This is an announcement for the paper “Asymptotic greediness of the Haar system in the spaces $L_p[0,1], 1<p<\infty$” by Fernando Albiachttps://arxiv.org/search?searchtype=author&query=Albiac%2C+F, José L. Ansorenahttps://arxiv.org/search?searchtype=author&query=Ansorena%2C+J+L, Pablo M. Bernáhttps://arxiv.org/search?searchtype=author&query=Bern%C3%A1%2C+P+M.
Abstract: Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant $C_g[\mathcal{H}^{(p)},L_p]$ of the (normalized) Haar system $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ for $1<p<\infty$. We will show that the superdemocracy constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ grows as $p^{\ast}=\max{p,p/(p-1)}$ as $p^*$ goes to $\infty$. Thus, since the unconditionality constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ is $p^*-1$, the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and unconditionality) yield that $p^{\ast}\lesssim C_g[\mathcal{H}^{(p)},L_p]\lesssim (p^{\ast})^{2}$. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, $C_g[\mathcal{H}^{(p)},L_p]\approx p^{\ast}$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.01528