Twisted bilinear spherical maximal functions

TL;DR

This paper establishes $L^p$ bounds for twisted bilinear spherical maximal functions in all dimensions, using slicing methods for higher dimensions and a specialized inequality for the one-dimensional case.

Contribution

It introduces new $L^p$ estimates for these maximal functions, employing slicing techniques in higher dimensions and a novel trilinear smoothing inequality in one dimension.

Findings

$L^p$ bounds are proven for all dimensions $d \\geq 1$.

Slicing methods are effective for dimensions $d \\geq 2$.

A new sublevel set estimate is developed for the one-dimensional case.

Abstract

We obtain $L^p-$estimates for the full and lacunary maximal functions associated to the twisted bilinear spherical averages given by \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;d\sigma(z_1,z_2),\;t>0,\] for all dimensions $d\geq1$. We show that the estimates for such operators in dimensions $d\geq2$ essentially relies on the method of slicing. The bounds for the lacunary maximal function in dimension one is more delicate and requires a trilinear smoothing inequality which is based on an appropriate sublevel set estimate in this context.