Notes on lattice bump Fourier multiplier operators on $L^2 \times L^2$

TL;DR

This paper characterizes the conditions under which certain bilinear Fourier multiplier operators, constructed from translated bump functions, are bounded from $L^2 imes L^2$ to $L^2$-based amalgam spaces, advancing understanding of such operators.

Contribution

It provides a complete characterization of the coefficients for which the bilinear operator is bounded, under specific conditions on the bump function.

Findings

Identifies conditions for boundedness of bilinear Fourier multipliers

Characterizes coefficients for bounded operators on $L^2$ spaces

Advances theory of lattice bump Fourier multipliers

Abstract

Given a smooth bump function, we consider the multiplier formed by taking the linear combination of the translations of the bump function and the corresponding bilinear Fourier multiplier operator. Under certain condition on the bump function, we give a complete characterization of the coefficients of the linear combination for which the corresponding bilinear operator defines a bounded operator from $L^2\times L^2$ to $L^2$-based amalgam spaces.