Abstract
In this short note, we give an easy proof of the following result: for \( n\ge 2, \) \(\underset{t\rightarrow 0}{\lim }\ \,e^{it\Delta }f\left( x+\gamma (t)\right) = f(x) \) almost everywhere whenever \( \gamma \) is an \( \alpha \)-Hölder curve with \( \frac{1}{2}\le \alpha \le 1 \) and \( f\in H^s({\mathbb {R}}^n) \), with \( s > \frac{n}{2(n+1)} \). This is the optimal range of regularity up to the endpoint.
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1 Introduction
Consider the linear Schrödinger equation on \( {\mathbb {R}}^n\times {\mathbb {R}}\), \( n\ge 1, \) given by
Its solution can be formally expressed as
It was first proposed by Carleson in 1980 [3] to find the values of \( s>0 \) for which
holds true for all functions \( f\in H^s({\mathbb {R}}^n) \). Carleson [3] proved this convergence when \( n=1 \) and \( s\ge \frac{1}{4}\). Later, in 2006, Dahlberg and Kenig [6] showed that (3) was false whenever \( s<\frac{1}{4}. \)
Many researchers have worked in this problem throughout the years. Authors such as Carbery, Cowling, Vega, Sjölin, Moyua, Vargas, Tao, Lee and Bourgain to name a few. More recently, the problem has been solved in higher dimensions, except for the endpoint. In 2016, Bourgain [1] proved the necessity of \( s\ge \frac{n}{2(n+1)} \) in order to have (3). In 2017, Du et al. [7] proved the sufficiency of the condition \( s>\frac{1}{3} \) when \( n=2. \) Later, in 2019, Du and Zhang [8] proved the sufficiency of \( s>\frac{n}{2(n+1)} \) for general \( n\ge 3. \) A more detailed history of the problem can be found in [8] and the references therein.
Take a solution of (1). Consider a set of curves \( \rho (x,t) = x + \gamma (t) \) that are bi-Lipschitz in \( x\in {\mathbb {R}}^n \) and \( \alpha \)-Hölder in \( t\in {\mathbb {R}}\). Cho et al. [4] proved in 2012 that \( u\left( \rho (x,t),t\right) \) converges to f(x) almost everywhere as \( t\rightarrow 0 \) in \( n=1 \) when \( s>\max \left\{ \frac{1}{2}-\alpha , \frac{1}{4} \right\} \). They also found this to be sharp up to the endpoint. Later, in 2021, Li and Wang [11] proved that convergence in dimension \( n = 2 \), for index \( \frac{1}{2}\le \alpha \le 1 \) and the range \( s > \frac{3}{8} \). In 2023, Cao and Miao [2] gave a proof for general dimension n, index \( \frac{1}{2}\le \alpha \le 1 \), and \( s>\frac{n}{2(n+1)} \). Their proof followed the argument presented in [8] and relied on techniques such as dyadic pigeonholing, broad-narrow analysis and induction on scales.
Our objective is to give an easy proof of the result in [2] without using the aforementioned techniques.
Fix \(0<\alpha \le 1\) and \(\tau \ge 1.\) We consider the family of curves,
The convergence result follows from the maximal bound below. Let \( B_r^n(x_0) \) denote the ball of radius \( r>0 \) centered at \( x_0\in {\mathbb {R}}^n \).
Theorem 1.1
Let \(n \ge 1\). Fix \( \frac{1}{2}\le \alpha \le 1 \) and \(\tau \ge 1 \). For any \(\varepsilon >0\), there exists a positive constant \(C_{\varepsilon ,\tau }\) such that, for every \( \gamma \in \Gamma _{\tau }^\alpha \),
holds for all f \( \in H^{\frac{n}{2(n+1)}+\epsilon }({\mathbb {R}}^n). \)
Remark 1.2
A change of variables shows that it is enough to consider the case \( \tau = 1 \). From now on we assume \( \tau = 1.\)
Then, we can reduce Theorem 1.1 as in [8]. We begin with a definition.
Definition 1.3
Fixed \(0<\alpha \le 1\) and \( R>1 \), we define
By Littlewood–Paley decomposition, the time localization lemma (e.g. Lemma 3.1 in Lee [9]) and parabolic rescaling, Theorem 1.1 can be reduced to the following Theorem 1.4.
Theorem 1.4
Let \(n \ge 1\) and \( \frac{1}{2}\le \alpha <1 \). For any \(\varepsilon >0\), there exists a constant \(C_{\varepsilon }\) such that, for all \( \gamma \in \Gamma ^{\alpha }\left( R^{-1}\right) \),
holds for all \(R \ge 1\) and all f with \({\text {supp}} \widehat{f} \subset A(1)=\left\{ \xi \in \mathbb {R}^n:|\xi | \sim 1\right\} \).
2 Intermediate Results
We consider the following result from [8].
Theorem 2.1
(Corollary 1.7 in [8]) Let \(n \ge 1\). For any \(\varepsilon >0\), there exists a constant \(C_{\varepsilon }\) such that the following holds for all \(R \ge 1\) and all f with \({\text {supp}} \widehat{f} \subset B_1^n(0)\). Suppose that \(X=\cup _k B_k\) is a union of lattice unit cubes in \(B_R^{n+1}(0)\). Let \(1 \le \beta \le n+1\) and
Then
We generalize the above result to include \(\alpha \)-Hölder curves.
