A deterministic counterexample for high dimensional $ L^2 L^{\infty} $ Strichartz estimates for the Wave equation

TL;DR

This paper provides a deterministic counterexample demonstrating the failure of certain high-dimensional $L^2 L^{ abla} $ Strichartz estimates for the Wave equation, which were previously shown to fail using probabilistic methods.

Contribution

It introduces a deterministic example that disproves the high-dimensional $L^2 L^{ abla} $ Strichartz estimates for the Wave equation, complementing prior probabilistic results.

Findings

Deterministic counterexample for $L^2 L^{ abla} $ estimates in $n \,\geq\, 4$ dimensions.

Confirms the failure of these estimates without probabilistic methods.

Highlights limitations of $L^2 L^{ abla} $ estimates in high-dimensional wave analysis.

Abstract

In this note we discuss the question of homogeneous $ L^2 L^{\infty} $ Strichartz estimates for the Wave equation in dimensions $ n \geq 4 $ raised by Fang and Wang and recently shown to fail by Guo, Li, Nakanishi and Yan using probability theory. We record a deterministic example for disproving this estimate.