Sobolev bounds and counterexamples for the second derivative of the maximal function in one dimension
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916573664509952 |
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| author | Weigt, Julian |
| author_facet | Weigt, Julian |
| contents | We investigate the question whether the $L^1(\mathbb R)$-norm of the second derivative of the uncentered Hardy-Littlewood maximal function can be bounded by a constant times the $L^1(\mathbb R)$-norm of the function itself. We give a positive answer for a class of functions that contains Sobolev functions on the real line which are decreasing away from the origin and even, and we provide a counterexample which is also decreasing away from the origin but not even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12631 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sobolev bounds and counterexamples for the second derivative of the maximal function in one dimension Weigt, Julian Classical Analysis and ODEs 42B25, 26A45 We investigate the question whether the $L^1(\mathbb R)$-norm of the second derivative of the uncentered Hardy-Littlewood maximal function can be bounded by a constant times the $L^1(\mathbb R)$-norm of the function itself. We give a positive answer for a class of functions that contains Sobolev functions on the real line which are decreasing away from the origin and even, and we provide a counterexample which is also decreasing away from the origin but not even. |
| title | Sobolev bounds and counterexamples for the second derivative of the maximal function in one dimension |
| topic | Classical Analysis and ODEs 42B25, 26A45 |
| url | https://arxiv.org/abs/2409.12631 |