An equivalence for weighted integrals of an analytic function and ist derivative

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Trace class criteria for Toeplitz and composition operators on small Bergman spaces (2014, unknown): “0 ω(s)ds = 1. We recall the Littlewood-Paley formula kfk p Ap v ≍ Z D |f (n )(z)| (1 −|z|)npv(z)dA(z) + nX−1 j=0 |f j (0)| , f ∈ H(D), valid for regular continuous weights v, 0 < p < ∞ and n ∈ N[15]. This implies k(Kω a ) (N)kp Ap v ≍ Z D |(Kω a ) (N+2)(z)|p(1 −|z|)2pv(z)dA(z) +|(Kω a ) (N)(0)”
Bergman projection induced by radial weight. (2019, unknown): “Different extensions of this equivalence as well as related partial results can be found in [2, 4, 5, 22].”
Integral Operators, Embedding Theorems and a Littlewood–Paley Formula on Weighted Fock Spaces (2013, methodology): “(2.9) gives (2.8). This ﬁnishes the proof. 3. A Littlewood-Paleyformula Our aim will be to obtain a Littlewood-Paley formula for a large class of weighted Fock spaces Fφ p , following the approach in [21]. Let us introduce some useful notation. For an entire function fand 0 ≤ r<∞, we set Mp(r,f) ”
Two weight inequality for Bergman projection (2014, background): “, and the proof is complete. 3.2. Proof of Theorem 1. We begin with proving (ii) for v∈ R continuous. In this case we have two advantages compared to the general case v∈ Db. First, the main result in [16] implies the Littlewood-Paley formula kfk p Ap v ≍ Z D |f (n )(z)| (1 −|z|)npv(z)dA(z) + nX−1 j=”