Symmetric and Zygmund measures in several variables

Doubtsov, Evgueni; Nicolau, Artur

doi:10.5802/aif.1881

Doubtsov, Evgueni ¹; Nicolau, Artur 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, Volume 52 (2002) no. 1, pp. 153-177.

Abstract
Résumé

Let $ω : (0, \infty) \to (0, \infty)$ be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure $μ \in ℝ^{n}$ is called $ω$ -Zygmund if there exists a positive constant $C$ such that $| μ (Q_{+}) - μ (Q_{-}) | \leq C ω (ℓ (Q_{+})) | Q_{+} |$ for any pair $Q_{+}, Q_{-} \subset ℝ^{n}$ of adjacent cubes of the same size. Similarly, $μ$ is called an $ω$ - symmetric measure if there exists a positive constant $C$ such that $| μ (Q_{+}) / μ (Q_{-}) - 1 | \leq C ω (ℓ (Q_{+}))$ for any pair $Q_{+}, Q_{-} \subset ℝ^{n}$ 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 $\int_{0} ω^{2} (t) t^{- 1} d t < \infty$ 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.

Soit $ω : (0, \infty) \to (0, \infty)$ une fonction de jauge suffisamment régulière. On dit qu’une mesure signée $μ$ sur $ℝ^{n}$ est $ω$ -Zygmund s’il existe une constante positive $C$ telle que $| μ (Q_{+}) - μ (Q_{-}) | \leq C ω (ℓ (Q_{+})) | Q_{+} |$ pour chaque paire $Q_{+}, Q_{-} \subset ℝ^{n}$ de cubes adjacents de même taille. De la même manière, on dit que $μ$ est une mesure $ω$ - symétrique s’il existe une constante positive $C$ telle que $| μ (Q_{+}) / μ (Q_{-}) - 1 | \leq C ω (ℓ (Q_{+}))$ pour chaque paire $Q_{+}, Q_{-} \subset ℝ^{n}$ de cubes adjacents de même taille, $ℓ (Q_{+}) = ℓ (Q_{-}) < 1$ . Nous caractérisons les mesures de Zygmund et les mesures symétriques en termes de leurs extensions harmoniques. Nous montrons aussi que la condition quadratique $\int_{0} ω^{2} (t) t^{- 1} d t < \infty$ commande l’existence de mesures $ω$ -Zygmund ( $ω$ -symétriques) singulières. Le cas de la dimension un est bien connu, cependant les démonstrations correspondantes utilisent des techniques d’analyse complexe.

MR Zbl

DOI: 10.5802/aif.1881

Classification: 28A15, 31B10
Keywords: doubling measures, Zygmund measures, harmonic extensions, quadratic condition
Mot clés : mesures doublantes, mesures de Zygmund, extensions harmoniques, condition quadratique

Author's affiliations:

Doubtsov, Evgueni ¹; Nicolau, Artur ²

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     author = {Doubtsov, Evgueni and Nicolau, Artur},
     title = {Symmetric and {Zygmund} measures in several variables},
     journal = {Annales de l'Institut Fourier},
     pages = {153--177},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {1},
     year = {2002},
     doi = {10.5802/aif.1881},
     zbl = {1037.31005},
     mrnumber = {1881575},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1881/}
}

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Doubtsov, Evgueni; Nicolau, Artur. Symmetric and Zygmund measures in several variables. Annales de l'Institut Fourier, Volume 52 (2002) no. 1, pp. 153-177. doi : 10.5802/aif.1881. https://aif.centre-mersenne.org/articles/10.5802/aif.1881/

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