Hardy-Littlewood inequalities for Ciesielski-Fourier series (Q2568541)

The Ciesielski systems are systems of uniformly bounded orthonormal functions, obtained from the spline systems of order \((m,k)\) in the same way as the Walsh system arises from the Haar system, namely by the same orthogonal transformation of dyadic blocks. The Hardy-Littlewood-Paley inequality established in \textit{R. E. A. C. Paley} [Stud. Math. 3, 226--238 (1931; Zbl 0003.35201)] states that \[ \Big( \sum_{n=1}^\infty n^{p-2} |\hat f(n)|^p\Big)^{1/p} \leq C_p \|f\|_{L_p},\qquad (1<p\leq 2) \] for Fourier coefficients \(\hat f(n)\) with respect to an arbitrary uniformly bounded system of orthonormal functions. The present paper, as a continuation of the author's previous work, extends this inequality to the Ciesielski systems on Hardy spaces \(H_p\) for \(0<p\leq2\), both, in the one- and two-parameter case. In the borderline cases of the one-parameter inequality, a weak estimate can be proved. Dual versions follow by the characterization of the dual of \(H_1\) as BMO.