Inequalities on Tent Spaces and Closed Range Integration Operators on spaces of Average Radial Integrability

Abstract

We deal with a reverse Carleson measure inequality for the tent spaces of analytic functions in the unit disc $\mathbb{D}$ of the complex plane. The tent spaces of measurable functions were introduced by Coifman, Meyer and Stein. Let $1\leq p,q < \infty$ and consider the positive Borel measure $d\mu(z) = \chi_{G}(z)\frac{dm(z)}{(1-|z|)}$ defined in terms of a measurable set $G \subseteq \mathbb{D}$ and of the area Lebesgue measure $dm(z)$ in $\mathbb{D}$. We prove a necessary and sufficient condition on $G$ in order to exist a constant $K>0$ such that $$ \int_{\mathbb{T}} \left(\int_{\Gamma(\xi)} |f(z)|^{p}\ d\mu(z) \right)^{q/p}\ |d\xi|\geq K \,\int_{\mathbb{T}} \left(\int_{\Gamma(\xi)} |f(z)|^{p}\ \frac{dm(z)}{1-|z|}\right)^{q/p}\ |d\xi|, $$ for any $f$ analytic in $\mathbb{D}$ with the property, the right term of the inequality above is finite. Here $\mathbb{T}$ stands for the unit circle and $\Gamma(\xi)$ is a non-tangential region with vertex at $\xi \in \mathbb{T}$. This work extends the study of D. Luecking on Bergman spaces to the analytic tent spaces. We apply our result to characterize the closed range property of the integration operator known as Pommerenke operator when acting on the average radial integrability spaces. T. Aguilar-Hern\'andez, M. Contreras and L. Rodr\'iguez-Piazza introduced these spaces for the first time in the literature. The Hardy and the Bergman spaces form part of this family.