We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g., with (alpha) metric). These are bi-Lipschitz neighborhoods domain manifold, constants controlling distortion size that depend only natural geometric properties manifold. The proof these results relies novel estimates, from above below, for kernel its gradient, as well their hold in non-smooth category, stable respect...

We show that if a subset $K$ in the Heisenberg group (endowed with Carnot-Carathéodory metric) is contained rectifiable curve, then it satisfies modified analogue of Peter Jones's geometric lemma. This quantitative version statement finite length curve has tangent at almost every point. condition complements work by Ferrari, Franchi, and Pajot (2007) except power 2 changed to 4. Two key tools we use proof are martingale argument like Schul as well new curvature inequality group.

Abstract A measure is 1-rectifiable if there a countable union of finite length curves whose complement has zero measure. We characterize Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 terms positivity the lower density and finiteness geometric square function, which loosely speaking, records an L gauge extent to admits approximate tangent lines, or rapidly growing ratios, along its support. In contrast with classical theorems Besicovitch, Morse Randolph, Moore, we do not...

We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with C α metric).These are bi-Lipschitz embedded balls domain manifold, distortion constants that depend only natural geometric properties manifold.The proof these results relies estimates, from above below, for kernel its gradient, as well their gradient.These estimates hold in non-smooth category, stable respect...

We show that a sufficient condition for subset E in the Heisenberg group (endowed with Carnot–Carathéodory metric) to be contained rectifiable curve is it satisfies modified analogue of Peter Jones’s geometric lemma. Our estimates improve on those [6], allowing any power r < 4 replace 2 Jones- \beta -number. This complements an open ended way our work [13], where we showed such estimate was necessary, however = 4.

We identify two sufficient conditions for locally finite Borel measures on $\mathbb {R}^n$ to give full mass a countable family of Lipschitz images {R}^m$. The first condition, extending prior result Pajot, is test in terms $L^p$ affine approximability measure $\mu$ satisfying the global regularity hypothesis \[ \limsup _{r\downarrow 0} \mu (B(x,r))/r^m <\infty \;\; \text {at $\mu $-a.e.~$x\in \mathbb {R}^n$} \] be $m$-rectifiable sense above. second condition an assumption growth rate...

For $d\geq 2$, we construct a doubling measure $\nu$ on $\mathbb {R}^d$ and rectifiable curve $\Gamma$ such that $\nu (\Gamma )>0$.

We prove a version of Peter Jones' analyst's traveling salesman theorem in class highly non-Euclidean metric spaces introduced by Laakso and generalized Cheeger-Kleiner.These are constructed as inverse limits graphs, include examples which doubling have Poincaré inequality.We show that set one these is contained rectifiable curve if only it quantitatively "flat" at most locations scales, where flatness measured with respect to so-called monotone geodesics.This provides first examination...

We discuss 1-Ahlfors-regular connected sets in a general metric space and prove that such are `flat' on most scales locations. Our result is quantitative, when combined with work of I. Hahlomaa, gives characterization 1-Ahlfors regular subsets curves spaces. generalization to the setting Analyst's (Geometric) Traveling Salesman theorems P. Jones, K. Okikiolu, G. David S. Semmes, it can be stated terms average Menger curvature.

We prove a quantitative version of the following statement. Given Lipschitz function f from k-dimensional unit cube into general metric space, one can be decomposed finite number BiLipschitz functions f|_{F_i} so that k-Hausdorff content f([0,1]^k\smallsetminus \cup F_i) is small. thus generalize theorem P. Jones [Lipschitz and bi-Lipschitz functions. Rev. Mat. Iberoamericana 4 (1988), no. 1, 115–121] setting \mathbb{R}^d to space. This positively answers problem 11.13 in “Fractured Fractals...

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum what is now known as \beta -numbers, numbers measuring flatness given scale and location. This work was generalized to \mathbb R^n by Okikiolu, Hilbert space second author, has many variants variety metric settings. Notably, 2005, Hahlomaa gave sufficient condition for subset be contained curve. We prove sharpest possible converse Hahlomaa’s theorem doubling curves, then...

We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. prove that it is always possible to decompose domain such a mapping pieces on which “behaves like projection mapping” along with “garbage set” <bold>arbitrarily small</bold> in an appropriate sense. Moreover, our control quantitative, i.e., independent both particular and space maps into. This improves theorem Azzam-Schul paper “Hard Sard”, answers question left open paper. The proof uses...

We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with $\mathcal{C}^α$ metric). These are bi-Lipschitz embedded balls domain manifold, distortion constants that depend only natural geometric properties manifold. The proof these results relies estimates, from above below, for kernel its gradient, as well their gradient. estimates hold in non-smooth category, stable...

The purpose of this note is to point out a simple consequence some earlier work the authors, “Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps”. For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, from Euclidean space into metric space, we give...

For a given connected set $\Gamma$ in $d-$dimensional Euclidean space, we construct $\tilde\Gamma\supset \Gamma$ such that the two sets have comparable Hausdorff length, and $\tilde\Gamma$ has property it is quasiconvex, i.e. any points $x$ $y$ can be via path, all of which $\tilde\Gamma$, length bounded by fixed constant multiple distance between $y$. Thus, for $K$ space as above to shortest containing $K$. Constants appearing here depend only on ambient dimension $d$. In case where...

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of spaces that are contained connected set finite Hausdorff length. do so by extending and improving results Peter Jones Kate Okikiolu for $\R^d$. Their formed the basis quantitative rectifiability prove version following statement: length (or subset one), characterized fact inside balls at most scales around points set, lies close straight line segment (which depends on ball). This...

We show that a sufficient condition for subset $E$ in the Heisenberg group (endowed with Carnot-Carath\'{e}odory metric) to be contained rectifiable curve is it satisfies modified analogue of Peter Jones's geometric lemma. Our estimates improve on those \cite{FFP}, by replacing power $2$ Jones-$\beta$-number any $r<4$. This complements (in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we showed such estimate was necessary, but $r=4$.

For $d\geq 2$, we construct a doubling measure $ν$ on $\R^d$ and rectifiable curve $Γ$ such that $ν(Γ)&gt;0$.