Integral Operators in the Localized Hölder Spaces with Variable
Exponent
As is known, the operator of fractional integration of order
establishes an isomorphism between the Hölder spaces
and
. We give a survey of some results on integral operators in weighted Hölder and generalized Hölder spaces, including the case of complex and imaginary order
. We also consider problems in the Hölder spaces of variable order. We pay special attention to the case where the order
at the fixed point
of
is higher than in other points. It is known that the singular integral operator does not preserve such a class. At the same time, for fractional integration the problem has a positive solution in the sense that the fractional integral at the point
has higher order
. Besides this, the Riesz potential on a ball and the singular integral operator on a closed smooth curve are also considered in the Hölder spaces of variable order.
