Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

Abstract

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L p(·) (Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L p(·) (R+, dμ) where dμ is an invariant measure on multiplicative group R+ = {r ∈ R : r > 0}.(2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L p(·) (Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on R+ and local invertibility of singular integral operators on R.(3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L p(·) (Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities. Mathematics Subject Classification (2000). Primary 47G30.

Operator Theory, Pseudo-Differential Equations, and Mathematical Physics

“…In the papers [143] (1997), [144] (2008) and [145] (2011) joint with the author of this article there were studied singular integral operators and also pseudodifferential in variable exponents Lebesgue spaces, including the case of composite Carleson curves. In particular, in [145] (2011) the Simonenko local principle was extended to the case of variable exponent Lebesgue spaces where the main challenge was the localization of the space itself.…”

Operator covariant transform and local principle

“…In 1965, Simonenko [22,23] pioneered the localization technique in the theory of operators. It still remains an important tool in this area, see for example [2,4,8,9,21,24]. Many questions addressed by this technique, e.g.…”

Pseudodifferential Operators on Variable Lebesgue Spaces

“…The study of pseudodifferential operators Op(a) with symbols in S 0 1,0 on socalled variable Lebesgue spaces was started by Rabinovich and Samko [23,24].…”