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

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