On Some properties of dyadic operators

Abstract

In this paper, the objects of our investigation are some dyadic operators, including dyadic shifts, multilinear paraproducts and multilinear Haar multipliers. We mainly focus on the continuity and compactness of these operators. First, we consider the continuity properties of these operators. Then, by the Fr\'echet-Kolmogorov-Riesz-Tsuji theorem, the non-compactness properties of these dyadic operators will be studied. Moreover, we show that their commutators are compact with \textit{CMO} functions, which is quite different from the non-compaceness properties of these dyadic operators. These results are similar to those for Calder\'on-Zygmund singular integral operators.