Bilinear spherical maximal function with fractal dilations

Abstract

In this paper, we investigate the $L^p-$boundedness of the bilinear spherical maximal function associated with a general set of dilations $E\subset {\mathbb{R}}_{+}$. We quantify the range of $L^p-$boundedness in terms of a dilation-invariant notion of the upper Minkowski dimension of the set $E$. A particular case of this study settles an open question of $L^p-$boundedness of the maximal lacunary bilinear spherical function in borderline cases $p_1=1$ or $p_2=1$ in dimension $d\ge 4$.

arXiv commentThe new idea in the previous version had a gap. The manuscript has been revised accordingly, and the present version is essentially the same as Version 1