AI 中文总结
本文研究由倍权诱导的平均径向可积空间上的Cesàro型算子，给出有界性的充要条件并刻画紧性与本质范数，将已有结果推广至完整参数范围。
AI 中文摘要
本文研究Cesàro型算子$$ \Ces_{\mu,\beta}f(z)=\int_{[0,1)}\frac{f(tz)}{(1-tz)^\beta}\\,d\mu(t), $$其中\\(\mu\\)是\\([0,1)\\)上的正Borel测度且\\(\beta>0\\)。对于\\(0<p,q<\infty\\)和径向倍权\\(\omega_1,\omega_2\\)，我们证明\\(\Ces_{\mu,\beta}:\AL_p^q(\omega_1)\to\AL_p^q(\omega_2)\\)有界当且仅当$$ \mu([0,1))+\sup_{1/2\le r<1}\frac{\mu([r,1))}{(1-r)^\beta}\left(\frac{\what\omega_2(r)}{\what\omega_1(r)}\right)^{1/p}<\infty, $$且其范数与该量相当。对于标准权，我们的结果将\cite{BlascoMas2026,GalanopoulosSiskakisZhao2025}推广到完整范围\\(0<p=q<\infty\\)。我们进一步刻画了由倍权诱导的平均径向可积空间上的紧性与本质范数。
英文摘要
In this paper, we study the Cesàro-type operator $$ \Ces_{μ,β}f(z)=\int_{[0,1)}\frac{f(tz)}{(1-tz)^β}\,dμ(t), $$ where \(μ\) is a positive Borel measure on \([0,1)\) and \(β>0\). For \(0<p,q<\infty\) and radial doubling weights \(ω_1,ω_2\), we prove that \(\Ces_{μ,β}:\AL_p^q(ω_1)\to\AL_p^q(ω_2)\) is bounded if and only if $$ μ([0,1))+\sup_{1/2\le r<1}\frac{μ([r,1))}{(1-r)^β}\left(\frac{\whatω_2(r)}{\whatω_1(r)}\right)^{1/p}<\infty, $$ and its norm is comparable to this quantity. For standard weights, our result extends \cite{BlascoMas2026,GalanopoulosSiskakisZhao2025} to the full range $0<p=q<\infty$. We further characterize compactness and the essential norm on average radial integrability spaces induced by doubling weights.
