Discrete bilinear operators and commutators

TL;DR

This paper studies the boundedness and compactness of discrete bilinear operators and their commutators, drawing parallels with continuous pseudodifferential operators and Calderón-Zygmund theory.

Contribution

It introduces discrete analogues of bilinear pseudodifferential operators and establishes their boundedness and compactness properties, expanding the understanding of discrete harmonic analysis.

Findings

Discrete bilinear operators exhibit boundedness similar to continuous cases.

Commutators of these operators are shown to be compact.

Connections to discrete Calderón-Zygmund operators are established.

Abstract

We discuss boundedness properties of certain classes of discrete bilinear operators that are similar to those of the continuous bilinear pseudodifferential operators with symbols in the H\"ormander classes $BS^{\omega}_{1, 0}$. In particular, we investigate their relation to discrete analogues of the bilinear Calder\'on-Zygmund singular integral operators and show compactness of their commutators.