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Hardy and BMO spaces on Weyl chambers

Paweł Plewa, Krzysztof Stempak

Abstract

Let $W$ be a finite reflection group associated with root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber distinguished by a choice of $R_+$, a set of positive roots. We define and investigate Hardy and BMO spaces on $C_+$ in the framework of boundary conditions given by a homomorphism $η\in\mathrm{Hom}(W,\hat{\mathbb{Z}}_2)$ which attaches the $\pm$ signs to the facets of $C_+$. Specialized to orthogonal root systems, atomic decompositions in $H^1_η$ and $h^1_η$ are obtained and the duality problem is also treated.

Hardy and BMO spaces on Weyl chambers

Abstract

Let be a finite reflection group associated with root system in . Let denote a positive Weyl chamber distinguished by a choice of , a set of positive roots. We define and investigate Hardy and BMO spaces on in the framework of boundary conditions given by a homomorphism which attaches the signs to the facets of . Specialized to orthogonal root systems, atomic decompositions in and are obtained and the duality problem is also treated.
Paper Structure (10 sections, 17 theorems, 105 equations, 1 figure)

This paper contains 10 sections, 17 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on finite reflection groups
  3. Function and distribution spaces on $\overline{C_+}$
  4. General procedure: function spaces
  5. General procedure: distribution spaces
  6. $\eta$-Hardy spaces on $\overline{C_+}$
  7. Atomic decomposition in $H^1_\eta$ and $h^1_\eta$ for orthogonal root systems
  8. $\eta$-$\mathrm{BMO}$ spaces on Weyl chambers and duality
  9. The case of orthogonal root systems
  10. Duality

Key Result

Proposition 3.1

The spaces $H^1_{-\Delta^+_\eta}(C_+)$ and $H^1_{(-\Delta^+_\eta)^{1/2}}(C_+)$ coincide with $H^1_\eta(C_+)$ with equivalence of norms. Analogous statement holds for the local spaces $h^1_{-\Delta^+_\eta}(C_+)$ and $h^1_{(-\Delta^+_\eta)^{1/2}}(C_+)$.

Figures (1)

  • Figure 1: The Whitney decomposition in the case $d=2$, $U=(0,\infty)\times \mathbb{R}$, and $n=2$, with an example of the cube $Q$.

Theorems & Definitions (38)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Definition 4.3
  • Lemma 4.4
  • proof
  • ...and 28 more