Boundedness of Commutators of Integral Operators of Fractional Type and Mα,LrlogL Maximal Operator in Variable Lebesgue Spaces

Abstract

In this paper we study the commutators of integral operator T in variable Lebesgue spaces Lp(·)(Rn) , with p(· ) ∈ K(Rn) ∩ N∞(Rn) , where T is the operator with kernel K(x,y)=k1(x-A1y)⋯km(x-Amy),A1, ⋯ , Am are invertible matrices and each ki satisfies certain fractional size condition Sn-αi,Ψi , and certain fractional Hörmander condition Hn-αi,Ψi , with α1+ ⋯ + αm= n- α , 0 ≤ α< n and Ψ i are Young functions. We obtain the maximal operator Mα,LrlogLλ , with 1≤r