Local and multilinear noncommutative de Leeuw theorems
Martijn Caspers, Bas Janssens, Amudhan Krishnaswamy-Usha, Lukas, Miaskiwskyi

TL;DR
This paper extends classical Fourier multiplier restriction theorems to noncommutative groups, providing explicit bounds and new multilinear results, including the first examples of bilinear multipliers on nonabelian groups.
Contribution
It generalizes de Leeuw's restriction theorem to noncommutative groups with explicit constants and develops multilinear Fourier multiplier theory, including the first bilinear examples on nonabelian groups.
Findings
Established a quantitative restriction inequality involving a constant c(U).
Provided explicit lower bounds for c in real reductive Lie groups.
Proved multilinear restriction, approximation, and periodization theorems for noncommutative groups.
Abstract
Let be a discrete subgroup of a locally compact unimodular group . Let be a -multiplier on with and let be the corresponding Fourier multiplier. Similarly, let be the Fourier multiplier associated to the restriction of to . We show that \[ c( {\rm supp}( m\vert_{\Gamma} ) ) \Vert T_{m \vert_\Gamma}: L_p(\widehat{\Gamma}) \rightarrow L_p(\widehat{\Gamma}) \Vert \leq \Vert T_{m }: L_p(\widehat{G}) \rightarrow L_p(\widehat{G}) \Vert, \] for a specific constant that is defined for every . The function quantifies the failure of to admit small almost -invariant neighbourhoods and can be determined explicitly in…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Operator Algebra Research · Noncommutative and Quantum Gravity Theories
