Relations between weighted Orlicz and $BMO_\phi$ spaces through fractional integrals

Abstract

summary: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