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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Lipschitz spaces on stratified groups
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by Steven G. Krantz PDF
Trans. Amer. Math. Soc. 269 (1982), 39-66 Request permission

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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • 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