# 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.

## Key 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.