summing up, we get $D_{1} = \partial _{r}$ and for all $j>1$ ,

$$ \begin{align*}D_{j} = \partial_{j} +2 i\varphi^{\prime} \tilde e_{j-1}^{\; a} \Sigma_{1(a+1)} + i \tilde \omega_{j-1}^{(a-1)(b-1)} \Sigma_{a,b}. \end{align*} $$

Since

$$ \begin{align*}\Sigma_{1,(a+1)} = \begin{pmatrix} \tilde \Sigma_{0,a} & (0) \\ (0) & \tilde \Sigma_{0,a} \end{pmatrix} \textrm{ and } \Sigma_{a,b} = \begin{pmatrix} \tilde \Sigma_{a-1,b-1} & (0) \\ (0) & \tilde \Sigma_{a-1,b-1} \end{pmatrix}, \end{align*} $$

we have

$$ \begin{align*}D_{j} = \begin{pmatrix} \tilde D_{j-1} + 2i\varphi^{\prime}\tilde e_{j-1}^{\; a} \tilde \Sigma_{0,a}& (0) \\ (0) & \tilde D_{j-1} +2i\varphi^{\prime}\tilde e_{j-1}^{\; a} \tilde \Sigma_{0,a} \end{pmatrix}, \end{align*} $$

where $\tilde D_{j}$ are covariant derivatives for spinor bundles over $\mathbb {K}^{2}$ .

We deduce

$$ \begin{align*} ie^{j}_{\; b} \alpha^{b} D_{j} = i\alpha_{1}\partial_{r} + i\frac1{\varphi} \tilde e^{j-1}_{\; b} \begin{pmatrix} (0) & \tilde \alpha^{b} \\ \tilde \alpha^{b} & (0) \end{pmatrix} &\begin{pmatrix} \tilde D_{j-1} + 2i\varphi'\tilde e_{j-1}^{\; a}\tilde \Sigma_{0,a}& (0) \\ (0) & \tilde D_{j-1} +2i\varphi'\tilde e_{j-1}^{\; a} \tilde\Sigma_{0,a} \end{pmatrix} \\ &= i\alpha_{1}\partial_{r} +\frac1{r} \begin{pmatrix} (0) & i \tilde e^{j}_{\; b}\tilde \alpha^{b} (\tilde D_{j} + i\tilde e_{j}^{\; a}\tilde \Sigma_{0,a}) \\ i \tilde e^{j}_{\; b}\tilde \alpha^{b} (\tilde D_{j} + i\tilde e_{j}^{\; a}\tilde \Sigma_{0,a}) &(0) \end{pmatrix}. \end{align*} $$

Using that $\tilde e^{j}_{\; b}\tilde e_{j}^{\; a} = \delta ^{a}_{\; b}$ and that $\tilde \alpha ^{a} \tilde \Sigma _{0,a} = -i \frac {n-1}4 \tilde \alpha ^{0} $ , we deduce

$$ \begin{align*}\mathcal{D} = m \alpha^{0} + i\alpha_{1}\partial_{r} + \begin{pmatrix} (0) & \frac1{\varphi(r)}\mathcal{D}_{\mathbb{K}^{n-1}} + i\frac{n-1}{2}\frac{\varphi^{\prime}(r)}{\varphi(r)} \tilde\alpha^{0} \\ \frac1{\varphi(r)}\mathcal{D}_{\mathbb{K}^{n-1}} + i\frac{n-1}{2}\frac{\varphi^{\prime}(r)}{\varphi(r)} \tilde\alpha^{0}&(0) \end{pmatrix}, \end{align*} $$

where $\mathcal {D}_{\mathbb {K}^{n-1}}$ is the Dirac operator on $\mathbb {K}^{n-1}$ . We thus get

(2.2) $$ \begin{align} \mathcal{D} = \begin{pmatrix} m & i\tilde\alpha^{0} \Big( \partial_{r} + \frac{n-1}{2}\frac{\varphi^{\prime}(r)}{\varphi(r)} \Big) + \frac1{\varphi(r)}\mathcal{D}_{\mathbb{K}^{n-1}} \\ i\tilde\alpha^{0} \Big( \partial_{r} + \frac{n-1}{2}\frac{\varphi^{\prime}(r)}{\varphi(r)} \Big) + \frac1{\varphi(r)}\mathcal{D}_{\mathbb{K}^{n-1}} & -m \end{pmatrix}. \end{align} $$

Now the key step (for us) consists of ensuring that the operator $\mathcal {D}_{\mathbb {K}^{n-1}}$ can in fact be diagonalized. In the case $\mathbb {K}^{n-1}$ being the two-dimensional unit sphere, this fact is classical and well-known; the eigenvalues and eigenfunctions are explicit (see, e.g., [Reference Thaller36] or [Reference Camporesi and Higuchi14]). In the general case, we can nevertheless evoke the following result, which can be found, for example, in [Reference Roe35, Theorem 5.27].

Let $\mu> 0$ be in the spectrum of $\mathcal {D}_{\mathbb {K}^{n-1}}$ ; we fix $(\psi _{\mu ,j})_{j}$ an orthogonal basis of the eigenspace of $\mathcal {D}_{\mathbb {K}^{n-1}}$ with eigenvalue $\mu $ . We set $\psi _{-\mu ,j} = i\tilde \alpha ^{0} \psi _{\mu ,j}$ . Since $\tilde \alpha ^{0}$ anticommutes with $\mathcal {D}_{\mathbb {K}^{n-1}}$ , we get that $\psi _{-\mu ,j}$ is an eigenfunction of $\mathcal {D}_{\mathbb {K}^{n-1}}$ with eigenvalue $-\mu $ . Note that for all $\mu $ in the spectrum of $\mathcal {D}_{\mathbb {K}^{n-1}}$ , since $(i\alpha ^{0})^{2} = -1$ , we have for $\mu> 0$ , $(1+i\tilde \alpha ^{0}) \psi _{\mu ,j} = \psi _{\mu ,j} + \psi _{-\mu ,j} $ and $(1-i\tilde \alpha ^{0})\psi _{-\mu ,j} = \psi _{-\mu ,j} - \psi _{\mu ,j}$ . Similarly,

$$ \begin{align*} (1-i\tilde \alpha^{0}) \psi_{\mu,j} = \psi_{\mu,j} - \psi_{-\mu,j} \quad \textrm{and } \quad (1-i\tilde \alpha^{0}) \psi_{-\mu,j} = \psi_{-\mu,j} + \psi_{\mu,j}. \end{align*} $$

Therefore, the family

$$ \begin{align*}\mathcal B = \left( \begin{pmatrix}\frac{1+i\tilde \alpha^{0}}{\sqrt 2} \psi_{\mu,j} \\0 \end{pmatrix},\begin{pmatrix} 0 \\ -\frac{1-i\tilde \alpha^{0}}{\sqrt 2} \psi_{\mu,j} \end{pmatrix} \right)_{\mu \in Sp(\mathcal{D}_{\mathbb{K}^{n-1}}),j} \end{align*} $$

forms an orthonormal basis of $L^{2} (\mathbb {K}^{n-1}, \mathbb {C}^{M})$ .

We deduce that we have the decomposition

$$ \begin{align*} L^{2}(\Sigma, \mathbb{C}^{M}) = \bigoplus_{\mu,j} \mathcal H_{\mu,j}, \end{align*} $$

where $\mathcal H_{\mu ,j}$ is the tensor product of $L^{2}_{\varphi }$ (the $L^{2}$ maps of $\mathbb {R}_{+}$ with measure $\varphi ^{2} dr$ ) and with values in $\mathbb {C}$ and the vector space generated by

$$ \begin{align*} \left( \begin{pmatrix}\frac{1+i\tilde \alpha^{0}}{\sqrt 2} \psi_{\mu,j} \\0 \end{pmatrix},\begin{pmatrix} 0 \\ -\frac{1-i\tilde \alpha^{0}}{\sqrt 2} \psi_{\mu,j} \end{pmatrix} \right). \end{align*} $$

In other words, any map $u\in L^{2}(\Sigma , \mathbb {C}^{M})$ may be written as

$$ \begin{align*} u(r,\omega) = \sum_{\mu,j} u_{\mu,j}^{+} (r) \begin{pmatrix}\frac{1+i\tilde \alpha^{0}}{\sqrt 2} \psi_{\mu,j}(\omega) \\0 \end{pmatrix} + u_{\mu,j}^{-} (r)\begin{pmatrix} 0 \\ -\frac{1-i\tilde \alpha^{0}}{\sqrt 2} \psi_{\mu,j}(\omega) \end{pmatrix}, \end{align*} $$

where $\omega \in \mathbb {K}^{n-1}$ , and $u_{\mu ,j}^{\pm } \in L^{2}_{\varphi }$ are such that

$$ \begin{align*} \sum_{\mu,j} \|u_{\mu,j}^{+} (r)\|_{L^{2}_{\varphi}}^{2} + \|u_{\mu,j}^{-} (r)\|_{L^{2}_{\varphi}}^{2} < \infty. \end{align*} $$

For any $\mu \in Sp(\mathcal {D}_{\mathbb {K}^{n-1}})$ and any j, and $f(r)$ a radial test function, we have

$$ \begin{align*} \mathcal{D} \Big[f \begin{pmatrix} (1+i\tilde \alpha^{0} ) \psi_{\mu,j} \\ 0 \end{pmatrix} \Big] &= \\ & mf \begin{pmatrix} (1+i\tilde \alpha^{0} ) \psi_{\mu,j} \\ 0 \end{pmatrix} + \left(\Big[ -\Big(\partial_{r} + \frac{n-1}{2}\frac{\varphi'(r)}{\varphi(r)} \Big) + \frac{\mu}{\varphi(r)} \Big] f \right) \begin{pmatrix} 0 \\ (1-i\tilde \alpha^{0} ) \psi_{\mu,j} \end{pmatrix} \end{align*} $$

and

$$ \begin{align*} \mathcal{D} \Big[f \begin{pmatrix} 0 \\ (1-i\tilde \alpha^{0} ) \psi_{\mu,j} \end{pmatrix} \Big] &=\\ &-mf \begin{pmatrix} 0 \\ (1-i\tilde \alpha^{0} ) \psi_{\mu,j} \end{pmatrix} +\left( \Big[ \Big( \partial_{r} + \frac{n-1}{2}\frac{\varphi'(r)}{\varphi(r)} \Big) + \frac{\mu}{\varphi(r)} \Big] f \right) \begin{pmatrix} (1+i\tilde \alpha^{0} ) \psi_{\mu,j} \\ 0 \end{pmatrix}. \end{align*} $$

We are thus left with studying the dispersion of the equation

(2.3) $$ \begin{align} i\partial_{t} F+h_{\mu} F=0, \end{align} $$

with

(2.4) $$ \begin{align} h_{\mu}= \begin{pmatrix} m & -\Big(\partial_{r} + \frac{n-1}{2}\frac{\varphi^{\prime}(r)}{\varphi(r)} \Big) + \frac{\mu}{\varphi(r)} \\\Big(\partial_{r} + \frac{n-1}{2}\frac{\varphi^{\prime}(r)}{\varphi(r)} \Big) + \frac{\mu}{\varphi(r)}& -m \end{pmatrix} \end{align} $$

for any $\mu \in Sp(\mathcal {D}_{\mathbb {K}^{n-1}})$ .

