Strichartz Estimates and Fourier Restriction Theorems on the Heisenberg Group

Abstract

This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group ${\mathop {\mathbb H}\nolimits }^d$ for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on ${\mathop {\mathbb H}\nolimits }^d$ is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. Our approach, inspired by the Fourier transform restriction method initiated in Tomas (Bull Am Math Soc 81: 477–478, 1975), is based on Fourier restriction theorems on ${\mathop {\mathbb H}\nolimits }^d$, using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.

Appendices

Appendix A: Proof of a technical result

We have used the following lemma due to Müller (see [38]), whose proof we sketch below.

Lemma 5.4

Defining for f, g in ${\mathcal S}(T^\star {\mathop {\mathbb R}\nolimits }^d)$ and $\lambda$ in ${\mathop {\mathbb R}\nolimits }\setminus \{0\}$ the $\lambda$-twisted convolution

$$\begin{aligned} (f \star _\lambda g )(Y)\buildrel \hbox { def}\over =\int _{T^\star {\mathop {\mathbb R}\nolimits }^d} f(Y-w) g(w) e^{2i\lambda \sigma (Y,w)}dw\, , \end{aligned}$$

(A.1)

with $\sigma$ defined in (1.9), the following estimate holds: there exists a positive constant $C_{d}$ such that

$$\begin{aligned} \Vert f \star _\lambda \widetilde{\mathcal W}(\ell ,\lambda ,\cdot ) \Vert _{L^{p}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\le C_{d} \, |\lambda |^{- \frac{2d}{p'}}\, \ell ^ {(d-1)(1-\frac{2}{p'})}\, \Vert f \Vert _{L^{p}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, , \end{aligned}$$

(A.2)

for all $1\le p \le 2$ and all integers $\ell \ge 1$, where the function $\widetilde{\mathcal W}(\ell ,\lambda ,Y)$ is defined by (3.13).

Proof of Lemma 5.4

Let us start by establishing that for f, g in ${\mathcal S}(T^\star {\mathop {\mathbb R}\nolimits }^d)$ and $1\le p \le 2$, the following estimate holds:

$$\begin{aligned} \Vert f \star _\lambda g \Vert _{L^{p'}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\le C_{p,d} \, |\lambda |^{- \frac{d}{p'}} \, \Vert f \Vert _{L^{p}(T^\star {\mathop {\mathbb R}\nolimits }^d)} \Vert g \Vert ^{\frac{2}{p'}} _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)} \Vert g \Vert ^{1-\frac{2}{p'}} _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, . \end{aligned}$$

By definition,

$$\begin{aligned} (f \star _\lambda g )(Y)= \int _{T^\star {\mathop {\mathbb R}\nolimits }^d} f(Y-w) g(w) e^{2i\lambda \sigma (Y,w)}dw\, , \end{aligned}$$

which easily implies by Young’s inequalities that

$$\begin{aligned} \Vert f \star _\lambda g \Vert _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\le \Vert f \Vert _{L^{1}(T^\star {\mathop {\mathbb R}\nolimits }^d)} \Vert g \Vert _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, . \end{aligned}$$

Therefore invoking the method of real interpolation, we are reduced to showing that

$$\begin{aligned} \Vert f \star _\lambda g \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\le C_{d} \, |\lambda |^{- \frac{d}{2}} \, \Vert f \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)} \Vert g \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, , \end{aligned}$$

and this follows by an easy dilation argument from the well-known fact that $(L^2(T^\star {\mathop {\mathbb R}\nolimits }^d), \star _1)$ is a Hilbert algebra (see for instance [32]).

Let us now focus on Estimate (A.2). We first recall that

$$\begin{aligned} \widetilde{\mathcal W}(\ell ,\lambda ,Y) = \sum _{|n|= \ell } {\mathcal L}_n(|\lambda |^{\frac{1}{2}} \, Y)\, , \end{aligned}$$

where for $Z=(Z_1, \ldots , Z_d)$ in $T^\star {\mathop {\mathbb R}\nolimits }^d$, we denote ${\mathcal L}_n(Z)= e^{-|Z|^2} \Pi ^d_{j=1} L_ {n_j}( 2 |Z_j|)$ with $L_ {n_j}$ the Laguerre polynomial of order $n_j$ and type 0.

