由Bergman核诱导的Cesàro算子
Cesàro operator induced by a Bergman kernel
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- Universitat de Valencia（瓦伦西亚大学）
- University of Eastern Finland（芬兰东部大学）
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中文总结 AI 辅助
本文研究由加权Bergman空间再生核诱导的Cesàro型算子在Hardy空间和加权Bergman空间上的有界性，利用尾积分或矩给出完全刻画，结果在标准权及Cauchy核情形下亦为新。
中文摘要 AI 辅助
设$\mu$为$[0,1)$上的正Borel测度，$\omega$为径向权。本文考虑由加权Bergman空间$A^2_{\omega}$的再生核$B^{\omega}$诱导的Cesàro型算子$C_{\mu,\omega}$，其定义为$$ C_{\mu,\omega}(f)(z)=\int_{0}^{1}f(tz)B^{\omega}_t(z)\\,d\mu(t), \quad z \in \mathbb{D}, $$其中$f$在$\mathbb{D}$上解析。在$\omega$满足自然加倍性质的假设下，我们研究$C_{\mu,\omega}$在若干解析函数空间（包括Hardy空间$H^p$和加权Bergman空间$A^p_{\nu}$）上的有界性。对于$0<p,q<\infty$和双侧加倍权$\nu$，我们完全刻画了$C_{\mu,\omega}: H^p \to H^q$与$C_{\mu,\omega}: A^p_{\nu} \to A^q_{\nu}$有界性的条件，该条件以诱导权与测度$\mu$的尾积分或矩的相互作用表述。所得结果中许多即使在标准权或Bergman再生核替换为Cauchy核的情形下也是新的。此外，我们还考虑了$C_{\mu,\omega}$在$H^{\infty}$、Korenblum空间和加权Hardy空间上的作用。
英文摘要
Let $μ$ be a positive Borel measure on $[0,1)$ and $ω$ a radial weight. In this paper we consider the Cesàro-type operator $C_{μ,ω}$ induced by the reproducing kernel $B^ω$ of the weighted Bergman space $A^2_ω$, given by $$ C_{μ,ω}(f)(z)=\int_{0}^{1}f(tz)B^ω_t(z)\,dμ(t), \quad z \in \mathbb{D}, $$ for functions $f$ analytic in $\mathbb{D}$. Under the assumption that $ω$ satisfies a natural doubling property, we study the boundedness of $C_{μ,ω}$ acting on several spaces of analytic functions, including Hardy spaces $H^p$ and weighted Bergman spaces $A^p_ν$. For $0<p,q<\infty$ and a two-sided doubling weight $ν$, we completely characterize when $C_{μ,ω}: H^p \to H^q$ and $C_{μ,ω}: A^p_ν \to A^q_ν$ are bounded in terms of the interplay of tail integrals or moments of the inducing weights and the measure $μ$. Many of the results obtained are new even in the setting of standard weights or when the Bergman reproducing kernel is replaced by the Cauchy kernel. In addition, we consider $C_{μ,ω}$ acting on $H^{\infty}$, Korenblum spaces and weighted Hardy spaces.
