Uniqueness in the Calderón problem and bilinear restriction estimates

Abstract

Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition between surfaces have played important roles. Since such advanced machineries usually provide strengthened estimates, it seems natural to attempt to utilize these estimates to improve the known results. In this paper, we make use of the sharp bilinear restriction estimates, which is due to Tao, and relax the regularity assumption on conductivity. We also consider the inverse problem for the Schrödinger operator with potentials contained in the Sobolev spaces of negative orders.

Introduction

For $d\ge 3$, let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded domain with Lipschitz boundary, and let $A(\mathrm{\Omega })$ denote the set of all functions $\gamma \in L^{\infty }(\mathrm{\Omega })$ satisfying $\gamma \ge c$ in Ω for some $c>0$. Throughout the paper, we assume $\gamma \in A(\mathrm{\Omega })$. For $f\in H^{1/2}(\partial \mathrm{\Omega })$ and $\gamma \in A(\mathrm{\Omega })$, we consider the Dirichlet problem:

$$\{\begin{array}{cccc}\hfill \mathrm{div}(\gamma \mathrm{\nabla }u)& =0\hfill & \hfill \text{in}& \mathrm{\Omega },\hfill \\ \hfill u& =f\hfill & \hfill \text{on}& \partial \mathrm{\Omega }.\hfill \end{array}$$

Let $\partial /\partial \nu$ denote the outward normal derivative on the boundary ∂Ω. The Dirichlet-to-Neumann map ${\mathrm{\Lambda }}_{\gamma }$ is formally defined by ${\mathrm{\Lambda }}_{\gamma }(f)=\gamma \frac{\partial u_f}{\partial \nu }{|}_{\partial \mathrm{\Omega }}$. Since the boundary value problem (1.1) has a unique solution $u_f\in H^1(\mathrm{\Omega })$ (for example, see [14, Theorem 2.52]), by the trace theorem and Green's formula, the operator can be formulated in the weak sense. Precisely, for $f\in H^{1/2}(\partial \mathrm{\Omega })$ and $g\in H^{1/2}(\partial \mathrm{\Omega })$,

$$({\mathrm{\Lambda }}_{\gamma }(f),g)=\underset{\mathrm{\Omega }}{\int }\gamma \mathrm{\nabla }{u}_f\cdot \mathrm{\nabla }vdx$$

where $v\in H^1(\mathrm{\Omega })$ and $v{|}_{\partial \mathrm{\Omega }}=g$. It is well known that ${\mathrm{\Lambda }}_{\gamma }$ is well defined and ${\mathrm{\Lambda }}_{\gamma }$ is continuous from $H^{1/2}(\partial \mathrm{\Omega })$ to $H^{-1/2}(\partial \mathrm{\Omega })$.

