Lp−Fourier inversion formula on certain locally compact groups

Wassim Nasserddine

doi:10.1016/j.crma.2019.06.011

Group theory/Harmonic analysis

L^p−Fourier inversion formula on certain locally compact groups

Presented by: the Editorial Board

Wassim Nasserddine ¹

¹ Université libanaise, Faculté des sciences – Section I, 2905-3901 Hadath-Beyrouth, Liban

Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 583-588.

Abstract
Résumé

Let G be a second countable locally compact group with type-I left regular representation, $\hat{G}$ its dual and $K = {(K_{π})}_{π \in \hat{G}}$ a specific measurable field of operators. In this paper, we investigate an inversion formula for $L^{p} (G)$ . Let $1 < p, r \leq 2$ , $\frac{1}{p} + \frac{1}{q} = \frac{1}{s} + \frac{1}{r} = 1$ , and $F_{p} : L^{p} \cap L^{1} (G) ⟶ L^{q} (\hat{G})$ be defined by $F_{p} (f) π = π (f) K_{π}^{\frac{1}{q}}$ . The map $F_{p}$ extends uniquely to a linear map of $L^{p} (G)$ into $L^{q} (\hat{G})$ , denoted by $F_{p}$ . Let ${\bar{F}}_{p}$ be the transpose of $F_{p}$ and $f \in L^{p} (G)$ . We prove that $\bar{f} \in {\bar{F}}_{r} [L^{r} (\hat{G})]$ if and only if $F_{p} (f) K^{\frac{1}{p} - \frac{1}{s}} \in L^{r} (\hat{G})$ , and, if that is the case, we have

\bar{f} = {\bar{F}}_{r} (K^{\frac{1}{p} - \frac{1}{s}} {[F_{p} (f)]}^{⁎}) .

Soit G un groupe localement compact à base dénombrable dont la représentation régulière gauche est de type I, $\hat{G}$ son dual et $K = {(K_{π})}_{π \in \hat{G}}$ un champ mesurable spécifique d'opérateurs. Dans cette note, on aborde une formule d'inversion pour $L^{p} (G)$ . Soit $1 < p, r \leq 2$ , $\frac{1}{p} + \frac{1}{q} = \frac{1}{s} + \frac{1}{r} = 1$ , et $F_{p} : L^{p} \cap L^{1} (G) ⟶ L^{q} (\hat{G})$ la fonction définie par $F_{p} (f) π = π (f) K_{π}^{\frac{1}{q}}$ . L'application $F_{p}$ se prolonge de manière unique en une application linéaire de $L^{p} (G)$ dans $L^{q} (\hat{G})$ , notée $F_{p}$ . Soit ${\bar{F}}_{p}$ la transposée de $F_{p}$ et $f \in L^{p} (G)$ . On prouve que $\bar{f} \in {\bar{F}}_{r} [L^{r} (\hat{G})]$ si et seulement si $F_{p} (f) K^{\frac{1}{p} - \frac{1}{s}} \in L^{r} (\hat{G})$ , et, si c'est le cas, on a

\bar{f} = {\bar{F}}_{r} (K^{\frac{1}{p} - \frac{1}{s}} {[F_{p} (f)]}^{⁎}) .

Received: 2019-02-22
Accepted: 2019-06-28
Published online: 2019-07-01

DOI: 10.1016/j.crma.2019.06.011

Author's affiliations:

Wassim Nasserddine ¹

¹ Université libanaise, Faculté des sciences – Section I, 2905-3901 Hadath-Beyrouth, Liban

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     author = {Wassim Nasserddine},
     title = {\protect\emph{L}\protect\textsuperscript{\protect\emph{p}}\ensuremath{-}Fourier inversion formula on certain locally compact groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {583--588},
     publisher = {Elsevier},
     volume = {357},
     number = {7},
     year = {2019},
     doi = {10.1016/j.crma.2019.06.011},
     language = {en},
}

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Wassim Nasserddine. L^p−Fourier inversion formula on certain locally compact groups. Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 583-588. doi : 10.1016/j.crma.2019.06.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.06.011/

References
Cited by

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[4] W. Nasserddine The Hausdorff–Young theorem for the matricial groups $G_{n m} = a x + b$ , Arch. Math., Volume 87 (2006), pp. 578-590

[5] W. Nasserddine Une caractérisation de l'algèbre de Fourier pour certains groupes localement compacts, C. R. Acad. Sci. Paris, Ser. I, Volume 355 (2017), pp. 543-548

[6] N. Tatsuuma Plancherel formula for non-unimodular locally compact groups, J. Math. Kyoto Univ., Volume 12 (1972), pp. 179-261

[7] M. Terp $L^{p}$ -Fourier transformation on non-unimodular locally compact groups, Adv. Oper. Theory, Volume 2 (2017) no. 4, pp. 547-583

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^☆ Ce projet a été financé avec le soutien de l'Université libanaise.

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