Boundedness and Fredholmness of Pseudodifferential Operators in Variable Exponent Spaces

Abstract

We prove a statement on the boundedness of a certain class of singular type operators in the weighted spaces L p(·) (R n , w) with variable exponent p(x) and a power type weight w, from which we derive the boundedness of pseudodifferential operators of Hörmander class S 0 1,0 in such spaces. This gives us a possibility to obtain a necessary and sufficient condition for pseudodifferential operators of the class OP S m 1,0 with symbols slowly oscillating at infinity, to be Fredholm within the frameworks of weighted Sobolev spaces H s,p(·) w (R n ) with constant smoothness s, variable p(·)-exponent, and exponential weights w. (2000). Primary 47G30. Mathematics Subject Classification

The maximal operator on weighted variable Lebesgue spaces

“…The first results of this kind we proved by Kokilashvili, Samko and their collaborators [25,40,41,42,43]; these results were more recently extended to classes of weights who oscillate between powers: see [1,21,22,23,24,26,27,37,38]. Other results in this direction have been proved by a number of authors; see, for example, [4,3,15,20,32,33].…”

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

“…Similar to the case of the constant p, the Fredholm theory of the mentioned operators in spaces related to L p(·) has also a big interest. With respect to one-dimensional singular integral operators in variable exponent Lebesgue spaces we refer, for instance, to [17][18][19][20][21][22][23][24][25][26]41].…”

“…In our paper [41] we proved the boundedness of pseudodifferential operators of the class OP S 0 1,0 acting in the variable exponent Lebesgue spaces L p(·) (R n ) and obtained the necessary and sufficient conditions of the Fredhom property of operators of the class OP S 0 1,0 with symbols slowly oscillating at infinity, in the spaces L p(·) (R n ). The proof of the sufficiency of conditions of the Fredholmness is more or less standard being based on the calculus of pseudodifferential operators, the boundedness theorems and the interpolation in the spaces L p(·) (R n ), while the proof of the necessity of those conditions meet big difficulties.…”

“…Applying results from [41] we prove that singular integral operators are bounded in L p(·) (Γ, w) and they are the local type operators in the Simonenko sense [46][47][48]. Consequently, for the investigation of the Fredholm property we can apply the Simonenko local principle.…”

Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights

“…It allows one to study not only piecewise continuous coefficients but also coefficients admitting discontinuities of slowly oscillating type. In this connection note that very recently V. Rabinovich and S. Samko [33] have started to study pseudodifferential operators in the setting of variable Lebesgue spaces. However, it seems that the method based on the Mellin technique does not allow to consider the case of arbitrary Carleson curves.…”