On the vector-valued Littlewood-Paley-Rubio de Francia inequality
Littlewood-Paley-Rubio de Francia inequality
The authors consider Banach spaces \(X\) having the property that NEWLINE\[NEWLINE\left(\int_0^1 \int_{\mathbb R} \|\sum_j r_j(t) S_{I_j} f(x)\|^p dt dx\right)^{1/p}\leq C\left(\int_{\mathbb R} \|f(x)\|^pdx\right)^{1/p}NEWLINE\]NEWLINE for some \(2\leq p<\infty\), where \(r_j\) are the Rademacher functions and \(S_I f(x)= \int_I \hat f(\xi)e^{i\xi x}d\xi\) for an interval \(I\) and where \(I_j\) is a collection of pairwise disjoint intervals in \(\mathbb R\).NEWLINENEWLINESuch a property is called the Littlewood-Paley-Rubio de Francia property, in short \(LPR_p\) (due to the extension of the Littlewood-Paley result to general intervals due to \textit{J. L. Rubio de Francia} [Rev. Mat. Iberoam. 1, No. 2, 1--14 (1985; Zbl 0611.42005)]. It is known (see the paper by \textit{T. P. Hytönen}, \textit{J. L. Torrea} and \textit{D. V. Yakubovich} [Proc. R. Soc. Edinb., Sect. A, Math. 139, No. 4, 819--832 (2009; Zbl 1173.42010)]) that this property implies the UMD property and type 2. In the present paper the authors show that for Banach lattices if the \(2\)-concavification of \(X\), denoted \(X_{(2)}\), is a UMD Banach lattice then \(X\) has the \(LPR_p\) property. Although it is still not known whether the property is independent of \(p\) it is proved by the authors that \(LPR_p\) implies \(LPR_q\) for any \(q\geq p\).
scientific article; zbMATH DE number 6078125