Vanishing Carleson Measures Associated with Families of Multilinear Operators

Abstract

In this paper we define a kind of vanishing Carleson measure on $\mathbb {R}^{n+1}_+$ and give its characterization by the compact property of some convolution operator. We also investigate the construction of vanishing Carleson measures generated by a family of the multilinear operators $\{\Theta _t\}_{t>0}$ and $CMO$ functions. As some applications of our results, we also give the boundedness and compactness for the paraproduct $\pi _{\vec {b}}$ associated with the family $\{\Theta _t\}_{t>0}$ on $L^2(\mathbb {R}^n)$, which is defined by

$$\begin{aligned} \pi _{\vec {b}}(f)(x)= \int _0^\infty \eta _t*\big ((\varphi _t*f)\Theta _t(b_1,\ldots ,b_m)\big )(x)\; \frac{dt}{t}. \end{aligned}$$

Further, for the linear case (i.e., $m=1$), we show that the paraproduct

$$\begin{aligned} B_b(f)(x)=\int _0^\infty (f*\varphi _t)(x)(b*\psi _t)(x)\frac{\alpha (t)}{t}dt, \end{aligned}$$

which was introduced by Coifman and Meyer, is also a compact operator on $L^2(\mathbb {R}^n)$ if $b\in CMO(\mathbb {R}^n)$ and $\alpha \in L^\infty (\mathbb {R}^n)$.