Characterizations of HΔN1(Rn) and BMOΔN(Rn) via weak factorizations and commutators

Ji Li; Brett D. Wick

doi:10.1016/j.jfa.2017.03.007

Characterizations of H_{Δ_N}¹(Rⁿ) and BMO_{Δ_N}(Rⁿ) via weak factorizations and commutators

Ji Li
, Brett D. Wick

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

31 Scopus citations

Abstract

This paper provides a deeper study of the Hardy and BMO spaces associated to the Neumann Laplacian Δ_N. For the Hardy space H_{Δ_N}¹(Rⁿ) (which is a proper subspace of the classical Hardy space H¹(Rⁿ)) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space BMO_{Δ_N}(Rⁿ) (which contains the classical BMO(Rⁿ)) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on L^∞(Rⁿ) functions and in terms of the behavior of the commutator with the Riesz transforms. The results obtained extend many of the fundamental results known for H¹(Rⁿ) and BMO(Rⁿ).

Original language	English
Pages (from-to)	5384-5416
Number of pages	33
Journal	Journal of Functional Analysis
Volume	272
Issue number	12
DOIs	https://doi.org/10.1016/j.jfa.2017.03.007
State	Published - Jun 15 2017

Keywords

Commutator
Hardy and BMO spaces
Neumann Laplacian
Riesz transforms

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10.1016/j.jfa.2017.03.007

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title = "Characterizations of HΔN1(Rn) and BMOΔN(Rn) via weak factorizations and commutators",

abstract = "This paper provides a deeper study of the Hardy and BMO spaces associated to the Neumann Laplacian ΔN. For the Hardy space HΔN1(Rn) (which is a proper subspace of the classical Hardy space H1(Rn)) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space BMOΔN(Rn) (which contains the classical BMO(Rn)) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on L∞(Rn) functions and in terms of the behavior of the commutator with the Riesz transforms. The results obtained extend many of the fundamental results known for H1(Rn) and BMO(Rn).",

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T1 - Characterizations of HΔN1(Rn) and BMOΔN(Rn) via weak factorizations and commutators

AU - Li, Ji

AU - Wick, Brett D.

PY - 2017/6/15

Y1 - 2017/6/15

N2 - This paper provides a deeper study of the Hardy and BMO spaces associated to the Neumann Laplacian ΔN. For the Hardy space HΔN1(Rn) (which is a proper subspace of the classical Hardy space H1(Rn)) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space BMOΔN(Rn) (which contains the classical BMO(Rn)) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on L∞(Rn) functions and in terms of the behavior of the commutator with the Riesz transforms. The results obtained extend many of the fundamental results known for H1(Rn) and BMO(Rn).

AB - This paper provides a deeper study of the Hardy and BMO spaces associated to the Neumann Laplacian ΔN. For the Hardy space HΔN1(Rn) (which is a proper subspace of the classical Hardy space H1(Rn)) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space BMOΔN(Rn) (which contains the classical BMO(Rn)) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on L∞(Rn) functions and in terms of the behavior of the commutator with the Riesz transforms. The results obtained extend many of the fundamental results known for H1(Rn) and BMO(Rn).

KW - Commutator

KW - Hardy and BMO spaces

KW - Neumann Laplacian

KW - Riesz transforms

UR - https://www.scopus.com/pages/publications/85016732851

U2 - 10.1016/j.jfa.2017.03.007

DO - 10.1016/j.jfa.2017.03.007

M3 - Article

AN - SCOPUS:85016732851

SN - 0022-1236

VL - 272

SP - 5384

EP - 5416

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 12

ER -

Characterizations of H_{Δ_N}¹(Rⁿ) and BMO_{Δ_N}(Rⁿ) via weak factorizations and commutators

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Keywords

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