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Transactions of the American Mathematical Society

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Lipschitz spaces on stratified groups


Author: Steven G. Krantz
Journal: Trans. Amer. Math. Soc. 269 (1982), 39-66
MSC: Primary 22E30; Secondary 22E25, 35H05, 46E35, 58G05
MathSciNet review: 637028
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Abstract: Let $ G$ be a connected, simply connected nilpotent Lie group. Call $ G$ stratified if its Lie algebra $ \mathfrak{g}$ has a direct sum decomposition $ \mathfrak{g} = {V_1} \oplus \cdots \oplus {V_m}$ with $ [{V_i},{V_j}] = {V_{i + j}}$ for $ i + j \leqslant m$, $ [{V_{i,}}{V_j}] = 0$ for $ i + j > m$. Let $ \{ {X_1}, \ldots ,{X_n}\} $ be a vector space basis for $ {V_1}$. Let $ f \in C(G)$ satisfy $ \vert\vert f(g\exp {X_i} \cdot )\vert\vert \in {\Lambda _\alpha }({\mathbf{R}})$, uniformly in $ g \in G$, where $ {\Lambda _\alpha }$ is the usual Lipschitz space and $ 0 < \alpha < \infty $. It is proved that, under these circumstances, it holds that $ f \in {\Gamma _\alpha }(G)$ where $ {\Gamma _\alpha }$ is the nonisotropic Lipschitz space of Folland. Applications of this result to interpolation theory, hypoelliptic partial differential equations, and function theory are provided.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0637028-6
Article copyright: © Copyright 1982 American Mathematical Society