# On Some properties of dyadic operators

## TL;DR

This paper investigates the continuity, compactness, and commutator properties of dyadic operators like shifts and paraproducts, revealing similarities to Calderón-Zygmund operators and highlighting novel compactness results with CMO functions.

## Contribution

It provides new insights into the compactness and continuity of dyadic operators and their commutators, extending known properties of Calderón-Zygmund operators to dyadic settings.

## Key findings

- Dyadic operators are continuous under certain conditions.
- Non-compactness of dyadic operators is established using the Fréchet-Kolmogorov-Riesz-Tsuji theorem.
- Commutators with CMO functions are shown to be compact.

## 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.