On the Littlewood-Paley theorem for arbitrary intervals

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 f k 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.