Paper detail

Dimension of divergence set of the wave equation

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence $\lim_{t\rightarrow 0} e^{it\sqrt{-Δ}} f(x) = f(x)$ 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 $d\ge4$. We also show that a Strichartz type estimate for $f\to e^{it\sqrt{-Δ}} f$ with the measure $ dt\,dμ(x)$ is essentially equivalent to the estimate for the spherical average of $\widehat μ$ 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.