Calderón's problem  Calderón's inverse conductivity problem concerns whether γ can be uniquely determined from ${\mathrm{\Lambda }}_{\gamma }$, that is to say, whether the map $\gamma \mapsto {\mathrm{\Lambda }}_{\gamma }$ is injective. The problem was introduced by Calderón [9] who showed uniqueness for the linearized problem. Afterwards, numerous works have been devoted to extending the function class $X(\mathrm{\Omega })\subset A(\mathrm{\Omega })$ for which the map $X(\mathrm{\Omega })\ni \gamma \mapsto {\mathrm{\Lambda }}_{\gamma }$ is injective ([47]). Kohn and Vogelius [26] showed that if ∂Ω is smooth and ${\mathrm{\Lambda }}_{\gamma }=0$ then γ vanishes to infinite order at ∂Ω provided that $\gamma \in C^{\infty }(\bar{\mathrm{\Omega }})$ (also, see [40]). Consequently, the mapping $\gamma \mapsto {\mathrm{\Lambda }}_{\gamma }$ is injective if we choose $X(\mathrm{\Omega })$ to be the space of analytic functions on $\bar{\mathrm{\Omega }}$. Sylvester and Uhlmann in their influential work [39] proved that γ is completely determined by ${\mathrm{\Lambda }}_{\gamma }$ if $\gamma \in C^2(\bar{\mathrm{\Omega }})$ for $d\ge 2$. They made use of the complex geometrical optics solutions which become most predominant tool not only in the Calderón problem but also in various related problems. Afterward, it has been shown that regularity on conductivity can be lowered further. The $C^2$ regularity assumption was relaxed to $C^{3/2+ϵ}$ by Brown [5]. Päivärinta, Panchenko, and Uhlmann [37] showed global uniqueness of conductivities in $W^{3/2,\infty }$, and results with conductivities in $W^{3/2,p}$, $p>2d$ were obtained by Brown and Torres [7]. Nguyen and Spirn [36] obtained a result with conductivities in $W^{s,3/s}$ for $3/2<s<2$ when $d=3$. In two dimensions, the problem has different nature and uniqueness of $L^{\infty }$ conductivity was established by Astala and Päivärinta [2]. Their result is an extension of the previous ones in [35], [8]. Recently, Cârstea and Wang [11] obtained uniqueness of unbounded conductivities. (See [1] and references therein for related results.) For $d\ge 3$, the regularity condition was remarkably improved by Haberman and Tataru [21]. By making use of Bourgain's $X^{s,b}$ type spaces, they proved uniqueness when $\gamma \in C^1(\bar{\mathrm{\Omega }})$, or $\gamma \in W^{1,\infty }(\mathrm{\Omega })$ with the assumption that ${\Vert \mathrm{\nabla }\log \gamma \Vert }_{L^{\infty }(\bar{\mathrm{\Omega }})}$ is small. This smallness assumption was later removed by Caro and Rogers [10].

As already mentioned before, for $d\ge 3$, most of the previous results were obtained under the assumption that γ has bounded gradient. Since the equation $\mathrm{div}(\gamma \mathrm{\nabla }u)=0$ can be rewritten as $\mathrm{\Delta }u+W\cdot \mathrm{\nabla }u=0$ with $W=\mathrm{\nabla }\log \gamma$, it naturally relates to the unique continuation problem for u satisfying $|\mathrm{\Delta }u|\le W|\mathrm{\nabla }u|$. Meanwhile, it is known that the unique continuation property holds with $W\in L_{loc}^d$ [48] and generally fails if $W\in L_{loc}^p$ for $p<d$ [25]. In this regards Brown [7] proposed a conjecture that uniqueness should be valid for $\gamma \in W^{1,d}(\mathrm{\Omega })$, but no counterexample which shows the optimality of this conjecture has been known yet. Recently, Brown's conjecture was verified by Haberman [19] for $d=3,4$, and he also showed that uniqueness remains valid even if ∇γ is unbounded when $d=5,6$. More precisely, he showed that $\gamma \mapsto {\mathrm{\Lambda }}_{\gamma }$ is injective if γ belongs to $W^{s,p}(\mathrm{\Omega })$ with $d\le p\le \infty$ and $s=1$ for $d=3,4$, and $d\le p<\infty$ and $s=1+\frac{d-4}{2p}$ for $d=5,6$.

For a given function q, let ${\mathcal{M}}_q$ be the multiplication operator $f\mapsto qf$ and let $O_d$ be the orthonormal group in ${\mathbb{R}}^d$. Most important part of the argument in Haberman [19] ([21]) is to show that there are sequences $\{U_j\}$ in $O_d$ and $\{{\tau }_j\}$ in $(0,\infty )$ such that

$$\underset{j\to \infty }{\lim }{\Vert {\mathcal{M}}_{(\mathrm{\nabla }f)\circ U_j}\Vert }_{X_{\zeta ({\tau }_j)}^{1/2}\to X_{\zeta ({\tau }_j)}^{-1/2}}=0$$

