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.
