Carleson measures for the area Nevanlinna spaces and applications

Let 1 ≤ p < ∞ and let μ be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space Np(μ) consists of all measurable functions h on D such that log+ |h| ∈ Lp(μ), and the area Nevanlinna space Nαp is the subspace consisting of all holomorphic functions, in Np((1−|z|2)αdv(z)), where α > −1 and ν is area measure on D. We characterize Carleson measures for Nαp, defined to be those measures μ for which Nαp ⊂ Np(μ). As an application, we show that the spaces Nαp 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.