On localization of Schr\"odinger means

TL;DR

This paper investigates the localization properties of Schr"odinger means in higher dimensions, providing insights into their behavior beyond the one-dimensional case.

Contribution

It extends the analysis of Schr"odinger means localization to higher dimensions, offering new theoretical understanding.

Findings

Localization properties characterized in higher dimensions

Extension of previous one-dimensional results

Theoretical framework for Schr"odinger means in multiple dimensions

Abstract

Localization properties for Schr\"odinger means are studied in dimension higher than one.

Equations

$$\hat{f}(\xi )=\underset{{\mathbb{R}}^n}{\int }{e}^{-i\xi \cdot x}f(x)dx,\xi \in {\mathbb{R}}^n.$$

$$S_{t}f(x)=\underset{{\mathbb{R}}^n}{\int }{e}^{i\xi \cdot x}{e}^{it|\xi {|}^2}\hat{f}(\xi )d\xi ,x\in {\mathbb{R}}^n,t\in \mathbb{R}.$$

$$K_t(x)=\frac{1}{t^{n/2}}K\left(\frac{x}{t^{1/2}}\right),x\in {\mathbb{R}}^n,t>0.$$

$$S_{t}f(x)=K_t\star f(x),t>0,$$

$$H_s=\{f\in {\mathcal{S}}^{\prime }:||f|{|}_{H_s}<\infty \},s\in \mathbb{R},$$

$$||f|{|}_{H_s}={\left({\int }_{{\mathbb{R}}^n}(1+|\xi {|}^2{)}^s|\hat{f}(\xi ){|}^{2}d\xi \right)}^{1/2}$$

$$\underset{t\to 0}{\lim }\frac{1}{2\pi }{S}_{t}f(x)=f(x)$$

$$\underset{t\to 0}{\lim }\frac{1}{(2\pi {)}^n}{S}_{t}f(x)=f(x)$$

$$\underset{t\to 0}{\lim }{S}_{t}f(x)=0$$

$$\underset{t\to 0}{\lim }{S}_{t}f(x)=0.$$

$$\check{f}(x)=(2\pi {)}^{-n}\underset{{\mathbb{R}}^n}{\int }{e}^{i\xi \cdot x}f(\xi )d\xi ,x\in {\mathbb{R}}^n,$$

$$G_v(x^{\prime })=R^{-(n-1)/4}\Phi (x^{\prime })\prod _{j=2}^n\left(\sum _{R/2D<l_j<R/D}{e}^{iD{l}_j{x}_j}\right),0<v<1,$$

$$h_v(x)=h_v(x_1,x^{\prime })=f_v(x_1)G_v(x^{\prime }),0<v<1.$$

$$\mathrm{supp}{f}_v\subset (-v,v).$$

$$S_t{h}_v(x_1,x^{\prime })=S_t{f}_v(x_1)S_t{G}_v(x^{\prime }).$$

$$t=\frac{x_1}{2R}+\tau$$

$$|S_t{f}_v(x_1)|\gtrsim |\check{g}(R^{1/2}{x}_1-2t{R}^{3/2})|=|\check{g}(2\tau R^{3/2})|\ge c_0.$$

$$|S_{t_k(x)}{G}_{v_k}(x^{\prime })|\gtrsim R^{-(n-1)/4}\prod _{j=2}^n\left|\sum _{l_j}{e}^{iD{l}_j{x}_j}{e}^{i{D}^2{l}_j^{2}t}\right|\ge c_0.$$

$$|t_k(x)|\sim \frac{1}{R_k}=v_k^2$$

$$h(x)=\sum _{k=K}^{\infty }{h}_{v_k}(x)=\sum _{k=K}^{\infty }{f}_{v_k}(x_1)G_{v_k}(x^{\prime }).$$

$$|S_{t_{k_j}(x)}{G}_{v_{k_j}}(x^{\prime })|\ge c_0\text{ and }|S_{t_{k_j}(x)}{f}_{v_{k_j}}(x_1)|\ge c_0.$$

$$|S_{t_{k_j}(x)}h(x)|\ge c_0$$

$${\hat{f}}_v({\xi }_1)=vg(v{\xi }_1+1/v)=R^{-1/2}g(R^{-1/2}{\xi }_1+R^{1/2})$$

$$R^{-1/2}|{\xi }_1|\ge R^{1/2}A,$$

$$|R^{-1/2}{\xi }_1+R^{1/2}|\ge B{R}^{-1/2}|{\xi }_1|$$

$$|{\xi }_1|\ge |{\xi }_1{|}^{1/2}{A}_1{R}^{1/2}.$$

$$|R^{-1/2}{\xi }_1+R^{1/2}|\ge B_1|{\xi }_1{|}^{1/2}$$

$$|{\hat{f}}_v({\xi }_1)|\le C|\xi {|}^{-N}\text{ for }|{\xi }_1|\ge AR,$$

$$\underset{|{\xi }_1|\ge AR}{\int }|{\hat{f}}_v({\xi }_1){|}^2(1+{\xi }_1^2{)}^{s}d{\xi }_1\le C{R}^{-N}$$

