Rubio de Francia proved a one-sided Littlewood-Paley inequality for the square function constructed from an arbitrary system of disjoint intervals. Later, Osipov proved a similar inequality for Walsh systems. We prove a similar inequality for more general Vilenkin systems. Bibliography: 11 titles.
The question of existence is treated for near-minimizers for the distance functional (or E-functional in the interpolation terminology) that are stable under the action of certain operators. In particular, stable near-minimizers for the couple (L 1 , L p ) are shown to exist when the operator is the projection on wavelets and these wavelets possess only some weak conditions of decay at infinity.
We prove a generalization of the Littlewood-Paley characterisation of the BMO space where the shifts of a Schwartz function are replaced by a family of functions with suitable conditions imposed on them. We also prove that a certain family of Triebel-Lizorkin spaces can be characterized in a similar way.
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