Remark 2.2. If we take $\psi _{\mu }$ an eigenfunction of $\mathcal {D}_{\mathbb {K}^{n-1}}$ on $\mathbb {K}^{n-1}$ with eigenvalue $\mu \neq 0$ and we suppose that $\theta =f(r) \begin {pmatrix} (1+i\tilde \alpha ^{0} ) \psi _{\mu } \\ 0 \end {pmatrix} $ is an eigenspinor of $\mathcal {D}^{2}$ with eigenvalue $\rho ^{2}\neq 0$ , then we have that f satisfies the following ODE

$$ \begin{align*}f^{\prime\prime}+\frac{n-1}r+\left[\rho^{2}-\left(\mu^{2}-\mu-\frac{n^{2}-4n+3}4\right)\frac1{r^{2}}\right]f=0, \end{align*} $$

which has solutions $f(r)=\gamma ^{c} J_{\pm \nu ^{+}}(\rho r)$ , where $c=(2-n)/2$ , $\nu ^{+}=|2\mu -1|/2$ and $J_{\nu ^{+}}$ is the standard Bessel function of order $\nu ^{+}$ . Analogously, assuming that now $\theta =f(r) \begin {pmatrix} 0 \\ (1-i\tilde \alpha ^{0} ) \psi _{\mu } \end {pmatrix} $ is an eigenspinor of $\mathcal {D}^{2}$ with eigenvalue $\rho ^{2}\neq 0$ , we see that f satisfies the ODE

$$ \begin{align*}f^{\prime\prime}+\frac{n-1}r+\left[\rho^{2}-\left(\mu^{2}+\mu-\frac{n^{2}-4n+3}4\right)\frac1{r^{2}}\right]f=0, \end{align*} $$

which has solutions $f(r)=r^{c} J_{\pm \nu ^{+}}(\rho r)$ , with c as before and $\nu ^{+}=|2\mu +1|/2$ . These equations recall the connection with the Klein-Gordon equation, which has now been brought to the ‘radial’ level. In particular, in [Reference Ben-Artzi, Cacciafesta, de Suzzoni and Zhang5], these equations are the starting point in order to prove the crucial local smoothing estimates for the Klein-Gordon equation; nevertheless, we stress once again the fact that the argument of deducing dispersive estimates for the Dirac flow from the corresponding Klein-Gordon ones does not work for free, as indeed the Laplace operator that comes into play when squaring the Dirac operator is the spinorial one (and not the standard scalar one that we dealt with in [Reference Ben-Artzi, Cacciafesta, de Suzzoni and Zhang5]).

We can explicitly write down the positive and negative eigenspinors of the operator $\mathcal {D}_{\mathbb {K}^{n-1}}$ , when we are calling ‘positive’ (respectively, ‘negative’) the ones corresponding to Bessel functions of positive (respectively, negative) order. It can then be shown that both positive and negative ones fall in the domain of $\mathcal {D}_{\mathbb {K}^{n-1}}$ . The negative ones, though, correspond to eigenvalues $\mu $ of $\mathcal {D}_{\mathbb {K}^{n-1}}$ such that $|\mu |\leq 1/2$ (this can be seen by studying the asymptotic behaviours of the Bessel functions). Finally, negative solutions in the domain of $\mathcal {{D}}_{\mathbb {K}^{n-1}}$ prevent the operator $\mathcal {D}_{\mathbb {K}^{n-1}}$ from being self-adjoint. This is why we need the assumption $|\mu |> 1/2$ .

2.2 The squaring trick and weighted spinors

We now introduce weighted spinors, the main goal being transforming the system equation (2.3) into a system of wave equations on $\mathbb {R}^{n}$ perturbed by a radial, electric potential in order to exploit the existing theory to obtain dispersive estimates. This strategy has been already employed in [Reference Pierfelice33, Reference Banica and Duyckaerts2, Reference D’Ancona and Zhang21] in different contexts (the Schrödinger equation on Damek-Ricci spaces and on spherically symmetric manifolds and equivariant wave maps, respectively) and in the predecessor of this paper, [Reference Cacciafesta and de Suzzoni11], to deal with the local-in-time case.

Take $\sigma : \mathbb {R}_{+} \rightarrow \mathbb {R}_{+}$ such that for all $r>0$ ,

$$ \begin{align*}\sigma (r) = \frac{r}{\varphi (r)}, \end{align*} $$

where $\varphi (r)$ satisfies the assumptions of Theorem 1.4 and write, for $n\geq 3$ ,

$$ \begin{align*}\sigma_{n} = \sigma^{(n-1)/2}. \end{align*} $$

Lemma 2.3. The map $\sigma $ prolonged by continuity at $0$ is $\mathcal C^{1}$ and the map

$$ \begin{align*} \frac{\sigma^{\prime}}{\sigma} \end{align*} $$

is bounded on $(0,\infty )$ .

Proof. Indeed, for $r\geq 0$ ,

$$ \begin{align*} \sigma^{\prime}(r) = \Big( \frac1{r} - \frac{\varphi^{\prime}(r)}{\varphi(r)}\Big) \sigma. \end{align*} $$

The map $\sigma $ at $0$ converges to $1$ and we have, as $r\downarrow 0$ , writing $a= \frac {\varphi ^{\prime \prime }(0)}{2}$ ,

$$ \begin{align*} \frac{\sigma(r)-1}{r} = \frac1{r} \Big( \frac{r}{r + ar^{2} +o(r^{2})} -1\Big) \rightarrow - a, \end{align*} $$

hence $\sigma ^{\prime }(0) = -a$ . What is more, as $r\rightarrow 0$ ,

$$ \begin{align*} \sigma^{\prime}(r) = \frac1{\varphi(r)} - \frac{r\varphi^{\prime}(r)}{\varphi^{2}} = \frac1{r}(-ar +o(r)) \rightarrow -a. \end{align*} $$

Finally, since $\sigma \rightarrow 1 $ at $0$ and $\sigma>0$ , we deduce that $\frac {\sigma ^{\prime }}{\sigma }$ is continuous on $[0,\infty )$ .

Finally,

$$ \begin{align*} \frac{\sigma^{\prime}(r)}{\sigma(r)} = \frac1{r} - \frac{\varphi^{\prime}}{\varphi}, \end{align*} $$

which ensures its boundedness.

Lemma 2.4. The multiplication by $\sigma _{n}$ is an isometry from $L^{2}_{r}$ to $L^{2}_{\varphi }$ . What is more, the multiplication by $\sigma _{n}$ is an isomorphism from $H^{1}_{r}$ to $H^{1}_{\varphi }$ that satisfies

$$ \begin{align*} \Big[ 1+ c_{\varphi} \frac{n-1}{2} \Big]^{-1} \|f\|_{H^{1}_{r}} \leq \|\sigma_{n}f\|_{H^{1}_{\varphi}} \leq \Big[ 1+ c_{\varphi} \frac{n-1}{2}\Big] \|f\|_{H^{1}_{r}} \end{align*} $$

with $c_{\varphi } = \|\frac {\sigma ^{\prime }}{\sigma }\|_{L^{\infty }((0,\infty ))}$ . In particular, by interpolation, we get, for all $s\in [0,1]$ ,

$$ \begin{align*} \Big[ 1+ c_{\varphi} \frac{n-1}{2}\Big]^{-s} \|f\|_{H^{s}_{r}} \leq \|\sigma_{n}f\|_{H^{s}_{\varphi}} \leq \Big[ 1+ c_{\varphi} \frac{n-1}{2} \Big]^{s} \|f\|_{H^{s}_{r}}. \end{align*} $$

Proof. The fact that $\sigma _{n}$ is an isometry at the $L^{2}$ -level follows by the definition of the norms, as indeed

$$ \begin{align*} \|\sigma_{n}f\|_{L^{2}_{\varphi}}^{2} = \int \sigma_{n}^{2} f^{2} \varphi^{n-1}dr = \int r^{n-1}f^{2} dr = \|f\|_{L^{2}_{r}}^{2}. \end{align*} $$

We now estimate $\|\sigma _{n} f\|_{H^{1}_{\varphi }}$ . We have, by the isometry in $L^{2}$ ,

$$ \begin{align*} \|\sigma_{n} f\|_{H^{1}_{\varphi}} \leq \|f\|_{H^{1}_{r}} + \|\sigma_{n}^{\prime} \sigma_{n}^{-1}f\|_{L^{2}_{r}}. \end{align*} $$

A direct computation yields

$$ \begin{align*} \sigma_{n}^{\prime} \sigma_{n}^{-1} = \frac{n-1}{2}\frac{\sigma^{\prime}}{\sigma}. \end{align*} $$

By Hölder’s inequality, we get

$$ \begin{align*} \|\sigma_{n}^{\prime} \sigma_{n}^{-1}f\|_{L^{2}_{r}}\leq \frac{n-1}{2} c_{\varphi} \|f\|_{L^{2}_{r}}. \end{align*} $$

We now estimate $\|\sigma _{n}^{-1} g\|_{H^{1}_{r}}$ . We have, by isometry in $L^{2}$ ,

$$ \begin{align*} \|\sigma_{n}^{-1} g\|_{H^{1}_{r}} \leq \|g\|_{H^{1}_{\varphi}} + \|(\sigma_{n}^{-1})^{\prime}\sigma_{n}g\|_{L^{2}_{\varphi}}. \end{align*} $$

We have

$$ \begin{align*} (\sigma_{n}^{-1})^{\prime}\sigma_{n} = -\frac{\sigma_{n}^{\prime}}{\sigma_{n}} =- \frac{n-1}{2}\frac{\sigma^{\prime}}{\sigma}. \end{align*} $$

We use Hölder’s inequality to get

$$ \begin{align*} \|(\sigma_{n}^{-1})^{\prime}\sigma_{n}g\|_{H^{1}_{\varphi}}\leq \frac{n-1}{2} c_{\varphi} \|g\|_{L^{2}_{\varphi}}. \end{align*} $$

This concludes the proof.

Lemma 2.5. We have

(2.5) $$ \begin{align} h_{\mu,n}:=\sigma_{n}^{-1} h_{\mu} \sigma_{n} = \begin{pmatrix} m & -\partial_{r} -\frac{n-1}{2r} + \frac{\mu}{\varphi} \\ \partial_{r} +\frac{n-1}{2r} + \frac{\mu}{\varphi} & -m \end{pmatrix}. \end{align} $$

Proposition 2.6. Let $s \in [-1,1]$ and $p,q \geq 1$ . If $e^{-it h_{\mu ,n}}$ is a continuous operator from $H_{r}^{1/2}$ to $ L^{p}(\mathbb {R}, W^{s,q}_{r})$ , then $e^{-it h_{\mu }}$ is a continuous operator from $H_{\varphi }^{1/2}$ to $\sigma _{n}^{1-2/q} L^{p}(\mathbb {R}, W^{s,q}_{\varphi })$ and

$$ \begin{align*}\big\| e^{-it h_{\mu}} \big\|_{H_{\varphi}^{1/2}\rightarrow \sigma_{n}^{1-2/q} L^{p}(\mathbb{R}, W^{s,q}_{\varphi})} \leq C_{\varphi} \big\| e^{-it h_{\mu,n}} \big\|_{H_{r}^{1/2}\rightarrow L^{p}(\mathbb{R}, W^{s,q}_{r})} \end{align*} $$

with a constant $C_{\varphi }$ that does not depend on $\mu $ .

This is a consequence of the fact that

$$ \begin{align*}e^{-it h_{\mu}} = \sigma_{n} e^{-it h_{\mu,n}} \sigma_{n}^{-1}, \end{align*} $$

of Lemma 2.4 and of the following lemma.

Lemma 2.7. The multiplication by $\sigma _{n}$ is a continuous operator from

$$ \begin{align*}L^{p}(\mathbb{R}, W^{s,q}_{r}) \end{align*} $$

to

$$ \begin{align*}\sigma_{n}^{1-2/q}L^{p}(\mathbb{R},W^{s,q}_{\varphi}) \end{align*} $$

for any $s\in [-1,1]$ and any $q\in (1,\infty )$ .

