LECTURES ON LIPSCHITZ ANALYSIS

JUHA HEINONEN

1. Introduction

m nA function f :A→R , AR , is said to be L-Lipschitz, L 0, if

(1.1) |f(a) f(b)| L|a b|

foreverypairofpointsa,b∈A. WealsosaythatafunctionisLipschitz

if it is L-Lipschitz for some L.

The Lipschitz condition as given in (1.1) is a purely metric condi-

tion; it makes sense for functions from one metric space to another.

In these lectures, we concentrate on the theory of Lipschitz functions

in Euclidean spaces. In Section 2, we study extension problems and

Lipschitz retracts. In Section 3, we prove the classical di erentiability

theoremsofRademacherandStepanov. InSection4, webrie ydiscuss

Sobolev spaces and Lipschitz behavior; another proof of Rademacher’s

theorem is given there based on the Sobolev embedding. Section 5 is

the most substantial. Therein we carefully develop the basic theory of

at di erential forms of Whitney. In particular, we give a proof of the

fundamental duality between at chains and at forms. The Lipschitz

invariance of at forms is also discussed. In the last section, Section 6,

we discuss some recent developments in geometric analysis, where at

forms are used in the search for Lipschitz variants of the measurable

Riemann mapping theorem.

DespitetheEuclideanframework,thematerialintheselecturesshould

be of interest to students of general metric geometry. Many basic re-

nsults about Lipschitz functions de ned on subsets of R are valid in

great generality, with similar proofs. Moreover, uency in the classical

theory is imperative in analysis and geometry at large.

Lipschitz functions appear nearly everywhere in mathematics. Typ-

ically, the Lipschitz condition is rst encountered in the elementary

theory of ordinary di erential equations, where it is used in existence

theorems. In the basic courses on real analysis, Lipschitz functions

appear as examples of functions of bounded variation, and it is proved

Lectures at the 14th Jyv askyal Summer School in August 2004.

Supported by NSF grant DMS 0353549 and DMS 0244421.

12 JUHA HEINONEN

that a real-valued Lipschitz function on an open interval is almost ev-

erywhere di erentiable. Among more advanced topics, Lipschitz anal-

ysis is extensively used in geometric measure theory, in partial di er-

ential equations, and in nonlinear functional analysis. The Lipschitz

condition is one of the central concepts of metric geometry, both -

nite and in nite dimensional. There are also striking applications to

topology. Namely, every topological manifold outside dimension four

admits a unique Lipschitz structure, while such a manifold may have

no smooth or piecewise linear structures or it may have many such.

On a more practical side, questions about Lipschitz functions arise in

image processing and in the study of internet search engines, for ex-

ample. Finally, even when one considers rougher objects, such as func-

tions in various Sobolev spaces or quasiconformal mappings, vestiges

of Lipschitz behavior are commonly found in them, and the theory is

applicable.

In many ways, the Lipschitz condition is more natural, and more

ubiquitous,thansaytheconditionofin nitesmoothness. Forexample,

families of Lipschitz functions are often (pre-)compact, so that Arzel a-

Ascoli type arguments can be applied. Compactness in the smooth

context is typically more complicated.

Some of the preceding issues will be studied in these lectures in more

detail, while others will only brie y be alluded to. Many important

topics are not covered at all.

References to the topics advertized in this introduction include [18],

[50], [17], [80], [20], [59], [16], [25], [63], [5], [67], [64], [47], [73], [62],

[43].

n1.1. Notation. Our notation is standard. Euclidean n-spaceR , n

1, is equipped with the distance

nX

2 1/2|x y| := ( (x y ) )i i

i=1

unless otherwise stipulated. The Lebesgue n-measure of a measurable

nset E R is denoted by |E|, and integration against Lebesgue mea-

sure by Z

f(x)dx.

