Smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups

Details Smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups Smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups

2008-09-16

arxiv.org/abs/0809.2806v1

Mathematics

Functional Analysis

Scientific paper

Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier algebra of $G$. We show that the anti-diagonal $\check{\Delta}_G=\{(g,g^{-1})\in G\times G \mid g\in G\}$ is both a set of local smooth synthesis and a set of local weak synthesis of degree at most $[\frac{n}{2}]+1$ for $A(G\times G)$. We achieve this by using the concept of the cone property in \cite{ludwig-turowska}. For compact $G$, we give an alternative approach to demonstrate the preceding results by applying the ideas developed in \cite{forrest-samei-spronk}. We also present similar results for sets of the form $HK$, where both $H$ and $K$ are subgroups of $G\times G\times G\times G$ of diagonal forms. Our results very much depend on both the geometric and the algebraic structure of these sets.

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