Weak and strong type $A_1$-$A_\infty$ estimates for sparsely dominated operators

TL;DR

This paper establishes new weighted bounds for operators with sparse domination, improving previous results by removing certain conditions and providing optimality results for strong and weak type estimates.

Contribution

It introduces novel techniques for weighted bounds of sparsely dominated operators, including removal of H"ormander conditions and optimality analysis of bounds.

Findings

Proves weighted strong type bounds for a range of p

Establishes weighted weak type bounds with mixed A1-A_infinity estimates

Shows optimality of the obtained bounds

Abstract

We consider operators $T$ satisfying a sparse domination property \[ |\langle Tf,g\rangle|\leq c\sum_{Q\in\mathscr{S}}\langle f\rangle_{p_0,Q}\langle g\rangle_{q_0',Q}|Q| \] with averaging exponents $1\leq p_0<q_0\leq\infty$. We prove weighted strong type boundedness for $p_0<p<q_0$ and use new techniques to prove weighted weak type $(p_0,p_0)$ boundedness with quantitative mixed $A_1$-$A_\infty$ estimates, generalizing results of Lerner, Ombrosi, and P\'erez and Hyt\"onen and P\'erez. Even in the case $p_0=1$ we improve upon their results as we do not make use of a H\"ormander condition of the operator $T$. Moreover, we also establish a dual weak type $(q_0',q_0')$ estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.