ISSN 0001-4346, Mathematical Notes, 2016, Vol. 100, No. 1, pp. 38–48. © Pleiades Publishing, Ltd., 2016. Original Russian Text © A. N. Karapetyants, S. G. Samko, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 1, pp. 47–57. Mixed Norm Bergman–Morrey-type Spaces on the Unit Disc A. N. Karapetyants 1,2* and S. G. Samko 3** 1 Southern Federal University, Rostov-on-Don, Russia 2 Don State Technical University, Rostov-on-Don, Russia 3 Universidade do Algarve, Portugal Received February 17, 2016 Abstract—We introduce and study the mixed-norm Bergman–Morrey space A q;p,λ (D), mixed- norm Bergman–Morrey space of local type A q;p,λ loc (D), and mixed-norm Bergman–Morrey space of complementary type ∁ A q;p,λ (D) on the unit disk D in the complex plane C. The mixed norm Lebesgue–Morrey space L q;p,λ (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 (D) is defined as the subspace of analytic functions in L q;p,λ (D). Two other spaces A q;p,λ loc (D) and ∁ A q;p,λ (D) are defined similarly by using the local Morrey L p,λ loc (I )-norm and the complementary Morrey ∁ L 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. DOI: 10.1134/S000143461607004X Keywords: Bergman–Morrey-type space, mixed norm. 1. INTRODUCTION Starting with the papers by Bergman [1] and Dzherbashyan [2], [3], the spaces of analytic functions which are p-integrable with respect to a sigma-finite measure on a connected open set in the complex plane C or in C n have been intensively studied by a number of authors (see the books [4]–[10] and the references therein). The study of Toeplitz operators as well as algebras of Toeplitz operators acting in Bergman spaces served as an important objective for developing the entire theory of such spaces. The knowledge of the structural properties of these spaces, in particular, is very useful in studying Toeplitz-type operators on such spaces. More recent advances in the theory of space of analytic functions are connected with the study of Bergman-type spaces and other spaces such as analytic Besov spaces, Q-spaces, Lipschitz, Bloch, BMOA, and their numerous analogues and generalizations. The variety of the definitions and ap- proaches used to define and study such spaces allows to characterize them from different points of views, and still these spaces have a significant interplay among themselves, since they draw much from the classical theory of Bergman and even of Hardy spaces. A major issue that passes through all the above is that the boundary behavior of a function from the space under consideration or the boundary behavior of the corresponding symbol of a Toeplitz operator is the most important point. The introduction of a mixed norm is the natural generalization of the classical Bergman space, which, in particular, allows to distinguish between variables and, hence, to specify the boundary behavior of functions with more accuracy. * E-mail: karapetyants@gmail.com ** E-mail: ssamko@ualg.pt 38