Sharp bounds for the commutators with variable kernels of fractional differentiations and BMO Sobolev spaces

For 0 < gamma < 1 and b is an element of I-gamma (BMO), we introduce a new class of commutators with fractional differentiations and variable kernels, which is defined by [b, T-gamma]f(x) = integral(Rn) Omega(x, x - y)/vertical bar x - y vertical bar(n+gamma)(b(x) - b(y))f(y)dy. In this paper, we give the sharp L-2 norm inequalities for the rough operators [b, T-gamma] with Omega(x, z') is an element of L-infinity(R-n) x L-q(Sn-1)(q > 2(n - 1)/n) satisfying the mean zero value condition in its second variable in the sense that the exponent q > 2(n - 1)/n is optimal. If strengthen the smoothness of Omega(x, z') in its second variable, we prove weight norm inequalities for these operators. Our results recover a previous result of Murray and extend a previous result of Calderon. (C) 2014 Elsevier Ltd. All rights reserved.