Proof. First of all, the norm in the t variable is not relevant in the proof; hence we only prove that the multiplication by $\sigma _{n}$ is continuous from $W^{s,q}_{r}$ to $\sigma _{n}^{1-2/q} W_{\varphi }^{s,q}$ for $q\in (1,\infty )$ and $s\in [-1,1]$ . This is equivalent to proving that the multiplication by $\sigma _{n}^{2/q}$ is continuous from $W_{r}^{s,q}$ to $W_{\varphi }^{s,q}$ .

For non-negative s, by interpolation, we can reduce the proof to the cases $s=0,1$ .

For negative s, by duality, the continuity of the multiplication by $\sigma _{n}^{2/q}$ from $W_{r}^{s,q}$ to $W_{\varphi }^{s,q}$ is implied by the continuity of the multiplication by $\sigma _{n}^{-2/q^{\prime }}$ from $W_{\varphi }^{-s,q^{\prime }}$ to $W_{r}^{-s,q^{\prime }}$ , where $q^{\prime }$ is the conjugated exponent of q.

Therefore, it sufficient to prove the following, for all $q\in (1,\infty )$ :

1. 1. the multiplication by $\sigma _{n}^{2/q}$ is an isometry from $L^{q}_{r}$ to $L^{q}_{\varphi }$ ,

2. 2. the multiplication by $\sigma _{n}^{2/q}$ is continuous from $W_{r}^{1,q}$ to $W_{\varphi }^{1,q}$ ,

3. 3. the multiplication by $\sigma _{n}^{-2/q}$ is continuous from $W_{\varphi }^{1,q}$ to $W_{r}^{1,q}$ .

(1) Let $f\in L^{q}_{r}$ ; we have by definition

$$ \begin{align*} \|\sigma_{n}^{2/q}f\|_{L^{q}_{\varphi}}^{q} = \int_{0}^{\infty} \sigma_{n}^{2} \varphi^{n-1}|f|^{q}, \end{align*} $$

and using the definition of $\sigma _{n}$ ,

$$ \begin{align*} \|\sigma_{n}^{2/q}f\|_{L^{q}_{\varphi}}^{q} = \int_{0}^{\infty} r^{n-1}|f(r)|^{q} dr = \|f\|_{L^{q}_{r}}^{q}. \end{align*} $$

(2) From (1), it is sufficient to prove that for all $f\in W_{r}^{1,q}$ , we have

$$ \begin{align*} \|\partial_{r} (\sigma_{n}^{2/q} f)\|_{L^{q}_{\varphi}} \lesssim \|f\|_{W_{r}^{1,q}}. \end{align*} $$

By the Leibniz rule, we have

$$ \begin{align*} \|\partial_{r} (\sigma_{n}^{2/q} f)\|_{L^{q}_{\varphi}} = \|\sigma_{n}^{2/q}\Big(\frac{n-1}{q}\frac{\sigma^{\prime}}{\sigma} f + \partial_{r} f \Big) \|_{L^{q}_{\varphi}}, \end{align*} $$

and from (1), we get

$$ \begin{align*} \|\partial_{r} (\sigma_{n}^{2/q} f)\|_{L^{q}_{\varphi}} = \|\frac{n-1}{q}\frac{\sigma^{\prime}}{\sigma} f + \partial_{r} f \|_{L^{q}_{r}}. \end{align*} $$

We conclude by using the fact that $\frac {\sigma ^{\prime }}{\sigma }$ is bounded.

(3) From (1), it is sufficient to prove that

$$ \begin{align*} \|\partial_{r} (\sigma_{n}^{-2/q} f)\|_{L^{q}_{r}} \lesssim \|f\|_{W_{\varphi}^{1,q}}. \end{align*} $$

By the Leibniz rule, we have

$$ \begin{align*} \|\partial_{r} (\sigma_{n}^{-2/q} f)\|_{L^{q}_{r}} = \|\sigma_{n}^{-2/q}\Big(-\frac{n-1}{q}\frac{\sigma^{\prime}}{\sigma} f + \partial_{r} f \Big) \|_{L^{q}_{r}}, \end{align*} $$

and from (1), we get

$$ \begin{align*} \|\partial_{r} (\sigma_{n}^{-2/q} f)\|_{L^{q}_{r}} = \|-\frac{n-1}{q}\frac{\sigma^{\prime}}{\sigma} f + \partial_{r} f \|_{L^{q}_{\varphi}}. \end{align*} $$

We conclude by using the fact that $\frac {\sigma ^{\prime }}{\sigma }$ is bounded.

As recalled in the introduction, the (massless) Dirac operator has been constructed as some square root of the Laplacian; in other words, every solution to the free Dirac equation on $\mathbb {R}^{n}$ satisfies a system of decoupled free wave/Klein-Gordon equations. This point of view can be carried at the ‘radial’ level:

Lemma 2.8. Let $V \in C^{2}(0,\infty )$ , and let

$$ \begin{align*} h_{V,n} = \begin{pmatrix} m & -\Big( \partial_{r} + \frac{n-1}{2r} \Big) + V \\ \Big( \partial_{r} + \frac{n-1}{2r} \Big) + V & -m \end{pmatrix}. \end{align*} $$

Then we have

$$ \begin{align*} h_{V,n}^{2} = \begin{pmatrix} m^{2} + H_{c_{-}} & 0 \\ 0 & m^{2} + H_{c_{+}}\end{pmatrix} \end{align*} $$

with

(2.6) $$ \begin{align} H_{c_{\pm}} = -\Big( \partial_{r}^{2} + \frac{n-1}{r}\partial_{r}\Big) +c_{\pm} \end{align} $$

and

$$ \begin{align*} c_{\pm} = -\frac{(n-1)(n-3)}{4r^{2}} + V^{2} \pm \partial_{r} V. \end{align*} $$

Remark 2.9. In other words, if $v= \begin {pmatrix} v_{+} \\ v_{-} \end {pmatrix}$ solves the equation

$$ \begin{align*}i\partial_{t} v = \tilde h_{\mu} v \end{align*} $$

with initial datum $v_{0} = \begin {pmatrix} v_{0,+} \\ v_{0,-} \end {pmatrix}$ , then $v_{+}$ and $v_{-}$ solve, respectively,

$$ \begin{align*}\partial_{t}^{2} v_{+} = -m^{2}v_{+} - H_{c_{-}}v_{+} \quad \textrm{ and }\quad \partial_{t}^{2} v_{-} = -m^{2}v_{-} -H_{c_{+}} v_{-} \end{align*} $$

with initial data

$$ \begin{align*}\begin{pmatrix} v_{+}(t=0) \\ \partial_{t} v_{+} (t=0) \end{pmatrix} = \begin{pmatrix} v_{0,+} \\ -i mv_{0,+} + i \Big( \partial_{r} + \frac1{r} - V \Big) v_{0,-} \end{pmatrix} \end{align*} $$

and

$$ \begin{align*}\quad \begin{pmatrix} v_{-}(t=0) \\ \partial_{t} v_{-}(t=0) \end{pmatrix} = \begin{pmatrix} v_{0,-} \\ im v_{0,-} -i \Big( \partial_{r} + \frac1{r} +V \Big) v_{0,+} \end{pmatrix}. \end{align*} $$

3.1 General Kato-smoothing

First of all, let us recall the following definition (see [Reference D’Ancona17, Definition 2.1]):

We prove the following proposition:

Proposition 3.2. Let $n\geq 3$ be the dimension, and let $c \in C^{1}((0,\infty ))$ and $r=|x|$ . Assume that $r^{2} c \in L^{\infty }$ and

(3.1) $$ \begin{align} \delta_{c} : = \min \Big[ \frac14, \inf \Big( cr^{2} + \frac{(n-2)^{2}}{4}\Big), \inf \Big( -r^{3}c^{\prime} -r^{2}c + \frac{(n-2)^{2}}{4}\Big)\Big]>0. \end{align} $$

Then the operator $H_{c}$ defined in equation (2.6) is positive on the Hilbert space $L^{2}(\mathbb {R}^{n})$ and the operator $|x|^{-1}$ (from $L^{2}(\mathbb {R}^{n})$ to $L^{2}(\mathbb {R}^{n})$ ) is $H_{c}$ super-smooth with constant $\delta _{c}^{-1}$ .

Proof. Because we have

$$ \begin{align*} \inf \Big( cr^{2} + \frac{(n-2)^{2}}{4}\Big) \geq \delta_{c}, \end{align*} $$

we get, for any $v \in C^{\infty }((0,\infty ))$ with compact support

$$ \begin{align*} \langle v,H_{c} v \rangle_{L^{2}} \geq \langle v,\Big(-\partial_{r}^{2} - \frac{n-1}{r}\partial_{r}\Big) v \rangle_{L^{2}} - \langle v,\frac{(n-2)^{2}}{4r^{2}}v \rangle_{L^{2}} + \delta_{c} \langle v,r^{-2}v \rangle_{L^{2}} \end{align*} $$

and by Hardy’s inequality, as we are in dimension $n\geq 3$ ,

$$ \begin{align*} \langle v,H_{c} v \rangle_{L^{2}} \geq \delta_{c} \langle v,r^{-2}v \rangle_{L^{2}}. \end{align*} $$

Therefore, $H_{c}$ is positive.

The fact that $|x|^{-1}$ is $H_{c}$ super-smooth is a consequence of [Reference D’Ancona and Zhang21, Theorem 3.3] with $a=\frac {n-1}{r}$ . Indeed, for v in the domain of $|x|^{-1}$ , write $f = |x|^{-1}v$ ; writing $R(\lambda +i\varepsilon )$ the resolvent of $H_{c}$ with $\varepsilon \neq 0$ , we have that $ R(\lambda +i\varepsilon ) |x|^{-1}v$ is the solution to

$$ \begin{align*} u^{\prime\prime} + \frac{n-1}{r}u^{\prime} + (\lambda +i\varepsilon) u -cu= -f \end{align*} $$

from which we deduce

$$ \begin{align*} \| \, |x|^{-1}u\|_{L^{2}(\mathbb{R}^{n})} \leq \delta_{c}^{-1} \|\,|x|f\|_{L^{2}(\mathbb{R}^{n})} = \delta_{c}^{-1}\|v\|_{L^{2}(\mathbb{R}^{n})}. \end{align*} $$

We get from this

$$ \begin{align*} \langle R(\lambda + i\varepsilon) |x|^{-1}v,|x|^{-1}v \rangle_{L^{2}} = \langle |x|^{-1}u,v \rangle_{L^{2}}\leq \| \, |x|^{-1}u\|_{L^{2}(\mathbb{R}^{n})}\|v\|_{L^{2}(\mathbb{R}^{n})}, \end{align*} $$

which yields

$$ \begin{align*} \langle R(\lambda + i\varepsilon) |x|^{-1}v,|x|^{-1}v \rangle_{L^{2}} \leq \delta_{c}^{-1}\|v\|_{L^{2}(\mathbb{R}^{n})}^{2} \end{align*} $$

and concludes the proof.

