Banach spaces of Lipschitz functions of an abstract metric space [PhD thesis]

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T h is d issertatio n h as b e e n m ic ro film e d e x a c tly as re c e iv e d

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JENKINS, Thomas Morton, 1935B A N A C H SPACES O F UPSCHITZ FUNCTIONS O F A N A B S T R A C T METRIC SPACE. Vnlo juu a v >■».«».«/>.-.»>>■«.j>j TJU T> 10CQ Mathematics

University Microfilms, Inc., A n n Arbor, Michigan

^Copyright by

THOMAS MORTON JENKINS 1968

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Banach Spaces of Lipschitz Functions on an Abstract Metric Space. By.' Thomas M? Jenkins

1967

A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy.

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Summary Given a metric space (X,d) and a positive number a < 1, let Lip(X,da ) denote the space of all bounded continuous com­ plex-valued functions f on X such that | |f | | =. d sup { jf*(x)-f (y) j/da (x,y) : x,y £ X, x ^ y] < oo. Lip(X,da ) is a Banach space under the norm

jj j | O

given by

J[f*j{ = IIiloo^ Ill’ll

C£

^

The closed linear subspace lip (X,d ) consists of those f in Lip(X,da ) which vanish at infinity and which satisfy the con­ dition

jf(x)-f(y)J/da (x,y) --- > 0

as

da (x,y) --- > 0.

A metric space (X,d) is said to satisfy the Lipschitz four-point property if every complex-valued function g defined on three points of X has a Lipschitz preserving extension to any fourth point of X; i.e., if |g ( x -)-g(x -)j < K d(x-,x.) -J-

J

J*

J

for i,j = 1,2,3 then g(x^) can be defined so that the inequality holds for i,j = 1,2,3,4. It has long been known that for every subset Y of an arbitrary metric space (X,d) and every real-valued function f £ Lip(Y,d), f has an extension f* £ Lip(X,d) such that 1 1 ^ 1 1 ^ * 1 I^ I Id and I I I too = ||f| 1^. We show that such extensions exist for every subset Y C X and every complex-valued f £ Lip(Y,d) if and only if (X,d) satisfies the Lipschitz four-point property. Our second main result states that if (X,d) satisfies the Lipschitz four-point property, if every closed bounded subset of X is compact and if 0 < a < 1, then Lip(X,da ) is isometrically isomorphic to the second dual of lip°(X,da ). A third result identifies the linear isometries of Lip(X,da ) onto itself and of lip°(X,da ) onto itself where we assume that 0 < a < 1 and that (X,d) is a compact, connected metric space with diam (X,da ) < 2. These results generalize some work of K. de Leeuw who proved the second result and the latter half of the third for the particular metric space (X,d) where X is a circle of unit circumference with distance measured along the circum­ ference.

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Table of Contents Summary § 1a _

TntrnHnptinn

± '

§ 2.

Preliminaries

5«

§3.

Extensions of Lipschitz functions which preserve the Lipschitz condition

10.

§

Lip a

as a second dual

19-

§ 5-

Isometries of Lipschitz spaces

35.

§ 6.

Some related results

53-

Bibliography

57.

Acknowledgments I wish to thank my advisor, Professor Charles E. Rickart, for his encouragement over the past several years and for many helpful suggestions in the preparation of this paper. am indebted to Dr. Edward J. Barbeau for many fruitful dis­ cussions as the results of this paper took shape. I also wish to express my appreciation for financial support from the National Science Foundation in the form of a Science Faculty Fellowship.

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I

1. 5 1-

Introduction.

0 < a < 1,

If

L i p (X ,da )

(X,d)

is a metric space and if

denotes the space of all bounded

continuous complex-valued functions

f

which satisfy a

Lipschitz condition with respect to the metric |f(x)-f(y)| < K(d(x,y))a

for all

Banach space with the norm max ( I If I loo* I |f I I a )

|| ||

where

da ;

i.e.,

x,y £ L

Lip(X,da )

given by

|jfJ J =

is a

||f|j a = sup { jf (x)-f (y) |/da (x,y) :

x,y 9 X, x i y}. We denote by those functions

f

lip(X,da ) in

the closed linear subspace of

Lip(X,da )

which satisfy the con­

dition |f(x)-f (y) |/da (x,y) --- > 0 When

(X,d)

subspace

consisting of all functions in

which vanish at infinity.

