If $m$ is a function on a commutative group $G$, one may define an associated Fourier multiplier $T_m$, which acts on functions on the dual group. If this $T_m$ is a bounded linear map 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 $H$? De Leeuw showed that this is indeed the case when $G=\mathbb{R}^n$, and others later extended this to all locally compact commutative groups. Moreover, the norm of the multiplier corresponding to the restricted symbol is bounded above by the norm of the original multiplier. 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 Neumann 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 the subgroup $H$”.

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 $m$. This constant measures the failure of the small invariant neighbourhood property, and can be explicitly estimated for real reductive Lie groups. We also prove non-commutative multilinear versions of the De Leeuw theorems, and use these to construct examples of multilinear multipliers on the Heisenberg group. I will outline these results in my talk, and if time permits, describe some possible extensions.

The video of this talk is available on the IISc Math Department channel.