One-sided Littlewood–Paley inequality in $ {\mathbb{R}^n} $ for 0 < p ≤ 2

Abstract

The one-sided Littlewood–Paley inequality for arbitrary collections of mutually disjoint rectangular parallelepipeds in $ {\mathbb{R}^n} $ for the L p -metric, 0 < p ≤ 2, is proved. The paper supplements author’s earlier work, which dealt with the situation of n = 2. That work was based on R. Fefferman’s theory, which makes it possible to verify the boundedness of certain linear operators on two-parameter Hardy spaces (i.e., Hardy spaces on the product of two Euclidean spaces, $ {H^p}\left( {{\mathbb{R}^{{d_1}}} \times {\mathbb{R}^{{d_2}}}} \right) $ ). However, Fefferman’s results are not applicable in the situation where the number of Euclidean factors is greater than 2. Here we employ the more complicated Carbery–Seeger theory, which is a further development of Fefferman’s ideas. This allows us to verify the boundedness of some singular integral operators on the multiparameter Hardy spaces $ {H^p}\left( {{\mathbb{R}^{{d_1}}} \times \cdots \times {\mathbb{R}^{{d_n}}}} \right) $ , which leads eventually to the required inequality of Littlewood–Paley type. Bibliography: 13 titles.