Commutators with fractional differentiation and new characterizations of BMO-Sobolev spaces

For b ∈ L_loc¹ (ℝⁿ) and α ∈ (0, 1), let D^α be the fractional differential operator and T be the singular integral operator. We obtain a necessary and sufficient condition on the function b to guarantee that [b, D^αT] is a bounded operator on a function space such as L ^p(ℝⁿ) and L ^p,λ(ℝⁿ) for any 1 < p < ∞. Furthermore, we establish a necessary and sufficient condition on the function b to guarantee that [b, D^αT] is a bounded operator from L^∞(ℝⁿ) to BMO(ℝⁿ) and from L¹(ℝⁿ) to L^1,∞(ℝⁿ). This is a new theory. Finally, we apply our general theory to the Hilbert and Riesz transforms.