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harmonic analysis on euclidean spaces

Journal of the Institute of Mathematics of Jussieu (1)

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MULTIFRACTAL ANALYSIS OF FUNCTIONS ON HEISENBERG AND CARNOT GROUPS
Part of
S. Seuret, F. Vigneron
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 16 / Issue 1 / February 2017

Published online by Cambridge University Press:

27 March 2015, pp. 1-38

Print publication:

February 2017
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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.