# Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

@article{Azzam2011HardSQ, title={Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps}, author={Jonas Azzam and Raanan Schul}, journal={Geometric and Functional Analysis}, year={2011}, volume={22}, pages={1062-1123} }

We prove a global implicit function theorem. In particular we show that any Lipschitz map $${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$$ (with n-dim. image) can be precomposed with a bi-Lipschitz map $${\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}$$ such that $${f \circ \bar{g}}$$ will satisfy, when we restrict to a large portion of the domain $${E \subset \mathbb{R}^{n} \times \mathbb{R}^{m}}$$ , that $${f \circ…

## 35 Citations

Morse--Sard theorem and Luzin $N$-property: a new synthesis result for Sobolev spaces

- Mathematics
- 2018

For a regular (in a sense) mapping $v:\mathbb{R}^n \to \mathbb{R}^d$ we study the following problem: {\sl let $S$ be a subset of $m$-critical a set $\tilde Z_{v,m}=\{{\rm rank} \nabla v\le m\}$ and…

An implicit function theorem for Lipschitz mappings into metric spaces

- Mathematics
- 2018

We prove a version of the implicit function theorem for Lipschitz mappings $f:\mathbb{R}^{n+m}\supset A \to X$ into arbitrary metric spaces. As long as the pull-back of the Hausdorff content…

Sets of Absolute Continuity for Harmonic Measure in NTA Domains

- Mathematics
- 2014

We show that if Ω is an NTA domain with harmonic measure ω and E⊆∂Ω is contained in an Ahlfors regular set, then ω|E≪ℋd|E$\omega |_{E}\ll \mathcal {H}^{d}|_{E}$. Moreover, this holds quantitatively…

Uniform domains with rectifiable boundaries and harmonic measure

- Mathematics
- 2015

We assume that $\Omega \subset \mathbb{R}^{d+1}$, $d \geq 2$, is a uniform domain with lower $d$-Ahlfors-David regular and $d$-rectifiable boundary. We show that if $\mathcal{H}^d|_{\partial \Omega}$…

Heat flow and quantitative differentiation

- MathematicsJournal of the European Mathematical Society
- 2019

For every Banach space $(Y,\|\cdot\|_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\in (0,\infty)$ with the following property. Suppose that $n\in \mathbb{N}$…

Approximate tangents, harmonic measure, and domains with rectifiable boundaries

- Mathematics
- 2016

We show that if $E \subset \mathbb R^d$, $d \geq 2$ is a closed and weakly lower Ahlfors-David $m$--regular set, then the set of points where there exists an approximate tangent $m$-plane, $m \leq…

Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator

- Mathematics
- 2015

In this paper, we propose a method for estimating the Sobolev-type embedding constant from W1,q(Ω)$W^{1,q}(\Omega)$ to Lp(Ω)$L^{p}(\Omega)$ on a domain Ω⊂Rn$\Omega\subset\mathbb{R}^{n}$…

Harmonic measure and approximation of uniformly rectifiable sets

- Mathematics
- 2015

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, be a uniformly rectifiable set of dimension $n$. We show $E$ that has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew…

Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$

- Mathematics
- 2012

We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect…

## References

SHOWING 1-10 OF 54 REFERENCES

REGULAR MAPPINGS BETWEEN DIMENSIONS

- Mathematics
- 2000

The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by "regular mappings"…

Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space

- Mathematics
- 2007

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number of…

An area formula in metric spaces

- Mathematics
- 2010

We present an area formula for continuous mappings between metric spaces, under minimal regularity assumptions. In particular, we do not require any notion of differentiability. This is a consequence…

Rectifiable metric spaces: local structure and regularity of the Hausdorff measure

- Mathematics
- 1994

We consider the question whether the "nice" density behaviour of Hausdorff measure on rectifiable subsets of Euclidian spaces preserves also in the general metric case. For this purpose we show the…

Solution of the Plateau problem form-dimensional surfaces of varying topological type

- Mathematics
- 1960

We use a definition due to J. F. Adams: DEFINITION. Let G be a compact Abelian group. Let S be a closed set in N-dimensional Euclidean space and A a closed subset of S. Let m be a non-negative…

Lectures on Lipschitz analysis

- Mathematics
- 2005

(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric…

The coarea formula for metric space valued maps

- Mathematics
- 2009

In this thesis we prove the coarea formula for Lipschitz maps defined on R and taking values in Hm-σ-finite metric spaces. We do this by first defining a coarea factor for almost every point in the…

Fitting a Cm-Smooth Function to Data

- Computer Science, Mathematics
- 2005

This paper and in [20] exhibits algorithms for constructing such an extension function F, and for computing the order of magnitude of its C norm.

Quantitative rectifiability and Lipschitz mappings

- Mathematics
- 1993

The classical notion of rectifiability of sets in R n is qualitative in nature, and in this paper we are concerned with quantitative versions of it. This issue arises in connection with L p estimates…