Refinements of the 2-dimensional Strichartz estimate on the maximum wave packet

TL;DR

This paper improves the 2D homogeneous Strichartz estimate for Schrödinger equations by refining bounds on the maximum size of a wave packet, using diverse mathematical techniques.

Contribution

It introduces new refinements of the 2D Strichartz estimate for wave packets, employing arithmetic, polynomial partitioning, and decoupling methods.

Findings

Refinements are optimal for certain p ranges.

Different proof techniques are applicable for various cases.

Examples demonstrate the sharpness of the results.

Abstract

The Strichartz estimates for Schr\"{o}dinger equations can be improved when the data is spread out in either physical or frequency space. In this paper we give refinements of the 2-dimensional homogeneous Strichartz estimate on the maximum size of a single wave packet. Different approaches are used in the proofs, including arithmetic approaches, polynomial partitioning, and the $l^2$ Decoupling Theorem, for different cases. We also give examples to show that the refinements we obtain cannot be further improved when $2 \leq p \leq 4$ and $p = 6$.