arXiv subjects
Publications and source records attributed to Ibrahim Fofana.
We establish a weighted inequality for fractional maximal and convolution type operators, between weak Lebesgue spaces and Wiener amalgam type spaces on $ \mathbb R $ endowed with a measure which needs not to be doubling.
The class of Banach spaces $(L^{q},L^{p}) ^α(X,d,μ)$, $1\leq q\leq α\leq p\leq \infty ,$ introduced in \cite{F1} in connection with the study of the continuity of the fractional maximal operator of Hardy-Littlewood and of the Fourier transformation in the case $% X=\mathbb{R}^{n}$ and $μ$ is the Lebesgue measure, was generalized in \cite{FFK} to the setting of homogeneous groups. We generalize it here to spaces of homogeneous type and we prove that the results obtained in \cite{FFK} such as relations between these spaces and Lebesgue spaces, weak Lebesgue and Morrey spaces, remain true.
We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,β}$ of Hardy-Littlewood and the fractional integral $I_γ$. These inequalities are established between $(L^{q},L^{p}) ^α(X,d,μ)$ spaces (which are super spaces of Lebesgue spaces $L^α(X,d,μ)$, and subspaces of amalgams $(L^{q},L^{p})(X,d,μ)$) and in the setting of space of homogeneous type $(X,d,μ)$. The conditions on the weights are stated in terms of Orlicz norm.
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 $(L^{q},L^{p}) ^α(G)$.