Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates

Abstract

We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for $n\ge 3$, ${\lim }_{t\to 0}{e}^{it\Delta }f(x)=f(x)$ 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.

14 pages. v2: 4 figures, minor updates based on referee suggestions. arXiv admin note: text overlap with arXiv:1612.08946