Approaching bilinear multipliers via a functional calculus

TL;DR

This paper introduces a spectral theorem-based framework for bilinear multiplier operators, establishing new multiplier theorems and fractional Leibniz rules applicable to various discrete and radial settings.

Contribution

It develops a unified spectral calculus approach for bilinear multipliers, extending classical results to discrete Laplacians, Dunkl operators, and Jacobi expansions.

Findings

Proved Coifman-Meyer type multiplier theorems for the framework.

Established fractional Leibniz rules within this setting.

Applied theory to discrete Laplacian, Dunkl, and Jacobi multipliers.

Abstract

We propose a framework for bilinear multiplier operators defined via the (bivariate) spectral theorem. Under this framework we prove Coifman-Meyer type multiplier theorems and fractional Leibniz rules. Our theory applies to bilinear multipliers associated with the discrete Laplacian on $\mathbb{Z}^d,$ general bi-radial bilinear Dunkl multipliers, and to bilinear multipliers associated with the Jacobi expansions.