Define now for $n \in \mathop {\mathbb N}\nolimits ^d$ the operator $T_n$ on $L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)$ by

$$\begin{aligned} T_n f\buildrel \hbox { def}\over =f \star _\lambda {\mathcal L}_n(|\lambda |^{\frac{1}{2}} \cdot )\, ,\end{aligned}$$

(A.3)

so that

$$\begin{aligned} Tf= f \star _\lambda \widetilde{\mathcal W}(\ell ,\lambda ,\cdot )= \sum _{|n|= \ell } T_n f \, . \end{aligned}$$

(A.4)

Then using the fact that the Laguerre polynomials $(L_k)_{k \in \mathop {\mathbb N}\nolimits }$ are pairwise orthogonal on $[0, \infty [$ with respect to the measure $e^{-x} dx$, we infer that the family of operators $(T_n)_{|n|= \ell }$ is also pairwise orthogonal, and thus denoting by $\Vert T \Vert$ the norm of T defined by (A.4) as an operator on $L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)$, we can conclude that

$$\begin{aligned} \Vert T \Vert _{{\mathcal L}(L^2(T^\star {\mathop {\mathbb R}\nolimits }^d))} = \max _{|n|= \ell }\Vert T_n \Vert _{{\mathcal L}(L^2(T^\star {\mathop {\mathbb R}\nolimits }^d))} \, .\end{aligned}$$

(A.5)

Since $\Vert {\mathcal L}_n(|\lambda |^{\frac{1}{2}} \cdot )\Vert _{L^2(T^\star {\mathop {\mathbb R}\nolimits }^d)} = |\lambda |^{- \frac{d}{2}} \, \Vert {\mathcal L}_n\Vert _{L^2(T^\star {\mathop {\mathbb R}\nolimits }^d)}$, one obtains

$$\begin{aligned} \Vert T_n \Vert _{{\mathcal L}(L^2(T^\star {\mathop {\mathbb R}\nolimits }^d))} \lesssim |\lambda |^{- d} \, . \end{aligned}$$

In view of (A.5), this implies that

$$\begin{aligned} \Vert f \star _\lambda \widetilde{\mathcal W}(\ell ,\lambda ,\cdot ) \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\lesssim |\lambda |^{- d} \, \Vert f \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, . \end{aligned}$$

Finally, by definition

$$\begin{aligned} \Vert \widetilde{\mathcal W}(\ell ,\lambda ,\cdot ) \Vert _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\lesssim \ell ^{d-1} \, , \end{aligned}$$

which ensures that

$$\begin{aligned} \Vert f \star _\lambda \widetilde{\mathcal W}(\ell ,\lambda ,\cdot ) \Vert _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\lesssim \ell ^{d-1} \, \Vert f \Vert _{L^{1}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, , \end{aligned}$$

and completes the proof of Estimate (A.2) by interpolation. $\square$

Appendix B: The inhomogeneous case

Denoting by $(\mathcal U(t))_{t\in {\mathop {\mathbb R}\nolimits }}$ the solution operator of the Schrödinger equation on the Heisenberg group, namely $\mathcal U(t)u_0$ is the solution of $(S_{\mathop {\mathbb H}\nolimits })$ with $f=0$ at time t associated with the data $u_0$, then similarly to the euclidean case $(\mathcal U(t))_{t\in {\mathop {\mathbb R}\nolimits }}$ is a one-parameter group of unitary operators on $L^2({\mathop {\mathbb H}\nolimits }^d)$. Moreover, the solution to the inhomogeneous equation

$$\begin{aligned} \left\{ \begin{array}{c} i\partial _t u -\Delta _{\mathop {\mathbb H}\nolimits }u = f\\ u_{|t=0} = 0\,, \end{array} \right. \end{aligned}$$

writes

$$\begin{aligned} u(t, \cdot )= -i \int ^t_0 \mathcal U(t-t') f(t', \cdot ) dt' ,\end{aligned}$$

(B.1)

Let us check that it satisfies, for all admissible pairs (p, q) in ${\mathcal A}^{\text{ S }}$,

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \lesssim \Vert f \Vert _{L_t^1 H^{\sigma }({\mathop {\mathbb H}\nolimits }^d)} \end{aligned}$$

(B.2)

with

Let us first assume that, for all t, the source term $f(t, \cdot )$ is frequency localized in in the unit ball ${\mathcal B}_1$ in the sense of Definition 3.1, and recall that according to the results of Sect. 5.2, if g is frequency localized in a unit ball, then for all $2\le p \le q \le \infty$

$$\begin{aligned} \Vert \mathcal U(t)g\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \lesssim \Vert g \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \, . \end{aligned}$$