E

nOpen and closed balls in R are denoted by B(x,r) and B(x,r), re-

nspectively; here x ∈ R and r > 0. If we need to emphasize the

ndimension of the underlying space, we write B (x,r). We also write

n n n 1 n nB := B (0,1) and S := ∂B . The closure of a set E R is E,

c nand the complement E :=R \E.LECTURES ON LIPSCHITZ ANALYSIS 3

Other standard or self-explanatory notation will appear.

1.2. Acknowledgements. I thank the organizers of the 14th Jyv as-

kyl aSummerSchool, especiallyProfessorsTeroKilpeal inenandRaimo

N akki,forinvitingmetogivetheselectures. IamgratefultoEeroSaks-

man for many illuminating conversations about the Whitney theory. I

also thank Ole Jacob Broch, Bruce Kleiner, and Peter Lindqvist for

some useful information, and Bruce Hanson, Leonid Kovalev, Seppo

Rickman and Jussi V ais aal for carefully reading the manuscript and

for their comments.

2. Extension

m nEvery Lipschitz function f : A→R , AR , can be extended to

n ma Lipschitz function F : R → R . This means that F is Lipschitz

andF|A =f. In this section, we o er three proofs of this fundamental

result,anddiscusstherelatedproblemofLipschitzretracts. Thedevel-

opment of this section reveals the great exibility aorded by Lipschitz

functions; they can be glued, pasted, and truncated without impairing

the Lipschitz property.

We begin with some preliminaries.

2.1. Distance functions and quasiconvexity. Distance functions

aresimplebutimportantexamplesofLipschitzfunctions. Thedistance

ncan be taken either to a point x ∈R ,0

(2.1) x7→ dist(x,x ) :=|x x |,0 0

nor more generally to a set E R ,

(2.2) x7→ dist(x,E) := inf{|x a| :a∈E}.

That dist(,x ) is 1-Lipschitz is a direct consequence of the triangle0

inequality. It is similarly straightforward to check from the de nitions

that the function dist(,E) in (2.2) is 1-Lipschitz, but it is worthwhile

to record the following general fact.

Lemma 2.1. Let {f : i ∈ I} be a collection of L-Lipschitz functionsi

nf :A→R, AR . Then the functionsi

x7→ inff (x), x∈A,i

i∈I

and

x7→ supf (x), x∈A,i

i∈I

are L-Lipschitz on A, if nite at one point.4 JUHA HEINONEN

The proof of Lemma 2.1 is an easy exercise.

NotethatthesetE in(2.2)isnotassumedtobeclosed. Ontheother

hand, we have that dist(x,E) = dist(x,E). Therefore, one typically

considers closed sets E in this connection. More generally, every L-

mLipschitz function f : A → R extends to an L-Lipschitz function

de ned on the closure A, simply by uniform continuity.

Lipschitzcondition(1.1)isglobal;itrequirescontrolovereachpairof

points a,b in A. Sometimes we only have local information. There is a

simple but useful lemma which shows that under special circumstances

local information can be turned into global.

nA set A R is said to be C-quasiconvex, C 1, if every pair of

points a,b∈A can be joined by a curve in A such that

(2.3) length() C|a b|.

WealsosaythatAisquasiconvexifitisC-quasiconvexforsomeC 1.

By the length of a curve we mean as usual the quantity,

N 1X

length() := sup |(t ) (t )|,i+1 i

i=0

where the supremum is taken over all partitions 0 = t < t < <0 1

nt = 1 for a curve : [0,1]→R .N

mA function f :A→R is called locally L-Lipschitz if every point in

A has a neighborhood on which f is L-Lipschitz.

n mLemma 2.2. If A R is C-quasiconvex and f : A → R is locally

L-Lipschitz, then f is CL-Lipschitz.

We leave the straightforward proof of this lemma to the reader. Now

consider the “slit plane”,

2A :={(r,) : 0<r <∞, < <}R ,

in polar coordinates. The function

2(r,)7→ (r,/2), A→R ,

is locally 1-Lipschitz, but not globally Lipschitz. This example shows

the relevance of quasiconvexity in the situation of Lemma 2.2.

