The Journal of Geometric Analysis Volume 8, Number 5, 1998
Continuity of the Integral as a Function of the Domain B y Jenny Harrison
ABSTRACT. We present here the fundamentals of a theory of domains that offers unifying techniques and terminology for a number of different fields. Using direct, geometric methods', we develop integration over p-dimensional domains in n-dimensional Euclidean space R n, replacing the method of parametrization of a domain with the method of approximation in Banach spaces. We prove basic results needed for a theory of integration - - continuity of the integral as a function of its domain and integrand (Corollary 4.8) and a generalization of Stokes's theorem (Corollary 4.14).
1. Introduction In 1949, Whitney [15] initiated the study of how the integral of a fixed differential form w defined in a neighborhood of a domain M in N n varies as a function of the position of M. The concept of a domain of integration that emerges extends beyond the usual notion for polyhedra to limits of polyhedra in Banach spaces. Whitney defined the sharp and fiat norms for his study. However, it turns out that the boundary operator is not continuous w.r.t, the sharp norm and the Banach space of fiat forms is not closed under the Hodge star operator. (See Example 4.3.) Thus, Green's theorem cannot be stated for either sharp or fiat domains and there is no Laplace operator A defined for either of them. Our approach bifurcates from Whitney's starting with his first definitions. The one parameter family of norms of this paper are a blend of Whitney's fiat norm [14] and Lebesgue's L 1 norm [12] (see Sections 2 and 4). Daniell [2] completed step functions w.r.t the L 1 norm to create the Banach space of L 1 functions. Lebesgue theory was satisfying in that all naturally arising integrable functions were represented in this Banach space. The author's search has been for Banach space representatives of all naturally arising domains. The Banach spaces of domains introduced in this paper are denoted ~@a, r ~ Z, r > - 1, 0 < a _< 1, and are taken with respect to norms IPIr,~, initially defined for polyhedra P. The mass of a polyhedral chain or p-dimensional Hansdorff measure coincides with [P [0. When the dimension p is understood, we write A r'a = ~4~~ and call its elements p-chainlets. We identify the integer r with (r - 1, 1) and order pairs (r, a) lexicographically. (Throughout the paper we assume 0 < o~ < 1. The reader is advised to set a = 1 for the first reading.) For (r, a) > 0, the C r'~ differentiability class of a differential form is defined in the standard way, with norm lifo IIc .... The exterior derivative of a smooth differential form is the standard analytic
9 1998 The Journal of Geometric Analysis ISSN 1050-6926
770
Jen.y Harrison
definition. The Banach space of differential p-forms in R n with bounded norm [Icollcr,~ is denoted denotes Lipschitz forms. We let B ~ denote the Banach space of continuous forms with bounded C O n o r m )
13~c~ = B r'a. For example,/31 = B ~
The basic inequality of integration pco <
l[@lc01PIo
for all polyhedra P and all continuous co e B ~ extends to higher norn~s and differentiability:
Continuity estimate.
I f P is a polyheclral p-chain and co E BIt~" then
~
co _< (p-t- 1)llcollcr,~lPlr,~
for all (r, or) > O.
(See Corollary 4.8.) This allows us to define the integral over chainlets if the integrand is smooth. If A e A~ ~, then A = Ei~__i Pi where the Pi are polyhedral p-chains and the series converges w.r.t the (r, o0-norm, (r, o~) > 0. F~C/
For co ~ Bp , define tO
We note that a limit point in a Banach space always has an equivalent expression as a telescoping infinite series. Conversely, every infinite series is a limit of its sequence of partial sums. The series version is more natural for chainlets for a number of reasons. First of all, it gives a chainlet an expression that closely resembles that of a simplicial chain. Also, the series version matches well with Fourier series expansion of integrands reducing all integration to integration of trigonometric functions over cubes with rational vertices. Furthermore, it is natural for basic examples such as the Van Koch snowflake, for instead of "erasing and pasting" one takcs formal sums of simplexes. Finally, in examples such as the graph of an L 1 function and Dirac delta functions, the series expression reveals the deep role of dipoles in the theory of integration. (See the examples in Section 2.1.) The boundary operator O on polyhedral chains is continuous w.r.t, the (r, o0-norms. Theorem 2.6.) We can immediately deduce the following: Stokes's Theorem for Chainlet Domains.
