Sharp bounds for fractional operator with $L^{\alpha ,r^{\prime }}$-Hörmander conditions

Abstract

In this paper we prove the sharp boundedness for a fractional type operator given by a kernel that satisfy a $L^{\alpha ,r^{\prime }}$-Hörmander conditions and a fractional size condition, where $0<\alpha <n$ and $1<r^{\prime }\le \infty$. To prove this result we use a new appropriate sparse domination which we provide in this work. For the case $r^{\prime }=\infty$ we recover the sharp boundedness for the fractional integral, $I_{\alpha }$, proved in [Lacey, M. T., Moen, K., Pérez, C., Torres, R. H. (2010). Sharp weighted bounds for fractional integral operators. Journal of Functional Analysis, 259(5), 1073-1097.]

17 pages, fixed some typos and added more examples