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Abstract. We show that the maximal Nevanlinna counting function and theCarleson function of analytic self-maps of the unit disk are equivalent, up to constants.

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We characterize the compactness of composition operators; in term of generalized Nevanlinna counting functions, on a large class of Hilbert spaces of analytic functions, which can be viewed between the Bergman and the Dirichlet spaces1

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The Hardy spaces of Dirichlet series denoted by H p (p ≥ 1) have been studied in [12] when p = 2 and in [3] for the general case. In this paper we study some L p -generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces denoted A p and B p . We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings between these spaces and "Littlewood-Paley" formulas when p = 2. We also show that the B p spaces have properties similar to the classical Bergman spaces of the unit disk while the A p spaces have a different behavior.

We give a criterion of weak compactness for the operators on the Morse-Transue spaceMΨ, the subspace of the Orlicz spaceLΨgenerated byL∞.