Weighted norm inequality for the singular integral with variable kernel and fractional differentiation

Abstract

For 0≤γ≤1, let Dγ be the fractional differentiation operator and T be the singular integral operator with variable kernel, which is defined byTf(x)=∫RnΩ(x,x−y)|x−y|nf(y)dy. Let T⁎ and T♯ denote the adjoint of T and the pseudo-adjoint of T respectively. Let T1T2 denote the product of T1 and T2, T1∘T2 denote the pseudo-product of T1 and T2. In this paper, we give the weighted norm inequalities for commutators of these singular integral operators and the fractional differentiation operator Dγ. Finally, we extend all results to the Morrey spaces.