1 results
Sharp convergence for sequences of Schrödinger means and related generalizations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 25 September 2023, pp. 1-17
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- Article
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For decreasing sequences $\{t_{n}\}_{n=1}^{\infty }$
converging to zero and initial data $f\in H^s(\mathbb {R}^N)$
, $N\geq 2$
, we consider the almost everywhere convergence problem for sequences of Schrödinger means ${\rm e}^{it_{n}\Delta }f$
, which was proposed by Sjölin, and was open until recently. In this paper, we prove that if $\{t_n\}_{n=1}^{\infty }$
belongs to Lorentz space ${\ell }^{r,\infty }(\mathbb {N})$
, then the a.e. convergence results hold for $s>\min \{\frac {r}{\frac {N+1}{N}r+1},\,\frac {N}{2(N+1)}\}$
. Inspired by the work of Lucà-Rogers, we construct a counterexample to show that our a.e. convergence results are sharp (up to endpoints). Our results imply that when $0< r<\frac {N}{N+1}$
, there is a gain over the a.e. convergence result from Du-Guth-Li and Du-Zhang, but not when $r\geq \frac {N}{N+1}$
, even though we are in the discrete case. Our approach can also be applied to get the a.e. convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.
