Espaces de fonctions à moyenne fractionnaire intégrable sur les groupes localement compacts

Abstract

Let $G$ be a locally compact group which is $σ$-compact, endowed with a left Haar measure $λ.$ 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 $L_{(q,p)}^π(G),$ 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 $(L^{q},L^{p}) ^α(G)$ of the space $% L_{(q,p)}^π(G)$. 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 $L^{α,+\infty}(G)$ is a complex subspace of $(L^{q},L^{p}) ^α(G)$.