A Hardy–Littlewood theorem for Bergman spaces

Abstract

We study positive weight functions ω(z) on the unit disk D such that D |f (z)| p ω(z) dA(z) < ∞ if and only ifˆD (1 − |z| 2) p |f ′ (z)| p ω(z) dA(z) < ∞, where f is analytic on D and dA is area measure on D. We obtain some conditions on ω that imply the equivalence above, and we apply our conditions to several important classes of weights that have appeared in the literature before. D |f (z) − f (0)| p dA α (z) ∼ˆD(1 − |z| 2) p |f ′ (z)| p dA α (z) for f ∈ H(D). See [6, 14]. It is then natural to ask for conditions on finite positive Borel measures µ on D such that D |f (z) − f (0)| p dµ(z) ∼ˆD(1 − |z| 2) p |f ′ (z)| p dµ(z)

Bergman projection induced by radial weight

“…where ω rβs pzq " ωpzqp1 ´|z|q β . Different extensions of this equivalence as well as related partial results can be found in [2,4,5,22]. The question for which radial weights the above equivalence is valid has been a known open problem for decades, and gets now completely answered by our next result.…”

Double integral estimates for Besov type spaces and their applications

“…This kind of estimates is usually called Littlewood-Paley estimates. We refer to [1,2,3,8,18,20] for the study of Littlewood-Paley estimates for A p ω ; that is, to characterize weight functions ω such that…”

“…The following Littlewood-Paley estimates for A p ω [8] will be useful in this paper. For a given weight ω, the conditions appeared in the following theorems are easy to verify.…”

“…where the comparison constants are independent of ζ, u and η. These facts allow us to apply a Littlewood-Paley estimate for Bergman spaces with nonradial weights ω ζ (u) = (1 − |u| 2 ) τ /|1 − ζu| τ −σ (see Theorem A and its proof in [8]). We get…”

“…where C is a positive constant independent of ζ. Using Theorem B and checking its proof in [8], we get that there exits another positive constant C independent of ζ such that…”