Functional Analysis and Applications Seminar 

24/09/2004, 14:00 — 15:00 — Room P3.10, Mathematics Building
Nikolai Karapetyants, Rostov State University, Russia

Integral Operators in the Localized Hölder Spaces with Variable Exponent

As is known, the operator of fractional integration of order $\alpha$ establishes an isomorphism between the Hölder spaces $H^{\mu }$ and $H^{\mu +\alpha }$. 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 $\alpha$. We also consider problems in the Hölder spaces of variable order. We pay special attention to the case where the order $\mu$ at the fixed point $x_0$ of $[0,1]$ 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 $x_0$ has higher order $\mu +\alpha$. 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.