Generalized Bloch spaces, Integral means of hyperbolic harmonic mappings in the unit ball

TL;DR

This paper explores the properties of hyperbolic harmonic mappings in the unit ball, establishing conditions for their inclusion in Bloch spaces, relating their integral means to gradients, and characterizing their boundedness via the quasihyperbolic metric.

Contribution

It introduces new criteria for hyperbolic harmonic mappings to belong to generalized Bloch spaces and links integral means with gradients, extending classical results to hyperbolic harmonic context.

Findings

Necessary and sufficient conditions for Bloch space membership

Relationship between integral means and gradients of mappings

Characterization of boundedness using quasihyperbolic metric

Abstract

In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball $\mathbb{B}^{n}$ in $\IR^n$ $(n\geq 2)$. Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space $\mathcal{B}(\mathbb{B}^{n})$ and the generalized Bloch space $\mathcal{L}_{\infty,\omega}\mathcal{B}_{\alpha,\mathrm{a}}^{0}(\mathbb{B}^{n})$, respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's related ones in the setting of hyperbolic harmonic mappings. Finally, we characterize the weak uniform boundedness property of hyperbolic harmonic mappings in terms of the quasihyperbolic metric.