One-sided Littlewood–Paley inequality in $\mathbb R^n$ for $0<p\le2$

We prove the one-sided Littlewood–Paley inequality for arbitrary collections of mutually disjoint rectangular parallelepipeds in $\mathbb R^n$ for the $L^p$-metric, $0<p\le2$. The paper supplements the 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 $H^p(\mathbb R^{d_1}\times\mathbb R^{d_2})$). 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. It allows us to verify the boundedness of some singular integral operators on the multiparameter Hardy spaces $H^p(\mathbb R^{d_1}\times\cdots\times\mathbb R^{d_n})$, which leads eventually to the required inequality of Littlewood–Paley type. Bibl. – 13 titles.