Mixed norm Bergman–Morrey-type spaces on the unit disc

Abstract

We introduce and study the mixed-norm Bergman–Morrey space A^q;p,λ\((\mathbb{D})\), mixednorm Bergman–Morrey space of local type A_loc^q;p,λ, and mixed-norm Bergman–Morrey space of complementary type ^CA^q;p,λ\((\mathbb{D})\) on the unit disk D in the complex plane C. Themixed norm Lebesgue–Morrey space L^q;p,λ\((\mathbb{D})\) is defined by the requirement that the sequence of Morrey L^p,λ(I)-norms of the Fourier coefficients of a function f belongs to l^q (I = (0, 1)). Then, A^q;p,λ\((\mathbb{D})\) is defined as the subspace of analytic functions in L^q;p,λ\((\mathbb{D})\). Two other spaces A q;p,λ loc \((\mathbb{D})\) and ^CA^q;p,λ\((\mathbb{D})\) are defined similarly by using the local Morrey L_loc^p,λ(I)-norm and the complementary Morrey ^CL^p,λ(I)-norm respectively. The introduced spaces inherit features of both Bergman and Morrey spaces and, therefore, we call them Bergman–Morrey-type spaces. We prove the boundedness of the Bergman projection and reveal some facts on equivalent description of these spaces.