Dimension of divergence set of the wave equation

2021/02/25 by Ham, Seheon, Ko, Hyerim, Lee, Sanghyuk · 2 citations
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2102.12701

Abstract

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence lim_{t→ 0} e^it√(-Δ) f(x) = f(x) fails when f ∈ H^s(\mathbb R^d). We especially prove the conjecture raised by Barceló, Bennett, Carbery and Rogers \citeBBCR for d=3, and improve the previous results in higher dimensions d≥4. We also show that a Strichartz type estimate for f→ e^it√(-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.

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• Weighted wave envelope estimates for the parabola
• Maximal estimates for orthonormal systems of wave equations

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