Abstract
Localization and convergence almost everywhere of Schrödinger means are studied.
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1 Introduction
For \(f\in L^{2}({\mathbb {R}}^{n})\), \(n \geqslant 1\) and \(a > 1\) we set
and
For \(a=2\) and f belonging to the Schwartz class \({\mathscr {S}}({\mathbb {R}}^{n})\) we set \(u(x,t) = S_{t}f(x)/(2\pi )^{n}\). It then follows that \(u(x,0) = f(x)\) and u satisfies the Schrödinger equation \(i{\partial u}/{\partial t} = \Delta u\).
We introduce Sobolev spaces \(H_{s} = H_{s}({\mathbb {R}}^{n})\) by setting
where
In the case n = 1 it is well-known (see Sjölin [7] and Vega [9] and in the case \(a = 2\) Carleson [3] and Dahlberg and Kenig [4]) that
almost everywhere if \(f\in H_{1/4}\). Also it is known that \(H_{1/4}\) can not be replaced by \(H_{s} \) if \(s < 1/4\).
Assuming \(n\geqslant 2\) and \(a=2\) Bourgain [1] has proved that (1) holds almost everywhere if \(f\in H_{s}\) and \(s>1/2-1/4n\). On the other hand Bourgain [2] has proved that \(s\geqslant n/2(n+1)\) is necessary for convergence for \(a=2\) and all \(n \geqslant 2\). In the case \(n=2\) and \(a = 2\), Du, Guth, and Li [5] proved that the condition \(s>1/3\) is sufficient. Recently Du and Zhang [6] proved that the condition \(s > n/2(n+1)\) is sufficient for \(a = 2\) and all \(n \geqslant 3\).
In the case \(a > 1\), \(n = 2\), it is known that (1) holds almost everywhere if \(f \in H_{1/2}\) and in the case \(a> 1\), \(n\geqslant 3\), convergence has been proved for \(f \in H_{s}\) with \(s>1/2\) (see [7] and [9]).
If \(f\in L^{2}({\mathbb {R}}^{n})\) then \((2\pi )^{-n}S_{t}f \rightarrow f\) in \(L^{2}\) as \(t \rightarrow 0\). It follows that there exists a sequence \((t_{k})_{1}^{\infty }\) satisfying
such that
almost everywhere.
We shall here study the problem of deciding for which sequences \((t_{k})_{1}^{\infty }\) one has
almost everywhere if \(f \in H_{s}\). We have the following result.
Theorem 1
Assume \(n\geqslant 1\) and \(a>1\) and \(s>0\). We assume that (2) holds and that \(\sum _{k = 1}^{\infty } t_{k}^{2s/a} < \infty \) and \(f\in H_{s}({\mathbb {R}}^{n})\). Then
for almost every x in \({\mathbb {R}}^{n}\).
Now assume \(n = 1\), \(a > 1\), and \(0 \leqslant s < 1/4\). In Sjölin [8] we studied the problem if there is localization or localization almost everywhere for the above operators \(S_{t}\) and the functions \(f \in H_{s}\) with compact support, that is, do we have
everywhere or almost everywhere in \({\mathbb {R}}^{n}\backslash \)(suppf)?
It is proved in [8] that there is no localization or localization almost everywhere of this type for \(0\leqslant s < 1/4\). In fact the following theorem was proved in Sjölin [8].
Theorem A
There exist two disjoint compact intervals I and J in \({\mathbb {R}}\) and a function f which belongs to \(H_{s}\) for all \(s < 1/4\), with the properties that (suppf) \(\subset I\) and for every \(x \in J\) one does not have
Let \(\omega \) be a continuous and decreasing function on \([0,\infty )\). Assume that \(\omega (t) \rightarrow 0\) as \(t \rightarrow \infty \). Set
where
We have the following result.
Theorem 2
The function f in theorem A can be chosen so that \(f \in H_{\omega }\).
Theorem 2 shows that the sufficient condition \(f\in H_{1/4}\) for convergence almost everywhere and localization almost everywhere of Schrödinger means is very sharp. In the case a = 2 Theorem 2 was obtained in 2009 (unpublished). After proving Theorem 2 we shall use Theorem 1 to make a remark on the Schrödinger means \(S_{t}f(x)\) for the function f which was constructed in [8] to prove Theorem A.
2 Proofs
In the proof of Theorem 1 we shall need the following lemma.
Lemma 1
Assume \(n \geqslant 1\), \(a>1\), \(0<s<1\), and \(0<\delta <1\). Set
Then one has
where the constant C does not depend on \(\delta \), and \(\Vert m\Vert _{\infty }\) denotes the norm of m in \(L^{\infty }({\mathbb {R}}^{n})\).
Proof of Lemma 1
We shall write \( A\lesssim B\) if there is a constant C such that \(A\leqslant CB\).
In the case \(|\xi | \geqslant \delta ^{-1/a}\) one has
Then assume \(0 \leqslant |\xi | \leqslant 1\). We obtain
In the remaining case \(1<|\xi |<\delta ^{-1/a}\) one obtains
and the proof of Lemma 1 is complete. \(\square \)
We shall then give the proof of Theorem 1.
