Precise Lipschitz Sandwich Theorems for Higher Order Lipschitz Functions

Core Concepts

When two Lip(γ) functions are close on a subset B, they remain close in a weaker Lip(η) sense throughout the larger set Σ, provided η < γ.

Abstract

The key insights and highlights of the content are:

The article investigates the consequences when two Lip(γ) functions, in the sense of Stein, are close throughout a subset B of their domain Σ.

The main result is the Lipschitz Sandwich Theorem 3.1, which states that if ψ and ϕ are Lip(γ) functions on Σ with bounded Lip(γ) norms, and ψ and ϕ coincide on a closed subset B that is a δ0-cover of Σ, then the Lip(η) norm of ψ - ϕ is bounded above by ε throughout Σ, provided η < γ.

The theorem requires that the closeness of ψ and ϕ on B is measured in a pointwise sense, rather than requiring the stronger condition that the Lip(γ) norm of ψ - ϕ on B is bounded.

The restriction that η < γ is shown to be sharp, as the result is false when η = γ.

The Lipschitz Sandwich Theorem has applications in cost-effective approximation of Lip(γ) functions, as shown in Section 4.

The article also establishes related results, such as the Single-Point Lipschitz Sandwich Theorem 3.7 and the Pointwise Lipschitz Sandwich Theorem 3.9, which consider the case when the subset B is a single point.

The proofs of the main results rely on careful estimates involving the Lip(γ) structure of the functions, as detailed in Sections 5-10.

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Key Insights Distilled From

Higher Order Lipschitz Sandwich Theorems

by Terry Lyons,... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.06849.pdf

Deeper Inquiries

How can the Lipschitz Sandwich Theorems be extended to non-Euclidean settings, such as Carnot groups or sub-Riemannian manifolds

The Lipschitz Sandwich Theorems can be extended to non-Euclidean settings, such as Carnot groups or sub-Riemannian manifolds, by adapting the definitions and concepts to fit the geometry and structure of these spaces. In the context of Carnot groups, a suitable definition of Lipschitz functions can be formulated that aligns with the underlying geometry of the group. This involves considering the appropriate norms on the tangent spaces and adapting the notion of Lipschitz continuity accordingly. Similarly, for sub-Riemannian manifolds, the Lipschitz Sandwich Theorems can be generalized by defining Lipschitz functions in a way that respects the sub-Riemannian structure and the constraints imposed by the distribution on the manifold.

Can the Lipschitz Sandwich Theorems be applied to problems in rough path theory or the study of the accessibility problem for rough differential equations

The Lipschitz Sandwich Theorems can be applied to problems in rough path theory and the study of the accessibility problem for rough differential equations by providing a framework for analyzing the regularity and closeness of functions in these contexts. In rough path theory, where the notion of Lipschitz continuity plays a crucial role in understanding the behavior of rough paths, the Lipschitz Sandwich Theorems can be used to establish relationships between different levels of regularity and approximation of rough paths. Similarly, in the study of the accessibility problem for rough differential equations, the theorems can help in determining the closeness of solutions and approximations, providing insights into the behavior of the system.

What are the implications of the Lipschitz Sandwich Theorems for the Whitney extension problem and the computation of optimal extensions

The implications of the Lipschitz Sandwich Theorems for the Whitney extension problem and the computation of optimal extensions are significant. These theorems provide a systematic way to extend functions defined on subsets to larger domains while ensuring certain regularity properties are preserved. In the context of the Whitney extension problem, the theorems offer a method to extend functions smoothly and efficiently, taking into account the Lipschitz continuity and closeness of functions. This can lead to improved approximations and extensions that maintain the desired regularity. Additionally, in the computation of optimal extensions, the theorems provide a framework for determining the best way to extend functions while controlling the error and maintaining the desired regularity properties.