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Littlewood-Paley theorem for arbitrary intervals: Weighted estimates

Authors:

S V Kislyakov

Affiliation:

(1) St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia

Abstract:

Let 1 < r < 2 and let b is a weight on ℝ such that $b^{ - \frac{1} {{r - 1}}}$ 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

$\left\| {\sum\nolimits_j {fj} } \right\|_{L^p \left( {\mathbb{R},{\kern 1pt} b} \right)} \leqslant C\left\| {\left( {\sum\nolimits_j {\left| {fj} \right|^2 } } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\|_{L^P \left( {\mathbb{R},{\kern 1pt} b} \right)}$

for 0 < p ≤ r. Bibliography: 9 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 180–198.

Keywords:

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