Asymptotic greediness of the Haar system in the spaces $L_p[0,1]$, $1<p<\infty$

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 $p^{\ast }$ goes to $\infty$. Thus, since the unconditionality constant of ${\mathcal{H}}^{(p)}$ in $L_p[0,1]$ is $p^{\ast }-1$, 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, $C_g[{\mathcal{H}}^{(p)},L_p]\approx p^{\ast }$.