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Superposition Operators, Hardy Spaces, And Dirichlet Type Spaces

Petros Galanopoulos, Daniel Girela, Maria Auxiliadora Marquez
Published 2018 · Mathematics
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Abstract For 0 p ∞ and α > − 1 the space of Dirichlet type D α p consists of those functions f which are analytic in the unit disc D and satisfy ∫ D ( 1 − | z | ) α | f ′ ( z ) | p d A ( z ) ∞ . The space D p − 1 p is the closest one to the Hardy space H p among all the D α p . Our main object in this paper is studying similarities and differences between the spaces H p and D p − 1 p ( 0 p ∞ ) regarding superposition operators. Namely, for 0 p ∞ and 0 s ∞ , we characterize the entire functions φ such that the superposition operator S φ with symbol φ maps the conformally invariant space Q s into the space D p − 1 p , and, also, those which map D p − 1 p into Q s and we compare these results with the corresponding ones with H p in the place of D p − 1 p . We also study the more general question of characterizing the superposition operators mapping D α p into Q s and Q s into D α p , for any admissible triplet of numbers ( p , α , s ) .
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