Lipschitz type, radial growth and Dirichlet type spaces on functions induced by certain elliptic operators

TL;DR

This paper extends known results on analytic functions to classes of functions related to elliptic operators, exploring their growth, properties, and Dirichlet energy integrals, with implications for function space theory.

Contribution

It generalizes key theorems from analytic function theory to broader classes associated with elliptic operators, including growth and energy integral analyses.

Findings

Extension of Dyakonov's theorem to elliptic operator-related functions

Generalization of Makarov and Korenblum's radial growth results

Discussion of Dirichlet type energy integrals and applications

Abstract

In this paper, we investigate properties of classes of functions related to certain elliptic operators. Firstly, we prove that a main result of Dyakonov (Acta Math. 178(1997), 143--167) on analytic functions can be extended to this more general setting. Secondly, we study the radial growth on these functions and the obtained results are generalizations of the corresponding results of Makarov (Proc. London Math. Soc. 51(1985), 369--384) and Korenblum (Bull. Amer. Math. Soc. 12(1985), 99--102). Finally, we discuss the Dirichlet type energy integrals on such classes of functions and their applications.