The distance function in (2.1) can be de ned by using the intrinsic

nmetric of a set. Let AR be a set such that every pair of points in

A can be joined by a curve of nite length in A. The intrinsic metric

in A is de ned asA

(2.4) (a,b) := inflength(),A

where the in mum is taken over all curves joining a and b in A.

Expression (2.4) indeed denes a metric in A, and A is quasiconvexLECTURES ON LIPSCHITZ ANALYSIS 5

if and only if the identity mapping between the two metrics is bi-

Lipschitz. Werecallherethatahomeomorphismbetweenmetricspaces

is bi-Lipschitz if it is Lipschitz and has a Lipschitz inverse.

The function

(2.5) x7→ dist (x,x ) := (x,x )A 0 A 0

is 1-Lipschitz with respect to the intrinsic metric; it is Lipschitz if A

is quasiconvex. We will return to quasiconvexity in connection with

Lipschitz retracts later in this section.

Finally, we say that a curve in a set A, joining two points a and b,

is an intrinsic geodesic if length() = (a,b).A

2.2. Extension theorems. We prove the important extension theo-

rems of McShane-Whitney and Kirszbraun.

Theorem2.3 (McShane-Whitneyextensiontheorem). Letf :A→R,

nAR , be an L-Lipschitz function. Then there exists an L-Lipschitz

nfunction F :R →R such that F|A =f.

Proof. Because the functions

f (x) :=f(a)+L|x a|, a∈A,a

nare L-Lipschitz onR , the function

nF(x) := inf f (x), F :R →R,a

a∈A

is L-Lipschitz by Lemma 2.1. It is obvious that F(a) =f(a) whenever

a∈A.

The extension F in Theorem 2.3 is the largest L-Lipschitz extension

nof f in the sense that if G :R →R is L-Lipschitz and G|A =f, then

G F. One can also nd the smallest Lhitz extension of f, by

setting

nF(x) := sup f(a) L|x a|, x∈R .

a∈A

m nCorollary 2.4. Let f :A→R , AR , be an L-Lipschitz function.

√

n mThen there exists an mL-Lipschitz function F :R →R such that

F|A =f.

Corollary 2.4 follows by applying Theorem 2.3 to the coordinate

√

functions of f. The multiplicative constant m in the corollary is in

fact redundant, but this is harder to prove.

m nTheorem 2.5 (Kirszbraun’s theorem). Let f : A → R , A R ,

be an L-Lipschitz function. Then there exists an L-Lipschitz function

n mF :R →R such that F|A =f.6

6

6 JUHA HEINONEN

Proof. By dividing the function f by L, we may assume that f : A→

mR is 1-Lipschitz.

To prove the theorem, the following is a key lemma.

mLemma 2.6. If f is an R -valued 1-Lipschitz function on a nite set

n n mF R , and ifx∈R , then there is an extension off to anR -valued

1-Lipschitz function on F ∪{x}.

To prove Lemma 2.6, we consider in turn the following assertion.

nLemma 2.7. Let {x ,...,x } be a nite collection of points in R ,1 k

mand let {y ,...,y } be a collection of points in R such that1 k

(2.6) |y y | |x x |i j i j

for all i,j ∈{1,...,k}. If r ,...,r are positive numbers such that1 k

k\

B(x,r ) =∅,i i

i=1

then

k\

B(y,r ) =∅.i i

i=1

Let us rst prove Lemma 2.6 by the aid of Lemma 2.7. Indeed, let

n mF ={x ,...,x }R , let f : F →R be a 1-Lipschitz map, and let1 k

nx∈R . Set r :=|x x| and y :=f(x ). By Lemma 2.7, there existsi i i i

ma point y∈R such that|y f(x )||x x| for each i. The desiredi i

extension is accomplished by settingf(x) =y. This proves Lemma 2.6

assuming Lemma 2.7.

