Maximal function and Carleson measures in Békollé-Bonami weights

Let $ω$ be a Békollé-Bonami weight. We give a complete characterization of the positive measures $μ$ such that $\int_{\mathcal H}|M_ωf(z)|^qdμ(z)\le C\left(\int_{\mathcal H}|f(z)|^pω(z)dV(z)\right)^{q/p}$ and $μ\left(\{z\in \mathcal H: Mf(z)>λ\}\right)\le \frac{C}{λ^q}\left(\int_{\mathcal H}|f(z)|^pω(z)dV(z)\right)^{q/p}$ where $M_ω$ is the weighted Hardy-Littlewood maximal function on the upper-half plane $\mathcal H$, and $1\le p,q<\infty$.

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