# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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## Lipschitz spaces on stratified groupsHTML articles powered by AMS MathViewer

by Steven G. Krantz
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