and ${\tau }_j\to \infty$ as $j\to \infty$. We refer the reader forward to Section 2 for the definition of the spaces $X_{\zeta (\tau )}^{1/2}$ and $X_{\zeta (\tau )}^{-1/2}$. If $f\in L^{\frac{d}{2}}$ and has compact support, it is not difficult to show ${\lim }_{\tau \to \infty }{\Vert {\mathcal{M}}_f\Vert }_{X_{\zeta (\tau )}^{1/2}\to X_{\zeta (\tau )}^{-1/2}}=0$, see Remark 5. However, ${\Vert {\mathcal{M}}_{\mathrm{\nabla }f}\Vert }_{X_{\zeta (\tau )}^{1/2}\to X_{\zeta (\tau )}^{-1/2}}$ does not behave as nicely as ${\Vert {\mathcal{M}}_f\Vert }_{X_{\zeta (\tau )}^{1/2}\to X_{\zeta (\tau )}^{-1/2}}$. This is also related to the failure of the Carleman estimate of the form ${\Vert e^{v\cdot x}\mathrm{\nabla }u\Vert }_{L^q}\le C{\Vert e^{v\cdot x}\mathrm{\Delta }u\Vert }_{L^p}$ when $d\ge 3$. See [24], [3], [48], [22]. To get around the difficulty averaged estimates over $O_d$ and τ were considered ([21], [36], [19]). In view of Wolff's work [48] it still seems plausible to expect (1.2) or its variant holds with $f\in L_{loc}^d$.

Restriction estimate  Let $S\subset {\mathbb{R}}^{d-1}$ ¹ be a smooth compact hypersurface with nonvanishing Gaussian curvature and let dμ be the surface measure on S. The estimate ${\Vert \hat{f}{|}_S\Vert }_{L^2(d\mu )}\lesssim {\Vert f\Vert }_{L^r({\mathbb{R}}^{d-1})}$, $r\le 2d/(d+2)$ is known as the Stein-Tomas theorem. The range of r is optimal since the estimate fails if $r>2d/(d+2)$. The restriction estimate can be rewritten in its adjoint form:

$${\Vert \widehat{fd\mu }\Vert }_{L^q({\mathbb{R}}^{d-1})}\le C(d,p,q,S){\Vert f\Vert }_{L^p(S,d\mu )}.$$

The restriction conjecture is to determine $(p,q)$ for which (1.3) holds. Even for most typical surfaces such as the sphere and the paraboloid, the conjecture is left open when $d\ge 4$. We refer the reader to [17], [18] for the most recent progress. There have been bilinear and multilinear generalizations of the linear estimate (1.3) under additional transversality conditions between surfaces ([43], [41], [4]), and these estimates played important roles in development of the restriction problem. To be precise, let $S_1,S_2\subset S$ be hypersurfaces in ${\mathbb{R}}^{d-1}$ and let $d{\mu }_1$, $d{\mu }_2$ be the surface measures on $S_1$, $S_2$, respectively. The following form of estimate is called bilinear (adjoint) restriction estimate:

$${\Vert \widehat{fd{\mu }_1}\widehat{gd{\mu }_2}\Vert }_{L^{q/2}({\mathbb{R}}^{d-1})}\le C{\Vert f\Vert }_{L^2(S_1,d{\mu }_1)}{\Vert g\Vert }_{L^2(S_2,d{\mu }_2)}.$$

Under certain condition between $S_1$ and $S_2$ the estimate (1.4) remains valid for some $q<\frac{2d}{d-2}$ with which (1.3) fails if $p=2$. (See Theorem 2.5 and [42], [29] for detailed discussion.)

By duality, in order to get estimate for ${\Vert {\mathcal{M}}_{\mathrm{\nabla }f}\Vert }_{X_{\zeta (\tau )}^{1/2}\to X_{\zeta (\tau )}^{-1/2}}$, we consider the bilinear operator ${\mathcal{B}}_{\mathrm{\nabla }f}$ which is given by

$$X_{\zeta (\tau )}^{1/2}\times X_{\zeta (\tau )}^{1/2}\ni (u,v)\mapsto {\mathcal{B}}_{\mathrm{\nabla }f}(u,v)=〈\mathrm{\nabla }fu,v〉.$$

Compared with the previous results the main new input in [19] was the $L^2$-Fourier restriction theorem for the sphere which is due to Tomas [45] and Stein [38] (Theorem 2.3). This is natural in that the multiplier which defines $X_{\zeta (\tau )}^{-1/2}$ has mass concentrated near the surface ${\mathrm{\Sigma }}^{\tau }$ given by (2.1) while the restriction estimate provides estimates for functions of which Fourier transform concentrates near hypersurface. The use of the bilinear restriction estimate instead of the linear one has a couple of obvious advantages. The bilinear restriction estimate not only has a wider range of boundedness but also naturally fits with the bilinear operator ${\mathcal{B}}_{\mathrm{\nabla }f}$.

