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Intrinsic Difference Quotients

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Harmonic Analysis, Partial Differential Equations and Applications

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

An alternative characterizations of intrinsic Lipschitz functions within Carnot groups through the boundedness of appropriately defined difference quotients is provided. It is also shown how intrinsic difference quotients along horizontal directions are naturally related with the intrinsic derivatives, introduced e.g. in Franchi et al. (Comm Anal Geom 11(5):909–944, 2003) and Ambrosio et al. (J Geom Anal 16:187–232, 2006) and used to characterize intrinsic real valued C 1 functions inside Heisenberg groups. Finally the question of the equivalence of the two conditions: (1) boundedness of horizontal intrinsic difference quotients and (2) intrinsic Lipschitz continuity is addressed in a few cases.

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Acknowledgements

The author is supported by University of Trento, by GNAMPA of the INdAM, by MAnET Marie Curie Initial Training Network Grant 607643-FP7-PEOPLE-2013-ITN and by MIUR, Italy.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, Via Sommarive 14, Povo, 38123, Trento, Italy

    Raul Paolo Serapioni

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  1. Raul Paolo Serapioni
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Correspondence to Raul Paolo Serapioni .

Editor information

Editors and Affiliations

  1. Dept. of Math, Rutgers University, Piscataway, New Jersey, USA

    Sagun Chanillo

  2. Department of Mathematics, University of Bologna, Bologna, Italy

    Bruno Franchi

  3. Dept. of Math., Univ. of Connecticut, Storrs CT, USA

    Guozhen Lu

  4. Department of Mathematics, University of Bilbao, Bilbao, Spain

    Carlos Perez

  5. Dept of Math and Stat, McMaster University, Hamilton, Ontario, Canada

    Eric T. Sawyer

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Serapioni, R.P. (2017). Intrinsic Difference Quotients. In: Chanillo, S., Franchi, B., Lu, G., Perez, C., Sawyer, E. (eds) Harmonic Analysis, Partial Differential Equations and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-52742-0_10

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