Bounds for lacunary bilinear spherical and triangle maximal functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916351687262208 |
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| author | Borges, Tainara Foster, Benjamin |
| author_facet | Borges, Tainara Foster, Benjamin |
| contents | We prove $L^p\times L^q\rightarrow L^r$ bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the Hölder relation $1/p+1/q=1/r$. The boundedness region that we get contains at least the interior of the Hölder boundedness region of the associated single scale bilinear averaging operator. In the case of the lacunary bilinear spherical maximal function in $d\geq 2$, we prove boundedness for any $p,q\in (1,\infty]^2$, which is sharp up to boundary; we then show how to extend this result to a more degenerate family of surfaces where some curvatures are allowed to vanish. For the lacunary triangle averaging maximal operator, we have results in $d\geq 7$, and the description of the sharp region will depend on a sharp description of the Hölder bounds for the single scale triangle averaging operator, which is still open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_12269 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounds for lacunary bilinear spherical and triangle maximal functions Borges, Tainara Foster, Benjamin Classical Analysis and ODEs We prove $L^p\times L^q\rightarrow L^r$ bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the Hölder relation $1/p+1/q=1/r$. The boundedness region that we get contains at least the interior of the Hölder boundedness region of the associated single scale bilinear averaging operator. In the case of the lacunary bilinear spherical maximal function in $d\geq 2$, we prove boundedness for any $p,q\in (1,\infty]^2$, which is sharp up to boundary; we then show how to extend this result to a more degenerate family of surfaces where some curvatures are allowed to vanish. For the lacunary triangle averaging maximal operator, we have results in $d\geq 7$, and the description of the sharp region will depend on a sharp description of the Hölder bounds for the single scale triangle averaging operator, which is still open. |
| title | Bounds for lacunary bilinear spherical and triangle maximal functions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2305.12269 |