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Zeitschrift für Analysis und ihre Anwendungen

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Volume 40, Issue 1, 2021, pp. 97–109
DOI: 10.4171/ZAA/1674

Published online: 2021-01-25

Principal Frequency of $p$-Sub-Laplacians for General Vector Fields

Michael Ruzhansky[1], Bolys Sabitbek[2] and Durvudkhan Suragan

(1) Ghent University, Belgium and Queen Mary University of London, UK
(2) Queen Mary University of London, UK, and Al Farabi Kazakh National University, Almaty, Kazakhstan

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet $p$-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

Keywords: $p$-sub-Laplacian, smooth manifold, principal frequency, Picone's identity, Caccioppoli inequality

Ruzhansky Michael, Sabitbek Bolys, Suragan Durvudkhan: Principal Frequency of $p$-Sub-Laplacians for General Vector Fields. Z. Anal. Anwend. 40 (2021), 97-109. doi: 10.4171/ZAA/1674