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

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 $1\le p_0<q_0\le \infty$. 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 Pérez and Hytönen and Pérez. Even in the case $p_0=1$ we improve upon their results as we do not make use of a Hörmander condition of the operator $T$. Moreover, we also establish a dual weak type $(q_0^{\prime },q_0^{\prime })$ 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.

Minor modifications. Version published in Journal of Geometric Analysis

References in corpus (3)

• A sparse domination principle for rough singular integrals
• Sparse bilinear forms for Bochner Riesz multipliers and applications
• Sparse bounds for maximal rough singular integrals via the Fourier transform

Cited by in corpus (1)

• Sparse domination for the lattice Hardy-Littlewood maximal operator