ON THE BOUNDEDNESS OF THE RIESZ POTENTIAL AND ITS COMMUTATOR’S IN THE GLOBAL MORREY TYPE SPACES WITH VARIABLE EXPONENTS

Abstract

The paper considers the global Morrey-type spaces GMp(.),θ(.),w(.)(Ω) with variable exponents p(.), θ(.), where Ω ⊂ Rn is an unbounded domain. The questions of boundedness of the Riesz potential and its commutator in these spaces are investigated. We give the conditions for variable exponents (p1(.), p2(.)), (θ1(.), θ2(.)) and on the functions (w1(.), w2(.)) under which the Riesz potential I α , will be bounded from GMp1(.),θ1(.),w1(.)(Ω) to GMp2(.),θ2(.),w2(.)(Ω). The same conditions are obtained for the boundedness of the commutator of the Riesz potential in these spaces. In the case when the exponents p, θ constant numbers, the questions of boundedness of the Riesz potential and its commutator in global Morrey spaces were previously studied by other authors. There are also well-known results on the boundedness of the Riesz potential in global Morrey-type spaces with variable exponents, when the domain Ω ⊂ Rn is bounded.