Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces

Abstract

The aim of the paper is twofold. First, we present a new general approach to the definition of a class of mixed norm spaces of analytic functions A q;X (D), 1 q < ∞ on the unit disc D. We study a problem of boundedness of Bergman projection in this general setting. Second, we apply this general approach for the new concrete cases when X is either Orlicz space or generalized Morrey space, or generalized complementary Morrey space. In general, such introduced spaces are the spaces of functions which are in a sense the generalized Hadamard type derivatives of analytic functions having l q summable Taylor coefficients.MSC 2010 : Primary 30H20; Secondary 46E30, 46E15

Estimates for singular numbers of Hausdorff–Zhu operators and applications

“…Now we turn to some related mixed norm spaces of analytic functions in the unit disc. The general theory of such spaces was developed in Karapetyants and Samko, 18 and therefore, we refer to this article to the detailed definitions and proofs; see also papers. [19][20][21][22]24 Given an analytic function on the unit disc 𝑓 its Fourier coefficients,…”

“…respectively. We refer to the already mentioned papers [18][19][20][21][22]24 for the precise statements and details. Similar to the above case, we observe the following statement.…”

Generalized Hölder type spaces of harmonic functions in the unit ball and half space

“…Studies of nonstandard spaces of holomorphic (harmonic) functions are in fact at the very beginning. We refer to the study of variable exponent spaces of holomorphic functions, Orlicz-holomorphic spaces, and Morrey-holomorphic spaces, including some their mixed norm versions, see [4], [5], [10], [11], [12], [13], [14], [15], [17]. A major interest in such spaces is due to the fact that in this way we include into consideration the spaces of functions with a general and nonstandard behavior near the boundary.…”