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October 15, 2025Open Access

Lipschitz spaces over non-porous sets

Key Points

The Lipschitz spaces $ ext{Lip}_0(M)$ and $ ext{Lip}_0( ext{R}^n)$ are shown to be linearly isomorphic.
Results hold for non-porous sets with positive $n$-dimensional Lebesgue measure.
The findings extend to Carnot groups when equipped with a Carnot-Carathéodory metric.
This work contributes to the understanding of functional analysis in non-porous geometric contexts.

Abstract

Let M be a subset of Rⁿ. If M is not porous, in particular if it has positive n-dimensional Lebesgue measure, we prove that the Lipschitz spaces Lip₀ (M) and Lip₀ (Rⁿ) are linearly isomorphic. The result also holds more generally if Rⁿ is replaced with a Carnot group equipped with its Carnot-Carathéodory metric.

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Cite This Study

Ramón J. Aliaga (Wed,) studied this question.

synapsesocial.com/papers/68ef858cc6a308ba06355565 https://doi.org/https://doi.org/10.48550/arxiv.2507.12119

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