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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 ([email protected]; [email protected])
F. Vigneron
Affiliation:
Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010, Créteil, France ([email protected]; [email protected])
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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 

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