Endpoint estimates for maximal operators associated to the wave equation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911154358452224 |
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| author | Cho, Chu-Hee Lee, Sanghyuk Li, Wenjuan |
| author_facet | Cho, Chu-Hee Lee, Sanghyuk Li, Wenjuan |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01686 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Endpoint estimates for maximal operators associated to the wave equation Cho, Chu-Hee Lee, Sanghyuk Li, Wenjuan Classical Analysis and ODEs Primary 35L05, Secondary 42B37 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. |
| title | Endpoint estimates for maximal operators associated to the wave equation |
| topic | Classical Analysis and ODEs Primary 35L05, Secondary 42B37 |
| url | https://arxiv.org/abs/2501.01686 |