Vanishing Carleson Measures Associated with Families of Multilinear Operators

In this paper we define a kind of vanishing Carleson measure on R-+(n+1) and give its characterization by the compact property of some convolution operator. We also investigate the construction of vanishing Carlesonmeasures 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((b) over right arrow) associated with the family {Theta(t)}(t>0) on L-2( R-n), which is defined by pi((b) over right arrow)(f)(x) = integral(infinity)(0) eta(t) & ((phi(t) * f) Theta(t) (b(1), ...,b(m))) ( x) dt/t. Further, for the linear case ( i. e., m = 1), we show that the paraproduct B-b (f)(x) = integral(infinity)(0) (f *phi(t))(x)(b * Psi(t))(x) alpha(t)/t dt, which was introduced by Coifman and Meyer, is also a compact operator on L-2( R-n) if b is an element of CMO(R-n) and alpha is an element of L-infinity(R-n).