(See
~t//d A I f co is a differential form in Nr+l'c~ "p-I
A~;~ , (r, at) _ O, then
P r o o f . Since do) ~ B~;c~the left-hand side is uniquely defined. By the definition of the integral over chainlets, Stokes's theorem for polyhedra and Lipschitz integrands, and continuity of the boundary operator we have
1Note that the spaces 13r'" are defined for (r, a) > 0 while the spaces .,4''~ are defined for (r, oe) > -1. In the sequel [7] we prove that dual spaces of cochains correspond to spaces of differential forms. Almgren [1] proved that mass continuous cochains correspond to forms, but these are not necessarily measurable. The Banach spaces ..4-1,~ are used to define the spaces .,4~, beginning an induction process.
771
Continuity of the Integral as a Function of the Domain
f A dw = f~
: fz
i=1 Pi
dc~
i=10Pi
do)= i=1
Pi
O)
i=1
f(z= O)=fO
[]
Pi) i=1
A less general extension of Stokes's theorem using norms defined using Euclidean coordinates was announced in [6]. Earlier works [4, 5, 11] extended the integral to codimension one domains. In sequels to this paper, we define three other basic continuous operators on chainlets A. The is dual to the pullback f * of differential forms, giving a change of variables theorem for chainlets [9]. The geometric Hodge star operator 9 is dual to the Hodge star operator 9 on forms [9] and the component operator re1 on chainlets is dual to the component operator on forms [8], each leading to new results. It follows that much of the algebraic theory of differential forms can be viewed as a geometric theory of chainlets. In [9] the theory is extended to ambient spaces of smooth Riemannian manifolds.
pushforward operator f ,
Relations to other fields
Differential topology. Domains permitted in Riemann integration such as compact, smooth submanifolds are examples of chainlets. Our theorem therefore extends the classic Stokes's theorem for smooth submanifolds. Hence, the norms of this paper lead to an extension of much of calculus to domains of chainlets. Since most parametrized curves in Euclidean space, in the sense of Baire, have no tangent vectors, these techniques significantly expand the extent of calculus in Euclidean space. What examples arise besides submanifolds? It is not required that a chainlet have locally Euclidean support. For example, we see below that some naturally arising sets such as the graph of an L 1 function, or a stable manifold of a Horseshoe support chainlets. Thus, we may treat them as submanifolds in many ways. We may find their boundaries, calculate flux across them, and integrate forms over them. In Section 2 we discuss four examples of chainlets from different areas of mathematics to show the broad applicability of the theory. Geometric measure theory. Another important influence on this work was de Rham's theory of currents. In a sequel [8] the author shows that the'.4 r,c~ norms identify all bounded currents as an infinite series of simplicial chains and therefore all "domains," given any definition of a domain in Euclidean space that permits integration of differential forms in B p satisfying linearity:
tfA and continuity
fAO)J ~
0 i f I[~ollcr ~ 0 for all r ___0.
In the preface to his book ([14], p. vii) Whitney wrote that his geometric integration theory had little in common with geometric measure theory. Federer agreed, writing ...the two books have very different aims; Whitney's book is directed to [real cohomology with general cochains, ours to (real and integral) homology with general chains.] ([3], p. 341).
772
Jenny Harrison
This paper and its sequels [7, 8] show that the theory of chainlets forms a bridge between the two theories of geometric measure and geometric integration.
Dynamical systems. Let f : M > M be a diffeomorphism of a smooth manifold. Eigenvalues of the induced mappings on homology and cohomology appear as eigenvalues of the induced mappings f , on chainlets and f * on cochains. Techniques of spectral analysis may be extended by working at the level of chains and cochains. A simple example is the "North Pole, South Pole" mapping f : S 2 ---+ S2 which has the North Pole N as a repellor and the South Pole S as an attractor. Let )~N and XS be the eigenvalues of D f at N and S, respectively. Then f induces a pushforward mapping f , on chainlets. The poles support chainlets AN and As that are nonzero "Dirac" masses concentrated at N and S, respectively, with AN, As ~ ,A~. (See Section 2.1 for a geometric representation of the Dirac mass at a point as a chainlet.) Furthermore, f , ( A N ) = ~.2AN and f , ( A s ) = ~.2As, so we can retrieve from the induced mapping on chains the eigenvalues of f . In contrast, AN and As are not cycles, and the induced mapping on homology does not have these eigenvalues. Let g denote a mapping of the two sphere containing a Smale horseshoe. It can be shown that the stable and unstable manifolds of g support chainlets Ms and Mu in ,A]. Again, the eigenvalues of Dg arise as eigenvalues of the induced mapping g,Ms = )~sMs, g~ Mu = )~u Mu. These cannot be recovered from the induced mapping on homology since all one-dimensional homology classes vanish.
