# Sobolev classes of Banach space-valued functions and quasiconformal mappings

@article{Heinonen2001SobolevCO, title={Sobolev classes of Banach space-valued functions and quasiconformal mappings}, author={Juha M. Heinonen and Pekka Koskela and Nageswari Shanmugalingam and Jeremy T. Tyson}, journal={Journal d’Analyse Math{\'e}matique}, year={2001}, volume={85}, pages={87-139} }

We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal…

## 182 Citations

### CHARACTERIZATIONS OF SOBOLEV CLASSES OF BANACH SPACE-VALUED FUNCTIONS ON METRIC MEASURE SPACE

- Mathematics
- 2018

In the paper, we investigate the Sobolev function classes on Euclidean space when the index is infinity and the ones of Banach space-valued functions on metric measure space when the index is…

### Sobolev Spaces and Quasiconformal Mappings on Metric Spaces

- Mathematics
- 2001

Heinonen and I have recently established a theory of quasiconformal mappings on Ahlfors regular Loewner spaces. These spaces are metric spaces that have sufficiently many rectifiable curves in a…

### Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces

- Mathematics
- 2021

We show that a mapping f : X → Y satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the Newton-Sobolev class N loc (X;Y ). Contrary to previous works, we…

### Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property

- Mathematics
- 2008

We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i.e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a…

### Quasiconformal mappings with Sobolev boundary values

- Mathematics
- 2002

We consider quasiconformal mappings in the upper half space R + of R , n ≥ 2, whose almost everywhere defined trace in R has distributional differential in L(R). We give both geometric and analytic…

### Cheeger Type Sobolev Spaces for Metric Space Targets

- Mathematics
- 2004

In this paper, we consider the natural generalization of Cheeger type Sobolev spaces to maps into a metric space. We solve Dirichlet problem for CAT(0)-space targets, and obtain some results about…

### Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients

- Mathematics
- 2015

Preface 1. Introduction 2. Review of basic functional analysis 3. Lebesgue theory of Banach space-valued functions 4. Lipschitz functions and embeddings 5. Path integrals and modulus 6. Upper…

## References

SHOWING 1-10 OF 48 REFERENCES

### Newtonian spaces: An extension of Sobolev spaces to metric measure spaces

- Mathematics
- 2000

This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also…

### Sobolev spaces on an arbitrary metric space

- Mathematics
- 1996

We define Sobolev space W1,p for 1<p≤∞ on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev…

### Quasiconformality and quasisymmetry in metric measure spaces.

- Mathematics
- 1998

A homeomorphism f : X → Y between metric spaces is called quasisymmetric if it satisfies the three-point condition of Tukia and Vaisala. It has been known since the 1960’s that when X = Y = R (n ≥…

### Quasiconformal maps in metric spaces with controlled geometry

- Mathematics
- 1998

This paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry. The principal message is that such a theory is both…

### Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

- Mathematics
- 1996

In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that…

### Geometric Nonlinear Functional Analysis

- Mathematics
- 1999

Introduction Retractions, extensions and selections Retractions, extensions and selections (special topics) Fixed points Differentiation of convex functions The Radon-Nikodym property Negligible sets…

### Sobolev met Poincaré

- Mathematics
- 2000

Introduction What are Poincare and Sobolev inequalities? Poincare inequalities, pointwise estimates, and Sobolev classes Examples and necessary conditions Sobolev type inequalities by means of Riesz…

### THE DIFFERENTIAL OF A QUASI-CONFORMAL MAPPING OF A CARNOT-CARATHEODORY SPACE

- Mathematics
- 1995

The theory of quasi-conformal mappings has been used to prove rigidity theorems on hyperbolic n space over the division algebras ℝ, ℂ, ℍ, and \({\Bbb O}\), by studying quasi-conformal mappings on…

### Analytic properties of quasiconformal mappings on Carnot groups

- Mathematics
- 1995

ANALYTIC PROPERTIES OF QUASICONFORMAL MAPPINGS ON CARNOT GROUPS t) S. K. Vodop~yanov and A. V. Greshnov UDC 512.813.52+517.548.2+517.518.23 In a series of recent articles, the properties of nilpotent…

### Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings

- Mathematics
- 1997

In this paper we shall show that the boundary 9Ip,q of the hyperbolic building Ip,q considered by M. Bourdon admits Poincare type inequalities. Then by using Heinonen-Koskela's work, we shall prove…