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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.03465 |
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| _version_ | 1866910010164903936 |
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| author | Ham, Seheon Kah, Jiwon Lee, Sanghyuk Li, Ji |
| author_facet | Ham, Seheon Kah, Jiwon Lee, Sanghyuk Li, Ji |
| contents | In his influential 1986 paper, Rubio de Francia established $L^p$ bounds for the maximal function generated by dilations of measures $μ$ whose Fourier transforms $\widehatμ$ satisfy specific decay condition. In the present work, we obtain results that complement his work in several directions. In particular, we obtain restricted weak-type endpoint bound on the maximal function and $L^p$--$L^q$ bounds on its local variant. We also investigate how Frostman's growth condition on the measure influences those maximal bounds. While a key feature of Rubio de Francia's result is that $L^p$ boundedness is determined solely by the decay order of $\widehatμ$, we show that the Frostman condition plays a significant role when the growth order exceeds $d-1$ or when $L^p$--$L^q$ estimates are considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03465 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Rubio de Francia's maximal theorem Ham, Seheon Kah, Jiwon Lee, Sanghyuk Li, Ji Classical Analysis and ODEs 42B25, 42B30, 28A78 In his influential 1986 paper, Rubio de Francia established $L^p$ bounds for the maximal function generated by dilations of measures $μ$ whose Fourier transforms $\widehatμ$ satisfy specific decay condition. In the present work, we obtain results that complement his work in several directions. In particular, we obtain restricted weak-type endpoint bound on the maximal function and $L^p$--$L^q$ bounds on its local variant. We also investigate how Frostman's growth condition on the measure influences those maximal bounds. While a key feature of Rubio de Francia's result is that $L^p$ boundedness is determined solely by the decay order of $\widehatμ$, we show that the Frostman condition plays a significant role when the growth order exceeds $d-1$ or when $L^p$--$L^q$ estimates are considered. |
| title | On Rubio de Francia's maximal theorem |
| topic | Classical Analysis and ODEs 42B25, 42B30, 28A78 |
| url | https://arxiv.org/abs/2602.03465 |