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On the Converse of Pansu's Theorem
Guido De Philippis1, Andrea Marchese2, Andrea Merlo3
1Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012 USA.
Archive for Rational Mechanics and Analysis
|December 23, 2024
Summary
This study generalizes Pansu's differentiability theorem for Lipschitz maps on Carnot groups. It demonstrates that if these maps are Pansu-differentiable for certain Radon measures, these measures must be absolutely continuous with respect to the Haar measure.
Area of Science:
- Geometric Measure Theory
- Analysis on Metric Spaces
- Harmonic Analysis
Background:
- Pansu's differentiability theorem is a fundamental result in the analysis of mappings between Carnot groups.
- Generalizing this theorem to broader classes of measures is crucial for understanding geometric properties of functions in these spaces.
- Radon measures provide a flexible framework for studying distributions and their properties.
Purpose of the Study:
- To extend Pansu's differentiability theorem to a more general setting involving Radon measures on Carnot groups.
- To investigate the relationship between Pansu-differentiability of Lipschitz maps and the properties of the underlying Radon measures.
- To establish conditions under which a Radon measure must be absolutely continuous with respect to the Haar measure.
Main Methods:
- Generalization of Pansu's differentiability theorem.
- Analysis of Lipschitz continuous mappings between Carnot groups.
- Characterization of Radon measures using properties of differentiability.
Main Results:
- A suitable generalization of Pansu's differentiability theorem is provided for Radon measures on Carnot groups.
- It is shown that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere with respect to a Radon measure, then this measure is absolutely continuous with respect to the Haar measure of the group.
Conclusions:
- The findings extend the applicability of Pansu's theorem to a wider class of measures.
- This result deepens the understanding of the interplay between differentiability properties of maps and the structure of measures on Carnot groups.
- Absolute continuity with respect to the Haar measure is identified as a necessary condition for Pansu-differentiability of Lipschitz maps for certain Radon measures.
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