Carleson Measures for the Area Nevanlinna Spaces and Applications

Let 1 ≤ p < ∞ and let μ be a positive finite 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 N α is the subspace of N p((1 − |z|2)αdν(z)), where α > −1 and ν is area measure on D, consisting of all holomorphic functions. We characterize Carleson measures for N α, defined to be those measures μ for which N α ⊂ Np(μ). One application is that the spaces N α are all 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.