Proposition 3.3. Under the same assumptions as Proposition 3.2, we have that $|x|^{-1} (H_{c}+\nu )^{-1/4} $ is $\sqrt {H_{c}+\nu }$ super-smooth for any $\nu \in \mathbb {R}^{+}$ with constant $C_{c}^{2} = (3+\pi )\delta _{c}^{-1}$ and, in particular, for all v in the domain of $(H_{c} + \nu )^{1/4}$ , we have that

$$ \begin{align*} \big \|\, |x|^{-1} e^{-it\sqrt{\nu + H_{c}}}v\big\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})} \leq C_{c} \|(H_{c}+\nu)^{1/4} v\|_{L^{2}(\mathbb{R}^{n})}. \end{align*} $$

Proposition 3.4. Under the assumptions of Proposition 3.2 and assuming $\nu $ nonnegative, set $U_{c}(t)$ to be the flow of the equation

(3.2) $$ \begin{align} \left \lbrace{\begin{array}{cc} \partial_{t}^{2} v + H_{c} v + \nu v = 0, & \\ v(t=0) = v_{0}, & \partial_{t} v(t=0) = v_{1}. \end{array}} \right. \end{align} $$

Set also $\mathcal X_{c,\nu }$ and $\mathcal H^{1/2}_{c,\nu }$ to be spaces respectively induced by the norms

$$ \begin{align*} \|(f,g)\|_{\mathcal X_{c,\nu}}^{2} = \|\,|x|^{-1} f\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}^{2} + \|\,|x|^{-1}(H_{c}+\nu)^{-1/2}g\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}^{2} \end{align*} $$

and

$$ \begin{align*} \|(v_{0},v_{1})\|_{\mathcal H^{1/2}_{c,\nu}}^{2} = \|(H_{c} +\nu)^{1/4}v_{0}\|_{L^{2}(\mathbb{R}^{n})}^{2} + \|(H_{c}+\nu)^{-1/4}v_{1}\|_{L^{2}(\mathbb{R}^{n})}^{2}. \end{align*} $$

Then we have for all $(v_{0},v_{1}) \in \mathcal H^{1/2}_{c,\nu }$ ,

$$ \begin{align*} \|U_{c}(t)(v_{0},v_{1})\|_{\mathcal X_{c,\nu}} \leq C_{c} \|(v_{0},v_{1})\|_{\mathcal H^{1/2}_{c,\nu}}. \end{align*} $$

Proof. The proof follows the usual lines, assuming, without loss of generality, that v is real and using the transform

$$ \begin{align*} U = v+ i(H_{c}+\nu)^{-1/2} \partial_{t} v. \\[-36pt]\end{align*} $$

3.2 Application to the Dirac equation with critical potentials

Proposition 3.5. Let $V \in C^{2}((0,\infty ))$ . Write $c_{\pm } = - \frac {(n-1)(n-3)}{4r^{2}} + V^{2} \pm V^{\prime }$ . Set $S_{V,n}(t)$ to be the flow of equation $i\partial _{t} u= h_{V,n}u$ with $h_{V,n}$ as in Lemma 2.8. Assume that

(3.3) $$ \begin{align} \delta_{V}^{\pm} : = \min \Big[ \frac14, \inf \Big( \frac14 + r^{2}(V^{2} \pm V^{\prime}) \Big),\inf \Big( \frac14 - r^{3} (2VV^{\prime} \pm V^{\prime\prime}) - r^{2}(V^{2}\pm V^{\prime}) \Big) \Big]>0. \end{align} $$

Then we have that for all $u_{0} \in \mathcal H^{1/2}_{c_{+},c_{-},m}$ the solution $S_{V,n}(t) u_{0}$ satisfies

$$ \begin{align*} \||x|^{-1} S_{V,n}(t) u_{0} \|_{L^{2}(\mathbb{R} \times \mathbb{R}^{n})} \leq 3(C_{c_{+}} + C_{c_{-}}) \|u_{0}\|_{\mathcal H^{1/2}_{c_{+},c_{-},m}}, \end{align*} $$

where $\mathcal H^{1/2}_{c_{+},c_{-},m}$ is the space induced by the norm

$$ \begin{align*} \left\Vert \begin{pmatrix} f\\g \end{pmatrix}\right\Vert{}_{\mathcal H^{1/2}_{c_{+},c_{-},m}} = \|(m^{2} + H_{c_{-}} )^{1/4} f\|_{L^{2}(\mathbb{R}^{n})} + \|(m^{2} + H_{c_{+}})^{1/4}g\|_{L^{2}(\mathbb{R}^{n})}. \end{align*} $$

Finally, we have $\delta _{c_{\pm }} = \delta _{V}^{\pm }$ .

Proof. Set

$$ \begin{align*} u_{0} = \begin{pmatrix} f_{0} \\ g_{0} \end{pmatrix} \quad \textrm{and}\quad S_{V,n}(t)(u_{0}) = \begin{pmatrix} f\\ g \end{pmatrix}. \end{align*} $$

Write $V_{\pm } = V \pm \Big ( \partial _{r} + \frac {n-1}{2r}\Big )$ . By a straightforward computation we get that $V_{-}V_{+} = H_{c_{-}}$ and $V_{+}V_{-} = H_{c_{+}}$ . Therefore, we recall that

(3.4) $$ \begin{align} h_{V,n}^{2} = \begin{pmatrix} m^{2} + H_{c_{-}} & 0 \\ 0 & m^{2} + H_{c_{+}} \end{pmatrix} \end{align} $$

and that f is the solution to $\partial _{t}^{2} f + H_{c_{-}}f + m^{2} f = 0$ with initial datum

$$ \begin{align*} f(t=0) = f_{0}\quad \textrm{and} \quad\partial_{t} f(t=0) = f_{1}: = -imf_{0} -iV_{-}g_{0}. \end{align*} $$

A direct computation yields $\delta _{V}^{\pm } = \delta _{c_{\pm }}$ and thus

$$ \begin{align*} \|\,|x|^{-1}f\|_{L^{2}_{t,x}} \leq C_{c_{-}} \Big( \|(m^{2} + H_{c_{-}})^{1/4}f_{0}\|_{L^{2}_{x}} + \|(m^{2} + H_{c_{-}})^{-1/4}f_{1}\|_{L^{2}_{x}}\Big) \end{align*} $$

We have that

$$ \begin{align*} \|(m^{2} + H_{c_{-}})^{-1/4}mf_{0}\|_{L^{2}} \leq \sqrt{|m|} \|f_{0}\|_{L^{2}} \end{align*} $$

since $m^{2} + H_{c_{-}} \geq m^{2}$ .

What is more,

$$ \begin{align*} \|(m^{2} + H_{c_{-}})^{-1/4}V_{-}g_{0}\|_{L^{2}}^{2} = \langle (m^{2} + H_{c_{-}})^{-1/4}V_{-}g_{0},(m^{2} + H_{c_{-}})^{-1/4}V_{-}g_{0} \rangle_{L^{2}}. \end{align*} $$

Since $m^{2} + H_{c_{-}} \geq H_{c_{-}}$ , we have

$$ \begin{align*} \|(m^{2} + H_{c_{-}})^{-1/4}V_{-}g_{0}\|_{L^{2}}^{2} \leq \langle H_{c_{-}}^{-1/4}V_{-}g_{0}, H_{c_{-}}^{-1/4}V_{-}g_{0} \rangle_{L^{2}}. \end{align*} $$

By taking adjoints, we get

$$ \begin{align*} \|(m^{2} + H_{c_{-}})^{-1/4}V_{-}g_{0}\|_{L^{2}}^{2} \leq \langle V_{+} H_{c_{-}}^{-1/2}V_{-}g_{0},g_{0} \rangle_{L^{2}}. \end{align*} $$

Let $A = V_{+} H_{c_{-}}^{-1/2}V_{-}$ . The operator A is positive and

$$ \begin{align*} A^{2} = V_{+}H_{c_{-}}^{-1/2}V_{-}V_{+} H_{c_{-}}^{-1/2} V_{-}. \end{align*} $$

Since $V_{-}V_{+} = H_{c_{-}}$ we get

$$ \begin{align*} A^{2} = V_{+} H_{c_{-}}^{-1/2}H_{c_{-}} H_{c_{-}}^{-1/2} V_{-} = V_{+}V_{-} = H_{c_{+}}. \end{align*} $$

Finally,

$$ \begin{align*} \|\,|x|^{-1}f\|_{L^{2}} \leq C_{c_{-}} \Big( 2\|(m^{2} + H_{c_{-}})^{1/4}f_{0}\|_{L^{2}} + \|(m^{2} + H_{c_{+}})^{1/4}g_{0}\|_{L^{2}}\Big). \end{align*} $$

With a similar computation, we get

$$ \begin{align*} \|\,|x|^{-1}g\|_{L^{2}} \leq C_{c_{+}} \Big( 2\|(m^{2} + H_{c_{+}})^{1/4}g_{0}\|_{L^{2}} + \|(m^{2} + H_{c_{-}})^{1/4}f_{0}\|_{L^{2}}\Big).\\[-36pt] \end{align*} $$

We can now exploit the powerful Rodnianski-Schlag argument (see [Reference Rodnianski and Schlag34]) to deduce Strichartz estimates from Proposition 3.5.

Proposition 3.6. Assume that $(p,q,m)$ is admissible, as in Definition 1.3. Then there exists a constant $C = C(p,q,m)$ such that for all $u_{0} \in H^{1/2}_{r} \cap \mathcal H^{1/2}_{c_{+},c_{-},m}$ , we have

(3.5) $$ \begin{align} \begin{aligned} \| S_{V,n}(t)u_{0}\|_{L^{p},W^{1/q-1/p,q}_{r}} &\leq C \Big( (1+\|rV\|_{L^{\infty}((0,\infty))})\|u_{0}\|_{H^{1/2}} \\ & + (\|r^{2}c_{+}\|_{L^{\infty}((0,\infty))} + \|r^{2}c_{-}\|_{L^{\infty}((0,\infty))})((\delta_{V}^{+})^{-1}+(\delta_{V}^{-})^{-1})\|u_{0}\|_{\mathcal H^{1/2}_{c_{+},c_{-},m}}\Big). \end{aligned} \end{align} $$

Proof. Let

$$ \begin{align*} S_{V,n}(t) u_{0} = \begin{pmatrix} f \\ g \end{pmatrix} \quad \textrm{and} \quad u_{0} = \begin{pmatrix} f_{0} \\ g_{0} \end{pmatrix}, \end{align*} $$

where, we recall, $S_{V,n}(t)$ is the flow of equation $i\partial _{t} u= h_{V,n}u$ with $h_{V,n}$ as in Lemma 2.8. We then have

$$ \begin{align*} f = V_{c_{-}}(t) (f_{0},f_{1}) = V_{0}(t) (f_{0},f_{1}) - \int_{0}^{t} \frac{\sin((m^{2}+H_{0})^{1/2}(t-\tau))}{(m^{2}+H_{0})^{1/2}}(c_{-} f(\tau))d\tau \end{align*} $$

with $f_{1} = -imf_{0} - i(V - (\partial _{r} + \frac {n-1}{2r}))g_{0}$ , and where $V_{c_{-}}(t)(f_{0},f_{1})$ is the solution to

$$ \begin{align*} \left\lbrace{\begin{array}{c} \partial_{t}^{2}f + m^{2}f+ H_{c_{-}} f = 0\\ f(t=0) =f_{0}, \partial_{t} f(t=0) = f_{1}. \end{array}} \right. \end{align*} $$

From standard arguments, we get the following estimate on f:

$$ \begin{align*} \|f\|_{L^{p},W^{1/q-1/p,q}_{r}} \leq C \Big( \|f_{0}\|_{H^{1/2}_{r}} + \|f_{1}\|_{H^{-1/2}_{r}} + \|r c_{-} f\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}\Big). \end{align*} $$

