Dimension of divergence set of the wave equation

Abstract

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence ${\lim }_{t\to 0}{e}^{it\sqrt{-\Delta }}f(x)=f(x)$ fails when $f\in H^s({\mathbb{R}}^d)$. We especially prove the conjecture raised by Barceló, Bennett, Carbery and Rogers \cite{BBCR} for $d=3$, and improve the previous results in higher dimensions $d\ge 4$. We also show that a Strichartz type estimate for $f\to e^{it\sqrt{-\Delta }}f$ with the measure $dtd\mu (x)$ is essentially equivalent to the estimate for the spherical average of $\hat{\mu }$ 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.

11 pages, 1 figure