发表机构
Taiyuan Normal University; Nanjing University of Aeronautics and Astronautics（太原师范学院; 南京航空航天大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出了分数阶Volterra型算子从加倍权Bergman空间到Hardy空间有界性与紧性的充要符号条件，通过Littlewood-Paley理论与约化方法，最终条件仅依赖参数β。
AI 中文摘要
对于\\(\omega\in\calD\\)，我们给出了Riemann-Liouville族\\(V^\varphi_{\alpha,\beta}:A^p_\omega\to H^q\\)在任意\\(0<p,q<\infty\\)及容许的\\(\alpha,\beta>0\\)下有界性和紧性的充分必要条件：当\\(\alpha\ge\beta\\)时无需任何额外权假设；当\\(\alpha<\beta\\)时，需满足条件\\(p(\beta-\alpha)<d_-(\omega)\\)，其中\\(d_-(\omega)\\)是临界反加倍指数。我们证明该不等式恰好等价于自然平移后的源权保持约化所需的双侧加倍和尾部几何性质；一个可积的对数例子表明端点情形失效。从幂权的推广并非形式上的，因为约化将\\(\omega\\)替换为\\(\omega(z)(1-|z|^2)^{p(\alpha-\beta)}\\)，而负平移可能破坏可积性。结合加倍Bergman空间的矩诱导Littlewood-Paley理论与精确的系数乘子比较，我们建立了所需的Riemann-Liouville传递，包括正整数平移阶时的有限维例外模态。一个有限余维的值域分解和分数阶\\(g\\)-函数约化将两个问题归结为单个加权面积映射问题。在容许参数集内，\\(\alpha\\)的影响被平移后的源权和有界模型修正吸收，因此最终符号条件仅依赖于\\(\beta\\)。阈值\\(p=2\\)和\\(p=q\\)给出逐点、Carleson、帐篷积分和非切向极大判据。在帐篷积分范围内，有界性已蕴含紧性。
英文摘要
For \(ω\in\calD\), we give necessary and sufficient symbol conditions for the boundedness and compactness of the Riemann--Liouville family\(V^φ_{α,β}:A^p_ω\to H^q\) for every \(0<p,q<\infty\) and admissible \(α,β>0\): without any additional weight assumption when \(α\geβ\), 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.
Comments46 pages
