Cesàro-type operators on mixed norm spaces

2025/07/28 by Óscar Blasco, Blasco, Óscar, Alejandro Mas +1
Mathematics · #30H20 #47B38 #Analytic and geometric function theory #Approximation Theory and Sequence Spaces #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2507.20586

openalex publication_date 2025/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a positive Borel measure μ on [0,1) and a parameter β>0, we consider the Cesàro-type operator \mathcal C_μ,β acting on the analytic function f(z)=∑_n=0^∞ a_n zⁿ on the unit disc of the complex plane \mathbb D, defined by \mathcal C_μ,β(f)(z)= ∑_n=0^∞ μ_n ( ∑_k=0ⁿ (Γ(n-k+β))/((n-k)! Γ(β)) a_k ) zⁿ = ∫₀¹ (f(tz))/((1-tz)^β) dμ(t), where μ_n=∫₀¹ tⁿ dμ(t). This operator generalizes the classical Cesàro operator (corresponding to the case where μ is the Lebesgue measure and β=1) and includes other relevant cases previously studied in the literature. In this paper we study the boundedness of \mathcal C_μ,β on mixed norm spaces H(p,q,γ) for 00. Our results extend and unify several known characterizations for the boundedness of Cesàro-type operators acting on spaces of analytic functions.

Citations

• Hankel matrices acting on the Dirichlet space
• A generalized Hilbert matrix acting on Hardy spaces
• Theory of Bergman Spaces in the Unit Ball of Cⁿ

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