Fractional Volterra-type operators from Bergman spaces with two-sided doubling weights to Hardy spaces
Abstract
For ω∈calD, we give necessary and sufficient symbol conditions for the boundedness and compactness of the Riemann--Liouville familyVφα,β:Apω→ Hq for every 0<p,q<∞ and admissible α,β>0: without any additional weight assumption when α≥β, and, when α<β, under the condition p(β−α)<d-(ω), where d-(ω) is the critical reverse-doubling exponent. We prove that this inequality is exactly equivalent to the naturally shifted source weight retaining the two-sided doubling and tail geometry required by the reduction; an integrable logarithmic example shows that the endpoint fails. The extension from power weights is not formal, because the reduction replaces ω by ω(z)(1−|z|2)p(α−β), and a negative shift may destroy even integrability. Combining moment-induced Littlewood--Paley theory for doubling Bergman spaces with an exact coefficient-multiplier comparison, we establish the required Riemann--Liouville transfer, including the finite-dimensional exceptional modes at positive integral shift orders. A finite-codimensional range decomposition and a fractional g-function reduction then reduce both questions to a single weighted area-map problem. Within the admissible parameter set, the effects of α are absorbed by the shifted source weight and bounded model corrections, so the final symbol conditions depend on β. The thresholds p=2 and p=q yield pointwise, Carleson, tent-integral, and non-tangential maximal criteria. In the tent-integral range, boundedness already implies compactness.
Related
- Compact linear combination of composition operators on weighted Bergman-Orlicz spaces induced by doubling weights
- Toeplitz and Hankel operators on Bergman spaces with doubling weights
- Generalized Volterra Companion Operators on Bergman Spaces over Convex Domains of Finite Type
- Cesàro-type operators on average radial integrability spaces induced by doubling weights
- Essential norms of fractional Volterra-type operators
BibTeX
@misc{zhu2026fractional,
title = {{Fractional Volterra-type operators from Bergman spaces with two-sided doubling weights to Hardy spaces}},
author = {Xiaolin Zhu and Feng Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15291},
eprint = {2609.15291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15291},
}
RIS
TY - UNPB
TI - Fractional Volterra-type operators from Bergman spaces with two-sided doubling weights to Hardy spaces
AU - Xiaolin Zhu
AU - Feng Guo
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15291
UR - https://arxiv.org/abs/2609.15291
AB - For ω∈calD, we give necessary and sufficient symbol conditions for the boundedness and compactness of the Riemann--Liouville familyV^φ_α,β:A^p_ω→ H^q for every 0<p,q<∞ and admissible α,β>0: without any additional weight assumption when α≥β, and, when α<β, under the condition p(β−α)<d_-(ω), where d_-(ω) is the critical reverse-doubling exponent. We prove that this inequality is exactly equivalent to the naturally shifted source weight retaining the two-sided doubling and tail geometry required by the reduction; an integrable logarithmic example shows that the endpoint fails. The extension from power weights is not formal, because the reduction replaces ω by ω(z)(1−|z|^2)^p(α−β), and a negative shift may destroy even integrability. Combining moment-induced Littlewood--Paley theory for doubling Bergman spaces with an exact coefficient-multiplier comparison, we establish the required Riemann--Liouville transfer, including the finite-dimensional exceptional modes at positive integral shift orders. A finite-codimensional range decomposition and a fractional g-function reduction then reduce both questions to a single weighted area-map problem. Within the admissible parameter set, the effects of α are absorbed by the shifted source weight and bounded model corrections, so the final symbol conditions depend on β. The thresholds p=2 and p=q yield pointwise, Carleson, tent-integral, and non-tangential maximal criteria. In the tent-integral range, boundedness already implies compactness.
ER -