Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-20T12:51:52.506Z Has data issue: false hasContentIssue false
  • English
  • Français

MULTIFRACTAL ANALYSIS OF FUNCTIONS ON HEISENBERG AND CARNOT GROUPS

Part of: Real functions Classical measure theory Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  27 March 2015

S. Seuret  and
F. Vigneron
S. Seuret
Affiliation:
Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010, Créteil, France (stephane.seuret@u-pec.fr; francois.vigneron@u-pec.fr)
F. Vigneron
Affiliation:
Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010, Créteil, France (stephane.seuret@u-pec.fr; francois.vigneron@u-pec.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we investigate the pointwise behaviors of functions on the Heisenberg group. We find wavelet characterizations for the global and local Hölder exponents. Then we prove some a priori upper bounds for the multifractal spectrum of all functions in a given Hölder, Sobolev, or Besov space. These upper bounds turn out to be optimal, since in all cases they are reached by typical functions in the corresponding functional spaces. We also explain how to adapt our proof to extend our results to Carnot groups.

Keywords

measure and integrationpartial differential equationsfunctional analysisharmonic analysis on Euclidean spaces

MSC classification

Primary: 28A80: Fractals 28A78: Hausdorff and packing measures
Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) 42C40: Wavelets and other special systems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 1 , February 2017 , pp. 1 - 38
Copyright
© Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balogh, Z., Durand Cartagena, E., Fässler, K., Mattila, P. and Tyson, J., The effect of projections on dimension in the Heisenberg group, Revista Mat. Iberoamericana 29(2) (2013), 381–432.Google Scholar
Balogh, Z., Fässler, K. and Mattila, P., Projection and slicing theorems in Heisenberg groups, Adv. Math. 231 (2012), 569–604.Google Scholar
Barral, J. and Seuret, S., Heterogeneous ubiquitous systems in ℝ d and Hausdorff dimension, Bull. Braz. Math. Soc. (N.S.) 38(3) (2007), 467–515.Google Scholar
Barral, J. and Seuret, S., Ubiquity and large intersections properties under digit frequencies constraints, Math. Proc. Cambridge Philos. Soc. 145(3) (2008), 527–548.CrossRefGoogle Scholar
Barral, J. and Seuret, S., A localized Jarnik–Besicovitch theorem, Adv. Math. 226(4) (2011), 3191–3215.Google Scholar
Beresnevich, V., Dickinson, D. and Velani, S., Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179(846) (2006).Google Scholar
Berhanu, S. and Pesenson, I., The trace problem for vector fields satisfying Hörmander’s condition, Math. Z. 231(1) (1999), 103–122.CrossRefGoogle Scholar
Bonfiglioli, A., Taylor formula for homogeneous groups and applications, Math. Z. 262(2) (2009), 255–279.Google Scholar
Bony, J.-M. and Chemin, J.-Y., Espaces fonctionnels associés au calcul de Weyl–Hörmander, Bull. Soc. Math. France 122(1) (1994), 77–118.Google Scholar
Cancelier, C. E., Chemin, J.-Y. and Xu, C. J., Calcul de Weyl et opérateurs sous-elliptiques, Ann. Inst. Fourier (Grenoble) 43(4) (1993), 1157–1178.Google Scholar
Chemin, J.-Y. and Xu, C. J., Sobolev embeddings in Weyl–Hörmander calculus, in Geometrical Optics and Related Topics (Cortona, 1996), Progress in Nonlinear Differential Equations and their Applications, Volume 32, pp. 79–93 (Birkhäuser, Boston, Boston, MA, 1997).CrossRefGoogle Scholar
Cygan, J., Subadditivity of homogeneous norms on certain nilpotent groups, Proc. Amer. Math. Soc. 83 (1981), 69–70.Google Scholar
Danielli, D., Garofalo, N. and Nhieu, D.-M., Non-doubling Ahlfors measures, perimeter measures and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces, Mem. Amer. Math. Soc. 182(857) (2006).Google Scholar
Ebert, S. and Wirth, J., Diffusive wavelets on groups and homogeneous spaces, Proc. Roy. Soc. Edinburgh Sect. A 141(3) (2011), 497–520.Google Scholar
Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13(2) (1975), 161–207.Google Scholar
Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Mathematical Notes, Volume 28 (Princeton University Press, Princeton, NJ, 1982).Google Scholar
Führ, H. and Mayeli, A., Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization, J. Funct. Spaces Appl. (2012), 523586.CrossRefGoogle Scholar
Furioli, G., Melzi, C. and Veneruso, A., Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth, Math. Nachr. 279(9–10) (2006), 1028–1040.Google Scholar
Geller, D. and Mayeli, A., Besov spaces and frames on compact manifolds, Indiana Univ. Math. J. 58(5) (2009), 2003–2042.Google Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford University Press, 2008).Google Scholar
Jaffard, S., On the Frisch-Parisi conjecture, J. Math. Pures Appl. (9) 79(6) (2000), 525–552.Google Scholar
Jaffard, S., Lashermes, B. and Abry, P., Wavelet leaders in multifractal analysis, in Wavelet Analysis and Applications, Applied and Numerical Harmonic Analysis, pp. 201–246 (Birkhäuser, Basel, 2007).Google Scholar
Korányi, A. and Reimann, H. M., Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111(1) (1995), 1–87.Google Scholar
Lemarié, P. G., Base d’ondelettes sur les groupes de Lie stratifiés, Bull. Soc. Math. France 117(2) (1989), 211–232.Google Scholar
Mustapha, S. and Vigneron, F., Construction of Sobolev spaces of fractional order with sub-Riemannian vector fields, Ann. Inst. Fourier (Grenoble) 57(4) (2007), 1023–1049.Google Scholar
Rigot, S., Counter example to the Besicovitch covering property for some Carnot groups equipped with their Carnot-Carathéodory metric. (English summary), Math. Z. 248(4) (2004), 827–848.CrossRefGoogle Scholar
Rigot, S., Isodiametric inequality in Carnot groups, Ann. Acad. Sci. Fenn. Math. 36(1) (2011), 245–260.Google Scholar
Saka, K., Besov spaces and Sobolev spaces on a nilpotent Lie group, Tohoku Math. J. (2) 31(4) (1979), 383–437.CrossRefGoogle Scholar
Sawyer, E. and Wheeden, R. L., Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114(4) (1992), 813–874.Google Scholar
Stein, E. M., Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies, Volume 63 (Princeton University Press, Princeton, NJ, 1970).Google Scholar
Stein, E. M., Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, in Monographs in Harmonic Analysis, Vol. III, Princeton Mathematical Series, Volume 43 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Triebel, H., Function spaces on Lie groups, the Riemannian approach, J. Lond. Math. Soc. (2) 35(2) (1987), 327–338.Google Scholar
Varadarajan, V. S., Lie Groups, Lie Algebras, and their Representations, Graduate Texts in Mathematics (Springer, 1984).CrossRefGoogle Scholar
Vigneron, F., The trace problem for Sobolev spaces over the Heisenberg group. (English summary), J. Anal. Math. 103 (2007), 279–306.CrossRefGoogle Scholar