Symmetric and Zygmund measures in several variables

Evgueni Doubtsov[1]; Artur Nicolau[2]

  • [1] St. Petersburg State University, Department of Mathematical analysis, Bibliotechnaya pl. 2, Staryi Petergof, 198904 St. Petersburg (Russie)
  • [2] Universitat Autonoma de Barcelona, Departament de Matemàtiques, 08193 Bellaterra, Barcelona (Espagne)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 1, page 153-177
  • ISSN: 0373-0956

Abstract

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Let ω : ( 0 , ∞ ) → ( 0 , ∞ ) be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure μ ∈ ℝ n is called ω -Zygmund if there exists a positive constant C such that | μ ( Q + ) - μ ( Q - ) | ≤ C ω ( ℓ ( Q + ) ) | Q + | for any pair Q + , Q - ⊂ ℝ n of adjacent cubes of the same size. Similarly, μ is called an ω - symmetric measure if there exists a positive constant C such that | μ ( Q + ) / μ ( Q - ) - 1 | ≤ C ω ( ℓ ( Q + ) ) for any pair Q + , Q - ⊂ ℝ n of adjacent cubes of the same size, ℓ ( Q + ) = ℓ ( Q - ) < 1 . We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition ∫ 0 ω 2 ( t ) t - 1 d t < ∞ governs the existence of singular ω -Zygmund ( ω -symmetric) measures. In the one- dimensional case, the results are well known, but complex analysis techniques are used at certain steps of the corresponding proofs.

How to cite

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Doubtsov, Evgueni, and Nicolau, Artur. "Symmetric and Zygmund measures in several variables." Annales de l’institut Fourier 52.1 (2002): 153-177. <http://eudml.org/doc/115971>.

@article{Doubtsov2002,
abstract = {Let $\omega :(0,\infty )\rightarrow (0,\infty )$ be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure $\mu \in \{\mathbb \{R\}\}^n$ is called $\omega $-Zygmund if there exists a positive constant $C$ such that $\vert \mu (Q_\{+\})- \mu (Q_\{-\})\vert \le C\omega (\ell (Q_\{+\}))\vert Q_\{+\}\vert $ for any pair $Q_+,Q_\{-\}\subset \{\mathbb \{R\}\}^n$ of adjacent cubes of the same size. Similarly, $\mu $ is called an $\omega $- symmetric measure if there exists a positive constant $C$ such that $\vert \mu (Q_+)/\mu (Q_\{-\})-1\vert \le C\omega (\ell (Q_\{+\}))$ for any pair $Q_+,Q_\{- \}\subset \{\mathbb \{R\}\}^n$ of adjacent cubes of the same size, $\ell (Q_\{+\})=\ell (Q_\{-\})&lt;1$. We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition $\int _0\omega ^2(t)t^\{-1\}dt&lt;\infty $ governs the existence of singular $\omega $-Zygmund ($\omega $-symmetric) measures. In the one- dimensional case, the results are well known, but complex analysis techniques are used at certain steps of the corresponding proofs.},
affiliation = {St. Petersburg State University, Department of Mathematical analysis, Bibliotechnaya pl. 2, Staryi Petergof, 198904 St. Petersburg (Russie); Universitat Autonoma de Barcelona, Departament de Matemàtiques, 08193 Bellaterra, Barcelona (Espagne)},
author = {Doubtsov, Evgueni, Nicolau, Artur},
journal = {Annales de l’institut Fourier},
keywords = {doubling measures; Zygmund measures; harmonic extensions; quadratic condition; symmetric measure},
language = {eng},
number = {1},
pages = {153-177},
publisher = {Association des Annales de l'Institut Fourier},
title = {Symmetric and Zygmund measures in several variables},
url = {http://eudml.org/doc/115971},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Doubtsov, Evgueni
AU - Nicolau, Artur
TI - Symmetric and Zygmund measures in several variables
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 1
SP - 153
EP - 177
AB - Let $\omega :(0,\infty )\rightarrow (0,\infty )$ be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure $\mu \in {\mathbb {R}}^n$ is called $\omega $-Zygmund if there exists a positive constant $C$ such that $\vert \mu (Q_{+})- \mu (Q_{-})\vert \le C\omega (\ell (Q_{+}))\vert Q_{+}\vert $ for any pair $Q_+,Q_{-}\subset {\mathbb {R}}^n$ of adjacent cubes of the same size. Similarly, $\mu $ is called an $\omega $- symmetric measure if there exists a positive constant $C$ such that $\vert \mu (Q_+)/\mu (Q_{-})-1\vert \le C\omega (\ell (Q_{+}))$ for any pair $Q_+,Q_{- }\subset {\mathbb {R}}^n$ of adjacent cubes of the same size, $\ell (Q_{+})=\ell (Q_{-})&lt;1$. We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition $\int _0\omega ^2(t)t^{-1}dt&lt;\infty $ governs the existence of singular $\omega $-Zygmund ($\omega $-symmetric) measures. In the one- dimensional case, the results are well known, but complex analysis techniques are used at certain steps of the corresponding proofs.
LA - eng
KW - doubling measures; Zygmund measures; harmonic extensions; quadratic condition; symmetric measure
UR - http://eudml.org/doc/115971
ER -

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