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Littlewood–Paley theorem for arbitrary intervals: weighted estimates
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 180-198 Cet article a éte moissonné depuis la source Math-Net.Ru

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Suppose $1 and $b$ is a weight on $\mathbb R$ such that $b^{-\frac1{r-1}}$ satisfies the Muckenhoupt condition $A_{r'/2}$ ($r'$ is the exponent conjugate to $r$). If $f_j$ are functions whose Fourier transforms are supported on mutually disjoint intervals, then $$ \Bigl\Vert\sum_j f_j\Bigr\Vert_{L^p(\mathbb R,b)}\le C\Bigl\Vert\Bigl(\sum_j|f_j|^2\Bigr)^{1/2}\Bigr\Vert_{L^p(\mathbb R,b)} $$ for $0. Bibl. – 9 titles. 
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S. V. Kislyakov. Littlewood–Paley theorem for arbitrary intervals: weighted estimates. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 180-198. http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a7/

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