User changyu guo - MathOverflow

Is there any known condition for the following property?

2012-12-05T14:48:25Z

For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the one-dimensional Hausdorff measure of $f^{-1}(E)$ is zero whenever $E$ is of zero one-dimensional Hausdorff measure zero. Note that f is not assumed to be a homeomorphism.

Answer by Changyu Guo for Books about Capacity theory

2012-11-01T13:05:15Z

Maz'ya's book contains a fruitful treatment of Capacity and Weighted capacity and its relation with Sobolev spaces theory, in particular the (weighted) Sobolev inequality or Poincare inequality. Heinonen's book contains the treatment of modulus and capacity in metric setting.

Maz'ya, Vladimir Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. xxviii+866 pp.

2.Heinonen, Juha Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001. x+140 pp.

3.Heinonen, Juha; Kilpeläinen, Tero; Martio, Olli Nonlinear potential theory of degenerate elliptic equations. Unabridged republication of the 1993 original. Dover Publications, Inc., Mineola, NY, 2006. xii+404 pp.

Answer by Changyu Guo for Applications of the notion of of Gromov-Hausdorff distance

2012-10-19T07:19:57Z

Stephen Keith used the Gromov-Hausdorff convergence to study the existence of (measurable) differentiable structure on metric measure spaces that supports a Poincare inequality or K-Lip-lip condition.

Juha Heinonen, Jeff Cheeger and Stephen keith also used this method as a standard blow up argument in related questions.

Heinonen, Juha; Keith, Stephen Flat forms, bi-Lipschitz parameterizations, and smoothability of manifolds. Publ. Math. Inst. Hautes Études Sci. No. 113 (2011), 1–37.

Keith, Stephen A differentiable structure for metric measure spaces. Adv. Math. 183 (2004), no. 2, 271–315.

J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (3) (1999) 428–517.

J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below. I, II, III, J. Differential Geom.

Is there a criterion for a map to be a local homeomorphis?

2012-10-12T07:52:12Z

Is there a known criterion for a map to be a local homeomorphism or a.e. local homeomorphism?

Of course, we may add some topological assumptions or analytic assumptions for the mapping, e.g. sense-preserving (means that the local index/degree is positive), discrete (means the fiber of the map is totally disconnected), open (means that it maps open set to open set). It seems that proper analytic assumptions are needed as we know that (non-constant) analytic map in the plane are local homeomorphism.

On differentiability relative to a n-rectifiable subset of \mathbb{R^N}

2012-10-05T11:58:35Z

Let S be a n-rectifiable subset of \mathbb{R}^N , we define the differentiability of a funtion f:S \to \mathbb{R} at a point x_0 in S as in Federer's book, where he called differentiable relative to S at x_0. Are there any known condition to ensure that f is differentiable relative to S at H^n-a.e. points in S?

Here we suppose that S is equipped with a metric d such that it is Ahlfors n-regular in Hausdorff measure, meaning that the Hausdorff n-measure of balls with radius r in S is comparable to r^n. That is, We may view S itself as an n-regular n-rectifiable metric measure space.

Answer by Changyu Guo for On differentiability relative to a n-rectifiable subset of \mathbb{R^N}

2012-10-11T10:03:10Z

I found some results on this topic, except Federer's book "geometric measure theory",

Ambrosio, Luigi; Kirchheim, Bernd Rectifiable sets in metric and Banach spaces. Math. Ann. 318 (2000), no. 3, 527–555.
Keith, Stephen Measurable differentiable structures and the Poincaré inequality. Indiana Univ. Math. J. 53 (2004), no. 4, 1127–1150.

A closed connected component in a topological space does not contain any path-connected subset?

2012-10-08T06:25:08Z

Does there exist such a non-trivial closed connected component U of some connected topological space X or a non-trivial connected topological space X that do not contain any non-trivial path-connected subset.

The answer is negative if the space is assumed to be connected and locally path-connected. Since every component of a connected and locally path-connected space is path connected.

Added after some useful comments: If we assume that the space X is actually a metric space (together with the metric topology), then can it possible to contain non-trivial path-connected subset. Note that i assume that any component of the metric space X is non-trivial (not a point).

Answer by Changyu Guo for Rellich-Kondrachov compactness theorem in arbitrary smooth metric measure spaces

2012-10-01T06:59:59Z

Please take a look at Theorem 8.1 (Chapter 8) in the following paper Hajłasz, Piotr; Koskela, Pekka Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp.

Can we extend a a.e. Lipschitz map defined on a closed subset of \bR^N to the whole space such that it is still a.e. Lipschitz

2012-09-28T06:03:06Z

I have the following question: If $A$ is a metrically oriented $n$-dimensional subset of $\mathbb{R}^N$ and $f$ is a continuous map from $A$ to $\mathbb{R}^M$ . We know that $\mathrm{Lip} f < +\infty$ $L^N$-almost everywhere, can we then continuously extends f to the whole $\mathbb{R}^N$ such that $\mathrm{Lip} f < +\infty$ $L^N$-almost everywhere? Here $\mathrm{Lip}$ is the local lipschitz constant of $f$.

Answer by Changyu Guo for Conformal maps in higher dimensions

2012-09-28T11:39:47Z

I think this is a good reference for it. Iwaniec, Tadeusz; Martin, Gaven Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001. xvi+552.

Answer by Changyu Guo for A Fact Of Quasiconformal Map

2012-09-28T06:13:54Z

A simple way is the use the modulus of curve family argument, see Väisälä's book lecture notes on n-dimensional quasiconformal mappings.

Comment by Changyu Guo

2012-12-06T08:11:07Z

Yes, you are right. It is better to use the Hausdorff measure H^1. \Omega is a domain in \bR^n and f is not necessarily to be a homeomorphism here.

Comment by Changyu Guo

2012-10-12T06:26:07Z

You need injectivity for the mapping f.

Comment by Changyu Guo

2012-10-11T12:45:05Z

could you please point out where is the reference for the above-mentioned result for surfaces?

Comment by Changyu Guo

2012-10-08T09:14:16Z

To Mark Grant: thanks for your reminder. I should also say that the connected component U or the space X in question is also non-trivial.

Comment by Changyu Guo

2012-10-08T08:38:41Z

To Goldstern: what do you mean by countable connected space? Is it a subclass of connected topological space?

Comment by Changyu Guo

2012-10-08T08:34:59Z

To Mark Grant: Yes, i should add non-trivial path-connected subset-

Comment by Changyu Guo

2012-10-01T08:15:32Z

Thanks. I will take a look at the paper.

Comment by Changyu Guo

2012-09-28T06:27:42Z

To "Ricky Demer": the term "metrically oriented" is from J.Heinonen and S.Rickman's paper "Geometric branched covers between generalized manifolds. Duke Math. J. 113 (2002)", it basically says that as A itself is a very good metric measurable space with n-dimensional Hausdorff measure (e.g. supports a (1,1)-Poincare inequality, n-Ahlfors regular, n-rectifiable), so Lipf<\infty L^n-a.e. means that as a metric measure space itself, Lipf is finite in Hausdorff n-measure a.e.. When you extend this map f, then f is defined on R^N which the corresponding measure should be L^N.

Comment by Changyu Guo

2012-09-28T06:10:47Z

Of course one can use the Tietze extension theorem to extend f continuously to the whole space. The problem then is that whether it keeps the a.e. finiteness of the Lipf.