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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1982-0637028-6
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 $||f(g\exp {X_i} \cdot )|| \in {\Lambda _\alpha }({\mathbf {R}})$, 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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Article copyright: © Copyright 1982 American Mathematical Society