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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.01686 |
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Table of Contents:
- We consider the $H^{s}$--$L^q$ maximal estimates associated to the wave operator \begin{equation*} e^{ it\sqrt{-Δ}}f(x) = \frac{1}{(2π)^d}\int_{\mathbb{R}^d} e^{i(x \cdot ξ\, + t|ξ|)} \widehat{f}(ξ\,) dξ. \end{equation*} Rogers--Villarroya proved $H^{s}$--$L^q$ estimates for the maximal operator $f\mapsto$ $\sup_{t} |e^{ it\sqrt{-Δ}}f|$ 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 $f\mapsto \sup_{t} |e^{ it\sqrt{-Δ}}f|$. We also prove that several different forms of the maximal estimates considered by Rogers--Villarroya are basically equivalent to each other.