Maximal theorems for weighted analytic tent and mixed norm spaces

TL;DR

This paper investigates maximal theorems for weighted analytic tent and mixed norm spaces, establishing boundedness of maximal operators, density of polynomials, and conditions for the Bergman projection's boundedness.

Contribution

It introduces new maximal inequalities for weighted tent and mixed norm spaces and characterizes the boundedness of the Bergman projection via a Bekollé-Bonami type condition.

Findings

Boundedness of the non-tangential maximal operator on weighted spaces.

Density of polynomials in the analytic weighted spaces.

Boundedness of the Bergman projection characterized independently of q.

Abstract

Let $\omega$ be a radial weight, $0<p,q<\infty$ and $\Gamma(\xi)=\left\{z\in\mathbb{D}:|\arg z-\arg\xi|<(|\xi|-|z|)\right\}$ for $\xi\in\overline{\mathbb{D}}$ . The average radial integrability space $L^q_p(\omega)$ consists of complex-valued measurable functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|^q_{L^q_p(\omega)}=\frac{1}{2\pi}\int_{0}^{2\pi}\left(\int_{0}^{1}|f(re^{i\theta})|^p\omega(r)r\,dr\right)^{\frac{q}{p}}d\theta <\infty,$$ and the tent space $T^q_p(\omega)$ is the set of those $f$ for which $$\|f\|^q_{T_{p}^{q}(\omega)}=\frac{1}{2\pi}\int_{\partial{\mathbb{D}}}\left(\int_{\Gamma(\xi)}|f(z)|^p\omega(z)\frac{dA(z)}{1-|z|}\right)^{\frac{q}{p}}\,|d\xi|<\infty.$$ Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in $\mathbb{D}$. It is shown that the non-tangential maximal operator $$f\mapsto N(f)(\xi)=\sup_{z\in\Gamma(\xi)}|f(z)|,\quad \xi\in \mathbb{D},$$ is bounded from $AL^q_p(\omega)=L^q_p(\omega)\cap\mathcal{H}(\mathbb{D})$ and $AT^q_p(\omega)=T^q_p(\omega)\cap\mathcal{H}(\mathbb{D})$ to $L^q_p(\omega)$ and $T^q_p(\omega)$, respectively. These pivotal inequalities are used to establish further results such as the density of polynomials in $AL^q_p(\omega)$ and $AT^q_p(\omega)$, and the identity $AL^q_p(\omega)=AT^q_p(\omega)$ for weights admitting a one-sided integral doubling condition. It is also shown that the boundedness of the classical Bergman projection $P_\gamma$, induced by the standard weight $(\gamma+1)(1-|z|^2)^{\gamma}$, on $L^q_p(\omega)$ and $T^q_p(\omega)$ with $1<q,p<\infty$ is independent of $q$, and is described by a Bekoll\'e-Bonami type condition.