  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.
