Lipschitz-type spaces of pluriharmonic mappings

Abstract

In this paper, we discuss the pluriharmonic mappings in the n-dimensional complex space Cn. Several characterizations for pluriharmonic mappings to be in Lipschitz-type spaces are given, which are generalizations of the corresponding results for harmonic functions. Our proofs are related to the corresponding results of Gehring and Martio, Lappalainen, Mateljevic, Dyakonov and Pavlovic.

Distortion of Quasiregular Mappings and Equivalent Norms on Lipschitz-Type Spaces

“…The following three theorems in [9] are a generalization of the corresponding one in [15]. Theorem 28.…”

“…In Section 3, we also consider Lipschitz-type spaces of pluriharmonic mappings and extend some results from [9].…”

“…In Section 2, we give a short proof of Lemma 4, using a version with the average Jacobian instead of , and we also characterize bi-Lipschitz mappings with respect to quasihyperbolic metrics by Jacobian and the average Jacobian; see Theorems 8,9, and 10 and Proposition 13. Gehring and Martio [2] extended HL-result to the class of uniform domains and characterized the domains with the property that functions which satisfy a local Lipschitz condition in for some always satisfy the corresponding global condition there.…”

Lipschitz Spaces and Bounded Mean Oscillation of Harmonic Mappings

“…Recently, many authors have considered this topic and generalised Dyakonov's results to quasiconformal mappings and real harmonic functions in several variables for some special majorant ω(t) = t α , where α > 0 (see [1,15,23,26,27,[29][30][31]). For the general majorant ω to holomorphic mappings and pluriharmonic mappings in the unit ball, see [7,15,33].…”

Lipschitz-Type and Bloch-Type Spaces of Pluriharmonic Mappings in a Hilbert Space

“…Later, Pavlović [3] came up with a relatively simple proof of the results of Dyakonov. For the generalizations of this topic, we refer to [4][5][6]. In this paper, we consider the corresponding problem in the case of P(Ω).…”