Journal of Mathematical Sciences

, Volume 139, Issue 2, pp 6417–6424 | Cite as

On the Littlewood-Paley theorem for arbitrary intervals

  • S. V. Kislyakov
  • D. V. Parilov
Article

Abstract

We extend the results of Rubio de Francia and Bourgain by showing that, for arbitrary mutually disjoint intervals Δk ⊂ ℤ+, arbitrary p ∈, (0, 2], and arbitrary trigonometric polynomials fk with supp \(\hat f_k \subset \Delta _k \), we have
$$\left\| {\sum\limits_k {f_k } } \right\|_{H^p (\mathbb{T})} \leqslant a_p \left\| {\left( {\sum\limits_k {\left| {f_k } \right|} ^2 } \right)^{1/2} } \right\|_{L^p (\mathbb{T})} $$
. The method is a development of that by Rubio de Francia. Bibliography: 9 titles.

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. V. Kislyakov
    • 1
  • D. V. Parilov
    • 1
  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia

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