Espaces de fonctions \`a moyenne fractionnaire int\'egrable sur les groupes localement compacts

TL;DR

This paper provides a new, more manageable definition of certain function spaces on locally compact groups, extending classical spaces of integrable mean to non-abelian groups and exploring their subspace relationships.

Contribution

It introduces an equivalent, more manageable definition of the Banach space $L_{(q,p)}^{ heta}(G)$ and studies subspaces related to functions with integrable mean on non-abelian groups.

Findings

New definition of $L_{(q,p)}^{ heta}(G)$ spaces

Extension of integrable mean spaces to non-abelian groups

Identification of complex subspace relationships

Abstract

Let $G$ be a locally compact group which is $\sigma $-compact, endowed with a left Haar measure $\lambda .$ Denote by $e$ the unit element of $G$, and by $B$ an open relatively compact and symmetric neighbourhood of $e$. For every $(p,q) $ belonging to $[ 1 ; +\infty ] ^{2}$, we give an equivalent and a priori more manageable definition of the Banach space $L_{(q,p)}^{\pi}(G),$ defined by R. C. Busby and H. A. Smith in \cite% {1}. In the case $G$ is a group of homogeneous type, we look at the subspaces $(L^{q},L^{p}) ^{\alpha}(G)$ of the space $% L_{(q,p)}^{\pi}(G)$. Theses subspaces are extensions to non abelian groups of the spaces of functions with integrable mean, defined by I. Fofana in \cite{2}. Finally we show that $L^{\alpha ,+\infty}(G)$ is a complex subspace of $(L^{q},L^{p}) ^{\alpha}(G)$.