Sharp bounds for fractional operator with $L^{\alpha,s}$-H\"ormander conditions

Abstract

In this paper we provide the sharp bound for a fractional type operator given by a kernel satisfying the$L^{alpha,s}$-H"ormander condition and fractional size condition, $0<alpha<n$ and $1< sleq infty$. To prove this result we use a new appropriate sparse domination provided in this work. Examples of these operators include the fractional rough operators. For the case $s=infty$ we recover the sharp bound of the fractional integral, $I_{alpha}$, proved in [Lacey, M. T., Moen, K., P´erez, C., Torres, R. H. (2010). Sharp weighted bounds for fractional integral operators. Journal of Functional Analysis, 259(5), 1073-1097].