1 Introduction

Consider the linear Schrödinger equation on \( {\mathbb {R}}^n\times {\mathbb {R}}\), \( n\ge 1, \) given by

$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t u(x,t) - \Delta _x u (x,t) = 0,\\ u(x,0) = f(x). \end{array}\right. } \end{aligned}$$
(1)

Its solution can be formally expressed as

$$\begin{aligned} e^{it\Delta }f \left( x\right) = \int _{{\mathbb {R}}^n} e^{2\pi i x\cdot \xi } e^{2\pi it|\xi |^2} {\widehat{f}}(\xi ) d\xi . \end{aligned}$$
(2)

It was first proposed by Carleson in 1980 [3] to find the values of \( s>0 \) for which

$$\begin{aligned} \lim _{t\rightarrow 0} e^{it\Delta }f \left( x\right) = f(x), \quad \text {a.e.}\quad x\in {\mathbb {R}}^n, \end{aligned}$$
(3)

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,

$$\begin{aligned} \Gamma _{\tau }^\alpha := \left\{ \gamma :[0,1] \rightarrow {\mathbb {R}}^n : \text {for all } t,t'\in [0,1],\; |\gamma (t)-\gamma (t')| \le \tau |t-t'|^\alpha \right\} . \end{aligned}$$
(4)

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 \( \gamma \in \Gamma _{\tau }^\alpha \),

$$\begin{aligned} \left\| \sup _{0<t<1}\left| e^{it\Delta }f \left( x + \gamma (t)\right) \right| \right\| _{L^2\left( B^{n}_1(0)\right) } \le C_{\varepsilon ,\tau } \Vert f\Vert _{ H^{\frac{n}{2(n+1)} +\epsilon } \left( {\mathbb {R}}^n\right) }, \end{aligned}$$
(5)

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

$$\begin{aligned} \Gamma ^{\alpha }\left( R^{-1}\right) := \left\{ \gamma :[0,R^{-1}] \rightarrow {\mathbb {R}}^n : \text {for all } t,t'\in [0,R^{-1}],\; |\gamma (t)-\gamma (t')| \le |t-t'|^\alpha \right\} . \end{aligned}$$
(6)

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 \( \gamma \in \Gamma ^{\alpha }\left( R^{-1}\right) \),

$$\begin{aligned} \left\| \sup _{0<t \le R}\left| e^{i t \Delta } f \left( x + R\gamma \left( \frac{t}{R^2}\right) \right) \right| \right\| _{L^2\left( B^n_R(0)\right) } \le C_{\varepsilon } R^{\frac{n}{2(n+1)}+\varepsilon } \Vert f\Vert _2. \end{aligned}$$
(7)

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

$$\begin{aligned} \phi := \phi _{X,\beta }:= \max _{\begin{array}{c} B_r^{n+1}(x') \subset B_R^{n+1}(0) \\ x^{\prime } \in \mathbb {R}^{n+1}, r \ge 1 \end{array}} \frac{\#\left\{ B_k: B_k \subset B^{n+1}_r\left( x^{\prime } \right) \right\} }{r^\beta } . \end{aligned}$$
(8)

Then

$$\begin{aligned} \left\| e^{i t \Delta } f\right\| _{L^2(X,dx dt)} \le C_{\varepsilon } \phi ^{\frac{1}{n+1}} R^{\frac{\beta }{2(n+1)}+\varepsilon }\Vert f\Vert _2. \end{aligned}$$
(9)

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

$$\begin{aligned} \left\| e^{i t \Delta } f\left( x + R\gamma \left( \frac{t}{R^2}\right) \right) \right\| _{L^2(X, dx dt)} \le C_{\varepsilon } \phi ^{\frac{1}{n+1}} R^{\frac{\beta }{2(n+1)}+\varepsilon }\Vert f\Vert _2. \end{aligned}$$
(10)

Proof of Theorem 2.2

Denote

$$\begin{aligned} \theta (t) := \theta _{R } (t) := R\gamma \left( \frac{t}{R^2}\right) . \end{aligned}$$
(11)

We begin with

$$\begin{aligned} \left\| e^{it\Delta }f \left( x + R\gamma \left( \frac{t}{R^2}\right) \right) \right\| _{L^2(X, dx dt)}^2&= \sum _{k} \int _{B_k} \left| e^{it\Delta }f \left( x+\theta (t)\right) \right| ^2 dx dt. \end{aligned}$$
(12)

