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

For 0<γ<10<γ<1 and b∈Iγ(BMO), we introduce a new class of commutators with fractional differentiations and variable kernels, which is defined by [b,Tγ]f(x)=∫RnΩ(x,x−y)|x−y|n+γ(b(x)−b(y))f(y)dy. In this paper, we give the sharp L2L2 norm inequalities for the rough operators [b,Tγ][b,Tγ] with Ω(x,z′)∈L∞(Rn)×Lq(Sn−1)Ω(x,z′)∈L∞(Rn)×Lq(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)/nq>2(n−1)/n is optimal. If strengthen the smoothness of Ω(x,z′)Ω(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 Calderón.