In this paper we aim to improve Haberman's results by making use of the bilinear restriction estimate (1.4) for the elliptic surfaces (see Definition 2.2 and Theorem 2.5). However, the bilinear estimates outside of the range of the $L^2$ restriction estimate are only true under the extra separation condition between the supports of Fourier transforms of the functions (see Corollary 2.6). Such estimates cannot be put in use directly. This leads to considerable technical involvement. The following is our main result.

Theorem 1.1

Let $d=5,6$ and Ω be a bounded domain with Lipschitz boundary. Then the map $\gamma \mapsto {\mathit{\Lambda }}_{\gamma }$ is injective if $\gamma \in W^{s,p}(\mathit{\Omega })\cap A(\mathit{\Omega })$ for $s>s_d(p)$, where

$$s_d(p)=\{\begin{array}{cc}1+\frac{d-5}{2p}\hfill & \mathit{\text{if}}d+1\le p<\infty ,\hfill \\ 1+\frac{d^2-5d+6-p}{2p(d-1)}\hfill & \mathit{\text{if}}d\le p<d+1.\hfill \end{array}$$

Here, $W^{s,p}(\mathit{\Omega })$ is the Sobolev-Slobodeckij space (cf. Fig. 1).

In particular, uniqueness holds for $\gamma \in W^{s,5}(\mathrm{\Omega })\cap A(\mathrm{\Omega })$ if $d=5$ and $s>\frac{41}{40}$, and for $\gamma \in W^{s,6}(\mathrm{\Omega })\cap A(\mathrm{\Omega })$ if $d=6$ and $s>\frac{11}{10}$. Since $W^{s_d(p)+ϵ,p}\hookrightarrow ̸W^{1,\infty }$ if $ϵ>0$ is small enough, this result is not covered by the result in [10].

Even though our estimates are stronger than those in [19], the estimates do not immediately yield improved results in every dimension. As is to be seen later in the paper, our estimates for the low frequency part are especially improved but this is not the case for the high frequency part since we rely on the argument based on the properties of $X_{\zeta }^b$ spaces ([21], [19]).

The argument based on the complex geometrical optics solutions shows that the Fourier transforms of $q_i={\gamma }_i^{-1/2}\mathrm{\Delta }{\gamma }_i^{1/2}$, $i=1,2$, are identical as long as ${\mathrm{\Lambda }}_{{\gamma }_1}={\mathrm{\Lambda }}_{{\gamma }_2}$. As was indicated in [19] this approach has a drawback when we deal with less regular conductivity. In order to use the Fourier transform one has to extend ${\gamma }_1-{\gamma }_2\in W_0^{s,p}(\mathrm{\Omega })$ to the whole space ${\mathbb{R}}^d$ such that ${\gamma }_1-{\gamma }_2=0$ on ${\mathrm{\Omega }}^c$. Such extension is possible by exploiting the trace theorem ([30, Theorem 1]) but only under the condition $s-\frac{1}{p}\le 1$. This additional restriction allows new results only for $d=5,6$ in Theorem 1.1.² The same was also true with the result in [19]. However, as is mentioned in [19], if we additionally impose the condition $\partial {\gamma }_1/\partial \nu =\partial {\gamma }_2/\partial \nu$ on the boundary ∂Ω, then Theorem 1.1 can be extended to higher dimensions $d\ge 7$. In fact, by [30, Theorem 1], the restriction $s-\frac{1}{p}\le 1$ can be relaxed so that $s-\frac{1}{p}\le 2$, which is valid for $s>s_d(p)$ for $d\ge 7$ and $p\ge d$. See Remark 4 for the value of $s_d(p)$. However, the additional condition on the boundary is not known to be true under the assumption ${\mathrm{\Lambda }}_{{\gamma }_1}={\mathrm{\Lambda }}_{{\gamma }_2}$ for ${\gamma }_1,{\gamma }_2$ as in Theorem 1.1. (In [7] Brown and Torres proved that if ${\mathrm{\Lambda }}_{{\gamma }_1}={\mathrm{\Lambda }}_{{\gamma }_2}$, then $\partial {\gamma }_1/\partial \nu =\partial {\gamma }_2/\partial \nu$ on ∂Ω for ${\gamma }_1,{\gamma }_2\in W^{3/2,p}$, $p>2d$, and $d\ge 3$.)

