E. Harboure, O. Salinas, B. Viviani
Relations between weighted Orlicz and $BMO_{\phi }$ spaces through fractional integrals

Comment.Math.Univ.Carolinae 40,1 (1999) 53-69.

Abstract:We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_\alpha $ maps weak weighted Orlicz$-\phi $ spaces into appropriate weighted versions of the spaces $BMO_\psi $, where $\psi (t)=t^{\alpha /n}\phi ^{-1}(1/t)$. This generalizes known results about boundedness of $I_\alpha $ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha $ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_\alpha $ acting on weak$-L_\phi $ for $\phi $ of lower type equal or greater than $n/\alpha $, is the same as the one solving the problem for weak$-L^p$ with $p$ the lower index of Orlicz-Maligranda of $\phi $, namely $\omega ^{p'}$ belongs to the $A_1$ class of Muckenhoupt.

Keywords: theory of weights, Orlicz spaces, $BMO$ spaces, fractional integrals
AMS Subject Classification: Primary 42B25