Browsing by Author "Karapetyants, A. N."
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- Mixed norm Bergman-Morrey-type spaces on the unit discPublication . Samko, Stefan; Karapetyants, A. N.We introduce and study the mixed-norm Bergman-Morrey space A (q;p,lambda) , mixednorm Bergman-Morrey space of local type A (loc) (q;p,lambda) , and mixed-norm Bergman-Morrey space of complementary type (C) A (q;p,lambda) on the unit disk D in the complex plane C. Themixed norm Lebesgue-Morrey space L (q;p,lambda) is defined by the requirement that the sequence of Morrey L (p,lambda)(I)-norms of the Fourier coefficients of a function f belongs to l (q) (I = (0, 1)). Then, A (q;p,lambda) is defined as the subspace of analytic functions in L (q;p,lambda) . Two other spaces A q;p,lambda loc and (C) A (q;p,lambda) are defined similarly by using the local Morrey L (loc) (p,lambda) (I)-norm and the complementary Morrey (C) L (p,lambda)(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.
- Mixed norm variable exponent Bergman space on the unit discPublication . Karapetyants, A. N.; Samko, StefanWe introduce and study the mixed norm variable order Bergman space A(q,p(.)) (D), 1 <= q < infinity, 1 <= p(r) <= infinity, on the unit disc D in the complex plane. The mixed norm variable order Lebesgue/type space L-q,L-p(.) (D) is defined by the requirement that the sequence of the variable exponent L-p(.) (I)-norms of the Fourier coefficients of the function f belongs to l(q). Then A(q,p(.)) (D) is defined to be the subspace of L-q,L-p(.) (D) which consists of analytic functions. We prove the boundedness of the Bergman projection and reveal the dependence of the nature of such spaces on possible growth of variable exponent p(r) when r -> 1 from inside the interval I = (0, 1). The situation is quite different in the cases p(1) < infinity and p(1) = infinity. In the case p(1) < infinity we also characterize the introduced Bergman space A(2,p(.)) (D) as the space of Hadamards's fractional derivatives of functions from the Hardy space H-2(D). The case p(1) = infinity is specially studied, and an open problem is formulated in this case. We also reveal the conditions on the rate of growth of p(r) when r -> 1, when A(2,p(.)) (D) = H-2(D) isometrically, and when this is not longer true.
- On singular operators in vanishing generalized variable-exponent Morrey spaces and applications to Bergman-type spacesPublication . Karapetyants, A. N.; Rafeiro, H.; G. Samko, StefanWe give a proof of the boundedness of the Bergman projection in generalized variable-exponent vanishing Morrey spaces over the unit disc and the upper half-plane. To this end, we prove the boundedness of the Calderon-Zygmund operators on generalized variable-exponent vanishing Morrey spaces. We give the proof of the latter in the general context of real functions on R-n, since it is new in such a setting and is of independent interest. We also study the approximation by mollified dilations and estimate the growth of functions near the boundary.