If we had the endpoint bilinear restriction estimate (i.e., the estimate (1.4) with $q=\frac{2(d+1)}{d-1}$, see Remark 1), the argument in this paper would allow us to obtain the uniqueness result with $s=1$ and $p\ge 6$ when $d=5$, and with $s=1+1/p$ and $p\ge 8$ when $d=7$. Unfortunately the endpoint bilinear restriction estimate is still left open.

Inverse problem for the Schrödinger operator  For $d\ge 3$, let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded domain with $C^{\infty }$ boundary. We now consider the Dirichlet problem:

$$\{\begin{array}{cccc}\hfill \mathrm{\Delta }u-qu& =0\hfill & \hfill \text{ in }& \mathrm{\Omega },\hfill \\ \hfill u& =f\hfill & \hfill \text{ on }& \partial \mathrm{\Omega }.\hfill \end{array}$$

Let us set

$$H_c^{s,p}(\mathrm{\Omega })=\{q\in H^{s,p}({\mathbb{R}}^d):\mathrm{supp}q\subset \mathrm{\Omega }\}.$$

Here $H^{s,p}$ is the Bessel potential space, see Notations for its definition. Since $H^{s,p}$ is defined by Fourier multipliers, the space is more convenient for dealing with various operators which are defined by Fourier transform. If we disregard ϵ–loss of the regularity, the spaces $H^{s,p}$ and $W^{s,p}$ are essentially equivalent because $W^{s_1,p}\hookrightarrow H^{s_2,p}$ and $H^{s_1,p}\hookrightarrow W^{s_2,p}$ provided $s_1>s_2$. (See [46, Section 2.3] for more details.) Thus, the statement of Theorem 1.2 does not change if $H^{s,p}$ is replaced by $W^{s,p}$.

Let $q\in H_c^{s,p}(\mathrm{\Omega })$ with $s,p$ satisfying (1.8). We assume that zero is not a Dirichlet eigenvalue of $\mathrm{\Delta }-q$. Then, by the standard argument ([31]) we see that there is a unique solution $u_f\in H^1(\mathrm{\Omega })$ for every $f\in H^{\frac{1}{2}}(\partial \mathrm{\Omega })$. In fact, this can be shown by a slight modification of the argument in [27, Appendix A] (also see the proof of Lemma 6.4 where $\int quvdx$ is controlled while $q\in H_c^{s,p}(\mathrm{\Omega })$ and $u,v\in H^1(\mathrm{\Omega })$).

For $q\in H_c^{s,p}(\mathrm{\Omega })$ let ${\mathcal{L}}_q$ denote the Dirichlet-to-Neumann map given by

$$({\mathcal{L}}_{q}f,g)=\underset{\mathrm{\Omega }}{\int }\mathrm{\nabla }u\cdot \mathrm{\nabla }v+quvdx,$$

where u is the unique solution to (1.6) and $v\in H^1(\mathrm{\Omega })$ with $v{|}_{\partial \mathrm{\Omega }}=g$. As in the Calderón problem, one may ask whether $q\mapsto {\mathcal{L}}_q$ is injective. As is well known the problem is closely related to the Calderón problem. In fact, the Calderón problem can be reduced to the inverse problem for $\mathrm{\Delta }-q$ with $q={\gamma }^{-1/2}\mathrm{\Delta }{\gamma }^{1/2}$ (see [39]). The problem of injectivity of $q\mapsto {\mathcal{L}}_q$ was originally considered with $q\in H_c^{0,p}(\mathrm{\Omega })$, but it is not difficult to see that we may consider $q_1,q_2\in H_c^{s,p}(\mathrm{\Omega })$ with $s<0$. (See, for example, Brown-Torres [7].) Since u, $v\in H^1(\mathrm{\Omega })$, it is natural to impose $s\ge -1$. In fact, ${\mathcal{L}}_{q}f$ is well defined provided that $q\in H_c^{s,p}(\mathrm{\Omega })$ with

$$\max \{-2+\frac{d}{p},-1\}\le s.$$

The standard argument shows that ${\mathcal{L}}_q:H^{\frac{1}{2}}(\partial \mathrm{\Omega })\to H^{-\frac{1}{2}}(\partial \mathrm{\Omega })$ is continuous.

