Maximal theorems for weighted analytic tent and mixed norm spaces

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.