Biroli, Mosco: Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces

Biroli, Marco and Mosco, Umberto:
Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces (Diseguaglianze isoperimetriche e di Sobolev per forme di Dirichlet su spazi di tipo omogeneo)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 6 (1995), fasc. n.1, p. 37-44, (English)
pdf (927 Kb), djvu (219 Kb). | MR1340280 | Zbl 0837.31006

Sunto

Si provano risultati di immersione locale del tipo Sobolev e Morrey per forme di Dirichlet su spazi di tipo omogeneo. I risultati si applicano a certe classi generali di operatori subellitici e a operatori di Dirichlet su certi frattali come il «Sierpinski gasket». Si definiscono inoltre spazi BV e perimetri intrinseci e si ottengono per essi diseguaglianze isoperimetriche.

Referenze Bibliografiche

[1] M. BIROLI - U. MOSCO, Formes de Dirichlet et estimations structurelles dans les milieux discontinus. C.R. Acad. Sci. Paris, Série I, 313, 1991, 593-598. | MR 1133491 | Zbl 0760.49004

[2] M. BIROLI - U. MOSCO, A Saint-Venant principle for Dirichlet forms on discontinuous media. Preprint Series Univ. Bonn SBF 256 n. 224, 1992; revised Publ. Lab. An. Num. Univ. Paris VI, 1993, Ann. Mat. Pura Appl., in print. | Zbl 0851.31008

[3] M. BIROLI - U. MOSCO, Sobolev inequalities for Dirichlet forms on homogeneous spaces. In: C. BAIOCCHI J. L. LIONS (eds.), Boundary Value Problems for Partial Differential Equations and Applications. Research Notes in Applied Mathematics, Masson, 1993. | MR 1260455 | Zbl 0820.35035

[4] N. BURGER, Espace des fonctions à variation bornée sur un espace de nature homogène. C.R. Acad. Sci. Paris, Série I, 286, 1978, 139-142. | MR 467176 | Zbl 0368.46037

[5] R. R. COIFMAN - G. WEISS, Analyse harmonique sur certaines espaces homogènes. Lectures Notes in Mathematics, 242, Springer-Verlag, Berlin-Heidelberg-New York 1971. | MR 499948 | Zbl 0224.43006

[6] T. COULHON, Espaces de Lipschitz et inégalités de Poincaré. To appear. | Zbl 0859.58009

[7] E. FABES - C. KENIG - R. SERAPIONI, The local regularity of solutions of degenerate elliptic equations. Comm. in P.D.E., 7, 1982, 77-116. | fulltext (doi) | MR 643158 | Zbl 0498.35042

[8] C. L. FEFFERMAN - D. H. PHONG, Subelliptic eigenvalue problems. In: W. BECKNER et al. (eds.), Conference on Harmonic Analysis in Honor of A. Zygmund. Vol. 2, Wadsworth Math. Series, Chicago 1983, 590-606. | MR 730094 | Zbl 0503.35071

[9] G. B. FOLLAND - E. M. STEIN, Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton 1982. | MR 657581 | Zbl 0508.42025

[10] B. FRANCHI - E. LANCONELLI, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality. Comm. in P.D.E., 9, 1984, 1237-1264. | fulltext (doi) | MR 764663 | Zbl 0589.46023

[11] B. FRANCHI - S. GALLOT - R. L. WHEEDEN, Inégalités isopérimétriques pour les métriques dégénérées. C.R. Acad. Sci. Paris, Série I, 317, 1993, 651-654. | MR 1245092 | Zbl 0794.51011

[12] M. FUKUSHIMA, Dirichlet Forms and Markov Processes. North-Holland Math. Library, North-Holland, Amsterdam 1980. | MR 569058 | Zbl 0422.31007

[13] M. FUKUSHIMA - T. SHIMA, On a spectral analysis for the Sierpinski gasket. Potential An., 1, 1992, 1-35. | fulltext (doi) | MR 1245223 | Zbl 1081.31501

[14] D. JERISON, The Poincaré inequalities for vector fields satisfying Hörmander's condition. Duke Math. Jour., 53, 1986, 505-523. | fulltext mini-dml | Zbl 0614.35066

[15] J. KIGAMI, A harmonic calculus on the Sierpinski spaces. Japan Jour. Appl. Math., 6, 1989, 252-290. | fulltext (doi) | MR 1001286 | Zbl 0686.31003

[16] S. M. KOZLOV, Harmonization and homogenization on fractals. Comm. Math. Phys., 153, 1993, 339-357. | fulltext mini-dml | MR 1218305 | Zbl 0767.58033

[17] S. KUSUOKA, A diffusion process on fractals. In: K. ITÔ - N. IKEDA (eds.), Proceedings of the Taniguchi Symposium (Katata 1985), North-Holland, Amsterdam 1987, 251-274. | MR 933827 | Zbl 0645.60081

[18] N. LOHOUÉ, Estimées de type Hardy-Sobolev sur certaines variétés riemanniennes et les groupes de Lie. Jour. Func. An., 112, 1993, 121-158. | fulltext (doi) | MR 1207939 | Zbl 0773.58003

[19] G. LU, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander condition and applications. Riv. Iberoam., 8 (3), 1992, 367-440. | fulltext EuDML | fulltext (doi) | MR 1202416 | Zbl 0804.35015

[20] R. M. MACIAS - C. SEGOVIA, Lipschitz functions on spaces of homogeneous type. Advances in Math., 33, 1979, 257-270. | fulltext (doi) | MR 546295 | Zbl 0431.46018

[21] U. MOSCO - L. NOTARANTONIO, Homogeneous fractal spaces. To appear. | MR 1414498 | Zbl 0877.46028

[22] A. NAGEL - E. M. STEIN - S. WAINGER, Balls and metrics defined by vector fields I: Basic properties. Acta Math., 155, 1985, 103-147. | fulltext (doi) | MR 793239 | Zbl 0578.32044

[23] L. SALOFF-COSTE, Uniformly elliptic operators on Riemannian manifolds. Jour. Diff. Geom., 36, 1992, 417-450. | fulltext mini-dml | MR 1180389 | Zbl 0735.58032

[24] N. TH. VAROPOULOS, Sobolev inequalities on Lie groups and symmetric spaces. Jour. Func. An., 86, 1989, 19-40. | fulltext (doi) | MR 1013932 | Zbl 0697.22013

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche