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Mixed norm variable exponent Bergman space on the unit disc
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We 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.