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

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 . 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 $(L^q,L^p{)}^{\alpha }(G)$.

19 pages