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- W4362553560 abstract "The sub-Riemannian Heisenberg group is the simplest nonabelian example of a Carnot group. Suppose $$(gamma _k)_{0 le k le m}$$ is a collection of continuous function defined on a compact set $$K subset mathbb {R}$$ taking values in the sub-Riemannian Heisenberg group. When is there a horizontal $$C^{m}$$ curve $$Gamma $$ so that $$D^k Gamma |_K = gamma _k$$ for $$k = 0, 1,dots ,m$$ ? Such extensions are known as “Whitney extensions” due to the original work of Whitney for real valued mappings. This question was answered by the authors together with Andrea Pinamonti. Suppose in addition that $$gamma _m$$ is uniformly continuous with modulus of continuity $$omega $$ . When, then, is there a horizontal $$C^{m,omega }$$ curve $$Gamma $$ so that $$D^k Gamma |_K = gamma _k$$ for $$k = 0, 1,dots ,m$$ ? In this paper, we show that the hypotheses in the previous $$C^m$$ extension result are not sufficient in this case, and we provide new assumptions which are necessary and sufficient to guarantee the existence of such an extension." @default.
- W4362553560 created "2023-04-06" @default.
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- W4362553560 date "2023-04-04" @default.
- W4362553560 modified "2023-10-16" @default.
- W4362553560 title "A $$C^{m,omega }$$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group" @default.
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- W4362553560 doi "https://doi.org/10.1007/s12220-023-01233-w" @default.
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