Isometric renormings for greedy bases in Banach spaces, with applications to the Haar System in $L_p[0,1]$, $1<p<\infty$

TL;DR

This paper demonstrates that under certain conditions, Banach spaces can be renormed to make their bases isometrically greedy, with applications to classical systems like the Haar basis in $L_p$ spaces.

Contribution

It introduces a method to renorm Banach spaces so that their bases become isometrically greedy, resolving a long-standing open problem for the Haar system in $L_p$ spaces.

Findings

Haar system in $L_p$ can be made 1-greedy under an equivalent norm

Renormings preserve lattice 1-unconditionality and bidemocracy

Applicable to various bases in Besov spaces, mixed-norm spaces, and Schlumprecht space

Abstract

We investigate the problem of improving the greedy-type constant of a basis by means of an equivalent renorming of the ambient Banach space. Our main result shows that if a Banach space admits an unconditional and bidemocratic basis whose fundamental function satisfies certain regularity properties, then the space can be renormed so that the basis becomes isometrically greedy. The renorming simultaneously ensures lattice $1$-unconditionality, isometric bidemocracy, and allows prescribing the fundamental function up to a suitable regularization. As a principal application, we resolve a long-standing problem posed by Albiac--Wojtaszczyk in 2006 by proving that for each $1<p<\infty$ the $L_p$-normalized Haar system can be made $1$-greedy under an equivalent norm of $L_p$. Further applications include isometric greedy renormings for bases of Besov spaces, mixed-norm direct sums, and for a wide class of subsymmetric and conditional bases, including spreading models and the canonical basis of Schlumprecht space. These results show that isometric greedy renormings arise in far greater generality than previously known.