Full text

1. Introduction

Let $f$ belong to the Schwartz class $\mathcal{S}({\mathbb{R}}^{n})$ where $n\geq 1$. We define the Fourier transform $\widehat{f}$ by setting

[TABLE]

For $f\in\mathcal{S}({\mathbb{R}}^{n})$ we also set

[TABLE]

If we set $u(x,t)=S_{t}f(x)/(2\pi)^{n}$, then $u(x,0)=f(x)$ and $u$ satisfies the Schrödinger equation $i\partial u/\partial t=\Delta u$.

It is well-known that $e^{i|\xi|^{2}}$ has the Fourier transform $K(x)=ce^{-i|x|^{2}/4}$, where $c$ denotes a constant, and $e^{it|\xi|^{2}}$ has the Fourier transform

[TABLE]

One has $S_{t}f(x)=K_{t}\star f(x)$ for $f\in\mathcal{S}({\mathbb{R}}^{n})$ and $t>0$, and we set

[TABLE]

for $f\in L^{1}({\mathbb{R}}^{n})$. For $f\in L^{2}({\mathbb{R}}^{n})$ we define $S_{t}f$ by formula (1.1).

We introduce Sobolev spaces $H_{s}=H_{s}({\mathbb{R}}^{n})$ by setting

[TABLE]

where

[TABLE]

In the case $n=1$ it is well-known (see Carleson [3] and Dahlberg and Kenig [4]) that

[TABLE]

almost everywhere if $f\in H_{1/4}$. Also it is known that $H_{1/4}$ cannot be replaced by $H_{s}$ if $s<1/4$. In the case $n\geq 2$ Sjölin [6] and Vega [9] proved independently that

[TABLE]

almost everywhere if $f\in H_{s}$ and $s>1/2$. This result was improved by Bourgain [1], who proved that $f\in H_{s}({\mathbb{R}}^{n})$, $s>1/2-1/4n$, is sufficient for convergence almost everywhere.

In the case $n=2$, Du, Guth, and Li [5] have recently proved that the condition $s>1/3$ is sufficient. On the other hand Bourgain [2] has proved that $s\geq n/2(n+1)$ is necessary for convergence for all $n\geq 2$.

We shall here study localization of Schrödinger means and shall first state a result on localization everywhere (see Sjölin [7]).

Theorem A. Assume $n\geq 1$. If $f\in H_{n/2}({\mathbb{R}}^{n})$ and $f$ has compact support then

[TABLE]

for every $x\in{\mathbb{R}}^{n}\setminus(\mathrm{supp}f)$.

It is also proved in [7] that this result is sharp in the sense that $H_{n/2}$ cannot be replaced by $H_{s}$ with $s<n/2$.

We say that one has localization almost everywhere for functions in $H_{s}$ if for every $f\in H_{s}$ one has

[TABLE]

almost everywhere in ${\mathbb{R}}^{n}\setminus(\mathrm{supp}f)$.

In the case $n=1$ Sjölin and Soria proved that there is no localization almost everywhere for functions in $H_{s}$ if $s<1/4$ (see Sjölin [8]). In fact they proved that there exist two disjoint compact intervals $I$ and $J$ in ${\mathbb{R}}$ and a function $f$ which belongs to $H_{s}$ for every $s<1/4$, with the properties that $\mathrm{supp}f\subset I$ and for every $x\in J$ one does not have

[TABLE]

In the case $n\geq 2$ Sjölin and Soria also proved that one does not have localization almost everywhere for functions in $H_{s}({\mathbb{R}}^{n})$ if $s<1/4$.

We shall here improve this result and prove that there is no localization almost everywhere for functions in $H_{s}({\mathbb{R}}^{n})$ if $n\geq 2$ and $s<n/2(n+1)$. In fact we shall prove the following theorem.

Theorem 1.1**.**

If $n\geq 2$ and $s<n/2(n+1)$ there exist a function $f$ in $H_{s}({\mathbb{R}}^{n})\cap L^{1}({\mathbb{R}}^{n})$ and a set $F$ with positive Lebesgue measure such that $F\subset{\mathbb{R}}^{n}\setminus(\mathrm{supp}f)$ and for every $x\in F$ one does not have $\lim\limits_{t\to 0}S_{t}f(x)=0$.

To prove this result we shall combine the method in [8] with an estimate of Bourgain [2].

If $A$ and $B$ are numbers we write $A\lesssim B$ if there is a positive constant $C$ such that $A\leq CB$. If $A\lesssim B$ and $B\lesssim A$ we write $A\sim B$.

