Pointwise convergence of Schr\"odinger solutions and multilinear refined Strichartz estimates

TL;DR

This paper advances the understanding of pointwise convergence of Schrödinger solutions in fractal measures, establishing new almost everywhere convergence results for functions in certain Sobolev spaces using refined Strichartz estimates.

Contribution

It introduces new partial results on Schrödinger pointwise convergence for fractal measures and develops multilinear refined Strichartz estimates using decoupling and Kakeya techniques.

Findings

Almost everywhere convergence for $f \

in $H^s(\

for $s > (n+1)/2(n+2)$ in dimensions $n \\geq 3$,

Abstract

We obtain partial improvement toward the pointwise convergence problem of Schr\"odinger solutions, in the general setting of fractal measure. In particular, we show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ almost everywhere with respect to Lebesgue measure for all $f \in H^s (\mathbb{R}^n)$ provided that $s>(n+1)/2(n+2)$. The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya.