Dimension of divergence set of the wave equation

TL;DR

This paper investigates the Hausdorff dimension of divergence sets for the wave equation's solutions, proving a conjecture in three dimensions and extending results to higher dimensions, with implications for Strichartz estimates and Falconer distance problems.

Contribution

It proves the divergence set dimension conjecture for d=3 and extends results to higher dimensions, linking Strichartz estimates to spherical averages of measures.

Findings

Confirmed the divergence set dimension conjecture for three dimensions.

Extended divergence set results to dimensions d≥4.

Established the equivalence between Strichartz estimates and spherical averages.

Abstract

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence $\lim_{t\rightarrow 0} e^{it\sqrt{-\Delta}} f(x) = f(x)$ fails when $f \in H^s(\mathbb R^d)$. We especially prove the conjecture raised by Barcel\'o, Bennett, Carbery and Rogers \cite{BBCR} for $d=3$, and improve the previous results in higher dimensions $d\ge4$. We also show that a Strichartz type estimate for $f\to e^{it\sqrt{-\Delta}} f$ with the measure $ dt\,d\mu(x)$ is essentially equivalent to the estimate for the spherical average of $\widehat \mu$ which has been extensively studied for the Falconer distance set problem. The equivalence provides shortcuts to the recent results due to B. Liu and K. Rogers.