##### Department of Mathematics,

University of California San Diego

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### Math 243 - Functional Analysis Seminar

## Amudhan Krishnaswamy-Usha

#### Delft University of Technology

## Multilinear Fourier multipliers on non-commutative groups

##### Abstract:

For a function $m$ on the real line, its Fourier multiplier $T_m$ is the operator which acts on a function $f$ by first multiplying the Fourier transform of $f$ by $m$, and then taking the inverse Fourier transform of the product. These are well-studied objects in classical harmonic analysis. Of particular interest is when the Fourier multiplier defines a bounded operator on $L_p$. Fourier multipliers can be generalized to arbitrary locally compact groups. If the group is non-abelian, the $L_p$ spaces involved are now the non-commutative $L_p$ spaces associated with the group von Neumann algebra. Fourier multipliers also have a natural extension to the multilinear setting. However, their behaviour can differ markedly from the linear case, and there is much that is unknown even about multilinear Fourier multipliers on the reals. One question of interest is this: If $m$ is a function on a group $G$ which defines a bounded $L_p$ multiplier, is the restriction of m to a subgroup $H$ also the symbol of a bounded $L_p$ multiplier on $H$? De Leeuw proved that the answer is yes, when $G$ is $\mathbb{R}^n$. This was extended to the commutative case by Saeki and to the non-commutative case (provided the group $G$ is sufficiently nice) by Caspers, Parcet, Perrin and Ricard. In this talk, I will show how to extend these De Leeuw type theorems to multilinear Fourier multipliers on non-commutative groups. This is part of joint work with Martijn Caspers, Bas Janssens and Lukas Miaskiwskyi.

Host: David Jekel

### November 23, 2021

### 10:00 AM

### Please email djekel@ucsd.edu for Zoom details.

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