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 L p(·) (Rn), with p(·) ∈ K0(Rn)∩N∞(Rn), where T is the operator with kernel K(x, y) = k1(x − A1 y)... km(x − Am y), A1,..., Am are invertible matrices and each ki satisfies certain fractional size condition Sn−αi,i , and certain fractional Hörmander condition Hn−αi,i , with α1 +···+ αm = n −α, 0 ≤ α < n and i are Young functions. We obtain the maximal operator Mα,Lr log Lλ , with 1 ≤ r < α n and λ ≥ 0, and the commutators of T are bounded from L p(·) (Rn) into Lq(·) (Rn), for 1 q(·) = 1 p(·) − α n and certain p(·). Also, in the case α = 0 we obtain that the commutator of T satisfies a L log L-type endpoint estimate.