Denote \( (x_k,t_k)\) to be the center of \( B_k. \) Then,

$$\begin{aligned}&\le \sum _{k} \int _{t_k-1}^{t_k+1} \int _{B^n_1(x_k)} \left| e^{it\Delta }f \left( x+\theta (t)\right) \right| ^2 dxdt \end{aligned}$$
(13)
$$\begin{aligned}&= \sum _{k} \int _{t_k-1}^{t_k+1} \int _{B^n_1\left( x_k+\theta (t)\right) } \left| e^{it\Delta }f \left( y\right) \right| ^2 dydt. \end{aligned}$$
(14)

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,

$$\begin{aligned}&\le \sum _{k} \int _{t_k-1}^{t_k+1} \int _{B^n_3\left( x_k+\theta (t_k)\right) } \left| e^{it\Delta }f \left( y\right) \right| ^2 dydt \end{aligned}$$
(15)
$$\begin{aligned}&\le \int _{{\mathbb {R}}^n}\sum _k \chi _{B^{n+1}_4\left( x_k + \theta (t_k),t_k\right) }(y) \left| e^{it\Delta }f \left( y\right) \right| ^2dydt \end{aligned}$$
(16)
$$\begin{aligned}&\le C \int _{\bigcup B^{n+1}_4(x_k + \theta (t_k),t_k)} \left| e^{it\Delta }f \left( y\right) \right| ^2dydt, \end{aligned}$$
(17)

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

$$\begin{aligned} \left\| e^{it\Delta }f \left( x+\theta (t)\right) \right\| _{L^2(X, dx dt)}&\le C_n \left\| e^{it\Delta }f \left( x\right) \right\| _{L^2(Y,dx dt)}. \end{aligned}$$
(18)

Hence by Theorem 2.1,

$$\begin{aligned}&\le C_{\epsilon ,n} \phi _{Y,\beta }^{\frac{1}{n+1}}R^{\frac{\beta }{2(n+1)}+\epsilon } \Vert f\Vert _2. \end{aligned}$$
(19)

We claim that

$$\begin{aligned} \phi _{Y,\beta } \le c_n \phi _{X,\beta }, \end{aligned}$$
(20)

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,

$$\begin{aligned} \dfrac{\#\{Q_l : Q_l \subset B_r^{n+1}(y_0,s_0)\}}{r^{\beta }}\le & c_n \dfrac{ \#\left\{ B_k : B_k \subset B_{r+5}^{n+1} (y_0 -\theta (s_0), s_0) \right\} }{(r+5)^\beta } \cdot \dfrac{(r+5)^\beta }{r^\beta } \nonumber \\ \le & c_n \phi _{X,\beta }. \end{aligned}$$
(21)

\(\square \)

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,

$$\begin{aligned} \left| e^{it'\Delta }f(x')\right| \le \sum _{\mathfrak {l}\in {\mathbb {Z}}^n} \frac{1}{(1+|\mathfrak {l}|)^{n+1}} \left| e^{is'\Delta }f_\mathfrak {l}(y')\right| , \end{aligned}$$
(22)

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,

$$\begin{aligned} \left| e^{it\Delta }f(x+\theta (t))\right| \le \sum _{\mathfrak {l}\in {\mathbb {Z}}^n} \frac{1}{(1+|\mathfrak {l}|)^{n+1}} \left| e^{is\Delta }f_\mathfrak {l}(y+ \theta (s))\right| . \end{aligned}$$
(23)

Therefore, if \( |x-x_0| \le 1 \) and \( |t-t_0|\le 1 \), then,

$$\begin{aligned} \left| e^{it\Delta }f(x+\theta (t))\right| \le \sum _{\mathfrak {l}\in {\mathbb {Z}}^n} \frac{1}{(1+|\mathfrak {l}|)^{n+1}} \int _{t_0}^{t_0+1}\int _{B_1(x_0)}\left| e^{is\Delta }f_\mathfrak {l}(y+ \theta (s))\right| dyds. \end{aligned}$$
(24)

Proof of Theorem 1.4

For the sake of briefness, given \( (x,t)\in {\mathbb {R}}^{n+1} \) let us denote

$$\begin{aligned} E'f(x,t) := E'_{\gamma ,R} f(x,t) := e^{i t \Delta } f \left( x + R\gamma \left( \frac{t}{R^2}\right) \right) . \end{aligned}$$
(25)

