加权伯格曼空间的加布里埃尔问题和弗雷泽问题及其应用
Gabriel's and Frazer's problems for weighted Bergman spaces and their applications
AI总结：
研究加权伯格曼空间的加布里埃尔和弗雷泽问题，建立加权积分不等式，证明加布里埃尔不等式在加权调和伯格曼空间全范围成立，还研究弗雷泽问题，推导相关空间的加布里埃尔型和弗雷泽型不等式。
AI中文摘要：
在本文中，我们研究了解析和复值调和加权伯格曼空间的加布里埃尔问题和弗雷泽问题。具体而言，我们建立了形如\[ \int_C |f(z)|^p(1-|z|^2)^{\alpha+1}\,|dz| \leq K_{p,\alpha,C} \int_{\mathbb D}|f(z)|^p(1-|z|^2)^\alpha\,dA(z) \]的加权积分不等式，其中$f$是单位圆盘$\mathbb D$上的解析或复值调和函数，$C$是包含在$\mathbb{D}$中的任意凸曲线。相应问题最早由加布里埃尔针对解析哈代空间进行研究，该不等式对每个$0<p<\infty$都成立。相比之下，调和哈代空间的类似情况最近表明，当$0<p\le1$时不成立。我们通过在整个$0<p<\infty$范围内建立加布里埃尔不等式，证明了这种现象在加权调和伯格曼空间中不会出现。我们还进一步研究了圆以及两条相交直径的并集的弗雷泽问题。作为主要结果的重要应用，我们推导了解析和调和莫比乌斯不变空间$Q(n,p,\alpha)$和$Q_h(n,p,\alpha)$的加布里埃尔型和弗雷泽型不等式。
英文摘要：
In this paper, we investigate Gabriel's and Frazer's problems for analytic and complex-valued harmonic weighted Bergman spaces. More precisely, we establish weighted integral inequalities of the form \[ \int_C |f(z)|^p(1-|z|^2)^{α+1}\,|dz| \leq K_{p,α,C} \int_{\mathbb D}|f(z)|^p(1-|z|^2)^α\,dA(z), \] where $f$ is an analytic or complex-valued harmonic function on the unit disk $\mathbb D$ and $C$ is an arbitrary convex curve contained in $\mathbb{D}$. The corresponding problem was first studied by Gabriel [Proc. Lond. Math. Soc. 28 (1928), 121--127] for analytic Hardy spaces, where the inequality holds for every $0<p<\infty$. In contrast, the harmonic Hardy space analogue was recently shown to fail whenever $0<p\le1$. We prove that this phenomenon does not occur in the weighted harmonic Bergman setting by establishing Gabriel's inequality throughout the full range $0<p<\infty$. We further study Frazer's problem for circles and for the union of two intersecting diameters. As important applications of our main results, we derive Gabriel-type and Frazer-type inequalities for the analytic and harmonic Möbius invariant spaces $Q(n,p,α)$ and $Q_h(n,p,α)$.
