Approaching Bilinear Multipliers via a Functional Calculus

Area of Science:

• Harmonic Analysis
• Functional Analysis
• Operator Theory

Background:

• Bilinear multiplier operators are fundamental in harmonic analysis.
• Existing theories often lack a unified framework for diverse multiplier types.
• The spectral theorem provides a powerful tool for operator analysis.

Purpose of the Study:

• To develop a general framework for bilinear multiplier operators using the bivariate spectral theorem.
• To establish Coifman-Meyer type multiplier theorems within this new framework.
• To investigate the applicability of the framework to various specific multiplier contexts.

Main Methods:

• Definition of bilinear multiplier operators via the bivariate spectral theorem.
• Application of spectral theory to prove multiplier theorems.
• Derivation of fractional Leibniz rules using the established framework.

Main Results:

• A novel framework for analyzing bilinear multiplier operators.
• Proof of Coifman-Meyer type multiplier theorems for these operators.
• Demonstration of applicability to discrete Laplacian, bi-radial Dunkl multipliers, and Jacobi expansions.

Conclusions:

• The proposed framework unifies the study of various bilinear multipliers.
• The results extend existing multiplier theory and fractional calculus.
• The framework offers a versatile tool for future research in harmonic analysis.