Now we turn to the proof of Lemma 2.7. Put

|y y|i mG(y) := max , y∈R .

i=1,...,k ri

mThen G : R → R is a continuous function (Lipschitz, in fact) with

G(y) → ∞ as |y| → ∞. It follows that G achieves its minimum at a

mpoint w∈R , and we need to show that G(w) 1.

Towards a contradiction, assume that G(w) =:> 1. Let J denote

those indices j ∈{1,...,k} for which|w y | =r . Pick a pointj j

\

x∈ B(x ,r ),j j

j∈J

and consider the following two sets of directions,

x x y wj jn 1 0 m 1D :={ :j ∈J}S , D :={ :j ∈J}S .

|x x| |y w|j j6

LECTURES ON LIPSCHITZ ANALYSIS 7

It is easy to see from the de nitions, and from the contrapositive as-

0sumption, that the natural map D → D strictly decreases distances.

We therefore require the following additional lemma.

m 1Lemma 2.8. Let g : K →S be an L-Lipschitz map, L < 1, where

n 1K S is compact. Then g(K) is contained in an open hemisphere.

Before we prove Lemma 2.8, let us point out how Lemma 2.7 fol-

0lows from it. Indeed, the map between directions, D → D, strictly

decreases the distances, and so is L-Lipschitz for some L < 1 because

0thesetsinquestionare nite. Itfollowsthat D iscontainedinanopen

0 m 1hemisphere; sayD S ∩{x > 0}. But then by movingw slightlym

in the direction of the mth basis vector e , the value of the functionm

G decreases, contradicting the fact that G assumes its minimum at w.

It therefore su ces to prove Lemma 2.8. To do so, let C be the

m

convex hull of g(K) in B . We need to show that C does not contain

the origin. Thus, assume

g(v )++ g(v ) = 01 1 k k

for some vectors v ∈ K, and for some real numbers ∈ [0,1] suchi iP

kthat = 1. Because g is L-Lipschitz with L< 1, we have thatii=1

hg(v ),g(v )i>hv,v ii j i j

for every i =j. Thus, writing b :=v , we ndi i i

kX

hb ,bi< 0j i

i=1

for each j. But this implies

kX

h(b ++b ),(b ++b )i = hb ,bi< 0,1 k 1 k j i

i,j=1

which is absurd.

This completes our proof of Lemma 2.8, and hence that of Lemma

2.6. It remains to indicate how Kirszbraun’s theorem 2.5 follows from

Lemma 2.6.

We use a standard Arzel a-Ascoli type argument. Choose countable

ndensesets{a ,a ,...}and{b ,b ,...}inAandinR \A, respectively.1 2 1 2

nWe may assume that both of these sets are in nite. (If R \A is nite,

the extension is automatic; ifA is nite, the ensuing argument requires

only minor notational modi cations.) For each k = 1,2,..., we can

use Lemma 2.6 repeatedly so as to obtain a 1-Lipschitz map

mf :{a ,...,a ,b ,...,b }→Rk 1 k 1 k6

6

8 JUHA HEINONEN

suchthatf (a ) =f(a )foreveryi = 1,...,k. Thesequence(f (b ))k i i k 1

mR is bounded, and hence has a convergent subsequence, say (f 1(b )).1k

j

Similarly, from the mappings corresponding to this subsequence we

can subtract another subsequence, say (f 2), such that the sequencek

j

m(f 2(b ))R converges. Continuing this way, and nally passing to2kj

jthe diagonal sequence (g ), g :=f , we nd that the limitj j k

j

mg(c) := lim g (c)∈Rj

j→∞

exists for everyc∈C :={a ,a ,...}∪{b ,b ,...}. Moreover, g :C →1 2 1 2

mR is 1-Lipschitz, and g(a ) = f(a ) for each i = 1,2,.... Because Ci i

nis dense inR , and because{a ,a ,...} is dense in A, we have that g1 2

n mextends to a 1-Lipschitz mapR →R as required.