The injectivity of the mapping

$$H_c^{s,p}(\mathrm{\Omega })\ni q\mapsto {\mathcal{L}}_q$$

was shown with $s=0$, $p=\infty$ by Sylvester and Uhlmann [39]. The result was extended to include unbounded potential $q\in L^{\frac{d}{2}+ϵ}$ by Jerison and Kenig (see Chanillo [12]). The injectivity for q contained in the Fefferman-Phong class with small norm was shown by Chanillo [12] and the result for $q\in L^{\frac{d}{2}}$ was announced by Lavine and Nachman in [34].³ Their result was recently extended to compact Riemannian manifolds by Dos Santos Ferreira, Kenig, and Salo [13]. Also see [27] for extensions to the polyharmonic operators.

The regularity requirement for q can be relaxed. Results in this direction were obtained by Brown [5], Päivärinta, Panchenko, and Uhlmann [37], Brown and Torres [7] in connection with the Calderón problem. Those results can be improved to less regular q. In fact, Haberman's result implies that the injectivity holds with $q\in H^{-1,d}$ when $d=3,4$ (see [20]). It seems natural to conjecture that the same is true in any higher dimensions. Interpolating this conjecture with the result due to Lavine and Nachman [34] ($q\in L^{\frac{d}{2}}$) leads to the following:

Conjecture 1

Let $d/2\le p<\infty$, and Ω be a bounded domain with Lipschitz boundary. Suppose $q_1,q_2\in H_c^{s,p}(\mathit{\Omega })$ and ${\mathcal{L}}_{q_1}={\mathcal{L}}_{q_2}$. If $s\ge s_d^{⁎}(p):=\max \{-1,-2+\frac{d}{p}\}$, then $q_1=q_2$.

We define $r_d:[\frac{d}{2},\infty )\to \mathbb{R}$. For $3\le d\le 6$, set

$$r_d(p)=\{\begin{array}{cc}-1+\frac{d-5}{2p}\hfill & \text{if }p\ge d+1,\hfill \\ -\frac{3}{2}+\frac{d-2}{p}\hfill & \text{if }d+1>p\ge 4,\hfill \\ -2+\frac{d}{p}\hfill & \text{if }4>p\ge \frac{d}{2},\hfill \end{array}$$

and, for $d\ge 7$, set

$$r_d(p)=\{\begin{array}{cc}-1+\frac{d-5}{2p}\hfill & \text{if }p\ge \frac{d+9}{2},\hfill \\ -\frac{3}{2}+\frac{3d-1}{4p}\hfill & \text{if }\frac{d+9}{2}>p\ge \frac{3d+7}{6},\hfill \\ -\frac{12}{7}+\frac{6d}{7p}\hfill & \text{if }\frac{3d+7}{6}>p\ge \frac{d}{2}.\hfill \end{array}$$

The following is a partial result concerning Conjecture 1 when $d\ge 5$.

Theorem 1.2

Let $d\ge 3$ and $d/2\le p<\infty$. The mapping (1.9) is injective if $s>\max \{-1,r_d(p)\}$.

When $d=5$, Theorem 1.2 shows that $s>s_d^{⁎}(p)$ for $\frac{d}{2}\le p<4$ or $p\ge d+1$, where Conjecture 1 is verified except for the critical case $s=s_d^{⁎}(p)$. Similarly, when $d=6$ injectivity of (1.9) holds if $s>s_d^{⁎}(p)$ and $\frac{d}{2}\le p<4$. We illustrate our result in Fig. 2.

