The Boundedness for Commutator of Fractional Integral Operator With Rough Variable Kernel

Abstract

For b ∈ BMO(ℝn) and 0 < α ≤ 1/2, the commutator of the fractional integral operator TΩ,α with rough variable kernel is defined by $$ [b, T_{\Omega, \alpha}]f(x)= \int_{\mathbb{R}^n} \frac{\Omega(x,x-y)}{|x-y|^{n-\alpha}}(b(x)-b(y))f(y)dy. $$ In this paper the authors prove that the commutator [b, TΩ,α] is a bounded operator from \(L^{\frac{2n}{n+2\alpha}}(\mathbb{R}^n)\) to L2(ℝn). The result obtained in this paper is substantial improvement and extension of some known results.