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

TL;DR

This paper proves that the anti-diagonal in the Fourier algebra of a Lie group exhibits smooth and weak synthesis properties, with results depending on the group's dimension and structure, using geometric and algebraic methods.

Contribution

It establishes the smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups, extending previous methods and considering subgroup products.

Findings

Anti-diagonal is a set of local smooth synthesis for $A(G imes G)$.

Anti-diagonal is a set of local weak synthesis with degree at most $[rac{n}{2}]+1$.

Results depend on geometric and algebraic structures of the sets.

Abstract

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.