De-Höldering factorization
Abstract
We study a factorization notion for Lipschitz functions between metric spaces in which such a function is written as a composition of a Hölder function and a function that suitably ``undoes'' the Hölder regularity. We show simple ways to construct such ``de-Höldering'' factorizations. If the identity mapping on a metric space ℳ admits a de-Höldering factorization through a metric space 𝒵 that has a conical geodesic bicombing, then the class of metric spaces from which one can extend 𝒵-valued Lipschitz functions is shown to be contained in the corresponding class of metric spaces for ℳ-valued Lipschitz functions. As a quick consequence of these abstract permanence properties, we deduce that every L1-valued Lipschitz function from a subset of ℓ2 can be extended to a Lipschitz function that takes values in L1 and is defined on all of ℓ2, answering a 1992 question of Ball. By work of Makarychev and Makarychev, this implies that every weighted graph has a vertex cut sparsifier of size n and quality O(√(log n)), improving Moitra's 2009 bound. We also show that for every metric space 𝒵 that has a conical geodesic bicombing, any metric transform of a metric space ℳ has 𝒵-valued Lipschitz extension modulus at most a universal constant multiple of the 𝒵-valued Lipschitz extension modulus of ℳ itself, improving the 2002 bound of Brudnyi and Shvartsman.
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BibTeX
@misc{naor2026dehöldering,
title = {{De-Höldering factorization}},
author = {Assaf Naor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07564},
eprint = {2609.07564},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07564},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - De-Höldering factorization
AU - Assaf Naor
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.07564
UR - https://arxiv.org/abs/2609.07564
AB - We study a factorization notion for Lipschitz functions between metric spaces in which such a function is written as a composition of a Hölder function and a function that suitably ``undoes'' the Hölder regularity. We show simple ways to construct such ``de-Höldering'' factorizations. If the identity mapping on a metric space ℳ admits a de-Höldering factorization through a metric space 𝒵 that has a conical geodesic bicombing, then the class of metric spaces from which one can extend 𝒵-valued Lipschitz functions is shown to be contained in the corresponding class of metric spaces for ℳ-valued Lipschitz functions. As a quick consequence of these abstract permanence properties, we deduce that every L_1-valued Lipschitz function from a subset of ℓ_2 can be extended to a Lipschitz function that takes values in L_1 and is defined on all of ℓ_2, answering a 1992 question of Ball. By work of Makarychev and Makarychev, this implies that every weighted graph has a vertex cut sparsifier of size n and quality O(√(log n)), improving Moitra's 2009 bound. We also show that for every metric space 𝒵 that has a conical geodesic bicombing, any metric transform of a metric space ℳ has 𝒵-valued Lipschitz extension modulus at most a universal constant multiple of the 𝒵-valued Lipschitz extension modulus of ℳ itself, improving the 2002 bound of Brudnyi and Shvartsman.
ER -