A quantitative approach to weighted Carleson condition
Abstract
Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea 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. Pérez and E. Rela and very recently by M.
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.