Complete mapping criteria for generalized Cesàro operators between Hardy spaces
Abstract
Let μ be a finite positive Borel measure on [0,1) and let γ>0. We establish sharp mapping criteria for the generalized Cesàro operator 𝒞μ,γf(z) =∑n=0∞ μn (∑k=0n (Γ(n−k+γ))/(Γ(γ)(n−k)!)ak)zn, z∈ 𝔻, between Hardy spaces, including the source and target H∞ endpoints. For 0<p<q<∞, for 0<p=q<1, and for 0<p≤1 with q=∞, the boundedness of $ \mathcal C_{μ,γ}: H^p \to H^q$ is equivalent to μ being a (γ+1/p−1/q)-Carleson measure, without further restrictions on γ. For 1≤ q<p≤∞, set 1/r=1/q−1/p. In this range, boundedness and compactness are equivalent to ∫01 (μ([t,1))/((1−t)γ−1/r))r dt/(1−t)<∞. This condition is also equivalent to Fμ,γ∈ Hr and to ∑n≥0(n+1)rγ−2μnr<∞, where Fμ,α=𝒞μ,α(1) and μn=∫[0,1)tn dμ(t). For 1<p<∞, boundedness and compactness from Hp to H∞ are characterized by the shifted condition Fμ,γ+1∈ Hp', where p'=p/(p−1). We therefore obtain a complete boundedness classification of the generalized Cesàro operators 𝒞μ,γ between Hardy spaces Hp and Hq for the full range 0<p,q≤∞.
Related
- Vector-Carleson, Calderón, and Poisson-Atomic Criteria for Generalized Hilbert Operators on Hp, p>2
- Integral Operators on Fractional Cesàro--Morrey Spaces over ℚp
- On absolutely Cesàro bounded operators
- The Cesàro Operator, Rhaly operators, and the Toeplitz Algebra
- Cesàro-type operators on average radial integrability spaces induced by doubling weights
BibTeX
@misc{tang2026complete,
title = {{Complete mapping criteria for generalized Cesàro operators between Hardy spaces}},
author = {Pengcheng Tang and Huayou Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19725},
eprint = {2609.19725},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19725},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Complete mapping criteria for generalized Cesàro operators between Hardy spaces
AU - Pengcheng Tang
AU - Huayou Xie
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19725
UR - https://arxiv.org/abs/2609.19725
AB - Let μ be a finite positive Borel measure on [0,1) and let γ>0. We establish sharp mapping criteria for the generalized Cesàro operator 𝒞_μ,γf(z) =∑_n=0^∞ μ_n (∑_k=0^n (Γ(n−k+γ))/(Γ(γ)(n−k)!)a_k)z^n, z∈ 𝔻, between Hardy spaces, including the source and target H^∞ endpoints. For 0<p<q<∞, for 0<p=q<1, and for 0<p≤1 with q=∞, the boundedness of $ \mathcal C_{μ,γ}: H^p \to H^q$ is equivalent to μ being a (γ+1/p−1/q)-Carleson measure, without further restrictions on γ. For 1≤ q<p≤∞, set 1/r=1/q−1/p. In this range, boundedness and compactness are equivalent to ∫_0^1 (μ([t,1))/((1−t)^γ−1/r))^r dt/(1−t)<∞. This condition is also equivalent to F_μ,γ∈ H^r and to ∑_n≥0(n+1)^rγ−2μ_n^r<∞, where F_μ,α=𝒞_μ,α(1) and μ_n=∫_[0,1)t^n dμ(t). For 1<p<∞, boundedness and compactness from H^p to H^∞ are characterized by the shifted condition F_μ,γ+1∈ H^p', where p'=p/(p−1). We therefore obtain a complete boundedness classification of the generalized Cesàro operators 𝒞_μ,γ between Hardy spaces H^p and H^q for the full range 0<p,q≤∞.
ER -