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

Abstract

For $b\in L_{loc}^1\left({ℝ}^n\right)$ and $\alpha \in \left(0,1\right)$, let $D^{\alpha }$ 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 $\left[b,D^{\alpha }T\right]$ is a bounded operator on a function space such as $L^p\left({ℝ}^n\right)$ and $L^{p,\lambda }\left({ℝ}^n\right)$ for any $1<p<\infty$. Furthermore, we establish a necessary and sufficient condition on the function $b$ to guarantee that $\left[b,D^{\alpha }T\right]$ is a bounded operator from $L^{\infty }\left({ℝ}^n\right)$ to $BMO\left({ℝ}^n\right)$ and from $L^1\left({ℝ}^n\right)$ to $L^{1,\infty }\left({ℝ}^n\right)$. This is a new theory. Finally, we apply our general theory to the Hilbert and Riesz transforms.