Now, we can write

$$\begin{aligned}&\left\| \sup _{0<t \le R} \left| e^{i t \Delta } f \left( x + R\gamma \left( \frac{t}{R^2}\right) \right) \right| \right\| _{L^2\left( B^n_R(0)\right) } ^2 = \left\| \sup _{0<t \le R} |E'f(x,t)| \right\| _{L^2\left( B^n_R(0)\right) } ^2 \end{aligned}$$
(26)
$$\begin{aligned}&\quad = \int _{B^n_R(0)} \left( \sup _{\begin{array}{c} 0<t\le R \end{array} }|E'f(x,t)|^2 \right) dx \end{aligned}$$
(27)
$$\begin{aligned}&\quad = \sum _{\begin{array}{c} x_0\in {\mathbb {Z}}^n \\ |x_0|<R \end{array} } \int _{B^n_1(x_0)} \left( \sup _{0<t \le R} |E'f(x,t)|^2 \right) dx \end{aligned}$$
(28)
$$\begin{aligned}&\quad \le C_n \sum _{\begin{array}{c} x_0\in {\mathbb {Z}}^n \\ |x_0|<R \end{array} } \left[ \sup _{|x-x_0| \le 1} \left( \sup _{\begin{array}{c} 0 < t \le R \end{array}} |E'f(x,t)|^2 \right) \right] . \end{aligned}$$
(29)

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,

$$\begin{aligned} \left\| \sup _{0< t\le R}|E'f(x,t)|\right\| _{L^2(B^n_R(0))}^2&\lesssim \sum _{\begin{array}{c} x_0\in {\mathbb {Z}}^n \\ |x_0|<R \end{array} } \sup _{(x,t)\in B_1^{n+1}(x_0,\widetilde{t_0})} \left| E'f(x,t)\right| ^2, \end{aligned}$$
(30)

which is, by (24),

$$\begin{aligned}&\lesssim \sum _{\begin{array}{c} x_0\in {\mathbb {Z}}^n \\ |x_0|<R \end{array} } \left| \sum _{\mathfrak {l} \in \mathbb {Z}^n } \frac{1}{(1+|\mathfrak {l}|)^{N} } \int _{B^n_ 1\left( x_0\right) } \int _{\widetilde{t_0}}^{\widetilde{t_0}+1} \left| E'f_{\mathfrak {l} }(y,s) \right| d s d y \right| ^2. \end{aligned}$$
(31)

Therefore, denoting \( C_\mathfrak {l} = \frac{1}{(1+|\mathfrak {l}|)^{N}}\) and using Cauchy–Schwarz,

$$\begin{aligned}&\lesssim \sum _{\begin{array}{c} x_0\in {\mathbb {Z}}^n \\ |x_0|<R \end{array} } \sum _{\mathfrak {l} \in \mathbb {Z}^n } C_\mathfrak {l} \left\| E'f_{\mathfrak {l} }(y,t) \right\| ^2_{L^2(B^{n+1}_2(x_0,\widetilde{t_0}))} . \end{aligned}$$
(32)

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

$$\begin{aligned} \left\| \sup _{0< t\le R}|E'f(x,t)| \right\| _{L^2(B^n_R(0),dx)}^2&\lesssim \sum _{\mathfrak {l} \in \mathbb {Z}^n } C_{\mathfrak {l}} \left\| E'f_{\mathfrak {l} }(y,t) \right\| _{L^2(X)}^2. \end{aligned}$$
(33)

By Theorem 2.2, this is,

$$\begin{aligned}&\lesssim C_{\epsilon } \sum _{\mathfrak {l} \in \mathbb {Z}^n } C_{\mathfrak {l}} (\phi _{X,n})^{\frac{2}{n+1}} \Vert f_{\mathfrak {l} } \Vert _{L^2({\mathbb {R}}^n)}^2 R^{\frac{2n}{2(n+1)}+\epsilon }. \end{aligned}$$
(34)

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

$$\begin{aligned} \left\| \sup _{0< t\le R}|E'f(x,t)| \right\| _{L^2(B^n_R(0),dx)}^2 \lesssim C_{\epsilon } R^{\frac{2n}{2(n+1)}+\epsilon }\Vert f\Vert _{2}^2. \end{aligned}$$
(35)

\(\square \)