Littlewood-Paley inequalities for fractional derivative on Bergman spaces

2021/09/27 by Peláez, José Ángel, de la Rosa, Elena · 3 citations
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2109.12944

Abstract

For any pair (n,p), n∈ℕ and 01 such that inf_{0≤ r<1} \frac∫_r¹ω(s) ds∫_1-(1-r)/(k)¹ ω(s) ds>1. In this paper we extend this result to the setting of fractional derivatives. Being precise, for an analytic function f(z)=∑_n=0^∞ \widehatf(n) zⁿ we consider the fractional derivative D^μ(f)(z)=∑_n=0^∞ \frac\widehatf(n)μ_2n+1 zⁿ induced by a radial weight μ∈ D, where μ_2n+1=∫₀¹ r²ⁿ⁺¹μ(r) dr. Then, we prove that for any p∈ (0,∞), the Littlewood-Paley equivalence ∫_\mathbbD |f(z)|^p ω(z) dA(z)\asymp ∫_\mathbbD|D^μ(f)(z)|^p[∫_|z|¹μ(s) ds]^pω(z) dA(z) holds for any analytic function f in \mathbbD if and only if ω\inD. We also prove that for any p∈ (0,∞), the inequality ∫_\mathbbD |D^μ(f)(z)|^p [∫_|z|¹μ(s) ds]^pω(z) dA(z) \lesssim ∫_\mathbbD |f(z)|^p ω(z) dA(z) holds for any analytic function f in \mathbbD if and only if ω∈ \widehatD.

Cited by

• Duality of mixed norm spaces induced by radial one-sided doubling weight
• Fractional Volterra-type operator induced by radial weight acting on Hardy space
• Embedding theorems for Bergman-Zygmund spaces induced by doubling weights

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