Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.23831

Sewing on Thin Groupoids, Knitting, and Based Holonomy for Lipschitz Paths

Alexandre Reggiolli Teixeira
View original source ↗Read source document ↗Export metadata ↓

Abstract

An approximate action of the pair groupoid of a metric space determines an action of its Lipschitz-thin groupoid on complete extended metric fibers, uniquely characterized by its local sewing estimate. We prove this assertion under a superlinear three-point defect estimate and a Lipschitz bound for finite products. As a result, we obtain a positive proof of Curry and Manchon's sewing conjecture [2], with the necessary path-scaling correction to its estimate. Further restriction to isotropy constructs the based holonomy anticipated in their Remark 4.15 as a representation of based Lipschitz loops modulo thin equivalence. Rectangular comparison estimates give full relative Lipschitz-homotopy descent, in particular under their strong knitting hypothesis and under any three point estimate of total order greater than two. We prove the corresponding flatness and basepoint-covariance statements. An area model shows that the order threshold is sharp. We also separate the exponential estimate from the general knitting assumptions and exhibit a compact disk metric for which strong knitting does not imply invariance under merely continuous homotopy. As an application, for controlled fields on a metric space, we prove that the classical rough integral descends to the thin groupoid and admits substitution with a common rough controller.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexandre Reggiolli Teixeira. 2026-09-20. Sewing on Thin Groupoids, Knitting, and Based Holonomy for Lipschitz Paths. https://arxiv.org/abs/2609.23831

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Conditional expectation operators on $C(X)$

At the COSAEF conference in 2021, several participants asked the question whether a conditional expectation operator in the sense of Kuo, Labaushagne and Watson could be constructed in vector lattices other than $\mathcal{L}_p$ spaces and in particular in $C(X)$. This work answers positively to this question and participates in an old discussion on integrals in $C(X)$ space.

math.FA↗
arXiv2026

Fixed Point Rigidity of the Operator $Γ_pΠ_p^\ast$ and the LYZ Conjecture

We characterize the fixed points of the operator $Γ_pΠ_p^\ast$ for $n\geq 3$ and $1 0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak--Yang--Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa--Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces $\operatorname{vol}_n(Π_p^\ast K_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.

math.FA↗
arXiv2026

Logarithmic oscillatory multipliers and log-subdyadic square functions

We develop square-function estimates for Fourier multipliers whose local oscillation scale is \[ ρ(R)=\frac{R}{(\log R)^{γ-1}}, \qquad γ>1. \] This scale lies strictly between the dyadic scale and every fixed power-subdyadic scale at high frequency. For high-frequency symbols satisfying a localized Sobolev condition on balls of radius comparable to $ρ(R)$, we prove a pointwise square-function estimate and a weighted $L^2$ multiplier inequality. After adjoining a smooth compactly supported low-frequency part, we derive unweighted $L^p$ bounds. The weighted estimate is governed by a logarithmic geometric maximal operator which is strongly bounded above the critical $L^r$ threshold, satisfies weak type at the critical equality, and fails even weak type below it. As a model application, consider \[ L(ξ)=\frac12\log(e^2+|ξ|^2), \qquad m_{γ,β}(ξ)=L(ξ)^{-β}e^{iL(ξ)^γ}. \] For $p=2$, the associated multiplier is bounded on $L^2$ for every $β\geq0$. For $1 d(γ-1)\left|\frac12-\frac1p\right|. \] At the critical equality we obtain the corresponding Lorentz endpoint estimates.

math.FA↗