Endpoint estimates for maximal operators associated to the wave equation

2025/01/03 by C. Cho, Cho, Chu-Hee, Sang‐Hyuk Lee +3 · 2 citations
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems #Primary 35L05 #Secondary 42B37

paper · pdf · doi:10.48550/arxiv.2501.01686

openalex publication_date 2025/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the H^s--L^q maximal estimates associated to the wave operator e ^it√(-Δ)f(x) = (1)/((2π)^d)∫_ℝ^d e^{i(x ⋅ ξ + t|ξ|)} \widehatf(ξ ) dξ. Rogers--Villarroya proved H^s--L^q estimates for the maximal operator f↦ sup_t |e ^it√(-Δ)f| up to the critical Sobolev exponents s_c(q,d). However, the endpoint case estimates for the critical exponent s=s_c(q,d) have remained open so far. We obtain the endpoint H^s_c(q,d)--L^q bounds on the maximal operator f↦ sup_t |e ^it√(-Δ)f|. We also prove that several different forms of the maximal estimates considered by Rogers--Villarroya are basically equivalent to each other.

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