A l g e b r a i c topology. A number of results valid for homology and cohomology are valid at the level of chain and cochains. These include de Rham's theorem [7] and Poincar6 duality [10].
Lebesgue theory. Consider f : [a, b] --+ N, an L 1 function. The oriented graph Ps of a step function S is a simplicial chain. If Sn are step functions converging to f in the L 1 norm, we will see in Section 2 that {Fs,, } is a Cauchy sequence in the 1-norm. Thus, the graph of an L 1 function f supports a 1-chainlet F f E ,All. It follows from the generalized Stokes's theorem that
f bf
=
ydx . f
The right-hand side is a special case of a class of integrals related to the function f . One may also integrate other smooth forms over P f and consider other domains supported in the graph of f .
2. Chainlets Simplicial chains. A p~-dimensional simplex ~ is the convex hull Ha of p + 1 linearly independent vectors or vertices in ]Rn, with orientation induced by the ordering of the vertices. Its support spt(cr) is the pointset Ha. The p-direction of cr is the unit p-vector {a} whose spanning subspace is parallel to the subspace spanned by or, with the same orientation as o-. The mass of a simplex {r is its p-dimensional Hausdorff measure I{r10- We require that every simplex have diameter <1. A p-dimensional simplicial chain is a formal sum of p-simplexes with real coefficients. Multiplication of a simplex by - 1 reverses its orientation. The mass of a simplicial chain is defined
Continuity of the Integral as a Function of the Domain
773
as k
i~=laiffi o = ~i Its boundary is defined to be the simplicial chain
O
ai oi
=
ai Ooi 9 i=1
Its support is the pointset
Polyhedral chains. A simplicial chain is simple to work with as a formal sum. Its geometric meaning is less clear. Consider, for example, the sum S of a 2-simplex cr in the plane minus a smaller simplex r contained inside it, both positively oriented. We have seen the standard definitions of the support, boundary, and mass of S and they are not the same as its geometrical support, boundary, and mass. (See Figure 1.) One instinctively "erases" the region of overlap to find these geometrical concepts. This notion is formalized by passing to polyhedral chains, 2 defined as equivalence classes of simplicial chains: S ~ T i f f s co = fT COfor all infinitely smooth co. We write P = [S]. Polyhedral chains form a vector space with addition given by [S] § [T] = [S + T] and scalar multiplication t[s] = ITS].
FIGURE 1 Geometricsupport aridboundaryof a polyhedralchain. A basic example of a simplicial chain is the formal sum of simplexes subdividing a cube Q. We will denote such a simplicial chain simply by Q.
Boundary of a polyhedral chain.
Define the boundary of a polyhedral chain P by 0 P = [0S]
where S is any simplicial representative of P. By Stokes's theorem for simplexes, this definition is independent of the representative chosen.
2Whitney gives a purely geometricaldefinitionof a polyhedralchain, without mentioningforms.
Jenny Harrison
774
Support of a polyhedral chain. Polyhedral chains P and Q of dimension p are said to be nonoverlapping if spt(P) Cl spt(Q) has p-dimensional Hausdorff measure zero. A representative E L 1 aicri of a polyhedral chain P may be chosen with the ai ~ 0 and the O"i nonoverlapping. 3 For any such representative of P the set of points in the union of the closed simplexes ~ri is the same and is called the support spt(P) of P. If the support of P is contained in U C IRn, we say P is in U. We next define norms on the vector space of polyhedral chains that allow us to measure the proximity of two polyhedral chains. The first is given by mass.
Mass of a polyhedral chain.
