Fractional integrodifferentiation in Hölder classes of arbitrary order

Abstract

Holder classes of variable order n(x) are introduced and it is shown that the fractional integral Ia 0+ has Holder order n(x)+a (0 < a, n+, a + n+ < 1, n+ = supn(x)). The connection between the smoothness of image and the smoothness of density is an important problem in the theory of integral operators. In particular, the same problem for fractional integral operators I a ' in the interval (a;b) was thoroughly studied by G. H. Hardy and J. E. Littlewood