Carleson measures for the area Nevanlinna spaces and applications

Abstract

Let 1 <= p < infinity and let mu be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space N-p (mu) consists of all measurable functions h on D such that log(+) |h| is an element of L-p (mu), and the area Nevanlinna space N-alpha(p) is the subspace consisting of all holomorphic functions, in N-p((1-|z|(2))(alpha) dv(z)), where alpha > -1 and nu is area measure on D. We characterize Carleson measures for N-alpha(p), defined to be those measures mu for which N-alpha(p) subset of N-p(mu). As an application, we show that the spaces N-alpha(p) are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given.