Equivalence of norms of analytical functions on exterior of convex domain

Abstract

We study the spaces of functions holomorphic in the exterior of a bounded domain $D$ and vanishing at infinity. For each $\alpha >-\frac 12$ we introduce the integral weighted normed space $B_2^\alpha (G)$ with the weight $d^\alpha (z)$, where $d(z)$ denotes the distance from a point $z$ to the boundary of $G:=\mathbb{C} \setminus \overline{D}$. For $\alpha = - \frac 12$, the space $B_2^\alpha $ is chosen to be the Smirnov space. We prove that for a convex domain $D$, the norms in these spaces are equivalent to other norms defined in terms of the derivatives. For instance, the norm in the Smirnov space calculated as an integral with respect to the arc length over the boundary is equivalent to some norm defined by an integral with respect to the Lebesgue plane measure. In particular cases the proved results were obtained while studying the problem on describing the classes of Cauchy transforms of the functionals on the Bergman space on $D$. The general results may be applied in the study of Cauchy transforms of functionals on weighted Bergman spaces.