This completes the proof of Kirszbraun’s theorem 2.5.

Remark 2.9. (a)ThecruciallemmaintheprecedingproofofKirszbraun’s

theorem was Lemma 2.7. Gromov has asserted [19] an interesting vol-

ume monotonicity property that also can be used to derive Lemma 2.7.

Namely, assume that

B(x ,r ),...,B(x ,r ) and B(y ,r ),...,B(y ,r )1 1 k k 1 1 k k

nare closed balls inR , kn+1, such that|y y ||x x | for eachi j i j

i,j ∈{1,...,k}. Then

k k\ \

(2.7) | B(x,r )| | B(y,r )|.i i i i

i=1 i=1

It is easy to see that Lemma 2.7 follows from this assertion, thus pro-

viding another route to Kirszbraun’s theorem.

(b) The preceding proof of Kirszbraun’s theorem 2.5 works the same

n mwhen one replacesR by an arbitrary separable Hilbert space, andR

by an arbitrary nite dimensional Hilbert space. Standard proofs of

Kirszbraun’s theorem typically use Zorn’s lemma (in conjunction with

Lemma 2.7 or a similar auxiliary result). The preceding Arzel a-Ascoli

argument does not work for in nite-dimensional targets.

2.3. Exercises. (a) Let {B : i ∈ I} be an arbitrary collection ofi

closed balls in a Hilbert space with the property that

\

B =∅i

i∈F

for every nite subcollection F I. Prove that

\

B =∅.i

i∈ILECTURES ON LIPSCHITZ ANALYSIS 9

(Remember that bounded closed convex sets are compact in the weak

topology of a Hilbert space.)

(b) Prove Kirszbraun’s theorem in arbitrary Hilbert spaces.

It is a problem of considerable current research interest to determine

for which metric spaces Kirszbraun’s theorem remains valid. There

are various variants on this theme. One can consider special classes

of source spaces and target spaces, or even special classes of subspaces

from where the extension is desired. Moreover, the Lipschitz constant

may be allowed to change in a controllable manner. It would take us

toofara eldtodiscusssuchgeneraldevelopments(referencesaregiven

in the Notes to this section), but let us examine a bit further the case

of subsets of Euclidean spaces.

m2.4. Lipschitz retracts. A set Y R is said to have the Lipschitz

extension property with respect to Euclidean spaces, or the property, for short, if for every Lipschitz function f : A →

n nY, A R , extends to a Lipschitz function F : R → Y. Note

that we are asking for the mildest form of extension, with no control

of the constants. In applications, a more quantitative requirement is

often necessary. Sets with the Lipschitz extension property can be

characterized as Lipschitz retracts of Euclidean spaces.

mA set Y R is said to be a (Euclidean) Lipschitz retract if there is

maLipschitzfunction :R →Y suchthat(y) =y forally∈Y. Such

a function is called a Lipschitz retraction (onto Y). We also say that

mY is a Lipschitz retract ofR in this case. Note that ifY is a Lipschitz

m MretractofsomeR ,thenitisaLipschitzretractofeveryR containing

Y. Thus the term “Euclidean Lipschitz retract” is appropriate.

A Lipschitz retract is necessarily closed, as it is the preimage of zero

under the continuous map y 7→ (y) y. Therefore it is no loss of

generality to consider only closed sets in the ensuing discussion.

mProposition 2.10. A closed set Y R has the Lipschitz extension

mproperty if and only if Y is a Lipschitz retract of R .

Proof. If Y has the Lipschitz extension property, then Y is a Lipschitz

mretract of R , for the identity function Y → Y must have a Lipschitz

m mextension to R . On the other hand, if : R → Y is a Lipschitz

nretraction and if f : A → Y, A R , is a Lipschitz function, then

n nF :R → Y provides a Lipschitz extension of f, where F :R →

mR is an extension guaranteed by the McShane-Whitney extension

theorem.