Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon

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Chapter 18

Analysis on Metric Spaces

Stephen Semmes

It is well known that a lot of standard analysis works in the very general

setting of "spaces of homogeneous type." There is a lot of analysis, related to

Sobolev spaces and differentiability properties of functions, which does not

work at this level of generality. There are nontrivial results about when it

does work.

We understand very little about geometry, just a few special cases.

18.1

Introduction

Let (M, d(x, y)) be a metric space. For the record, this means that M is a

nonempty set, d(x, y) is a nonnegative symmetric function on M x M, which

vanishes exactly on the diagonal and which satisfies the triangle inequality

d(x,z) < d(x,y) + d(y,z)

(18.1)

for all x, y, z E M.

We all know that a lot of analysis, of continuity and convergence and so

forth, makes sense on abstract metric spaces. On the other hand, a lot of

analysis makes sense on abstract measure spaces, spaces with a measure but

no metric, analysis of integration and LP spaces and so forth. And then there

is a lot of analysis that makes sense on Euclidean spaces which does not carry

over to either of these abstract contexts because they do not have enough

structure. Coifman and Weiss [1], 12] had the marvelous realization that one

could go a long way with only a modest compatibility between measure and

Research partially supported by an NSF grant.

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Chapter 18: Analysis on Metric Spaces

metric. If (M, d(x, y)) is a metric space and 1 is a nonnegative Borel measure

on M, not identically zero, then we say that p is a doubling measure if there

is a constant C > 0 so that

p(B(x, 2r)) < C p(B(x, r))

(18.2)

for all x E M and r > 0, where B(x, r) denotes the (open) ball in M with

center x and radius r. In this case we call the triple (M, d(x, y), p) a space

of homogeneous type. We have both a metric and a measure, and we can

formulate many ideas in analysis in this setting. Examples include Euclidean

spaces as well as many familiar fractals, like Cantor sets and snowflakes.

For example, if f is a locally integrable function on M, then we say that

x is a Lebesgue point for f if

lim

If (y) - f (x) I dp(y) = 0.

(18.3)

r--.O

11(B(x, r))

f(x,r)

This concept makes sense as soon as we have both a measure and a metric,

and it turns out that the doubling condition is strong enough to ensure that

almost every point in M is a Lebesgue point for any given locally integrable

function.

This is true for practically the same reason that it is true on

Euclidean spaces; a suitable covering lemma of the Vitali variety works on

spaces of homogeneous type, and then the rest of the argument goes through.

To quiet the nervous voices in the back of my mind, let me assume also

that (M, d(x, y)) is complete. This ensures that closed and bounded subsets

of M are compact (in the presence of a doubling measure), and that obviates

some technical issues (like the Borel regularity of the measure).

Not only does one have Lebesgue points on spaces of homogeneous type,

but one also has working theories of Hardy spaces, BMO, and singular in-

tegral operators. Specific examples of interesting singular integral operators

on general spaces of homogeneous type are harder to come by. There do

not seem to be anything like Riesz transforms in general; they require more

geometric structure. One can build analogues of imaginary-order fractional

integral operators, but they are less interesting than Riesz transforms.

What about Lipschitz functions and Sobolev spaces? One of the great

features of analysis on Euclidean spaces is the special structure of Lipschitz

functions and Sobolev spaces, e.g., differentiability almost everywhere, vari-

ous mechanisms to control a function in terms of its gradient (Sobolev embed-

dings, Poincare inequalities, isoperimetric inequalities). Are there reasonable

versions of these results for general spaces of homogeneous type? The short

answer is no. One can see this with Cantor sets and snowflakes, for instance.

For the record, a Lipschitz function on a metric space (M, d(x, y)) means

a (real-valued) function f such that

If (x)

- f(y)I <_ Cd(x,y)

(18.4)

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287

for all x, y E M. If we replace d(x, y) by d(x, y)° for some a E (0,1), then

we say that f is Holder continuous of order a. In Euclidean analysis there

is an enormous difference 'between Lipschitz functions and Holder continu-

ous functions, and more generally between integer orders of smoothness and

fractional orders of smoothness. At the integers there is much more rigidity

and structure. At the level of spaces of homogeneous type one does not see

this distinction. Indeed, one is often feeling free on spaces of homogeneous

type to replace a metric d(x, y) with d(x, y)' for any s > 0, which will not

even be a metric when s > 1 (only a quasimetric).

