Maximal theorems for weighted analytic tent and mixed norm spaces

Let ω be a radial weight, 0<p,q<∞ and Γ(ξ)=:| z-ξ|<(|ξ|-|z|)\ for ξ̄ . The average radial integrability space Lqₚ(ω) consists of complex-valued measurable functions f on the unit disc D such that \|f\|qLqₚ(ω)=1/2π∫₀2π(∫₀¹|f(reiθ)|ᵖω(r)r\,dr)q/pdθ <∞, and the tent space Tqₚ(ω) is the set of those f for which \|f\|q_Tₚq(ω)=1/2π∫_∂D(∫Γ(ξ)|f(z)|ᵖω(z)dA(z)/1-|z|)q/p\,|dξ|<∞. Let H(D) denote the space of analytic functions in D. It is shown that the non-tangential maximal operator f↦ N(f)(ξ)=z∈Γ(ξ)|f(z)|, ξ∈ D, is bounded from ALqₚ(ω)=Lqₚ(ω)(D) and ATqₚ(ω)=Tqₚ(ω)(D) to Lqₚ(ω) and Tqₚ(ω), respectively. These pivotal inequalities are used to establish further results such as the density of polynomials in ALqₚ(ω) and ATqₚ(ω), and the identity ALqₚ(ω)=ATqₚ(ω) for weights admitting a one-sided integral doubling condition. Further, it is shown that any of the Littlewood-Paley formulas holds if and only if ω admits a two-sided integral doubling condition. It is also shown that the boundedness of the classical Bergman projection P_γ, induced by the standard weight (γ+1)(1-|z|²)γ, on Lqₚ(ω) and Tqₚ(ω) with 1<q,p<∞ is independent of q, and is described by a Bekoll\'e-Bonami type condition.