On a variant of bilinear spherical maximal function

Abstract

We obtain $L^p-$estimates for the full and lacunary maximal functions associated to the bilinear spherical averages given by \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;dσ(z_1,z_2),\;t>0,\] for all dimensions $d\ge 1$. We show that the estimates for such operators in dimensions $d\ge 2$ essentially rely on the method of slicing. The bounds for the lacunary maximal function in dimension one are more delicate and require a trilinear smoothing inequality, which is based on an appropriate sublevel set estimate in this context.

Revised the manuscript based on the comments and suggestions by referees. The title has been modified suitably. 28 pages, 4 figures