Symmetric and Zygmund measures in several variables
Annales de l'Institut Fourier, Volume 52 (2002) no. 1, pp. 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: 10.5802/aif.1881
Classification: 28A15, 31B10
Keywords: doubling measures, Zygmund measures, harmonic extensions, quadratic condition
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)
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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/

[1] A.B. Aleksandrov; J.M. Anderson; A. Nicolau Inner functions, Bloch spaces and symmetric measures, Proc. London Math. Soc., Volume 79 (1999), pp. 318-352 | DOI | MR | Zbl

[2] J.M. Anderson; J.L. Fernandez; A.L. Shields Inner functions and cyclic vectors in the Bloch space, Trans. Amer. Math. Soc., Volume 323 (1991) no. 1, pp. 429-448 | DOI | MR | Zbl

[3] C. Bishop Bounded functions in the little Bloch space, Pacific J. Math., Volume 142 (1990), pp. 209-225 | MR | Zbl

[4] J. Brossard Intégrale d'aire et supports d'une mesure positive, C.R.A.S. Paris, Ser. I Math., Volume 296 (1983), pp. 231-232 | MR | Zbl

[5] L. Carleson On mappings, conformal at the boundary, J. d'Analyse Math., Volume 19 (1967), pp. 1-13 | DOI | MR | Zbl

[6] J.J. Carmona; J. Donaire On removable singularities for the analytic Zygmund class, Michigan Math. J., Volume 43 (1996), pp. 51-65 | DOI | MR | Zbl

[7] S.Y.A. Chang; J.M. Wilson; T.H. Wolff Some weighted norm inequalities concerning the Schrödinger operator, Comment. Math. Helv., Volume 60 (1985), pp. 217-246 | DOI | MR | Zbl

[8] J. Garcí a-Cuerva; J.L. Rubio; de Francia Weighted Norm Inequalities and Related Topics, Math. Studies, 116, North-Holland, 1985 | MR | Zbl

[9] P.L. Duren; H.S. Shapiro; A. Shields Singular measures and domains not of Smirnov type, Duke Math. J., Volume 33 (1966), pp. 247-254 | DOI | MR | Zbl

[10] R.A. Fefferman; C.E. Kenig; J. Pipher The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math., Volume 134 (1991), pp. 65-124 | DOI | MR | Zbl

[11] F.P. Gardiner; D.P. Sullivan Symmetric structures on a closed curve, American J. Math., Volume 114 (1992), pp. 683-736 | DOI | MR | Zbl

[12] J.P. Kahane Trois notes sur les ensembles parfaits linéaires, Enseignement Math., Volume 15 (1969), pp. 185-192 | MR | Zbl

[13] J.G. Llorente Boundary values of harmonic Bloch functions in Lipschitz domains: a martingale approach, Potential Analysis, Volume 9 (1998), pp. 229-260 | DOI | MR | Zbl

[14] N.G. Makarov Probability methods in the theory of conformal mappings, Leningrad Math. J., Volume 1 (1990), pp. 1-56 | MR | Zbl

[15] G. Piranian Two monotonic, singular, uniformly almost smooth functions, Duke Math. J., Volume 33 (1966), pp. 255-262 | DOI | MR | Zbl

[16] W. Smith Inner functions in the hyperbolic little Bloch class, Michigan Math. J., Volume 45 (1998) no. 1, pp. 103-114 | DOI | MR | Zbl

[17] E. Stein Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970 | MR | Zbl

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