Symmetric and Zygmund measures in several variables

Doubtsov, Evgueni; Nicolau, Artur

Annales de l'Institut Fourier, Tome 52 (2002) no. 1, pp. 153-177.

Soit $\omega :(0,\infty )\to (0,\infty )$ une fonction de jauge suffisamment régulière. On dit qu’une mesure signée $\mu$ sur ${ℝ}^n$ est $\omega$-Zygmund s’il existe une constante positive $C$ telle que $|\mu \left(Q_{+}\right)-\mu \left(Q_{-}\right)|\le C\omega \left(\ell \left(Q_{+}\right)\right)\left|Q_{+}\right|$ pour chaque paire $Q_{+},Q_{-}\subset {ℝ}^n$ 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 $|\mu \left(Q_{+}\right)/\mu \left(Q_{-}\right)-1|\le C\omega \left(\ell \left(Q_{+}\right)\right)$ pour chaque paire $Q_{+},Q_{-}\subset {ℝ}^n$ de cubes adjacents de même taille, $\ell \left(Q_{+}\right)=\ell \left(Q_{-}\right)<1$. Nous caractérisons les mesures de Zygmund et les mesures symétriques en termes de leurs extensions harmoniques. Nous montrons aussi que la condition quadratique ${\int }_0{\omega }^2\left(t\right)t^{-1}dt<\infty$ 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.

Let $\omega :(0,\infty )\to (0,\infty )$ be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure $\mu \in {ℝ}^n$ is called $\omega$-Zygmund if there exists a positive constant $C$ such that $|\mu \left(Q_{+}\right)-\mu \left(Q_{-}\right)|\le C\omega \left(\ell \left(Q_{+}\right)\right)\left|Q_{+}\right|$ for any pair $Q_{+},Q_{-}\subset {ℝ}^n$ of adjacent cubes of the same size. Similarly, $\mu$ is called an $\omega$- symmetric measure if there exists a positive constant $C$ such that $|\mu \left(Q_{+}\right)/\mu \left(Q_{-}\right)-1|\le C\omega \left(\ell \left(Q_{+}\right)\right)$ for any pair $Q_{+},Q_{-}\subset {ℝ}^n$ of adjacent cubes of the same size, $\ell \left(Q_{+}\right)=\ell \left(Q_{-}\right)<1$. We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition ${\int }_0{\omega }^2\left(t\right)t^{-1}dt<\infty$ 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.

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