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 super-democracy constant of $\mathcal {H}^{(p)}$ in $L_{p}[0,1]$ grows as $p^{*}=\max \{p,p/(p-1)\}$ 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 $p^{*}\lesssim C_{g}[\mathcal {H}^{(p)},L_p]\lesssim (p^{*})^{2}$. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, $C_{g}[\mathcal {H}^{(p)},L_p]\approx p^{*}$. Our work answers a question that was raised by Hytonen  (2015).

Appendix: Summary of the Most Commonly Employed Constants

Symbol	Name of constant	Ref. equation
$C_{a}$	Symmetry for largest coeffs. constant	(1.10)
$C_{{g}}$	Greedy constant	(1.3)
$\Delta$	Democracy constant	(1.7)
$\Delta _{{b}}$	Bi-democracy constant	(1.13)
$\Delta _{{d}}$	Disjoint-democracy constant	(1.7)
$\Delta _{{s}}$	Superdemocracy constant	(1.9)
$\Delta _{{sb}}$	Super bi-democratic constant	(2.2)
$\Delta _{{sd}}$	Disjoint-superdemocracy constant	(1.9)
$K_{{su}}$	Suppression unconditional constant	(1.4)
$K_{{u}}$	Lattice unconditional constant	(1.5)