Gabriel's and Frazer's problems for weighted Bergman spaces and their applications

Abstract

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 $0<p\le 1$. 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 $Q(n,p,\alpha )$ and $Q_h(n,p,\alpha )$.

14 pages