Sometimes in analysis the special nature of the integer orders of smooth-

ness is , a nuisance, but there is a lot of geometry there that we should not

forget. The concept of Lipschitz functions has the wonderful property that

it makes sense on any metric space and enjoys interesting rigidity properties

on some metric spaces, like Euclidean spaces. Lipschitz functions are nicer

than Riesz transforms in that one always has them even if they do not always

have interesting structure.

Here is a special feature of analysis on Euclidean spaces.

Let f be a

compactly supported function on R, or at least a function which tends to 0

at oo. Then

in-1

IVf(y)I dy

(18.5)

f (x)I

Jxn Ix - y

for all x E R". This is true, for instance, if f is C', and follows from a well-

known identity. It can be proved by computing f (x) in terms of an integral

of V f over each ray emanating from x, and then averaging over the rays.

(See [15].)

It holds more generally for Lipschitz functions, or functions in

Sobolev spaces, by approximation arguments. (In this case we should take

V f in the sense of distributions.)

If f is not continuous we may have to

settle for (18.5) holding almost everywhere, etc.

This inequality captures a nontrivial amount of Euclidean geometry. Once

we have it we can derive Sobolev embeddings through estimates on the po-

tential operator

h (9) (x) = f 1

Ix -

yI"_1

9(y) dy.

(18.6)

The usual estimates for this potential operator (as in [151) work much more

generally than simply on Euclidean spaces; they depend only on fairly crude

considerations of metric and measure.

It is (18.5) which contains subtle

information about Euclidean geometry, and there is nothing like it for spaces

of homogeneous type in general.

Let us try to formulate the idea of (18.5) in a general setting. The main

point is to have a generalization of IVf(y)I, some quantity which controls

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Chapter 18: Analysis on Metric Spaces

the infinitesimal oscillation of f. There are a number of ways to do this,

or one can make discrete versions that avoid some technicalities. For the

present purposes the convenient choice is the following notion of generalized

gradient, for which we follow Hajlasz, Heinonen, and Koskela.

Definition 18.1. If (M, d(x, y)) is a metric space and f and g are two Borel

measurable functions on M, with f real-valued and g taking values in [0, oo],

then we say that 9 is a generalized gradient of f if

If (7(a)) - f(y(b))I 5

f g(-y(t)) dt

(18.7)

whenever 'y :

[a, b] -4 M is 1-Lipschitz, i.e., d(7(s), y(t)) < Is - tI for all

s, t E [a, b].

For instance, if f is a Lipschitz function on M, then

g(x) = lim inf sup

I f (y) - f (x) I (18.8)

r-+0

YEB(x,r)

is always a generalized gradient for f.

To formulate a general version of (18.5) we also need a measure, for which

we shall use Hausdorff measure. Let (M, d(x, y)) and n > 0 be given, and let

A be a subset of M. For each d > 0 set

Hb (A) = inf

(diam E2)" : {E;} is a sequence of sets in M

which covers A and satisfies diam E; < 6 for all j},

and then define the n-dimensional Hausdorff measure of A by

H"(A)

aHa (A).

(18.9)

The limit exists because of monotonicity in 6. The natural generalization of

(18.5) to a general metric space (M, d(x, y)) is then

If W1 < Cfm

d(x,y)"-'

Ig(y)I dH"(y)

(18.10)

for all x E M whenever f is a function on M with compact support and

g is a generalized gradient of f in the sense of Definition 18.1.

(For this

formulation we need M to be unbounded and there are other formulations

for the bounded case.)

This is not to say that such an estimate holds in general. When it is true

it contains real information about the geometry of M. To avoid trivialities

one should require something about the Hausdorff dimension of M though;

e.g., one should not take n stupidly too small so that the right-hand side is

always infinite.

When does (18.10) hold? I can prove the following.

Semmes 289

Theorem 18.2. Let (M, d(x, y)) be a metric space and n be a positive in-

teger. The estimate (18.10) holds for some C and all x, f, g as above if M

satisfies the following five conditions:

(i) M is complete and unbounded;

(ii) M is doubling, which means that every ball in M can be covered by a

bounded number of balls of half the radius;

(iii) every bounded subset of M has finite H"-measure;

(iv) M is a topological manifold of dimension n;

(v) M is locally linearly contractible, which means that there is a constant

k > 1 so that if B is any ball in M then B can be contracted to a

point inside of kB (the ball with the same center as B but k times the

radius).

One could say that the preceding assumptions on M are sufficient to imply

that M has some Euclidean behavior. Note that M = R" satisfies all these

conditions, and indeed much better. Condition (v) is probably the trickiest

of the conditions to understand. It says that M is locally contractible, with a

scale-invariant estimate. The requirement that M be a topological manifold

implies already that M is locally contractible, but (iv) is not quantitative;

it is (v) which provides the quantitative link between the metric and the

topology. It prevents cusps and long thin tubes, for instance. The doubling

condition is the only other part of the hypotheses which comes with a bound.

