A quantitative approach to weighted Carleson Condition

TL;DR

This paper develops quantitative weighted estimates for a maximal operator related to Carleson conditions, extending classical results and providing new bounds in the two-weight setting and for the Poisson integral.

Contribution

It introduces quantitative versions of weighted estimates for a maximal operator, extending classical results and deriving new bounds in the two-weight setting.

Findings

Quantitative estimates for the operator \\mathcal{M} are established.

Sufficient conditions for boundedness in the two-weight setting are derived.

New quantitative bounds for the Poisson integral are obtained.

Abstract

Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea in the 80's for the operator \[ \mathcal{M}f(x,t)=\sup_{x\in Q,\,l(Q)\geq t}\frac{1}{|Q|}\int_{Q}|f(x)|dx \qquad x\in\mathbb{R}^{n}, \, t \geq0 \] are obtained. As a consequence, some sufficient conditions for the boundedness of $\mathcal{M}$ in the two weight setting in the spirit of the results obtained by C. P\'erez and E. Rela and very recently by M.T. Lacey and S. Spencer for the Hardy-Littlewood maximal operator are derived. As a byproduct some new quantitative estimates for the Poisson integral are obtained.