In 1961 K. de Leeuw ClO] Lip(R-psa ) modulo 1 n

da (x,y) --- > 0.

is not assumed to be compact, we often need the

lip °(X,da )

lip(X,da )

as

where

0 < a < 1

proved two results for and

with the usual distance

is an integer}).

R-j_

is the real line

(s(x,y) = inf { jx-y+nj:

(As is common, we often refer to

as a circle with distance equal to arc length).

(R^,s)

We prove

similar results for a wide class of abstract metric spaces. Specifically de Leeuw showed that for Lip(R 2 »sa ) of

0 < a < 1,

is isometrically isomorphic with the second dual

lip(R-^,sa ).

every metric space

We prove the same result (Theorem 4.1) for (X,d)

in which closed' bounded sets are

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2. compact and which satisfies a condition we call the Lipschitz four-point property (Def. 3.4). not be assumed if

a < 1/2

The latter condition need

or if

(X,d)

satisfies the

Euclidean four-point property (Def. 3.3).

Moreover, for

(real) spaces of real-valued Lipschitz functions, the result is valid without the Lipschitz four-point property. De Leeuw*s proof involves constructing a sequence of approximations to a given function with

Fejer kernels.

different method.

h

by

convoluting

h

The general case calls for an entirely

Our construction involves extending the

domain of a Lipschitz function without increasing the Lipschitz constant.

This is always possible for real valued functions,

while in the complex case, the extra conditions mentioned above come into play.

§ 3 is devoted to this extension pro­

blem. Next consider a distance preserving map space

(Y, ^>)

number

\

(*)

onto a metric space

with

|>.| = 1.

for every

a

lip(X,da )

onto lip(Y,

circle lip(R^,sa )

with

and

f€C(X), T

of

0 < a < 1. ^»a ).

0 < a < 1,

of a metric

and acomplex

Then the equation

Tf(y) * Xf(?y),

defines a linear isometry

(X,d)

?

y € Y,

Lip(X,da ) Furthermore,

onto T

Lip(Y,^=>a ) maps

De Leeuw showed that for the every linear isometry

T

;

of

onto itself has the form (*=).

We show in Theorem 5.2 that if

X

and

Y

are compact

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3. and if one of the spaces satisfies certain connectedness properties, then every xsoiustry of either Lip(Y,

or of

lip(X,da )

form (=!=) where the map smaller than 2. for

?

onto

Lip{X,d )

lip(Y,^>^)

onto

has the

preserves distances which are

The latter result contains de L e euw9s result

lipff^ ,sa ). A problem similar to the isometry problem was considered

by D.R. Sherbert whose Ph.D. dissertation was devoted to the study of III

Lip(X,d)

III given by

as a Banach algebra with the norm

|||f||| = I |fI 1^+I |fI ld -

showed that every algebra homomorphism Lip(Y,

)

of

Lip(£,d)

into

is of the form

Tf(y) = f ( ? y ) , where

T

Sherbert [13]

: Y

>

f€Lip(X,d),

X

< * ( ? ■ y±, ^ y 2 )
y2 € y ,

for some positive constant K. if and only if

y£Y,

?“ is onto.

Furthermore T

is one-to-one

T is onto if and only if

satisfies the additional condition (°*yl ’y 2* - yl> ^ y 2 * ’ for some

positive constant K .

yl ’y2 Though not

€ Y» explicitly stated

by Sherbert, it follows from his proof that if isometric algebra-isomorphism, then for all

y1 ,Y2 ^ y *

T

is an

^ yi>y2^ = ?'y 2^

While Sherbert9s result is properly set

in a Banach algebra framework, our theorem involves a linear

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4isometry of Banach spaces with no assumption that the map preserves multiplication.

We have not succeeded in character­

izing the linear-homomorphisms of Lipschitz spaces.

This

appears to be more difficult than the linear-isomorphism result. Since the Banach algebra norm j j J |J J || ||,

is equivalent to

many of Sherbertvs results carry over to our situation.

For our purposes, however,

II II

is more appropriate.