Condition (iv) implies that M has topological dimension n, and hence

Hausdorff dimension > n. From (iii) we conclude that the Hausdorff dimen-

sion is exactly n. We do not demand good control on the Hausdorff measure

here, but bad behavior would mean that the Hausdorff measure is too large

and we would pay for that in (18.10) (in the right side being large).

Theorem 18.2 is proved in [13]. In fact, one has approximately the same

structure as for Euclidean spaces, namely a large family of curves that em-

anate from a given point x and that are fairly well distributed in mass. These

curves play the role of the rays which emanate from a given point in a Eu-

clidean space. It is not so easy to produce the curves directly; instead they

are found in the fibers of some special mappings with controlled Lipschitz

behavior and topological nondegeneracy. In this outline form the methods

are the same as in [4], which treated the case of geometries obtained by

perturbing Euclidean spaces via "metric doubling measures." In the present

situation the construction of the special mappings is more involved because

of the absence of a background Euclidean structure. See also [14] for a ver-

sion of [4] in which the basic principles are implemented more clearly and

with more room for maneuver.

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Chapter 18: Analysis on Metric Spaces

Part of the point here is that there are plenty of examples of spaces with

good geometric properties but no good parameterization.

(See [11], [12].)

Theorem 18.2 and its proof show that many natural constructions that would

be easy if there were good parameterizations can still be carried out when

such parameterizations fail to exist.

There are many variations on this theme in [13], e.g., localized control on

the oscillation of a function in terms of its generalized gradient, finding plenty

of curves that are not too long, concluding Sobolev or Poincare inequalities

under suitable measure-theoretic assumptions. Let us forget all of that and

instead play with the conceptual meaning of (18.10).

Let us put ourselves more firmly in the mode of spaces of homogeneous

type and require that (M, d(x, y)) be Ahlfors regular of dimension n, which

means that it is complete and that there is a constant C > 0 so that

C-'r" < H"(B(x,r)) <Cr" (18.11)

for all x E M and r > 0. This implies that M is doubling and unbounded, and

that H" is a doubling measure on M. Under this assumption the right side

of (18.10) behaves (as an integral operator) in practically the same manner

as on R.

Notice that we really need to have some concept of dimension in this story

of (18.10), because we need to know what number to put in the denominator

of the integrand in (18.10). This is not true of the usual theory of spaces of

homogeneous type.

Under what conditions can we hope that something like (18.10) holds?

Let us consider the space (R", Ix - y {') as an example, s > 0, s i6 1. If s > I

then this is not a metric space, and it is not even close, because there are no

nonconstant Lipschitz functions on (R'3, Ix - y'') when s > 1. This reflects

an important point about metric spaces and the triangle inequality: there

are plenty of Lipschitz functions on metric spaces. When we ask for versions

of (18.10), it is reasonable to start by making the a priori assumption that

our function f is Lipschitz, in order to have some reasonable behavior. If

s < 1 then (W, Ix - yl') is a kind of snowflake, and there are no nonconstant

Lipschitz mappings from an interval into it. The concept of generalized

gradient is then vacuous. Thus the power s = 1 is determined uniquely by

the existence of nontrivial Lipschitz functions on llY" and nontrivial rectifiable

curves in R".

Is anything like (18.10) ever true for spaces whose geometry is not some-

how very close to Euclidean? Some kind of fractals perhaps? The ones

with the best hope are the ones with the most rectifiable curves inside

of them, but the usual examples (like the Sierpinski carpet) do not have

enough rectifiable curves in them. For metric spaces which can be realized

as subsets of Euclidean spaces it might be that something like (18.10) would

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291

force approximately-Euclidean behavior, but for general metric spaces this

is not true, because there are estimates like (18.10) for the Heisenberg group

equipped with its Carnot metric. These metric spaces are Ahlfors-regular,

but with a dimension one larger than the topological dimension.

So under what conditions is something like (18.10) true? One can ask

similar questions about Sobolev inequalities, isoperimetric inequalities, etc.,

but we may as well stick to (18.10). It is a nice property that we can formulate

very generally, and we know that once we have it we can do a lot through

arguments typical for spaces of homogeneous type.

It seems reasonable to me to conjecture that if one has something like

(18.10) on an Ahlfors-regular space of dimension n, then n has to be an

integer.

