تقرير
Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions
| العنوان: | Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions |
|---|---|
| المؤلفون: | Bhojak, Ankit, Choudhary, Surjeet Singh, Shrivastava, Saurabh, Shuin, Kalachand |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Classical Analysis and ODEs, 42B15, 42B25 |
| الوصف: | 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 $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;d\sigma(y_1,y_2)\right|,$$ 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, \[\mathcal M (f,g)(x)=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty)g(x+ty)\;d\sigma(y)\right|\] 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 $\mathcal M$. Comment: 26 pages, 4 figures and 2 tables. We would like to thank Yumeng Ou, Tainara Borges and Benjamin Foster for pointing out a gap in the proof of Theorem 1.5 in the previous version. The result and its proof are modified accordingly. Other results remain unchanged |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2310.00425 |
| رقم الأكسشن: | edsarx.2310.00425 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |