Multilinear transference of Fourier and Schur multipliers acting on noncommutative -spaces

Abstract Let G be a locally compact unimodular group, and let $\phi $ some function of n variables on . To such , one can associate multilinear Fourier multiplier, which acts -fold product the noncommutative $L_p$ -spaces group von Neumann algebra. One may also define an associated Schur Schatten classes $S_p(L_2(G))$ We generalize well-known transference results from linear case to case. In particular, we show that so-called “multiplicatively bounded $(p_1,\ldots ,p_n)$ -norm” multiplier is above by corresponding multiplicatively norm with equality whenever amenable. Furthermore, prove bilinear Hilbert transform not as vector-valued map $L_{p_1}(\mathbb {R}, S_{p_1}) \times L_{p_2}(\mathbb S_{p_2}) \rightarrow L_{1}(\mathbb S_{1})$ $p_1$ $p_2$ are $\frac {1}{p_1} + \frac {1}{p_2} = 1$ A similar result holds for certain Calderón–Zygmund-type operators. This in contrast nonvector-valued Euclidean