## Lie Groups and Representation Theory

Seminar information archive ～05/18｜Next seminar｜Future seminars 05/19～

Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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### 2022/04/05

17:30-18:30 Room #online (Graduate School of Math. Sci. Bldg.)

Estimate of the norm of the $L^p$-Fourier transform on compact extensions of locally compact groups

(Japanese)

**Junko INOUE**(Tottori University)Estimate of the norm of the $L^p$-Fourier transform on compact extensions of locally compact groups

(Japanese)

[ Abstract ]

The classical Hausdorff-Young theorem for locally compact abelian groups is generalized by Kunze for unimodular locally compact groups.

When the group $G$ is of type I, the abstract Plancherel theorem gives a decomposition of the regular representation into a direct integral of irreducible representations through the Fourier transform;

By the Hausdorff-Young theorem generalized by Kunze, for exponents $p$ $(1 < p \leq 2)$ and ${p'}=p/(p-1)$, the Fourier transform yields a bounded operator $\mathcal{F}^p:L^p(G)\to L^{p'}(\widehat{G})$, where $L^{p'}(\widehat{G})$ is the $L^{p'}$ space of measurable fields of the Schatten class operators on the unitary dual $\widehat{G}$ of $G$.

Under this setting, we are concerned with the norm $\|\mathcal{F}^p(G)\|$ of the $L^p$-Fourier transform $\mathcal{F}^p$.

Let $G$ be a separable unimodular locally compact group of type I,and $N$ be a type I, unimodular, closed normal subgroup of $G$. Suppose $G/N$ is compact. Then we show the inequality $\|\mathcal{F}^p(G)\|\leq\|\mathcal F^p(N)\|$ for $1< p \leq 2$.

This result is a joint work with Ali Baklouti

(J. Fourier Anal. Appl. 26 (2020), Paper No. 26).

The classical Hausdorff-Young theorem for locally compact abelian groups is generalized by Kunze for unimodular locally compact groups.

When the group $G$ is of type I, the abstract Plancherel theorem gives a decomposition of the regular representation into a direct integral of irreducible representations through the Fourier transform;

By the Hausdorff-Young theorem generalized by Kunze, for exponents $p$ $(1 < p \leq 2)$ and ${p'}=p/(p-1)$, the Fourier transform yields a bounded operator $\mathcal{F}^p:L^p(G)\to L^{p'}(\widehat{G})$, where $L^{p'}(\widehat{G})$ is the $L^{p'}$ space of measurable fields of the Schatten class operators on the unitary dual $\widehat{G}$ of $G$.

Under this setting, we are concerned with the norm $\|\mathcal{F}^p(G)\|$ of the $L^p$-Fourier transform $\mathcal{F}^p$.

Let $G$ be a separable unimodular locally compact group of type I,and $N$ be a type I, unimodular, closed normal subgroup of $G$. Suppose $G/N$ is compact. Then we show the inequality $\|\mathcal{F}^p(G)\|\leq\|\mathcal F^p(N)\|$ for $1< p \leq 2$.

This result is a joint work with Ali Baklouti

(J. Fourier Anal. Appl. 26 (2020), Paper No. 26).