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.
Categories: math.CA