Lacunary Series in Dirichlet-Type Spaces and Pseudoanalytic Extension

Under some mild conditions on the weight function ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}, we characterize lacunary series in Dirichlet-type spaces. Moreover, we also obtain a new characterization of Dirichlet-type spaces in terms of pseudoanalytic extension. Two applications are also given.