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

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.