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

TL;DR

This paper establishes sharp bounds for a fractional operator with kernels satisfying specific Hormander conditions, using a novel sparse domination technique, extending known results for fractional integrals.

Contribution

Introduces a new sparse domination approach to prove sharp boundedness of fractional operators under Hormander conditions, generalizing previous fractional integral bounds.

Findings

Proves sharp boundedness for fractional operators with Hormander conditions.

Develops a new sparse domination method for these operators.

Recovers known bounds for fractional integrals as a special case.

Abstract

In this paper we prove the sharp boundedness for a fractional type operator given by a kernel that satisfy a $L^{\alpha,r'}$-H\"ormander conditions and a fractional size condition, where $0<\alpha<n$ and $1< r'\leq \infty$. To prove this result we use a new appropriate sparse domination which we provide in this work. For the case $r'=\infty$ we recover the sharp boundedness for 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.]