Maximal function and Carleson measures in Békollé-Bonami weights

Abstract: Let $\omega$ be a B\'ekoll\'e-Bonami weight. We give a complete characterization of the positive measures $\mu$ such that $$\int_{\mathcal H}|M_\omega f(z)|^qd\mu(z)\le C\left(\int_{\mathcal H}|f(z)|^p\omega(z)dV(z)\right)^{q/p}$$ and $$\mu\left({z\in \mathcal H: Mf(z)>\lambda}\right)\le \frac{C}{\lambda^q}\left(\int_{\mathcal H}|f(z)|^p\omega(z)dV(z)\right)^{q/p}$$ where $M_\omega$ is the weighted Hardy-Littlewood maximal function on the upper-half plane $\mathcal H$, and $1\le p,q<\infty$.