Theorem 2.2
Let \(n \ge 1\) and \( \frac{1}{2}\le \alpha \le 1 \).
For any \(\varepsilon >0\), there exists a constant \(C_{\varepsilon }\) such that the following holds for any \(R \ge 1\), every \( \gamma \in \Gamma ^{\alpha }\left( R^{-1}\right) \) and all f with \({\text {supp}} \widehat{f} \subset B_1^n(0)\). Suppose that \(X=\cup _k B_k\) is a union of lattice unit cubes in \(B_R^{n+1}(0)\). Let \(1 \le \beta \le n+1\) and \(\phi \) be given by (8). Then
Proof of Theorem 2.2
Denote
We begin with
Denote \( (x_k,t_k)\) to be the center of \( B_k. \) Then,
Recall that \( \gamma \in \Gamma ^{\alpha }\left( R^{-1}\right) \), and \( \alpha \ge \frac{1}{2} \). Thus, if \( t\in (t_k-1,t_k+1), \) then \( \left| \theta (t)-\theta (t_k)\right| \le 1 \). Hence,
for some \( C>0. \) Note that, if \( B_4(x_k + \theta (t_k), t_k) \cap B_4(x_i + \theta (t_i), t_i) \ne \emptyset , \) then \( |t_k-t_i|\le 8. \) Since \( \alpha \ge \frac{1}{2}, \) we have \( |\theta (t_k)-\theta (t_i)| \le 8 \). Hence, \( |x_k-x_i| \le 16. \) Therefore, the balls \( \{B_4(x_k+\theta (t_k),t_k)\}_k \) have finite overlap \( C = C_n \).
Define \( Y = \bigcup _l Q_l \) to be the minimal union of lattice unit cubes satisfying that \( \bigcup _k B_4^{n+1}(x_k + \theta (t_k), t_k)\subset Y. \) We have proven that
Hence by Theorem 2.1,
We claim that
for some \( C>0. \) This would conclude the proof.
To prove (20), note that, if \( Q_l \subset B_r^{n+1}(y_0,s_0) \), \( r\ge 1, \) and \( Q_l\cap B_4^{n+1}(x_k + \theta (t_k),t_k) \ne \emptyset \), then \( B_k = B_1^{n+1}(x_k,t_k) \subset B_{r+5}^{n+1} (y_0-\theta (s_0), s_0). \)
Therefore,
\(\square \)
3 Proof of Theorem 1.4
Before the proof, let us introduce a stability property of the Schrödinger operator. More general versions of the following appeared in an article of Tao [12] from 1999 and an article of Christ [5] from 1988.
Suppose that \( {\widehat{f}} \) is supported inside a ball of radius 1.
If \( |x'-y'| \le 4 \) and \( |t'-s'|\le 4, \) then,
where \( \widehat{f_\mathfrak {l}}(\xi ) = e^{2\pi i\mathfrak {l}\xi }{\widehat{f}}(\xi )\).
Now, fix \( \alpha \ge 1/2 \) and \( \gamma \in \Gamma ^{\alpha }\left( R^{-1}\right) \). Define \( \theta \) as in (11). Whenever \( |x-y|\le 2 \) and \( |t-s|\le 2 \), we have that \( \left| x+\theta (t) - ( y + \theta (s)) \right| \le 4. \) Thus,
Therefore, if \( |x-x_0| \le 1 \) and \( |t-t_0|\le 1 \), then,
Proof of Theorem 1.4
For the sake of briefness, given \( (x,t)\in {\mathbb {R}}^{n+1} \) let us denote
Now, we can write
For each \( x_0 \), there exists \(\widetilde{t_0} = \widetilde{t_0}(x_0)\in {\mathbb {Z}}\cap [0,R] \) such that the supremum on each term of the above sum is almost attained inside \( B^{n+1}_1(x_0,\widetilde{t_0}) \). Therefore,
which is, by (24),
Therefore, denoting \( C_\mathfrak {l} = \frac{1}{(1+|\mathfrak {l}|)^{N}}\) and using Cauchy–Schwarz,
Let us choose \( X = \bigcup _{\begin{array}{c} x_0\in {\mathbb {Z}}^n \\ |x_0|<R \end{array} } B^{n+1}_2\left( x_0,\widetilde{t_0}\right) \). The above lets us deduce that
By Theorem 2.2, this is,
Recall that, given \( x_0\in {\mathbb {Z}}^n\cap B_R^n(0), \) we have chosen exactly one \( \widetilde{t_0}\in {\mathbb {Z}}\cap [0,R] \). Consequently, \( \phi _{X,n} \le 1 \). Therefore, the above inequalities yield
\(\square \)
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Javier Minguillon was supported by Grant PID2022-142202NB-I00/AEI/10.13039/501100011033.
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Minguillón, J. A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum. J Geom Anal 34, 333 (2024). https://doi.org/10.1007/s12220-024-01755-x
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DOI: https://doi.org/10.1007/s12220-024-01755-x