Maximal theorems for weighted analytic tent and mixed norm spaces

Abstract

Let $\ddot{I}‰$ be a radial weight, $0<p,q<\infty$ and $\hat{I}“(\hat{I}¾)=\left\{z\in \mathbb{D}:|\arg z-\arg \hat{I}¾|<(|\hat{I}¾|-|z|)\right\}$ for $\hat{I}¾\in \overline{\mathbb{D}}$ . The average radial integrability space $L_p^q(\ddot{I}‰)$ consists of complex-valued measurable functions $f$ on the unit disc $\mathbb{D}$ such that $$\Vert f{\Vert }_{L_p^q(\ddot{I}‰)}^q=\frac{1}{2\ddot{I}€}{\int }_0^{2\ddot{I}€}{\left({\int }_0^1|f(r{e}^{i\hat{I}¸}){|}^p\ddot{I}‰(r)rdr\right)}^{\frac{q}{p}}d\hat{I}¸<\infty ,$$ and the tent space $T_p^q(\ddot{I}‰)$ is the set of those $f$ for which $$\Vert f{\Vert }_{T_p^q(\ddot{I}‰)}^q=\frac{1}{2\ddot{I}€}{\int }_{\partial \mathbb{D}}{\left({\int }_{\hat{I}“(\hat{I}¾)}|f(z){|}^p\ddot{I}‰(z)\frac{dA(z)}{1-|z|}\right)}^{\frac{q}{p}}|d\hat{I}¾|<\infty .$$ Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in $\mathbb{D}$. It is shown that the non-tangential maximal operator $$f\mapsto N(f)(\hat{I}¾)=\underset{z\in \hat{I}“(\hat{I}¾)}{\sup }|f(z)|,\hat{I}¾\in \mathbb{D},$$ is bounded from $A{L}_p^q(\ddot{I}‰)=L_p^q(\ddot{I}‰)\cap \mathcal{H}(\mathbb{D})$ and $A{T}_p^q(\ddot{I}‰)=T_p^q(\ddot{I}‰)\cap \mathcal{H}(\mathbb{D})$ to $L_p^q(\ddot{I}‰)$ and $T_p^q(\ddot{I}‰)$, respectively. These pivotal inequalities are used to establish further results such as the density of polynomials in $A{L}_p^q(\ddot{I}‰)$ and $A{T}_p^q(\ddot{I}‰)$, and the identity $A{L}_p^q(\ddot{I}‰)=A{T}_p^q(\ddot{I}‰)$ for weights admitting a one-sided integral doubling condition. It is also shown that the boundedness of the classical Bergman projection $P_{\hat{I}}^3$, induced by the standard weight $({\hat{I}}^3+1)(1-|z{|}^2{)}^3$, on $L_p^q(\ddot{I}‰)$ and $T_p^q(\ddot{I}‰)$ with $1<q,p<\infty$ is independent of $q$, and is described by a Bekollé-Bonami type condition.