This estimate can be obtained by combining standard Strichartz estimates for Klein-Gordon (see, e.g., [Reference D’Ancona and Fanelli19]), the Christ-Kiselev lemma ( $p>2$ ) and local smoothing on $e^{i\sqrt {m^{2}+H_{0}}t}$ (by, for example, taking the dual form of estimate (3.7) in [Reference D’Ancona17]). Indeed, we have

$$ \begin{align*} \|V_{0}(t) (f_{0},f_{1})\|_{L^{p},W^{1/q-1/p,q}_{r}} \lesssim \|f_{0}\|_{H^{1/2}_{r}} + \|f_{1}\|_{H^{-1/2}_{r}} \end{align*} $$

due to the Strichartz inequality for the Klein-Gordon equation. We also have that

$$ \begin{align*} \big \| \int_{0}^{\infty} \frac{\sin((m^{2}+H_{0})^{1/2}(t-\tau))}{(m^{2}+H_{0})^{1/2}}(c_{-} f(\tau)) &d\tau \big\|_{L^{p},W^{1/q-1/p,q}_{r}} \\ &\lesssim \big \| \int_{0}^{\infty} \frac{\sin((m^{2}+H_{0})^{1/2}(t-\tau))}{(m^{2}+H_{0})^{1/2}}(c_{-} f(\tau))d\tau \big\|_{H^{1/2}(\mathbb{R}^{n})}. \end{align*} $$

Since $H_{0}=\sqrt {-\Delta }$ and the Laplacian commute, we get

$$ \begin{align*} \big \| \int_{0}^{\infty} \frac{\sin((m^{2}+H_{0})^{1/2}(t-\tau))}{(m^{2}+H_{0})^{1/2}}(c_{-} f(\tau))&d\tau \big\|_{H^{1/2}(\mathbb{R}^{n})} \\ &= \big \| \int_{0}^{\infty} \sin((m^{2}+H_{0})^{1/2}(t-\tau))(c_{-} f(\tau))d\tau \big\|_{H^{-1/2}(\mathbb{R}^{n})}. \end{align*} $$

We then use the dual form of local smoothing to get

$$ \begin{align*} \big \| \int_{0}^{\infty} \sin((m^{2}+H_{0})^{1/2}(t-\tau))(c_{-} f(\tau))d\tau \big\|_{H^{-1/2}(\mathbb{R}^{n})}\lesssim \||x| c_{-} f\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}. \end{align*} $$

Now, exploiting the Christ-Kiselev lemma, we get that since

$$ \begin{align*} F \mapsto \int_{0}^{\infty} \frac{\sin((m^{2}+H_{0})^{1/2}(t-\tau))}{(m^{2}+H_{0})^{1/2}}(|x|^{-1}F(\tau))d\tau \end{align*} $$

is continuous from $L^{2}(\mathbb {R}\times \mathbb {R}^{n})$ to $L^{p},W^{1/q-1/p,q}_{r}$ and since $p>2$ , so is

$$ \begin{align*} F \mapsto \int_{0}^{t} \frac{\sin((m^{2}+H_{0})^{1/2}(t-\tau))}{(m^{2}+H_{0})^{1/2}}(|x|^{-1}F(\tau))d\tau. \end{align*} $$

We get that

$$ \begin{align*} \big \| \int_{0}^{t} \frac{\sin((m^{2}+H_{0})^{1/2}(t-\tau))}{(m^{2}+H_{0})^{1/2}}(c_{-} f(\tau))d\tau \big\|_{L^{p},W^{1/q-1/p,q}_{r}} \lesssim \|r c_{-} f\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}. \end{align*} $$

Because of the Hardy inequality, we have

$$ \begin{align*} \|f_{1}\|_{H^{-1/2}_{r}} \lesssim \|f_{0}\|_{L^{2}} + (1+\|rV\|_{L^{\infty}}) \|g_{0}\|_{H^{1/2}_{r}}. \end{align*} $$

Besides,

$$ \begin{align*} \|r c_{-} f\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}\leq \|r^{2}c_{-}\|_{L^{\infty}} \|r^{-1}f\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}. \end{align*} $$

Because of local smoothing on $S_{V,n}(t)$ , we get

$$ \begin{align*} \|r^{-1}f\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})} \lesssim (C_{c_{+}} + C_{c_{-}}) \|u_{0}\|_{\mathcal H^{1/2}_{c_{+},c_{-},m}}. \end{align*} $$

A similar computation on g yields the result.

3.3 Application to the Dirac equation in curved manifolds

In this section, we set $V=V_{\mu } = \frac {\mu }{\varphi }$ , $\delta _{\pm }(\mu ) = \delta _{V_{\mu }}^{\pm }$ and

$$ \begin{align*} c_{\pm}(\mu) = -\frac{(n-1)(n-3)}{4r^{2}} + V_{\mu}^{2} \pm V_{\mu}^{\prime} = -\frac{(n-1)(n-3)}{4r^{2}} + \mu \frac{\mu \mp \varphi^{\prime} }{\varphi^{2}}. \end{align*} $$

Finally, we set $H_{\pm } (\mu ) = H_{c_{\pm }}(\mu )$ . Here, we are assuming that $\varphi $ satisfies assumptions (A0)-(A1).

Lemma 3.7. For any $|s|\leq 1$ , we have the bound

$$ \begin{align*}\|(m^{2} + H_{\pm}(\mu))^{s/2}v\|_{L^{2}(\mathbb{R}^{n})}\lesssim_{\varphi,m} (1+\mu^{2})^{s/2} \|v\|_{ H^{s}(\mathbb{R}^{n})}. \end{align*} $$

In particular, we have

$$ \begin{align*} \|u_{0}\|_{\mathcal H^{1/2}_{c_{+}(\mu),c_{-}(\mu),m}} \lesssim_{\varphi,m} \sqrt{|\mu|} \|u_{0}\|_{H^{1/2}_{r,n}}. \end{align*} $$

Proof. We have that, for any $\mu $ ,

(3.6) $$ \begin{align} c_{\pm}(\mu)(r)\leq \frac{C_{\varphi} \mu^{2}}{r^{2}}, \quad |c_{\pm}(\mu)(r)|\leq \frac{(n-1)(n-3)}{4r^{2}} + \frac{C_{\varphi} \mu^{2}}{r^{2}} \end{align} $$

due to the assumption in equation (1.13) with

$$ \begin{align*} C_{\varphi} = \max \Big( \| \frac{r^{2}}{\varphi^{2}}\|_{L^{\infty}((0,\infty))}, 2\|\frac{r^{2} \varphi ^{\prime}}{\varphi^{2}}\|_{L^{\infty}((0,\infty))}\Big). \end{align*} $$

As done in [Reference D’Ancona and Zhang21, Section 2], the result follows from the application of Hardy inequality and interpolation in a standard way. We omit the details.

We deduce from Proposition 3.6 the following result.

Proposition 3.9. Assume that $(p,q,m)$ is admissible, as in Definition 1.3. Then there exists a constant $C = C(p,q,m,\varphi )$ such that for all $u_{0} \in H^{1/2}_{r} $ , we have

(3.7) $$ \begin{align} \| S_{V_{\mu},n}(t)u_{0}\|_{L^{p},W^{1/q -1/p,q}_{r}} \leq C |\mu|^{5/2}((\delta_{V_{\mu}}^{+})^{-1/2}+(\delta_{V_{\mu}}^{-})^{-1/2})\|u_{0}\|_{H^{1/2}_{r}}. \end{align} $$

Proof. The estimate in equation (3.7) is a direct consequence of Proposition 3.6, Lemma 3.7 and the bounds

$$ \begin{align*} \|rV_{\mu}\|_{L^{\infty}} \lesssim_{\varphi} |\mu|, \quad \|r^{2}c_{\pm}\|_{L^{\infty}} \lesssim_{\varphi} \mu^{2}. \end{align*} $$

The bound on c is due to equation (3.6). For the bound on $V_{\mu }$ , we recall that

$$ \begin{align*} V_{\mu} = \frac{\mu}{\varphi} \end{align*} $$

and thus

$$ \begin{align*} \|rV_{\mu}\|_{L^{\infty}} \leq |\mu|\; \|\frac{r}{\varphi}\|_{L^{\infty}}.\\[-36pt] \end{align*} $$

By interpolation, we get the following:

Corollary 3.10. Assume that $(p,q,m)$ is admissible, as in Definition 1.3, and let $\varepsilon>0$ . There exists $C = C(p,q,m,\varphi ,\varepsilon )$ such that for all $u_{0} \in H^{1/2}_{r} $ , we have

(3.8) $$ \begin{align} \|S_{V_{\mu},n}(t)u_{0}\|_{L^{p},W^{1/q-1/p,q}_{r}} \leq C |\mu|^{5/p+\varepsilon}((\delta_{V}^{+})^{-1/2}+(\delta_{V}^{-})^{-1/2})^{2/p + \varepsilon}\|u_{0}\|_{H^{1/2}_{r}}. \end{align} $$

Exploiting Proposition 2.6, we eventually get Theorem 1.4. In Proposition 2.6, we used the notation $ e^{-ith_{\mu ,n}} $ for $S_{V_{\mu },n}(t)$ and $\sigma _{n}(r) = \Big (\frac {r}{\varphi (r)}\Big )^{(n-1)/2}$ . Hence, this proposition allows to pass from Strichartz estimates for $S_{V_{\mu },n}(t)$ to Strichartz estimates for the operator $h_{\mu }$ .

4.1 Assumptions

Let us assume that the infimum of the positive part of the spectrum of the Dirac operator on $\mathbb {K}^{n-1}$ , denoted by $\mu _{0}$ , is strictly bigger than $\frac 12$ , and that $\varphi $ is asymptotically flat: in other words, that

$$ \begin{align*} \varphi = r(1 + \varphi_{1}) \end{align*} $$

with the following assumption on $\varphi _{1}$ :

• • $\varphi _{1}$ is non-negative and bounded,

• • $A_{\varphi } = \|\varphi _{1} + r\varphi _{1}^{\prime }\|_{\infty }$ and

  $$ \begin{align*} B_{\varphi} = \|r\varphi_{1}^{\prime} + (1+\varphi_{1})(\varphi_{1} + r\varphi_{1}^{\prime})\|_{\infty} + \|2r^{2} (\varphi_{1}^{\prime})^{2} + (1+\varphi_{1})r^{2}\varphi_{1}^{\prime\prime}\|_{\infty} \end{align*} $$
  are well-defined,

• •

  $$ \begin{align*} \max(A_{\varphi},B_{\varphi}) \left\lbrace{\begin{array}{cc} \leq 1 & \textrm{if } \mu_{0} \geq 2\\ < \min(\frac14 + \mu_{0}^{2} -\mu_{0} , \frac18) & \textrm{otherwise}\end{array}} \right.. \end{align*} $$

4.2 Asymptotically flat manifolds are admissible

In this subsection, we prove Proposition 1.12: if $\varphi (r)$ satisfies the assumptions above, the condition in equation (1.13) is satisfied, and therefore the Strichartz estimates proved in Theorem 1.4 hold. The only thing we need to prove is the following:

Lemma 4.1. Under the above assumptions on $\varphi _{1}$ , we have for all $\mu \geq \mu _{0}$ ,

$$ \begin{align*} \delta_{\pm}(\mu) \geq \left\lbrace{\begin{array}{cc} \frac14 & \textrm{if }\mu_{0} \geq 2,\\ \min(\frac14 + \mu_{0}^{2} -\mu_{0} , \frac18) - \max (A_{\varphi},B_{\varphi}) & \textrm{otherwise.} \end{array}} \right.. \end{align*} $$

Proof. We have

$$ \begin{align*} I(r):= \frac14 + r^{2}(V_{\mu} \pm V_{\mu}^{\prime}) = \frac14 + \frac{\mu^{2}\mp \mu}{(1+\varphi_{1})^{2}} \mp \mu\frac{\varphi_{1} + r\varphi_{1}^{\prime}}{(1+\varphi_{1})^{2}}. \end{align*} $$

Therefore,

$$ \begin{align*} I(r) \geq \frac14 + \frac{\mu^{2}-\mu}{(1+\varphi_{1})^{2}} -\mu \frac{A_{\varphi}}{(1+\varphi_{1})^{2}}. \end{align*} $$

Case 1: $\mu \geq 2$ , we have since $\mu ^{2} - \mu \geq \mu $ ,

$$ \begin{align*} I(r) \geq \frac14 + \frac{\mu}{(1+\varphi_{1})^{2}} (1-A_{\mu}) \end{align*} $$

and since $A_{\mu } \leq 1$ , we have $I(r)\geq \frac 14$ .

