A quantitative approach to weighted Carleson Condition

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.