Organization of the paper  In Section 2, we recall basic properties of the spaces $X_{\zeta }^b$, and obtain estimates which rely on $L^2$ linear and $L^2$ bilinear restriction estimates. In Section 3 which is the most technical part of the paper, we make use of the bilinear restriction estimates and a Whitney type decomposition to get the crucial estimates while carefully considering orthogonality among the decomposed pieces. In Section 4 we take average of the estimates from Section 3 over dilation and rotation, which allows us to exploit extra cancellation due to frequency localization. Combining the previous estimates together we prove key estimates in Section 5. We prove Theorem 1.1 and Theorem 1.2 in Section 6.

Notations  We list notations which are used throughout the paper.

• For $A,B>0$ we write $A\lesssim B$ if $A\le CB$ with some constant $C>0$. We also use the notation $A\simeq B$ if $A\lesssim B$ and $B\lesssim A$.

• The orthogonal group in ${\mathbb{R}}^d$ is denoted by $O_d$.

• Let $\tau >0$. For a function $f:{\mathbb{R}}^d\to \mathbb{C}$ and a matrix $U\in O_d$ we define $f_{\tau U}(x):={\tau }^{-d}f({\tau }^{-1}Ux)$. If U is the identity matrix, we denote $f_{\tau U}$ by $f_{\tau }$.

• The Fourier and inverse Fourier transforms: For an integrable function $u:{\mathbb{R}}^k\to \mathbb{C}$, we write $\mathcal{F}u(\xi )=\hat{u}(\xi )={\int }_{{\mathbb{R}}^k}{e}^{-ix\cdot \xi }u(x)dx$ and ${\mathcal{F}}^{-1}u(\xi )={(2\pi )}^{-k}\mathcal{F}u(-\xi )$.

• For a measurable function a with polynomial growth, let $a(D)f={\mathcal{F}}^{-1}(a\mathcal{F}f)$.

• For $E\subset {\mathbb{R}}^k$ and $x\in {\mathbb{R}}^k$, we write $\mathrm{dist}(x,E)=\inf \{|x-y|:y\in E\}$.

• For $E\subset {\mathbb{R}}^k$ and $\delta >0$, we denote by $E+O(\delta )$ the δ-neighborhood of E in ${\mathbb{R}}^k$, i.e., $E+O(\delta )=\{x\in {\mathbb{R}}^k:\mathrm{dist}(x,E)<\delta \}$.

• We set ${\mathbb{S}}^{k-1}=\{x\in {\mathbb{R}}^k:|x|=1\}$. Also, for $a\in {\mathbb{R}}^k$ and $r>0$, $B_k(a,r)=\{x\in {\mathbb{R}}^k:|x-a|<r\}$.

• If $e_1$ and $e_2$ are a pair of orthonormal vectors in ${\mathbb{R}}^d$, we write ${\xi }_1=\xi \cdot e_1$, ${\xi }_2=\xi \cdot e_2$.

• For $\xi \in {\mathbb{R}}^d$ we sometimes write $\xi =({\xi }_1,\tilde{\xi })\in \mathbb{R}\times {\mathbb{R}}^{d-1}$, $\xi =({\xi }_1,{\xi }_2,\overline{\xi })\in \mathbb{R}\times \mathbb{R}\times {\mathbb{R}}^{d-2}$.

• For $s\ge 0$ and $p\in [1,\infty ]$, we denote by $H^{s,p}$ the Bessel potential space $\{\phi \in {\mathcal{S}}^{\prime }:{(1+|D{|}^2)}^{\frac{s}{2}}\phi \in L^p\}$ which is endowed with the norm ${\Vert \phi \Vert }_{H^{s,p}}={\Vert {(1+|D{|}^2)}^{\frac{s}{2}}\phi \Vert }_{L^p}$.

• For $s<0$ and $p\in (1,\infty )$, we denote by $H^{s,p^{\prime }}(\mathrm{\Omega })$, $p^{\prime }=\frac{p}{p-1}$, the dual space of $H_0^{-s,p}(\mathrm{\Omega })$.

• We use $〈\cdot ,\cdot 〉$ and $(\cdot ,\cdot )$ to denote the inner product and the bilinear pairing between distribution and function, respectively.