Title: 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 Fréchet-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 Calderón-Zygmund singular integral
operators.