We introduce the inverse Fourier transform by setting

[TABLE]

for $f\in L^{1}({\mathbb{R}}^{n})$.

Also $B(x;r)$ denotes a ball with center $x$ and radius $r$.

2. Proof of the theorem

We start by taking $v_{1}$ such that $0<v_{1}<1$ and set $v_{k}=\varepsilon_{k}v_{k-1}^{\mu}$ for $k=2,3,4,...,$ where $\varepsilon_{k}=2^{-k}$ and $\mu=\max(n,2+n/4)$. Then one has $v_{k}<2^{-k}$ for $k\geq 2$ and $(v_{k})_{1}^{\infty}$ is a decreasing sequence tending to zero.

We then choose $g\in\mathcal{S}({\mathbb{R}})$ such that $\operatorname{supp}\widecheck{g}\subset(-1,1)$, $\widecheck{g}(x_{1})=1$ for $|x_{1}|<1/2$, and set $f_{v}(x_{1})=e^{-ix_{1}/v^{2}}\widecheck{g}(x_{1}/v)$, $0<v<1$, $x_{1}\in{\mathbb{R}}$.

The functions $f_{v}$ were used in Sjölin [8] to study the localization problem in the case $n=1$. Also let $\Phi\in\mathcal{S}({\mathbb{R}}^{n-1})$ have $\operatorname{supp}\widehat{\Phi}\subset B(0;1)$ and $\Phi(0)=1$. We then take $R=1/v^{2}$ and set

[TABLE]

where $x^{\prime}=(x_{2},...,x_{n})\in{\mathbb{R}}^{n-1}$, $l=(l_{2},...,l_{n})\in{\mathbb{Z}}^{n-1}$, and $D=R^{(n+2)/2(n+1)}$. We may also assume that $\Phi(x^{\prime})=\psi(x_{2})...\psi(x_{n})$ for some $\psi\in\mathcal{S}({\mathbb{R}})$.

We then set

[TABLE]

In [2] Bourgain studies functions similar to $h_{v}$. However, in [2] our function $\widecheck{g}$ is replaced by a function $\varphi$ with the property that $\widehat{\varphi}$ has compact support. In our argument it will be important that $\widecheck{g}$ has compact support so that

[TABLE]

We then observe that

[TABLE]

It is proved in Bourgain [2], p.394, that if one assumes $|x|<c$ and $|t|<c/R$ and sets

[TABLE]

with $|\tau|<R^{-3/2}/10$, then

[TABLE]

We then take $v=v_{k}$ for $k=1,2,3,...$, and apply an estimate in [2], p.395, namely that there exists a set $E_{k}\subset B(0;1)$ such that for every $x\in E_{k}$ there exists $t=t_{k}(x)$ such that

[TABLE]

Also one has $mE_{k}\geq c_{1}>0$, where $m$ denotes Lebesgue measure.

We then choose $\delta>0$ so small that if $F_{k}=E_{k}\cap\{x;\ |x_{1}|>\delta/2\}$ then one has $mF_{k}\geq c_{1}/2=c_{2}$ for $k=1,2,3,...$ . We may assume that $\delta<1$.

We then set $F=\bigcap\limits_{k=1}^{\infty}\left(\bigcup\limits_{j=k}^{\infty}F_{j}\right)$ so that $F$ is the set of all $x$ which belong to infinitely many $F_{k}$. Since the sets $\bigcup\limits_{j=k}^{\infty}F_{j}$ decrease as $k$ increases, one obtains $mF\geq c_{0}$.

It also follows from (2.2) that $|S_{t_{k}(x)}f_{v_{k}}(x)|\geq c_{0}$ for $x\in F_{k}$. From (2.1) one also concludes that

[TABLE]

where $R_{k}=1/v_{k}^{2}$.

We now choose $K$ so large that $v_{K}<\delta/4$ and set

[TABLE]

One has $F\subset{\mathbb{R}}^{n}\setminus(\operatorname{supp}h)$ since $\operatorname{supp}h\subset\{x;\ |x_{1}|<\delta/4\}$.

If $x=(x_{1},x^{\prime})\in F$ there exists a sequence $(k_{j})_{j=1}^{\infty}$ such that $x\in F_{k_{j}}$ and

[TABLE]

We shall prove that

[TABLE]

for $j$ large, and also that $h\in H_{s}\cap L^{1}$ for $s<n/2(n+1)$. It follows that one does not have localization almost everywhere in $H_{s}$ if $s<n/2(n+1)$.

