Lipschitz estimates for commutators of singular integral operators associated with the sections

Abstract

Abstract Let H be Monge-Ampère singular integral operator, b ∈ L i p F β $b\in Lip_{\mathcal{F}}^{\beta}$ , and 1 / q = 1 / p − β $1/q=1/p-\beta$ . It is proved that the commutator [ b , H ] $[b,H]$ is bounded from L p ( R n , d μ ) $L^{p}(\mathbb{R}^{n},d\mu)$ to L q ( R n , d μ ) $L^{q}(\mathbb{R}^{n},d\mu)$ for 1 < p < 1 / β $1< p<1/\beta$ and from H F p ( R n ) $H^{p}_{\mathcal{F}}(\mathbb{R}^{n})$ to L q ( R n , d μ ) $L^{q}(\mathbb{R}^{n},d\mu)$ for 1 / ( 1 + β ) < p ≤ 1 $1/(1+\beta)< p\leq1$ . For the extreme case p = 1 / ( 1 + β ) $p=1/(1+\beta)$ , a weak estimate is given.

Keywords

• Hardy spaces
• commutator
• Lipschitz function
• sections