Fractional Bessel integrals and derivatives on semi-axes

In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group property, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag-Leffler function

Mixed norm variable exponent Bergman space on the unit disc

We introduce and study the mixed norm variable order Bergman space A(q,p(.)) (D), 1 1 from inside the interval I = (0, 1). The situation is quite different in the cases p(1) 1, when A(2,p(.)) (D) = H-2(D) isometrically, and when this is not longer true