Theorem 18.2 says that we can see a phenomenon like (18.10) through

quantitative topology. The proof is very much for spaces approximately Eu-

clidean; it was very important that the topological and Hausdorff dimensions

coincide. The method is completely ineffective for the Heisenberg group.

There are a lot of aspects of analysis for which Euclidean spaces and

the Heisenberg group are very similar. This includes (18.10) and the dif-

ferentiability almost everywhere of Lipschitz mappings. (See [9].) It is not

so clear how to distinguish them in simple language; I mean language that

makes sense for Ahlfors-regular metric spaces for instance. One possibility

is that, while both Euclidean spaces and Heisenberg groups satisfy (18.10),

for Euclidean spaces this might be true in a much stabler way; i.e., so that

there is a much more generous notion of "approximately Euclidean" for which

(18.10) is preserved. This idea is illustrated by Theorem 18.2, where we see

that (18.10) can be captured by relatively loose conditions. In Theorem 18.2

one need not specify much structure that is approximately Euclidean, even

though in the end these conditions imply that the structure is more like Eu-

clidean than is obvious at first.

It is not clear that Heisenberg geometry is

so stable, that relatively loose conditions can still capture some nontrivial

aspect of the geometry, as in (18.10).

In fact I suspect that Heisenberg geometry is not so stable, precisely

because it is not so closely connected to topology the way that Euclidean

geometry is. Roughly speaking, one expects something like Euclidean geom-

etry when the topological dimension and the Hausdorff dimension coincide.

One can make precise statements of this nature, as discussed in [10]. Note,

however, that Heisenberg geometry is much more stable than most "fractal"

geometries, because of results like the differentiability almost everywhere of

Lipschitz mappings.

For what other geometries is there good behavior for analysis almost like

there is on Euclidean spaces, like (18.10) for instance? This question makes

me a little nervous; maybe we are a little bit lucky to have come across

Heisenberg geometry, especially since it cannot appear inside a Euclidean

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Chapter 18: Analysis on Metric Spaces

space even up to bilipschitz equivalence, because of [9]. (See [11].) Maybe

there are many other geometries with similar properties but which are very

different from Euclidean and Heisenberg geometries. They might exist and we

just do not know where to look for them. Or maybe there are not so many

geometries that are truly different and for which (18.10) works, but then

how do we extract some common principle which includes both Euclidean

geometry and Heisenberg geometry and excludes the rest?

Incidentally, the only examples that I know of metric spaces which are

doubling but which do not admit bilipschitz parameterizations into some 1R'

are based on the Heisenberg group (or some Carnot group). The argument

uses the differentiability almost everywhere of real-valued Lipschitz functions

on the Heisenberg group, as in [9). It would be very nice to have other

examples, preferably more elementary or based on more direct principles.

What can we say about geometries as a family? How to deform geome-

tries, when are geometries very stable, so that relatively loose conditions

imply that two spaces have similar geometry? Or how to understand the va-

riety in geometry, the ways in which geometries are truly different, in terms

comprehensible to human beings? I know some examples, but I think that

we are blind to what really happens in the variety of geometry.

As to the stability of geometry, of Euclidean geometry, the story of uni-

form rectifiability of David and myself (as in [5], [6], [3), [10]) provides a

language in which we can say that relatively loose conditions on a space

imply that the geometry is approximately Euclidean. The older story of

ordinary rectifiability also does this in a less structured way.

Issues like these show up in the recent work of Heinonen and Koskela

[7], (8] too. There one asks for certain Sobolev-Poincare inequalities and one

makes conclusions about the geometry of mappings. More precisely, one asks

for bounds on functions whose gradient lies in L", where n is the dimension as

in (18.11). This is the conformally invariant case, and such bounds are weaker

than the usual range of Sobolev-Poincare inequalities, where one wants to

control functions even when the gradient lies in LP for p < n. Again one can

ask for which spaces such inequalities hold, how stable are the conditions

under which they hold, and again the answer is not so clear. The conditions

arising in [7), [8].are much closer to the edge of invariance; it is basically

a matter of knowing when nontrivial continua can be seen by the invariant

Sobolev space of functions whose gradient lies in L'. Perhaps they can be

tied to geometry in a more delicate way.

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293

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tains Espaces Homogenes, Lecture Notes in Math. 242, Springer-Verlag,

1971.

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[3] G. David, Wavelets and Singular Integrals on Curves and Surfaces, Lec-

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quasiconformal mappings, in Analysis and Partial Differential Equations,

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[5] , Singular Integrals and Rectifiable Sets in R": au-deld des graphes

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[12] , Good metric spaces without

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