Case 2: $\mu \in [1,2)$ , we have since $\varphi _{1}\geq 0$ and $\mu ^{2} - \mu \geq 0$ ,

$$ \begin{align*} I(r) \geq \frac14 - 2 A_{\varphi} \geq \frac18 - A_{\varphi}, \end{align*} $$

which is positive since $A_{\varphi } < \frac 18$ .

Case 3: $\mu \in [\mu _{0},1)$ , we have, since $\varphi _{1}> 0$ and $\mu ^{2} - \mu \geq \mu _{0}^{2} - \mu _{0}$ ,

$$ \begin{align*} I(r) \geq \frac14 +\mu_{0}^{2} - \mu_{0} - A_{\varphi}, \end{align*} $$

which is positive.

Set

$$ \begin{align*} Q_{\pm}(r) = \frac14 - r^{3}(2V_{\mu} V_{\mu}^{\prime} \pm V_{\mu}^{\prime\prime}) -r^{2} (V_{\mu}^{2}\pm V_{\mu}^{\prime}). \end{align*} $$

We have

$$ \begin{align*} Q_{\pm}(r) = \frac14 + \frac{\mu^{2} \mp \mu}{(1+\varphi_{1})^{2}} + \frac{\mu^{2} \mp \mu}{(1+\varphi_{1})^{3}}f(r) \mp \frac{\mu}{(1+\varphi_{1})^{3}}g(r) \end{align*} $$

with

$$ \begin{align*} f(r) = r\varphi_{1}^{\prime} + (1+\varphi_{1})(\varphi_{1} + r\varphi_{1}^{\prime})\quad \textrm{and}\quad g(r) = 2r^{2} (\varphi_{1}^{\prime})^{2} + (1+\varphi_{1}) r^{2} \varphi_{1}^{\prime\prime}. \end{align*} $$

Case 1: We consider $Q_{+} (r)$ . We have

$$ \begin{align*} Q_{+}(r) = \frac14 + \frac{\mu^{2} + \mu}{(1+\varphi_{1})^{2}} + \frac{\mu^{2} + \mu}{(1+\varphi_{1})^{3}}f(r) + \frac{\mu}{(1+\varphi_{1})^{3}}g(r), \end{align*} $$

hence

$$ \begin{align*} Q_{+}(r) \geq \frac14 + \frac{\mu^{2} + \mu}{(1+\varphi_{1})^{2}} ( 1 - B_{\varphi}), \end{align*} $$

and since $B_{\varphi } \leq 1$ , we have $Q_{+}(r) \geq \frac 14$ .

Case 2: We consider $Q_{-}(r)$ . We have

$$ \begin{align*} Q_{-}(r) = \frac14 + \frac{\mu^{2} -\mu}{(1+\varphi_{1})^{2}} + \frac{\mu^{2} - \mu}{(1+\varphi_{1})^{3}}f(r) - \frac{\mu}{(1+\varphi_{1})^{3}}g(r). \end{align*} $$

Case 2.1: $\mu \geq 2$ , we have $\mu ^{2} - \mu \geq \mu $ , hence

$$ \begin{align*} Q_{-}(r) \geq \frac14 + \frac{\mu^{2}-\mu}{(1+\varphi_{1})^{2}} (1-B_{\varphi}), \end{align*} $$

and since $B_{\varphi } \leq 1$ , we have $Q_{-}(r)\geq \frac 14$ .

Case 2.2: $\mu \in [1,2)$ . We have $\mu ^{2} - \mu \leq 2$ and $\mu \leq 2$ , hence

$$ \begin{align*} Q_{-}(r) \geq \frac14 -2 B_{\varphi}. \end{align*} $$

Finally, case 2.3: $\mu \in [\mu _{0},1)$ , we have $0>\mu ^{2}-\mu \leq \mu _{0}^{2} - \mu _{0}$ and $|\mu ^{2} - \mu |\leq 1$ hence

$$ \begin{align*} Q_{-}(r) \geq \frac14 +\mu_{0}^{2} - \mu_{0} - B_{\varphi}, \end{align*} $$

which concludes the proof.

4.3 Local smoothing in the asymptotically flat case

From this subsection, we assume that $\mathbb {K}^{n-1}$ is $\mathbb S^{n-1}$ . We have that the positive spectrum of the Dirac operator on the sphere is $\frac {n-1}{2} + \mathbb {N}$ . Hence, we see that in dimensions higher than $5$ , we have that $\delta _{\pm } (\mu ) \geq \frac 14$ for all $\mu $ in the spectrum. In any case, for a fixed $\varphi $ satisfying Assumptions in (A2), we have that $\delta $ is uniformly bounded in $\mu $ by the below.

For $\mu $ in the spectrum of the Dirac operator on the sphere, we write $\mathcal H_{\mu }$ the space generated by

$$ \begin{align*} \left\{ \begin{pmatrix} (1+i\tilde \alpha^{0}) \psi_{\mu} \\0 \end{pmatrix}, \begin{pmatrix} 0 \\ (1-i\tilde \alpha^{0}) \psi_{\mu} \end{pmatrix}, \quad \mathcal D_{\mathbb S^{n-1}} \psi_{\mu} = \mu \psi_{\mu} \right\}. \end{align*} $$

For $0\leq a <b $ , we set

$$ \begin{align*} \mathcal H_{a,b} = \bigoplus_{|\mu|\in [a,b]} \mathcal H_{\mu} \end{align*} $$

and $p_{a,b}$ the orthogonal projection onto $L^{2}_{r}\otimes \mathcal H_{a,b}$ .

We recall that $\sigma _{n}^{-1} \mathcal D_{\Sigma } \sigma _{n}$ is entirely described by the $h_{_{\mu },n}$ and thus commute with $p_{a,b}$ . We write $S_{n}(t)$ the flow of

$$ \begin{align*} i\partial_{t} - \sigma_{n}^{-1} \mathcal D_{\Sigma} \sigma_{n} = 0. \end{align*} $$

We deduce the following proposition.

Proposition 4.2. Let $u_{0} \in H^{1/2}(\mathbb {R}^{n})$ . We have

$$ \begin{align*} \|r^{-1}p_{a,b}S_{n}(t)u_{0}\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})} \lesssim_{m,\varphi,n} b^{1/2} \|p_{a,b} u_{0}\|_{H^{1/2}(\mathbb{R}^{n})}. \end{align*} $$

Proof. Let $u_{0,\mu }$ be the orthogonal projection of $u_{0}$ over $L^{2}_{r}\otimes \mathcal H_{\mu }$ and $u_{\mu } = S_{n}(t)u_{0,\mu }$ . Because the orthogonal projection over $L^{2}_{r}\otimes \mathcal H_{\mu }$ and $S_{n}(t)$ commute, we get

$$ \begin{align*} \|r^{-1}p_{a,b}S_{n}(t)u_{0}\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}^{2} = \sum_{|\mu| \in [a,b]} \|r^{-1}u_{\mu}\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})}^{2}. \end{align*} $$

From Proposition 3.5, we have

$$ \begin{align*} \|r^{-1}u_{\mu}\|_{L^{2}(\mathbb{R}\times \mathbb{R}^{n})} \leq 3(C_{c_{+}(\mu)} + C_{c_{-}(\mu)})\|u_{0,\mu}\|_{\mathcal H_{c_{+}(\mu),c_{-}(\mu),m}}, \end{align*} $$

where, by abuse of notation, we identified $u_{0,\mu }$ with

$$ \begin{align*} \sum_{j} \frac{f_{j}}{\sqrt 2} \begin{pmatrix} (1+i\tilde \alpha^{0}) \psi_{\mu,j} \\0 \end{pmatrix}+ \frac{g_{j}}{\sqrt 2} \begin{pmatrix} 0 \\ (1-i\tilde \alpha^{0}) \psi_{\mu,j} \end{pmatrix}, \end{align*} $$

where the (finite) family $(\psi _{\mu ,j})_{j}$ is an orthonormal basis of the eigenspace of $\mathcal D_{\mathbb S^{n-1}}$ associated to $\mu $ , and we identified $\|u_{0,\mu }\|_{\mathcal H_{c_{+}(\mu ),c_{-}(\mu ),m}}^{2}$ with

$$ \begin{align*} \sum_{j} \big \| \begin{pmatrix} f_{j} \\ g_{j} \end{pmatrix}\big\|_{\mathcal H_{c_{+}(\mu),c_{-}(\mu),m}}^{2}. \end{align*} $$

From Lemma 3.7, we have for all j

$$ \begin{align*} \big \| \begin{pmatrix} f_{j} \\g_{j} \end{pmatrix}\big\|_{\mathcal H_{c_{+}(\mu),c_{-}(\mu),m}} \lesssim_{m,\varphi} \sqrt{|\mu|} \big \| \begin{pmatrix} f_{j} \\g_{j} \end{pmatrix}\big\|_{H^{1/2}(\mathbb{R}^{n})}, \end{align*} $$

from which we deduce

$$ \begin{align*} \|u_{0,\mu}\|_{\mathcal H_{c_{+}(\mu),c_{-}(\mu),m}} \lesssim_{m,\varphi} \sqrt{|\mu|} \|u_{0,\mu}\|_{H^{1/2}(\mathbb{R}^{n})} \leq \sqrt b \|u_{0,\mu}\|_{H^{1/2}(\mathbb{R}^{n})}. \end{align*} $$

We conclude by using the fact that $C_{c_{+}(\mu )}$ and $C_{c_{-}(\mu )}$ are uniformly bounded in $\mu $ .

4.4 Restricted Strichartz estimates in the asymptotically flat case

In this subsection, we prove the following proposition.

Proposition 4.3. Let $0\leq a < b$ , and let $m,p,q$ be admissible. We have, for all $u_{0} \in H^{1/2}(\mathbb {R}^{n})$ and all $\varepsilon>0$ ,

$$ \begin{align*} \|p_{a,b} S_{n}(t) u_{0}\|_{L^{p}(\mathbb{R}, W^{s,q}(\mathbb{R}^{n}))}\left \lbrace{\begin{array}{cc} \lesssim_{m,\varphi, n,\varepsilon,p,q} b^{5/p+\varepsilon} \|p_{a,b}u_{0}\|_{H^{1/2}} & \textrm{if } n=3,m=0\\ \lesssim_{m,\varphi, n,p,q} b^{5/p} \|p_{a,b}u_{0}\|_{H^{1/2}} & \textrm{otherwise },\end{array}} \right. \end{align*} $$

with $s = \frac 1{q}-\frac 1{p}$ .

Proof. We prove that

$$ \begin{align*} \|p_{a,b} S_{n}(t) u_{0}\|_{L^{p}(\mathbb{R}, W^{s,q}(\mathbb{R}^{n}))} \lesssim_{m,\varphi, n,} b^{5/2} \|p_{a,b}u_{0}\|_{H^{1/2}} \end{align*} $$

for all admissible triplets $(m,p,q)$ and conclude by interpolation.

