Symmetric and Zygmund measures in several variables
Annales de l'Institut Fourier, Volume 52 (2002) no. 1, p. 153-177
Let ω:(0,)(0,) be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure μ n is called ω-Zygmund if there exists a positive constant C such that |μ(Q + )-μ(Q - )|Cω((Q + ))|Q + | for any pair Q + ,Q - 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|Cω((Q + )) for any pair Q + ,Q - n of adjacent cubes of the same size, (Q + )=(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 dt< 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,)(0,) une fonction de jauge suffisamment régulière. On dit qu’une mesure signée μ sur n est ω-Zygmund s’il existe une constante positive C telle que |μ(Q + )-μ(Q - )|Cω((Q + ))|Q + | pour chaque paire Q + ,Q - 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|Cω((Q + )) pour chaque paire Q + ,Q - n de cubes adjacents de même taille, (Q + )=(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 0 ω 2 (t)t -1 dt< 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.
DOI : https://doi.org/10.5802/aif.1881
Classification:  28A15,  31B10
Keywords: doubling measures, Zygmund measures, harmonic extensions, quadratic condition
@article{AIF_2002__52_1_153_0,
     author = {Doubtsov, Evgueni and Nicolau, Artur},
     title = {Symmetric and Zygmund measures in several variables},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {1},
     year = {2002},
     pages = {153-177},
     doi = {10.5802/aif.1881},
     mrnumber = {1881575},
     zbl = {1037.31005},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2002__52_1_153_0}
}
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/item/AIF_2002__52_1_153_0/

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