Local smoothing estimates for Schr\"odinger equations in modulation spaces

TL;DR

This paper advances the understanding of Schrödinger equations by establishing refined local smoothing estimates in modulation spaces, introducing a new randomized Strichartz estimate, and linking reverse function estimates to Strichartz estimates.

Contribution

It introduces novel local smoothing estimates in modulation spaces and connects reverse square function estimates with Strichartz estimates for Schrödinger equations.

Findings

Refined summability exponents for modulation spaces.

A new randomized Strichartz estimate in modulation spaces.

Reverse function estimates imply Strichartz estimates in modulation spaces.

Abstract

Motivated by a recent work of Schippa (2022), we consider local smoothing estimates for Schr\"{o}dinger equations in modulation spaces. By using the C\'{o}rdoba-Fefferman type reverse square function inequality and the bilinear Strichartz estimate, we can refine the summability exponent of modulation spaces. Next, we will also discuss a new type of randomized Strichartz estimate in modulation spaces. Finally, we will show that the reverse function estimate implies the Strichartz estimates in modulation spaces. From this implication, we obtain the reverse square function estimate of critical order.