Symmetric and Zygmund measures in several variables

[Mesures symétriques et mesures de Zygmund à plusieurs variables]

Doubtsov, Evgueni ; Nicolau, Artur

Annales de l'Institut Fourier, Tome 52 (2002), p. 153-177 / Harvested from Numdam

Résumé

Soit $\omega :(0,\infty )\rightarrow (0,\infty )$ une fonction de jauge suffisamment régulière. On dit qu’une mesure signée $\mu $ sur ${\mathbb {R}}^n$ est $\omega $-Zygmund s’il existe une constante positive $C$ telle que $\vert \mu (Q_{+})-\mu (Q_{-})\vert \le C\omega (\ell (Q_{+}))\vert Q_{+}\vert $ pour chaque paire $Q_{+},Q_{-}\subset {\mathbb {R}}^n$ de cubes adjacents de même taille. De la même manière, on dit que $\mu $ est une mesure $\omega $- symétrique s’il existe une constante positive $C$ telle que $\vert \mu (Q_+)/\mu (Q_{- })-1\vert \le C\omega (\ell (Q_{+}))$ pour chaque paire $Q_{+},Q_{-}\subset {\mathbb {R}}^n$ de cubes adjacents de même taille, $\ell (Q_{+})=\ell (Q_{-})<1$. 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\omega ^2(t)t^{-1}dt<\infty $ commande l’existence de mesures $\omega $-Zygmund ($\omega $-symétriques) singulières. Le cas de la dimension un est bien connu, cependant les démonstrations correspondantes utilisent des techniques d’analyse complexe.

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, $\ell (Q_{+})=\ell (Q_{-})<1$. 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<\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.

Détail

Publié le : 2002-01-01

MR 1881575 | Zbl 1037.31005

DOI : https://doi.org/10.5802/aif.1881
Classification:  28A15,  31B10
Mots clés: mesures doublantes, mesures de Zygmund, extensions harmoniques, condition quadratique

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