(B.3)

Taking advantage of (B.1), we have for all $s \in {\mathop {\mathbb R}\nolimits }$,

$$\begin{aligned} \Vert u(t, \cdot , s) \Vert _{L^{p}_{Y}} \le \int _{\mathop {\mathbb R}\nolimits }\Vert \mathcal U(t) \mathcal U(-t') f(t', \cdot , s) \Vert _{L^{p}_{Y}} dt' . \end{aligned}$$

Therefore, still for all s,

$$\begin{aligned} \Vert u(\cdot ,\cdot , s) \Vert _{L_t^{q} L^{p}_{Y}} \le \int _{\mathop {\mathbb R}\nolimits }\Vert \mathcal U(\cdot ) \mathcal U(-t') f(t', \cdot , s) \Vert _{L_t^{q} L^{p}_{Y}} dt' . \end{aligned}$$

Invoking (B.3), we deduce that

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le \int _{\mathop {\mathbb R}\nolimits }\Vert \mathcal U(-t') f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' . \end{aligned}$$

Since $\mathcal U(-t')$ is unitary on $L^{2}({\mathop {\mathbb H}\nolimits }^d)$, we readily gather that

$$\begin{aligned} \Vert u \Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le \int _{\mathop {\mathbb R}\nolimits }\Vert f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' .\end{aligned}$$

(B.4)

Now if for all t, $f(t, \cdot )$ is frequency localized in a ball of size $\Lambda$, then setting

$$\begin{aligned} f_{\Lambda }(t, \cdot ) \buildrel \hbox { def}\over =\Lambda ^{-2} f (\Lambda ^{-2} t, \cdot ) \circ \delta _{\Lambda ^{-1}} \end{aligned}$$

we find that on the one hand, $f_{\Lambda }(t, \cdot )$ is frequency localized in a unit ball for all t, and on the other hand that the solution to the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{c} i\partial _t u_{\Lambda } -\Delta _{\mathop {\mathbb H}\nolimits }u_{\Lambda } = f_{\Lambda }\\ u_{|t=0} = 0\,, \end{array} \right. \end{aligned}$$

writes

$$\begin{aligned} u_{\Lambda }(t,w)= u (\Lambda ^{-2} t, \cdot ) \circ \delta _{\Lambda ^{-1}} \,. \end{aligned}$$

Now by scale invariance, we have

$$\begin{aligned} \int _{\mathop {\mathbb R}\nolimits }\Vert f_{\Lambda }(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' = \Lambda ^{\frac{Q}{2}}\int _{\mathop {\mathbb R}\nolimits }\Vert f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' \end{aligned}$$

and

$$\begin{aligned} \Vert u_{\Lambda }\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} =\Lambda ^{\frac{2}{q} + \frac{2d}{p}} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \, . \end{aligned}$$

Consequently, we get

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le C \int _{\mathop {\mathbb R}\nolimits }\Lambda ^{\frac{Q}{2}- \frac{2}{q} - \frac{2d}{p}} \Vert f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' \, . \end{aligned}$$

Since $\displaystyle \frac{Q}{2}- \frac{2}{q} - \frac{2d}{p} \ge 0$, we have

$$\begin{aligned} \Lambda ^{\frac{Q}{2}- \frac{2}{q} - \frac{2d}{p}} \Vert f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} \lesssim \Vert f(t', \cdot ) \Vert _{{H^{\frac{Q}{2}- \frac{2}{q} - \frac{2d}{p}}({\mathop {\mathbb H}\nolimits }^d)}} \,, \end{aligned}$$

which completes the proof of (B.2).

In the case of the wave equation, we have seen in Paragraph 5.3 that when the Cauchy data $u_0$ and $u_1$ are frequency localized in a ring, then u the solution to the Cauchy problem $(W_{\mathop {\mathbb H}\nolimits })$ with $f=0$ reads, with the previous notations,

$$\begin{aligned} u(t, \cdot )= \mathcal U^+ (t) \gamma ^+ + \mathcal U^- (t) \gamma ^- \, . \end{aligned}$$

This allows to investigate the inhomogeneous wave equation by similar arguments than the Schrödinger equation dealt with above. The proof is left to the reader.