Functional Analysis and Applications Seminar 

17/09/2004, 14:00 — 15:00 — Room P3.10, Mathematics Building
Vakhtang M. Kokilashvili, A. Razmadze Institute, Tbilisi, Georgia

Weighted Boundedness of Integral Operators in the Variable Exponent Spaces of Homogeneous Type

The talk deals with the boundedness (compactness) criteria for various classical integral operators (and their generalizations) in weighted Banach spaces with non-standard growth. The study of these spaces and behaviour of integral transforms there have been stimulated by various problems of elasticity theory, fluid mechanics, calculus of variations and differential equations with non-standard growth. The talk focuses on weighted estimates in variable Lebesgue and Lorentz spaces for integral transforms defined both on the Euclidean space with Lebesgue measure and general measure spaces with quasimetrics. We present boundedness criteria for maximal functions, singular operator and potentials in weighted variable spaces with weights of power-exponential type. The solution of two weighted problems for fractional integrals with variable fractional order is presented. The trace inequality for the generalized potentials defined on spaces of homogeneous type is also treated in the variable Lebesgue spaces. We also give a Sobolev type theorem and its weighted version for fractional integrals on Carleson curves (the recent result jointly with S.Samko). An application to the Dirichlet problem for harmonic functions in "bad" domains within the framework of the variable Lebesgue spaces is given. The explicit formulas for the solution are given together with the complete picture of the influence of the geometry of the domain to the solvability of the problem.