Proof of Theorem 1
We may assume \(0<s<1\). We set
We have \(f\in H_{s}\) and we define g by taking
It then follows that \(g \in L^{2}({\mathbb {R}}^{n})\).
We have
and
Hence
where
According to Lemma 1 we have \(\Vert m\Vert _{\infty } \lesssim t_{k}^{s/a}\) and applying the Plancherel theorem we obtain
It follows that
and applying the theorem on monotone convergence one also obtains
We conclude that \(\sum _{1}^{\infty }|h_{k}|^2\) is convergent almost everywhere and hence \(\lim _{k\rightarrow \infty }h_{k}(x) = 0\) and
almost everywhere. \(\square \)
Now assume \(n = 1\) and \(a>1\). We set
and let K denote the Fourier transform of m so that \(K \in {\mathscr {S}}'({\mathbb {R}})\). According to Sjölin [8] p.142, \(K\in C^{\infty }({\mathbb {R}})\) and there exists a number \(\alpha \geqslant 0\) such that
For \(t>0\) it is then clear that
has the Fourier transform
It follows that \(S_{t}f(x)=K_{t}\star f(x)\) for \(f\in L^{2}({\mathbb {R}}^{m})\) with compact support. We let denote the inverse Fourier transform of g and choose \(g\in {\mathscr {S}}({\mathbb {R}})\) such that supp and . We set
According to [7], p.143, one has \({\widehat{f}}_{v}(\xi ) = vg(v\xi +1/v)\) and
We shall state three lemmas from [8].
Lemma 2
There exist positive numbers \(c_0\), \(\delta \) and \(v_0\) such that
for \(0<v<v_0\) and \(0< x <\delta \).
In the remaining part of this paper \(\delta \) and \(v_0\) are given by Lemma 2. We may also assume that \(\delta < 1\).
Lemma 3
For \(0<v<\min (v_0,\delta /4)\), \(0<t<1\), and \(\delta /2< x < \delta \) one has
where \(\gamma = (1+\alpha )/a > 0\).
Lemma 4
For \(0<v<\min (v_0,\delta /4)\), \(0<t<1\), and \(\delta /2<x<\delta \) one has
where \(\beta = 2a\).
We shall use these lemmas to prove Theorem 2.
Proof of Theorem 2
Now let \(v_1\) satisfy \(0<v_1<\min (v_0,\delta /4)\) and set \(\epsilon _k = 2^{-k}\), \(k=1,2,3,...\)
We also set \(\mu = \max ((2a-2)\gamma ,\beta /(2a-2))\) and choose \(v_k\), \(k = 2,3,4,...\), such that \(0<v_k\leqslant \epsilon _{k}v_{k-1}^{\mu }\) and
We then set \(f = \sum _{k=1}^{\infty }f_{v_k}\) and shall prove that \(f\in H_\omega \).
Arguing as in [8, pp. 145–147], it follows from Lemmas 2, 3, and 4 that with \(t_k(x) = xv_{k}^{2a-2}/a\) one has
for \(\delta /2<x<\delta \) and \(k\geqslant k_0\). Hence we do not have \(\lim _{t\rightarrow 0}S_tf(x)=0\) in the interval \((\delta /2,\delta )\). Taking \(I = [-v_1,v_1]\) and \( J\subset (\delta /2,\delta )\) we have supp\(f\subset I\) and for every \(x\in J\) one does not have \(\lim _{t\rightarrow 0}S_tf(x)=0\). Thus Theorem 2 follows. It remains to prove that \(f\in H_{\omega }\).
We have
where
and
It follows that
Hence
and
We have \(f = \sum _{1}^{\infty }f_{v_k}\) and it follows that
since \(v_k \leqslant \epsilon _k\).
We conclude that \(f\in H_\omega \) and the proof of Theorem 2 is complete. \(\square \)
Remark 1
In Sjölin [8] the function f in Theorem A is given by the formula
where \(v_k\) is defined by taking \(0<v_1<\min (v_0,\delta /4)\) and \(v_k = \epsilon _kv_{k-1}^{\mu }\) for \(k = 2,3,4,...\) Here \(\epsilon _k = 2^{-k}\) and \(\mu >0\) is given in the proof of Theorem 2. Also let the intervals I and J be defined as in the proof of Theorem 2. We then set \(t_k(x) = xv_k^{2a-2}/a\) for \(x\in J\) and \(k=1,2,3,...\)
It is proved in [8] that for every \(x_0 \in J\)
We now fix \(x_0 \in J\) and shall use Theorem 1 to prove that although (3) holds one also has
We have \(v_k<\epsilon _k\) and it follows that
and
for \(0<s<1/4\). Also \(f \in H_s\) for \(0<s<1/4\) and (4) follows from an application of Theorem 1.
Remark 2
In the case \(a=2\) one has \(\mu = 2\) and \(v_k = \epsilon _kv_{k-1}^{2}\), and we also have \(0<v_1<1/4\). It can be proved that it follows that
where d is a constant and \(d>2\).
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Communicated by Mieczysław Mastyło.
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Sjölin, P. Two Theorems on Convergence of Schrödinger Means. J Fourier Anal Appl 25, 1708–1716 (2019). https://doi.org/10.1007/s00041-018-9644-0
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DOI: https://doi.org/10.1007/s00041-018-9644-0