On existence of a universal function for Lp[0, 1] with p∈(0, 1)

We show that, for every number p ∈ (0, 1), there is g ∈ L1[0, 1] (a universal function) that has monotone coefficients ck(g) and the Fourier–Walsh series convergent to g (in the norm of L1[0, 1]) such that, for every f ∈ Lp[0, 1], there are numbers δk = ±1, 0 and an increasing sequence of positive integers Nq such that the series ∑ k=0+∞δkck(g)Wk (with {Wk} theWalsh system) and the subsequence σNq(α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{{N_q}}^{\left( \alpha \right)}$$\end{document}, α ∈ (−1, 0), of its Cesáro means converge to f in the metric of Lp[0, 1].