The norm of linear extension operators for C^m-1,1(ℝⁿ)

2021/09/08 by Jacob Carruth, Carruth, Jacob, Abraham Frei-Pearson +3
Mathematics · #Advanced Harmonic Analysis Research #Advanced Banach Space Theory #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2109.03770

Abstract

Fix integers m≥ 2, n≥ 1. We prove the existence of a bounded linear extension operator for C^m-1,1(\Rⁿ) with operator norm at most exp(γD^k), where D := \binomm+n-1n is the number of multiindices of length n and order at most m-1, and γ,k > 0 are absolute constants (independent of m,n,E). Upper bounds on the norm of this operator are relevant to basic questions about fitting a smooth function to data. Our results improve on a previous construction of extension operators of norm at most exp(γD^k 2^D). Along the way, we establish a finiteness theorem for C^m-1,1(\Rⁿ) with improved bounds on the involved constants.

Related

• The Whitney extension theorem in high dimensions
• Extension operators for smooth functions on compact subsets of the reals
• An example related to Whitney's extension problem for L^2,p(ℝ²) when 1
• Normal extensions
• Extension properties for monotone and sublinear operators