Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916091020705792 |
|---|---|
| author | Bhojak, Ankit Choudhary, Surjeet Singh Shrivastava, Saurabh Shuin, Kalachand |
| author_facet | Bhojak, Ankit Choudhary, Surjeet Singh Shrivastava, Saurabh Shuin, Kalachand |
| contents | In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;dσ(y_1,y_2)\right|,$$ in dimensions $d=1,2$ and as an application, we deduce sharp endpoint estimates for the multilinear spherical maximal function. We also prove $L^p-$estimates for the local spherical maximal function in all dimensions $d\geq 2$, thus improving the boundedness left open in the work of Jeong and Lee (https://doi.org/10.1016/j.jfa.2020.108629). We further study necessary conditions for the bilinear maximal function, \[\mathcal M (f,g)(x)=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty)g(x+ty)\;dσ(y)\right|\] to be bounded from $L^{p_1}(\mathbb R^2)\times L^{p_2}(\mathbb R^2)$ to $L^p(\mathbb R^2)$ and prove sharp results for a linearized version of $\mathcal M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_00425 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions Bhojak, Ankit Choudhary, Surjeet Singh Shrivastava, Saurabh Shuin, Kalachand Classical Analysis and ODEs 42B15, 42B25 In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;dσ(y_1,y_2)\right|,$$ in dimensions $d=1,2$ and as an application, we deduce sharp endpoint estimates for the multilinear spherical maximal function. We also prove $L^p-$estimates for the local spherical maximal function in all dimensions $d\geq 2$, thus improving the boundedness left open in the work of Jeong and Lee (https://doi.org/10.1016/j.jfa.2020.108629). We further study necessary conditions for the bilinear maximal function, \[\mathcal M (f,g)(x)=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty)g(x+ty)\;dσ(y)\right|\] to be bounded from $L^{p_1}(\mathbb R^2)\times L^{p_2}(\mathbb R^2)$ to $L^p(\mathbb R^2)$ and prove sharp results for a linearized version of $\mathcal M$. |
| title | Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions |
| topic | Classical Analysis and ODEs 42B15, 42B25 |
| url | https://arxiv.org/abs/2310.00425 |