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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Subadditivity of homogeneous norms on certain nilpotent Lie groups

Author: Jacek Cygan
Journal: Proc. Amer. Math. Soc. 83 (1981), 69-70
MSC: Primary 22E30; Secondary 35H05, 43A80
MathSciNet review: 619983
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Abstract: Let $ N$ be a Lie group with its Lie algebra generated by the left-invariant vector fields $ {X_1}, \ldots ,{X_k}$ on $ N$. An explicit fundamental solution for the (hypoelliptic) operator $ L = X_1^2 + \cdots + X_k^2$ on $ N$ has been obtained for the Heisenberg group by Folland [1] and for the nilpotent (Iwasawa) groups of isometries of rank-one symmetric spaces by Kaplan and Putz [2]. Recently Kaplan [3] introduced a (still larger) class of step-$ 2$ nilpotent groups $ N$ arising from Clifford modules for which similar explicit solutions exist. As in the case of $ L$ being the ordinary Laplacian on $ N = {{\mathbf{R}}^k}$, these solutions are of the form $ g \mapsto {\text{const}}{\left\Vert g \right\Vert^{2 - m}}$, $ g \in N$, where the "norm" function $ \left\Vert {} \right\Vert$ satisfies a certain homogeneity condition. We prove that the above norm is also subadditive.

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  • [1] G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373-376. MR 0315267 (47:3816)
  • [2] A. Kaplan and R. Putz, Boundary behavior of harmonic forms on a rank one symmetric space, Trans. Amer. Math. Soc. 231 (1977), 369-384. MR 0477174 (57:16715)
  • [3] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147-153. MR 554324 (81c:58059)

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Keywords: Analysis on nilpotent groups, gauges and homogeneous norms, (analytic-) hypoelliptic operators
Article copyright: © Copyright 1981 American Mathematical Society

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