First, we have

$$ \begin{align*} \sigma_{n}^{-1} \mathcal D_{\Sigma} \sigma_{n} = \mathcal D_{\mathbb{R}^{n}} + \mathcal V \end{align*} $$

with $\mathcal V$ the operator

$$ \begin{align*} \mathcal V = \Big(\frac1{\varphi} - \frac1{r}\Big) \begin{pmatrix} 0 & \mathcal D_{\mathbb S^{n-1}} \\ \mathcal D_{\mathbb S^{n-1}} & 0 \end{pmatrix}. \end{align*} $$

Writing $u = S_{n}(t) u_{0}$ , we get that u satisfies

$$ \begin{align*} \partial_{t}^{2} u + (\mathcal D_{\mathbb{R}^{n}} + \mathcal V)^{2} u = 0 \end{align*} $$

with initial data $u(t=0) = u_{0}$ and $\partial _{t} u(t=0) =: u_{1} = -i(\mathcal D_{\mathbb {R}^{n}} + \mathcal V)u_{0}$ .

We have

$$ \begin{align*} (\mathcal D_{\mathbb{R}^{n}} + \mathcal V)^{2} = \mathcal D_{\mathbb{R}^{n}}^{2} + \mathcal W = m^{2} - \Delta_{\mathbb{R}^{n}} +\mathcal W \end{align*} $$

with

$$ \begin{align*} \mathcal W = \{\mathcal V,\mathcal D_{\mathbb{R}^{n}}\} + \mathcal V^{2}. \end{align*} $$

By the Rodnianski-Schlag argument that we previously used, we get

$$ \begin{align*} \|p_{a,b} u\|_{L^{p},W^{s,q}(\mathbb{R}^{n})} \lesssim_{n,p,q} \|p_{a,b}u_{0}\|_{H^{1/2}(\mathbb{R}^{n})} + \|p_{a,b}u_{1}\|_{H^{-1/2}(\mathbb{R}^{n})} + \|r\mathcal W p_{a,b} u\|_{L^{2}(\mathbb{R}^{n+1})}. \end{align*} $$

By the commutativity of $p_{a,b}$ and $\mathcal D_{\mathbb {R}^{n}} + \mathcal V$ , we get

$$ \begin{align*} \|p_{a,b}u_{1}\|_{H^{-1/2}} = \|(\mathcal D_{\mathbb{R}^{n}} + \mathcal V)p_{a,b} u_{0}\|_{H^{-1/2}} \end{align*} $$

and since $r(\frac 1{\varphi } - \frac 1{r})$ is bounded, by Hardy’s inequality, we get

$$ \begin{align*} \|p_{a,b}u_{1}\|_{H^{-1/2}} \lesssim_{n,\varphi}\|p_{a,b} u_{0}\|_{H^{1/2}}. \end{align*} $$

For the other part, we use that

$$ \begin{align*} \|r\mathcal W p_{a,b} u\|_{L^{2}(\mathbb{R}^{n+1})} \leq \|p_{a,b}r\mathcal W r p_{a,b}\|_{L^{2}\rightarrow L^{2}}\|r^{-1}p_{a,b}u\|_{L^{2}(\mathbb{R}^{n+1})}. \end{align*} $$

It remains to use Proposition 4.2 and prove that $p_{a,b}r\mathcal W r p_{a,b}$ is a bounded operator from $L^{2}(\mathbb {R}^{n+1})$ to itself and compute the dependence of its norm in $a,b$ to conclude.

Because multiplication by a radial function and the Dirac operator on the sphere commute, we get that

$$ \begin{align*} p_{a,b}r\mathcal V^{2} r p_{a,b} = \Big(\frac{r}{\varphi}-1\Big)^{2}\begin{pmatrix} p_{a,b}\mathcal D_{\mathbb S^{n-1}}^{2} p_{a,b} & 0 \\ 0 & p_{a,b}\mathcal D_{\mathbb S^{n-1}}^{2} p_{a,b}\end{pmatrix} \end{align*} $$

and we deduce

$$ \begin{align*} \| p_{a,b}r\mathcal V^{2} r p_{a,b}\|_{L^{2}\rightarrow L^{2}} \leq \|\Big(\frac{r}{\varphi}-1\Big)^{2}\|_{\infty} b^{2}, \end{align*} $$

which is finite because of the assumptions on $\varphi $ .

What is more, we have

$$ \begin{align*} \mathcal D_{\mathbb{R}^{n}} = \begin{pmatrix} m & i\tilde \alpha^{0} \Big( \partial_{r} + \frac{n-1}{2r}\Big) + \frac1{r}\mathcal D_{\mathbb S^{n-1}} \\ i\tilde \alpha^{0} \Big( \partial_{r} + \frac{n-1}{2r}\Big) + \frac1{r}\mathcal D_{\mathbb S^{n-1}} & -m \end{pmatrix}. \end{align*} $$

We deduce

$$ \begin{align*} \{ \mathcal D_{\mathbb{R}^{n}}, \mathcal V\} = \begin{pmatrix} \mathcal L & 0 \\ 0 & \mathcal L \end{pmatrix} \end{align*} $$

with

$$ \begin{align*} \mathcal L = \{ i\tilde \alpha^{0} \Big( \partial_{r} + \frac{n-1}{2r}\Big)+ \frac1{r}\mathcal D_{\mathbb S^{n-1}},\Big( \frac1{\varphi} - \frac1{r}\Big)\mathcal D_{\mathbb S^{n-1}}\}. \end{align*} $$

We have that $i\tilde \alpha ^{0}$ and $\mathcal D_{\mathbb S^{n-1}}$ anticommute, that $\partial _{r}$ and $\mathcal D_{\mathbb S^{n-1}}$ commute and that the multiplication by a radial function commutes with $\mathcal D_{\mathbb S^{n-1}}$ . Hence we get

$$ \begin{align*} \mathcal L = i\tilde \alpha^{0} \mathcal D_{\mathbb S^{n-1}}\Big[\partial_{r} + \frac{n-1}{2r}, \frac1{\varphi} - \frac1{r}\Big] + 2 \Big( \frac1{\varphi} - \frac1{r}\Big)\frac1{r} \mathcal D_{\mathbb S^{n-1}}^{2}. \end{align*} $$

We deduce

$$ \begin{align*} p_{a,b}r\{\mathcal V, \mathcal D_{\mathbb{R}^{n}}\}rp_{a,b} =\begin{pmatrix} \mathcal L_{a,b} & 0 \\ 0 & \mathcal L_{a,b}\end{pmatrix} \end{align*} $$

with

$$ \begin{align*} \mathcal L_{a,b} = ip_{a,b} \tilde \alpha^{0}\mathcal D_{\mathbb S^{n-1}}p_{a,b}r^{2} \partial_{r} \Big( \frac1{\varphi} - \frac1{r}\Big) + 2 r \Big( \frac1{\varphi} - \frac1{r}\Big)p_{a,b}\mathcal D_{\mathbb S^{n-1}}^{2}p_{a,b}. \end{align*} $$

Because

$$ \begin{align*} r^{2} \partial_{r} \Big( \frac1{\varphi} - \frac1{r}\Big) = \frac{\varphi_{1}}{1+\varphi_{1}}- \frac{r\varphi_{1}^{\prime}}{(1+\varphi_{1})^{2}} \end{align*} $$

belongs to $L^{\infty }$ , and so does $r\Big ( \frac 1{\varphi } - \frac 1{r}\Big ) = \frac 1{1+\varphi _{1}} - 1$ , we get

$$ \begin{align*} \|p_{a,b}r\{\mathcal V, \mathcal D_{\mathbb{R}^{n}}\}rp_{a,b}\|_{L^{2}\rightarrow L^{2}} \lesssim_{\varphi} b^{2}. \end{align*} $$

This concludes the proof.

4.5 Setup for the Littlewood-Paley argument

In this subsection, we draw a link between the spherical harmonics and the eigenfunctions of the Dirac operator on the sphere.

Proposition 4.4. Let $\pi _{j}$ be the orthogonal projection on $\mathcal S_{j} \otimes L^{2}_{r} \otimes \mathbb {C}^{M}$ , where $\mathcal S_{j}$ are the spherical harmonics of degree in $[2^{j},2^{j+1})$ , and let $u \in L^{2}(\mathbb {R}^{n},\mathbb {C}^{M})$ . We have

$$ \begin{align*} \pi_{j} u = \pi_{j} p_{a_{j},b_{j}} u \end{align*} $$

with $a_{j} = \frac {n-1}{2} + 2^{j} - 1$ and $b_{j} = \frac {n-1}2 + 2^{j+1}$ .

Before proving this proposition, we prove the following short lemma.

Lemma 4.5. We have for all $\mu $ in the spectrum of the Dirac operator on the sphere $\mathbb S^{n-1}$ ,

$$ \begin{align*} \mathcal H_{\mu} \subseteq (\mathcal S_{|\mu|-\frac{n-1}{2}} \oplus \mathcal S_{|\mu| - \frac{n-1}{2} + 1} ) \otimes \mathbb{C}^{M}. \end{align*} $$

Proof. Let $\psi _{\mu }$ be an eigenfunction of $\mathcal D_{\mathbb S^{n-1}}$ with eigenvalue $\mu $ , and write

$$ \begin{align*} \Psi_{\mu}^{+} = \begin{pmatrix} (1+i\tilde \alpha^{0})\psi_{\mu} \\ 0 \end{pmatrix},\quad \Psi_{\mu}^{-}= \begin{pmatrix} 0 \\ (1-i\tilde \alpha^{0}) \psi_{\mu} \end{pmatrix}. \end{align*} $$

We have

$$ \begin{align*} \mathcal D_{\mathbb{R}^{n}}\Psi_{\mu}^{+} = \Big(\mu- \frac{n-1}{2}\Big) \frac1{r} \Psi_{\mu}^{-} \end{align*} $$

and thus

$$ \begin{align*} -\Delta_{\mathbb{R}^{n}}\Psi_{\mu}^{+} = \mathcal D_{\mathbb{R}^{n}}^{2} \Psi_{\mu}^{+} = \mathcal D_{\mathbb{R}^{n}} \Big(\mu- \frac{n-1}{2}\Big) \frac1{r} \Psi_{\mu}^{-} = \Big( \mu - \frac{n-1}{2}\Big)\Big( \mu + \frac{n-1}{2}-1\Big) \frac1{r^{2}} \Psi_{\mu}^{+}. \end{align*} $$

Because $\Psi _{\mu }^{+}$ does not depend on r, we deduce that it is a spherical harmonics of degree $\mu -\frac {n-1}{2}$ if $\mu>0$ and $-\mu -\frac {n-1}{2}+1$ otherwise.

The same type of computation yields

$$ \begin{align*} -\Delta_{\mathbb{R}^{n}} \Psi_{\mu}^{-} = \Big( \mu+ \frac{n-1}{2}\Big) \Big(\mu -\frac{n-1}{2}+1\Big)\frac1{r^{2}} \Psi_{\mu}^{-}, \end{align*} $$

hence $\Psi _{\mu }^{-}$ is a spherical harmonics of degree $\mu - \frac {n-1}{2}+1$ if $\mu>0$ and $-\mu -\frac {n-1}{2}$ otherwise.

In other words,

$$ \begin{align*} \mathcal H_{\mu} \subseteq (\mathcal S_{|\mu|-\frac{n-1}{2}} \oplus \mathcal S_{|\mu| - \frac{n-1}{2} + 1} ) \otimes \mathbb{C}^{M}.\\[-36pt] \end{align*} $$

Proof of Proposition 4.4. We have

$$ \begin{align*} \pi_{j} u = \sum_{\mu} \pi_{j} u_{\mu}, \end{align*} $$

where $u_{\mu }$ is the orthogonal projection of u over $\mathcal H_{\mu } \otimes L^{2}_{r}$ . If $|\mu |> b_{j}$ , then

$$ \begin{align*} |\mu| - \frac{n-1}2> 2^{j+1}, \end{align*} $$

hence $u_{\mu } $ is a combination of spherical harmonics of degree higher than $2^{j+1}$ , hence $\pi _{j} u_{\mu } =0$ .

