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Rubio de Francia proved the one-sided Littlewood--Paley inequality forarbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, his methodsare developed and employed to prove an analogue of such an inequality "beyondthe index $p=\infty$", i.e., for spaces of H\"older functions and BMO.

Littlewood--Paley--Rubio de Francia inequality in Morrey--Campanato spaces