Endpoint estimates for the fractal circular maximal function and related local smoothing
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909659943665664 |
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| author | Lee, Sanghyuk Roncal, Luz Zhang, Feng Zhao, Shuijiang |
| author_facet | Lee, Sanghyuk Roncal, Luz Zhang, Feng Zhao, Shuijiang |
| contents | Sharp $L^p$--$L^q$ estimates for the spherical maximal function over dilation sets of fractal dimensions, including the endpoint estimates, were recently proved by Anderson--Hughes--Roos--Seeger. More intricate $L^p$--$L^q$ estimates for the fractal circular maximal function were later established in the sharp range by Roos--Seeger, but the endpoint estimates have been left open, particularly when the fractal dimension of the dilation set lies in $[1/2, 1)$. In this work, we prove these missing endpoint estimates for the circular maximal function. We also study the closely related $L^p$--$L^q$ local smoothing estimates for the wave operator over fractal dilation sets, which were recently investigated by Beltran--Roos--Rutar--Seeger and Wheeler. Making use of a bilinear approach, we also extend the range of $p,q$, for which the optimal estimate holds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20390 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Endpoint estimates for the fractal circular maximal function and related local smoothing Lee, Sanghyuk Roncal, Luz Zhang, Feng Zhao, Shuijiang Classical Analysis and ODEs 35L05 (primary), 42B20, 28A80 (secondary) Sharp $L^p$--$L^q$ estimates for the spherical maximal function over dilation sets of fractal dimensions, including the endpoint estimates, were recently proved by Anderson--Hughes--Roos--Seeger. More intricate $L^p$--$L^q$ estimates for the fractal circular maximal function were later established in the sharp range by Roos--Seeger, but the endpoint estimates have been left open, particularly when the fractal dimension of the dilation set lies in $[1/2, 1)$. In this work, we prove these missing endpoint estimates for the circular maximal function. We also study the closely related $L^p$--$L^q$ local smoothing estimates for the wave operator over fractal dilation sets, which were recently investigated by Beltran--Roos--Rutar--Seeger and Wheeler. Making use of a bilinear approach, we also extend the range of $p,q$, for which the optimal estimate holds. |
| title | Endpoint estimates for the fractal circular maximal function and related local smoothing |
| topic | Classical Analysis and ODEs 35L05 (primary), 42B20, 28A80 (secondary) |
| url | https://arxiv.org/abs/2506.20390 |