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theory of weights; Orlicz spaces; $BMO$ spaces; fractional integrals
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.
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