From weak type weighted inequality to pointwise estimate for the decreasing rearrangement
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911773479665664 |
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| author | Agora, Elona Antezana, Jorge Baena-Miret, Sergi Carro, María J. |
| author_facet | Agora, Elona Antezana, Jorge Baena-Miret, Sergi Carro, María J. |
| contents | We shall prove pointwise estimates for the decreasing rearrangement of $Tf$, where $T$ covers a wide range of interesting operators in Harmonic Analysis such as operators satisfying a Fefferman-Stein inequality, the Bochner-Riesz operator, rough operators, sparse operators, Fourier multipliers, etc. In particular, our main estimate is of the form $$ (Tf)^*(t) \leq C\left( \frac 1t\int_0^tf^*(s)\,ds + \int_t^\infty \left( 1 + \log\frac st \right)^{- 1}φ\left(1 + \log\frac st \right) f^*(s)\,\frac{ds}s \right), $$ where $φ$ is determined by the Muckenhoupt $A_p$-weight norm behaviour of the operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05323 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From weak type weighted inequality to pointwise estimate for the decreasing rearrangement Agora, Elona Antezana, Jorge Baena-Miret, Sergi Carro, María J. Classical Analysis and ODEs 42B25, 46E30, 47A30 We shall prove pointwise estimates for the decreasing rearrangement of $Tf$, where $T$ covers a wide range of interesting operators in Harmonic Analysis such as operators satisfying a Fefferman-Stein inequality, the Bochner-Riesz operator, rough operators, sparse operators, Fourier multipliers, etc. In particular, our main estimate is of the form $$ (Tf)^*(t) \leq C\left( \frac 1t\int_0^tf^*(s)\,ds + \int_t^\infty \left( 1 + \log\frac st \right)^{- 1}φ\left(1 + \log\frac st \right) f^*(s)\,\frac{ds}s \right), $$ where $φ$ is determined by the Muckenhoupt $A_p$-weight norm behaviour of the operator. |
| title | From weak type weighted inequality to pointwise estimate for the decreasing rearrangement |
| topic | Classical Analysis and ODEs 42B25, 46E30, 47A30 |
| url | https://arxiv.org/abs/2402.05323 |