Generalized Hölder Spaces of Holomorphic Functions in Domains in the Complex Plane

Holomorphic Hölder‐type spaces and composition operators

“…Here, we recall the definitions and some properties of the spaces under investigation following the notation in Karapetyants and Samko. 34 We start with the generalized Hölder spaces with a prescribed modulus of continuity. Let ∶ [0, 2] → R + be a modulus of continuity.…”

“…Here, we treat the generalized Hölder‐type spaces of holomorphic functions with prescribed behaviour near the unit circle determined by a modulus of continuity ω = ω ( h ). These type of spaces attract an attention due to numerous applications, and we refer to the books, 25,30 references therein and papers 25,30‐38 , 39 . These spaces are closely related to their counterparts in some sense, which are the generalized Besov‐type spaces of functions with prescribed behaviour of their derivatives.…”

“…For a certain class of modulus of continuity, both spaces coincide, but in general, they are different. When studying the generalized Hölder‐type spaces, we follow ideas of the papers 40,41 where such type spaces first appeared and also the recent paper 34 (in which some other spaces and general domains were treated as well) under certain assumptions on the modulus ω .…”

“…Let us mention the well-known result, due to Hardy-Littlewood (see Duren 20 , theorem 5.1), which establishes that B (D) = A (D) for (t) = t for 0 < < 1. For general modulus, the connection between A (D) and B (D) was first studied in Bloom and Soares de Souza 41 (see also Karapetyants and Samko 34 and Blasco and Soares de Souza 40 ) under certain assumptions on the modulus . The following result was proved in Karapetyants and Samko 34 : In particular, if satisfies both conditions (2.1) and (2.2), then the spaces A (D) and B (D) coincide up to equivalence of norms.…”

“…For general modulus, the connection between A (D) and B (D) was first studied in Bloom and Soares de Souza 41 (see also Karapetyants and Samko 34 and Blasco and Soares de Souza 40 ) under certain assumptions on the modulus . The following result was proved in Karapetyants and Samko 34 : In particular, if satisfies both conditions (2.1) and (2.2), then the spaces A (D) and B (D) coincide up to equivalence of norms. We present in this paper general theorems concerning the inclusion between the spaces formulated in terms independent of the conditions on .…”