2016/11/30 by Πέτρος Γαλανόπουλος, Petros Galanopoulos, Daniel Girela +1 · 5 citations
Mathematics · #Algebraic and Geometric Analysis #Combinatorics #Dirichlet distribution #Functional analysis #Hardy space #Holomorphic and Operator Theory #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Meromorphic and Entire Functions #Operator (biology) #Pure mathematics #Space (punctuation) #Superposition principle #Symbol (formal) #Type (biology) #math.CV #msc:30H10 #msc:47B35
paper · pdf · doi:10.1016/j.jmaa.2018.03.044
published in Journal of Mathematical Analysis and Applications 463(2), 659-680 (Elsevier BV)
arxiv created 2018/02/15 · openalex publication_date 2018/03/22 · arxiv updated 2018/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For 0<p<∞ and α>-1 the space of Dirichlet type \mathcal Dpα consists of those functions f which are analytic in the unit disc \mathbb D and satisfy ∫\mathbb D(1-| z| )α| f^′ (z)| p dA(z)<∞ . The space \Dp is the closest one to the Hardy space Hp among all the \mathcal Dpα. Our main object in this paper is studying similarities and differences between the spaces Hp and \Dp (0<p<∞ ) regarding superposition operators. Namely, for 0<p<∞ and 0<s<∞ , we characterize the entire functions φ such that the superposition operator Sφ with symbol φ maps the conformally invariant space Qs into the space \Dp, and, also, those which map \Dp into Qs and we compare these results with the corresponding ones with Hp in the place of \Dp. We also study the more general question of characterizing the superposition operators mapping \mathcal Dpα into Qs and Qs into \mathcal Dpα, for any admissible triplet of numbers (p, α, s).