Maximal theorems for weighted analytic tent and mixed norm spaces

Abstract

Let ω be a radial weight, 0 < p, q < ∞ and Γ(ξ) = {z ∈ D : | arg z − arg ξ| < (|ξ| − |z|)} for ξ ∈ D. The average radial integrability space Lq p(ω) consists of complex-valued measurable functions f on the unit disc D such that ∥f ∥q Lq p(ω) = 1 2π 2πˆ 0 ⎛ ⎝ 1ˆ 0 |f (reiθ)|pω(r)r dr ⎞ ⎠ q p dθ < ∞, and the tent space T q p (ω) is the set of those f for which ∥f ∥q T q p (ω) = ˆ ∂D ⎛ ⎜ ⎝ ˆ 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) = sup z∈Γ(ξ) |f (z)|, ξ ∈ D, is bounded from ALq p(ω) = Lq p(ω) ∩ ℋ(D) and AT q p (ω) = T q p (ω) ∩ ℋ(D) to Lq p(ω) and T q p (ω), respectively. These piv- otal inequalities are used to establish further results such as the density of polynomials in ALq p(ω) and AT q p (ω), and the identity ALq p(ω) = AT q p (ω) for weights admitting a one-sided integral doubling condition. Further, it is shown that any of the Littlewood-Paley formulas ∥f ∥ALq p(ω) ≍ ∥f (k)(1 − | · |)k∥Lq p(ω) + k−1∑︂ j=0 |f (j)(0)|, f ∈ ℋ(D), ∥f ∥AT q p (ω) ≍ ∥f (k)(1 − | · |)k∥T q p (ω) + k−1∑︂ j=0 |f (j)(0)|, f ∈ ℋ(D), holds if and only if ω admits a two-sided integral doubling condition. It is also shown that the boundedness of the clas- sical Bergman projection Pγ , induced by the standard weight (γ + 1)(1 − |z|2)γ , on Lq p(ω) and T q p (ω) with 1 < q, p < ∞ is independent of q, and is described by a Bekollé-Bonami type condition.