We shall first estimate $||h_{v}||_{H_{s}}$. We begin by studying $f_{v}$ and $G_{v}$. According to Sjölin [8], p. 143, one has

[TABLE]

and for $|\xi_{1}|\geq AR$ we have

[TABLE]

where $A$ is a large constant. It follows that $|\xi_{1}|\geq AR$ implies

[TABLE]

and

[TABLE]

Hence

[TABLE]

and

[TABLE]

where $N$ is large. It follows that

[TABLE]

and it is also easy to see that

[TABLE]

We have

[TABLE]

and it follows that $\operatorname{supp}\widehat{G}_{v}\subset\{\xi^{\prime};\ |\xi^{\prime}|\sim R\}$ and

[TABLE]

Integrating we obtain

[TABLE]

We have $D=R^{(n+2)/2(n+1)}$ and $R/D=R^{n/2(n+1)}$ and hence

[TABLE]

and

[TABLE]

For $s>0$ one obtains

[TABLE]

It follows that

[TABLE]

From (2.4) and (2.6) one also obtains

[TABLE]

and hence

[TABLE]

It follows that

[TABLE]

Since $v_{k}<\varepsilon_{k}$ and

[TABLE]

it follows that $h\in H_{s}$ if $s<n/2(n+1)$.

We also need some estimates for $S_{t}f_{v}$ and $S_{t}G_{v}$. In Sjölin [8] (see Lemmas 3 and 4) it is proved that

[TABLE]

and

[TABLE]

for $0<v<\delta/4$, $|x_{1}|\geq\delta/2$, and $0<|t|<1$. Actually it is assumed in [8] that $t>0$ but the same proofs work for $t<0$.

We also have the following estimates for $S_{t}G_{v}$.

Lemma 2.1**.**

For $0<v<\delta/4$, $0<|t|<1$, and $x^{\prime}\in{\mathbb{R}}^{n-1}$ one has

[TABLE]

and

[TABLE]

**Proof. **Choose the integer $p$ so that $|4p-R/D|\leq 4$. Then one has

[TABLE]

and

[TABLE]

where $D_{p}$ denotes the Dirichlet kernel. Setting $y=Dx_{j}$ one obtains

[TABLE]

for every $a\in{\mathbb{R}}$. It follows that

[TABLE]

for every $a\in{\mathbb{R}}$.

Letting $Q$ denote the cube $[0,1]^{n-1}$ we obtain

[TABLE]

and hence

[TABLE]

We have

[TABLE]

where

[TABLE]

and it follows that

[TABLE]

and hence we obtain (2.9).

We also have

[TABLE]

and we get

[TABLE]

and the proof of Lemma 2.1 is complete. ∎

Multiplying (2.7) and (2.9) one obtains

[TABLE]

and combining (2.8) and (2.10) one gets

[TABLE]

where $\gamma=n/2$, $\beta=4+n/2$, $0<v<\delta/4$, $0<|t|<1$, and $|x_{1}|>\delta/2$.

We remark that it also follows from the above estimates that $h\in L^{1}({\mathbb{R}}^{n})$. In fact it is easy to see that $||f_{v}||_{1}\sim v$ and we have proved that

[TABLE]

and hence $||h_{v}||_{1}\lesssim v^{5/4}$. It follows that

[TABLE]

We shall now finish the proof of the following result.

Theorem 2.2**.**

For $x\in F$ one has

[TABLE]

for $j$ large. It follows that there is no localization almost everywhere of Schrödinger means for functions in $H_{s}({\mathbb{R}}^{n})$ if $s<n/2(n+1)$.

**Proof. **We have

[TABLE]

The first term on the right hand side is larger than a positive number $c_{0}$ and it suffices to prove that the second term is small. For simplicity we write $k$ instead of $k_{j}$ in the following formulas.

We have $0<v_{i}\leq v_{i-1}/2$ and $0<v_{i}\leq 2^{-i}$ and it follows that

[TABLE]

and one also has

[TABLE]

since $1/v_{i-1}\leq 1/2v_{i}$ implies $1/v_{i-1}^{\beta}\leq\frac{1}{2^{\beta}}\frac{1}{v_{i}^{\beta}}$.

For $i\geq k+1$ the inequality (2.11) and the formula (2.3) give

[TABLE]

and invoking (2.13) we obtain

[TABLE]

since $\mu\geq 2\gamma$ and $v_{k+1}=\varepsilon_{k+1}v_{k}^{\mu}\leq\varepsilon_{k+1}v_{k}^{2\gamma}$. For $K\leq i\leq k-1$ the inequality (2.12) gives

[TABLE]

and invoking (2.14) one obtains

[TABLE]

Since $\mu\geq\beta/2$ we get $v_{k}\leq\varepsilon_{k}v_{k-1}^{\beta/2}$ and $v_{k}^{2}\leq\varepsilon_{k}^{2}v_{k-1}^{\beta}$. Hence the sum in (2.15) is majorized by $C\varepsilon_{k}^{2}$. Since $\varepsilon_{k}\to 0$ as $k\to\infty$ the proof of the theorem is complete.

∎