Local and multilinear non-commutative de Leeuw theorems

If m is a function on a group G which defines a bounded Fourier multiplier on the $L_p$ space of the dual group, is the restriction of m to a subgroup H also the symbol of a bounded multiplier on the $L_p$ space of the dual group of H? The answer is yes, for commutative groups, as shown initially by De Leeuw for ${\mathbb R}^n$. For non-commutative groups, one may ask the same question by replacing `$L_p$ spaces of the dual group' with the non-commutative $L_p$ space of the group von Neuman algebra. Caspers, Parcet, Perrin and Ricard showed that the answer is still yes in the non-commutative case, provided G has something called the `small-almost invariant neighbourhood property with respect to H'.

In recent joint work with Martijn Caspers, Bas Janssens and Lukas Miaskiwskyi, we prove a local version of this result, which removes this restriction (for a price). We show that the norm of the $L_p$ Fourier multiplier for the subgroup is bounded by some constant depending only on the support of the symbol m. We also prove a multilinear versions of the De Leeuw theorems, and use these to construct examples of multilinear multipliers on the Heisenberg group. In this talk, I will outline these results, and if time permits, also explain how the constants in the local restriction theorem may be explicitly estimated for real reductive Lie groups.