This is a continuation of a talk I gave at the University of Delhi in $2015.$ Let $G$ be a separable locally compact unimodular group of type I, and $\widehat G$ the unitary dual of $G$ endowed with the Mackey Borel structure. We regard the Fourier transform $\mathcal F$ as a mapping of $L^1(G)$ to a space of $\mu$-measurable field of bounded operators on $\widehat G$ defined for $\pi\in\widehat G$ by $ L^1(G)\ni f\mapsto \mathcal Ff : \mathcal Ff(\pi)=\pi(f), $ where $\mu$ denotes the Plancherel measure of $G$. The mapping $f \mapsto \mathcal F f$ extends to a continuous operator $\mathcal F^p : L^p(G) \to L^q(\widehat G)$, where $p\geq 1$ is real number and $q$ its conjugate. We are concerned in this talk with the norm of the linear map $\mathcal F^p$. We first record some results on the estimate of this norm for some classes of solvable Lie groups and their compact extensions and discuss the sharpness problem. We look then at the case where $G$ is a separable unimodular locally compact group of type I. Let $N$ be a unimodular closed normal subgroup of $G$ of type I, such that $G/N$ is compact. We show that $\Vert \mathscr F^p(G)\Vert \leq \Vert \mathscr F^p(N )\Vert$. In the particular case where $G=K\ltimes N$ is defined by a semi-direct product of a separable unimodular locally compact group $N$ of type I and a compact subgroup $K$ of the automorphism group of $N$, we show that equality holds if $N$ has a $K$-invariant sequence $(\varphi_j)_j$ of functions in $L^1(N)\cap L^p(N)$ such that ${\Vert \mathscr F\varphi_j \Vert_q}/{\Vert \varphi_j \Vert_p}$ tends to $\Vert \mathscr F^p(N )\Vert$ when $j$ goes to infinity.

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

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Last updated: 15 Jul 2024