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

Fernando Albiac, José L. Ansorena, Miguel Berasategui, Pablo M. Berná

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.

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

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 -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 the -normalized Haar system can be made -greedy under an equivalent norm of . 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.
Paper Structure (9 sections, 35 theorems, 88 equations)

This paper contains 9 sections, 35 theorems, 88 equations.

Table of Contents

  1. Introduction and Background
  2. Preliminaries
  3. A unified renorming scheme for greedy-type bases
  4. Renorming spaces with greedy bases
  5. The Haar system in , .
  6. Direct sums and Besov spaces
  7. Renorming spaces with subsymmetric bases
  8. Renorming Banach spaces with conditional bases
  9. Open problems

Key Result

Theorem 1.1

There is no renorming of the dyadic Hardy space $H_1$ that makes the Haar basis $1$-greedy.

Theorems & Definitions (61)

  • Theorem 1.1: DOSZ2011*Corollary 1.3
  • Theorem 1.2: DOSZ2011*Corollary 1.2
  • Theorem 1.3: DKOSZ2014*Theorem 10
  • Theorem 1.4: DOSZ2011*Theorem 2.1 and DKOSZ2014*Theorem 4
  • Theorem 1.5: DKOSZ2014*Corollary 12
  • Lemma 2.1
  • proof
  • Theorem 2.2: AW2006*Theorem 3.4
  • Theorem 2.3: AlbiacAnsorena2017b*Theorem 2.3
  • Lemma 2.4: DKKT2003*Proposition 4.4
  • ...and 51 more