While

the theorems of sections 4 and 5 are an outgrowth of de Lee u w ?s work [10], our point of view and some of our techniques were strongly influenced by the work of Sherbert [13,14]. This introduction is followed by some definitions and preliminary results in § 2.

After considering the extension

problem in § 3, we treat the two main results in Sections 4 and 5. In § 6 references are given to other research on Lipschitz functions.

Several authors, by suppressing the con­

stant functions in various ways, have considered (often denoted by

as norm. In this d context, for example, it was known by 1956 (Arens and Eells [1]) that linear space.

H )

with

Lip(X,da )

Lip(X,d)

|| ||

is always the dual of a normed

The first recognition of

Lip(X,da )

as a

second dual appears to be de Leeuw*s result for the circle.

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5. S 2. f

Preliminaries.

For any real or complex-valued function

on a metric space

"Lip- norm"

| |f |

(X,d),

tne sup norm

||x||

and

are defined by

N f I loo = S U P

(x) | :x G X]

{

Ilf Ild = sup { |f(x)-f(y)|/d(x ,y ):x,y € X, x / y}. We define tions

Lip(X,d)

to be the class of complex-valued func­

f : X --- > C

(Actually,

j| |(^

functions).

for which both norms are finite. is a semi-norm since it vanishes on constant

Then every

f

in

Lip(X,d)

is a bounded con­

tinuous function satisfying a Lipschitz condition with respect to the metric

d.

Denote by

lip(X,d)

the subspace

lip(X,d) = {f € Lip(X,d) :

— > 0

as

d(x,y) — > 0}.

The condition is to be interpreted "uniformly"; i.e., for every

e > 0,

implies

there exists

6 > 0

jf(x)-f(y)|/d(x,y) < e.

such that

Unless

d(x,y) < 8

(X,d)

is compact,

this interpretation is not equivalent to assuming that the condition holds in a neighborhood of each point.

When

(X,d)

is not assumed to be compact, we have occasion to consider the subspace

lip °(X,d)

functions

f

f(x)

which we define to be the set of

in lip(X,d)

is small for

x

outside sufficiently large compact sets).

It is well known that the norm

|| ||

which vanish at infinity (i.e.,

given by

Lip(X,d)

is a Banach space with

S if || = I \f I 1^ V | |f | ld

notes the maximum of the real numbers

a

clear that

are closed linear

lip(X,d)

and

lip °(X,d)

and

b).

(avb It is

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de­

6. subspaces of If

Lip(X,d).

G < a < I,

then

(da (x,y) = Cd(x,y)]a ). the metric

da

for

aa

is also a metric on

X

A Lipschitz condition with respect to

0 < a < 1

is often referred to as a

«•

Holder condition with respect to the metric (X,d)

d.

Whenever

is a fixed metric space, we will abbreviate

(lip(X,da ), lip ° (X ,da )) subscript

R

by

Lip a

the norm

(lip a, lip°a).

The

will be used when we wish to consider spaces of

real-valued Lipschitz functions. lipR (X,da )

Lip(X,da )

and || ||

lipR°(X,da )

Then

Lip^(X,da ),

are (real) Banach spaces with

defined above.

Of course the real spaces are

contained in the corresponding complex spaces. We now state several results for reference later in this paper.

Throughout the rest of this section,

an arbitrary metric space. real-valued function

f,

(X,d)

For each real number the functions

the results of truncating

f

fa a

a

and

above and below at

denotes and f va

a.

are

The

results in the following lemma are obvious. Lemma 2.1.

If

f £ Lip^(X,d)

and if

a

is a real number,

then (1)

I If+a||d = ||f||d

(2)

||f/\a||d
s to approximate any function

by a sequence of functions

uniformly on compact sets and be the element of

(lip a) ""

through the canonical embedding of

gn

in

lim sup

lip°a

J|gn lI
F is onto and isometric. We now proceed to the first of our lemmas.

Lemma 4.2. I|F|| < Proof. r

If

F G lip°(X,da )**,

then

F G Lip(X,da )

and

jIF||. Let

F G (llp°a)**.

Since

9*A G ( l i p ^ ) *

for each

A

x G X,

F(x)

is well defined and

|F(x)| = |F(x )|