A note on bilinear spherical maximal functions

Ankit Bhojak, Surjeet Singh Choudhary, Saurabh Shrivastava, Kalachand Shuin

Published 2023-09-30Version 1

In this article we address endpoint issues for the bilinear spherical maximal functions. We 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 linearizarion of $\mathcal M$. We also obtain borderline restricted weak type estimates for the well studied operator $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty_1)g(x-ty_2)\;d\sigma(y_1,y_2)\right|,$$ in dimension one and as a consequence, deduce similar estimates for a multilinear analogue of $\mathfrak{M}$.