Multilinear transference on product spaces and applications

We establish transference results for multilinear Fourier multiplier operators from Euclidean spaces to product spaces of the form ${\mathbb{R}}^{d_1}\times {\mathbb{T}}^{d_2}$. We prove strong and weak-type transference principles that preserve the corresponding Euclidean operator bounds. As applications, we obtain bounds for certain multilinear operators, as well as for some multiplier operators arising naturally in PDEs.