Boundedness of Commutators of Integral Operators of Fractional Type and Ma,Lr log L Maximal Operator in Variable Lebesgue Spaces

In this paper we study the commutators of integral operator T in variable Lebesgue spaces L-p(center dot)(R-n), with p(center dot) is an element of K-0(R-n)boolean AND N-infinity(R-n), where T is the operator with kernel K(x, y) = k(1)( x - A(1)y)(center dot center dot center dot)k(m)(x - A(m) y), A1,(center dot center dot center dot), A(m) are invertible matrices and each ki satisfies certain fractional size condition Sn- (alpha i),(Psi i), and certain fractional Hormander condition H-n-alpha i,H-Psi i, with alpha(1) + center dot center dot center dot + alpha(m) = n - alpha, 0 <= alpha < n and Psi(i) are Young functions. We obtain the maximal operator M-alpha,(r)(L) log L-lambda, with 1 <= r < alpha\n and lambda >= 0, and the commutators of T are bounded from L-p(center dot)(R-n) into L-q(center dot)(R-n), for 1\q(center dot) = 1\p(center dot) - alpha\n and certain p(center dot). Also, in the case alpha = 0 we obtain that the commutator of T satisfies a L log L-type endpoint estimate.