If $|\mu | < a_{j}$ , then

$$ \begin{align*} |\mu| - \frac{n-1}{2}+1 < 2^{j}, \end{align*} $$

hence $u_{\mu }$ is a combination of spherical harmonics of degree lesser than $2^{j}$ ; we have $\pi _{j} u_{\mu } = 0$ , and therefore

$$ \begin{align*} \pi_{j} u = \sum_{|\mu| \in [a_{j},b_{j}]}\pi_{j} u_{\mu} = \pi_{j} p_{a_{j},b_{j}} u.\\[-36pt] \end{align*} $$

4.6 Proof of Theorem 1.10.

As done in [Reference Cacciafesta and de Suzzoni11], by relying on Littlewood-Paley theory on the sphere, we are able to prove Strichartz estimates for the Dirac equation with general initial conditions in the setting of spherically symmetric manifolds. As the proof is very similar, we omit some details.

Proposition 4.6. Let $m,p,q$ be admissible. Let $a,b>0$ be such that

$$ \begin{align*} \frac1{2a} + \frac5{pb} < 1. \end{align*} $$

We have for all $u_{0} \in H^{a,b}(\mathbb {R}^{n})$ ,

$$ \begin{align*} \|S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))} \lesssim_{n,\varphi,m,p,q,a,b} \|u_{0}\|_{H^{a,b}}. \end{align*} $$

Proof. We have by the Littlewood-Paley theory ( $q \in [2,\infty )$ )

$$ \begin{align*} \|S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))}^{2} \lesssim \sum_{j} \|\pi_{j} S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))}^{2}. \end{align*} $$

By Proposition 4.4, we have

$$ \begin{align*} \|\pi_{j} S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))}^{2} = \|\pi_{j} p_{a_{j},b_{j}} S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))}^{2}. \end{align*} $$

Again by Littlewood-Paley theory, we have

$$ \begin{align*} \|\pi_{j} p_{a_{j},b_{j}} S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))}^{2}\lesssim \|p_{a_{j},b_{j}} S_{n}(t) u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))}^{2}. \end{align*} $$

We apply Proposition 4.3, and we get

$$ \begin{align*} \|\pi_{j} p_{a_{j},b_{j}} S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))}^{2}\lesssim b_{j}^{10/p + \varepsilon} \|p_{a_{j},b_{j}}u_{0}\|_{H^{1/2}}^{2} \end{align*} $$

with $\varepsilon>0$ if $m=0$ and $n=3$ (and $0$ otherwise). From the inequality

$$ \begin{align*} xy \leq x^{c} + y^{d} \end{align*} $$

for any $x,y\in [1,\infty )$ and $\frac 1{c}+ \frac 1{d} \leq 1$ , we deduce

$$ \begin{align*} \|\pi_{j} p_{a_{j},b_{j}} S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))}^{2}\lesssim b_{j}^{(10/p + \varepsilon)d} \|p_{a_{j},b_{j}}u_{0}\|_{L^{2}}^{2} + \|p_{aj,b_{j}}u_{0}\|_{H^{c/2}}^{2}. \end{align*} $$

Because $[a_{j},b_{j}]$ is localized around $2^{j}$ , we get

$$ \begin{align*} \|S_{n}(t)u_{0}\|_{L^{p}(\mathbb{R},W^{s,q}(\mathbb{R}^{n}))} \lesssim \|u_{0}\|_{H^{c/2, (5/p + \varepsilon/2)d}}. \end{align*} $$

Setting $a = c/2$ and $b = (5/p + \varepsilon /2)d$ , the condition on c and d becomes

$$ \begin{align*} \frac1{2a} + \frac{5/p + \varepsilon/2}{b} \leq 1, \end{align*} $$

which is equivalent to the hypothesis of Proposition 4.6 by discussing the possible values of $\varepsilon $ .

We now extend Lemmas 2.4 and 2.7 to include the angular dependence.

Proof. The fact that the multiplication by $\sigma _{n}$ is an isometry from $L^{2}(\mathbb {R}^{n})$ to $L^{2}(\Sigma )$ is already present in Lemma 2.4.

We have for all $F \in \mathcal C^{\infty } (\Sigma , \mathbb {C}^{M})$ , writing

$$ \begin{align*} F = \begin{pmatrix} f \\ g \end{pmatrix} \end{align*} $$

with f and g in $\mathcal C^{\infty } (\Sigma , \mathbb {C}^{M/2})$

$$ \begin{align*} h^{ij}\langle D_{i} F, D_{j} F \rangle_{\mathbb{C}^{M}} = \langle \partial_{r} F ,\partial_{r} F \rangle_{\mathbb{C}^{M}} + \frac1{\varphi^{2}} \Big( \tilde h^{ij} \langle \mathbb D^{\varphi}_{i} f, \mathbb D_{j}^{\varphi} f \rangle_{\mathbb{C}^{M/2}} + \tilde h^{ij} \langle \mathbb D_{i}^{\varphi} g, \mathbb D_{j}^{\varphi} g \rangle_{\mathbb{C}^{M/2}}\Big), \end{align*} $$

where $\mathbb D^{\varphi }_{j} = \tilde D_{j} + 2i \varphi ^{\prime } \tilde e^{a}_{\; j} \tilde \Sigma _{0,a}$ with $\tilde D$ the covariant derivatives for spinors on the sphere and $\tilde h$ is the metric of the sphere.

The fact that

$$ \begin{align*} \|\langle \partial_{r} (\sigma_{n} F) ,\partial_{r} (\sigma_{n} F) \rangle_{\mathbb{C}^{M}}\|_{L^{1}(\Sigma)} \sim \|\partial_{r} F\|_{L^{2}(\mathbb{R}^{n})}^{2} \end{align*} $$

is due to Lemma 2.4.

We have

$$ \begin{align*} \mathbb D^{\varphi}_{j} = \mathbb D^{r}_{j} + 2i (\varphi^{\prime} -1) \tilde e^{a}_{\; j}\tilde \Sigma_{0,a}. \end{align*} $$

Thanks to the Cauchy-Schwarz inequality applied to the scalar product $x,y\mapsto h^{ij}x_{i}y_{j}$ , we get

(4.1) $$ \begin{align} \sqrt{\frac1{\varphi^{2}}\tilde h^{ij} \langle \mathbb D^{\varphi}_{i} ,\mathbb D^{\varphi}_{j} f \rangle_{\mathbb{C}^{M/2}}} \lesssim \frac{r}{\varphi} \sqrt{\frac1{r^{2}}\tilde h^{ij} \langle \mathbb D^{r}_{i} ,\mathbb D^{r}_{j} f \rangle_{\mathbb{C}^{M/2}}} + \frac{|\varphi^{\prime} -1|}{\varphi} \sqrt{\langle f,f \rangle_{\mathbb{C}^{M/2}}} \end{align} $$

and conversely

$$ \begin{align*} \sqrt{\frac1{r^{2}}\tilde h^{ij} \langle \mathbb D^{r}_{i} ,\mathbb D^{r}_{j} f \rangle_{\mathbb{C}^{M/2}} }\lesssim \frac{\varphi}{r} \sqrt{\frac1{\varphi^{2}}\tilde h^{ij} \langle \mathbb D^{\varphi}_{i} ,\mathbb D^{\varphi}_{j} f \rangle_{\mathbb{C}^{M/2}}} + \frac{|\varphi^{\prime} -1|}{r} \sqrt{\langle f,f \rangle_{\mathbb{C}^{M/2}}}. \end{align*} $$

Because $\mathbb D$ and $\sigma _{n}$ commute, we get

$$ \begin{align*} \mathbb D^{\varphi}_{j} (\sigma_{n} f) = \sigma_{n} \mathbb D^{r}_{j} f + \sigma_{n} 2i (\varphi^{\prime}-1)\tilde e^{a}_{\; j}\tilde \Sigma_{0,a}. \end{align*} $$

To ensure that

$$ \begin{align*} \big\| \frac1{\varphi^{2}}\tilde h^{ij} \langle \mathbb D^{\varphi}_{i} \sigma_{n} f, \mathbb D_{j}^{\varphi} \sigma_{n} f \rangle_{\mathbb{C}^{M/2}}\big\|_{L^{1}(\Sigma)} \end{align*} $$

it is thus sufficient to prove that $ \frac {\varphi }{r}$ , $\frac {r}{\varphi }$ and $\frac {\varphi ^{\prime }-1}{\varphi }$ are bounded. But $\varphi = r(1+\varphi _{1})$ with $\varphi _{1}$ non-negative, bounded, $O(r)$ in $0$ and thus that $\varphi _{1}^{\prime }$ is bounded, hence

$$ \begin{align*} \frac{\varphi}{r} = 1+\varphi_{1},\quad \frac{r}{\varphi} = \frac1{1+\varphi_{1}},\quad \frac{\varphi^{\prime}-1}{\varphi} = \frac{\varphi_{1}}{r(1+\varphi_{1})} + \frac{\varphi_{1}^{\prime}}{1+\varphi_{1}} \end{align*} $$

are bounded.

Proof. As in Lemma 2.7, we reduce our proof to the proof of, for all $q\in (1,\infty )$ ,

1. 1. the multiplication by $\sigma _{n}^{2/q}$ is an isometry from $L^{q}(\mathbb {R}^{n})$ to $L^{q} (\Sigma )$ ,

2. 2. the multiplication by $\sigma _{n}^{2/q}$ is continuous from $W^{1,q}(\mathbb {R}^{n})$ to $W^{1,q}(\Sigma )$ ,

3. 3. the multiplication by $\sigma _{n}^{-2/q}$ is continuous from $W^{1,q}(\Sigma )$ to $W^{1,q}(\mathbb {R}^{n})$ .

(1) The proof of (1) is similar to what we have already done in the proof of Lemma 2.7.

(2) With the same notations as in the proof of Lemma 4.7, and keeping in mind (2) in Lemma 2.7, it remains to prove (with a slight abuse of notation) that for all $f\in W^{1,q}(\mathbb {R}^{n})$ ,

$$ \begin{align*} \big\| \sqrt{\frac1{\varphi^{2}} \tilde h^{ij} \langle \mathbb D_{i}^{\varphi} (\sigma_{n}^{2/q} f), \mathbb D_{j}^{\varphi} (\sigma_{n}^{2/q} f) \rangle_{\mathbb{C}^{M/2}}}\big\|_{L^{q}(\Sigma)} \lesssim \|f\|_{W^{1,q}(\mathbb{R}^{n})}. \end{align*} $$

But because of (1) and the fact that $\sigma _{n}^{2/q}$ and $\mathbb D^{\varphi }$ commute, it sufficient to prove that

$$ \begin{align*} \big\| \sqrt{\frac1{\varphi^{2}} \tilde h^{ij} \langle \mathbb D_{i}^{\varphi} f, \mathbb D_{j}^{\varphi} f \rangle_{\mathbb{C}^{M/2}}}\big\|_{L^{q}(\mathbb{R}^{n})} \lesssim \|f\|_{W^{1,q}(\mathbb{R}^{n})}. \end{align*} $$

We now use the inequality in equation (4.1) and the fact that $\frac {r}{\varphi }$ and $\frac {\varphi ^{\prime } -1}{\varphi }$ are bounded to get the result.

(3) Similar to (2).

Therefore, combining Proposition 4.6 with Lemma 4.8 eventually yields the Proof of Theorem 1.10.