The boundedness of multilinear commutators from L^p (ℝⁿ) to Triebel-Lizorkin spaces

Abstract

Let $\vec b = (b_1 , \cdots ,b_m ),b_i \in \dot \Lambda _{\beta _i } (\mathbb{R}^n ),1 \leqslant i \leqslant m,0 < \beta _i < \beta ,0 < \beta < 1$,

$$[\vec b,T]f(x) = \int_{\mathbb{R}^n } {(b_1 (x) - b_1 (y)) \cdots (b_m (x) - b_m (y))K(x - y)f(y)} dy$$

, where K is a Calderón-Zygmund kernel. In this paper, we show that [$\vec b$, T] is bounded from L^p (ℝⁿ) to .F^{β, ∞}_p (ℝⁿ), as well as [$\vec b$, I_α] from L^p (ℝⁿ) to Ḟ^{β, ∞}_q (ℝⁿ), where $\frac{1}{q} = \frac{1}{p} - \frac{\alpha }{n}$.