For a polyhedral chain P define IPl0 = inf {ISr0}
where S is any simplicial representative of P. This is a norm and may be used to complete the space of polyhedral chains to obtain a Banach space A ~ Given polyhedral chains Pi and ai E I~, define
ai Pi t_i=i
= -~0
~
lim ai Pi k--+oc t"="t
where the limit is taken with respect to the mass norm, assuming the limit exists. If A = [Y~4ec=iai Pi ]0 .4 ~ we call Y ~ I ai Pi a O-decomposition of A with
A-~aiPi i=1
-+ 0 a s k -+ e c . 0
E x a m p l e . The polyhedron cr represented by a p-simplex can be written as an infinite series of cubes cr = [ ~ i ~ 1 Qi]o" That is, ~ i=1 Qi is a O-decomposition of a. If the cubes are chosen to have disjoint interiors, as in the Whitney decomposition, then [or 10 = F_,i~=l [Qi 10- Of course, 0or 5& [y~iec__1 O Q i ] o . See Figure 2. Much of the technical aspects of this paper concern the interplay of estimates involving cubes or simplexes. We show in Section 3 that cubes alone cannot capture the full theory, yet cubes usually offer simpler estimates than simplexes.
Integration of differential forms over polyhedral chains.
For a polyhedral p-chain
P in R n, and co a differential p-form of class C r,", define
where S is any simplicial representative of P. It is immediate that the integral is well defined.
3Since we are working with polyhedral rather than simplicial chains, we do not require that a subdivision be a simplicial complex.
Continuity of the Integral as a Function of the Domain
FIGURE 2
775
The Whitney decomposition of a simplex.
N o r m s o n polyhedral chains.
For a polyhedral p-chain P in R n and (r, a) > 0, define
,[Pllcr,~=sup{fp~o
:o~Br'=,llo~llcr,~<_l}.
Once one has defined the C r,~ norm on forms, this operator norm has elegant simplicity. It is generally straightforward to find a lower bound for II PIlcr,= by integrating any form over P. Upper bounds are much more difficult to find but are useful for proving a given domain is in the Banach space obtained upon completion. We show in the sequel [7] that the C r'~ operator norm is equivalent to the geometric r r,~ norm defined below. It is generally straightforward to find upper bounds for the A r,~ norm. Upper bounds for the .Ar,~ norm usually give sufficient information to apply the theory as they can often be used to show that sequences of polyhedral chains are Cauchy and hence converge to chainlets. The A r,~ norm is thus useful for analyzing a diverse collection of nontrivial, naturally arising examples. The coordinate-free definition of the A ~,~ norm in this paper is comparable in technical difficulty to the definition of the C r,~ norm on forms. 4 In a number of ways, the definitions are dual to each other. However, since the respective Banach spaces are not reflexive, there are distinct points of difference between the definitions that are not entirely dual. The r r'~ norm leads to a geometric definition of the differentiability class of a differential form without mentioning derivatives. The C r,c~operator norm leads to an analytic definition of the fractal complexity of a domain without using Hausdorff dimension or measure. It is worth noting that norms, unlike measures, are sensitive to orientation and multiplicity. Whenever these properties are important, one might replace functions and measures with differential forms and norms.
Weighted mass of a polyhedral chain. Fix 0 < a <_ 1. The p-dimensional cube Q with side length a is defined by IOl-~,= =
ap-l+c~
a-weighted mass of a
9
Qi be a 0-decomposition of Q into cubes Qi with side length ai and each IQi 10 < 1. p-l+a Observe that a p- l+c~ < ~ i ~ 1 ai . This allows us to define the a-weighted mass of a polyhedral
Let Q = ~ i = 1
4This is in contrast to the much more complicated, earlier version of the norms in [6].
776
Jenny Harrison
p-chain P by
IPI-I,~ = inf
]ail
IQil-l,c~
,
i=1
where the infimum is taken over all 0-decompositions of P into cubes, P = [ ~'~i=1 ai Qi]o with each
IOil0 - 1. Observe that the 1-weighted mass ]P I-l, 1 is the same as mass Ie 10. Weighted mass is a norm and may be used to complete the vector space of polyhedral p-chains to form a Banach space .A-1,~ = .Ap 1'". A version of this norm was defined in [4].
Dipoles.
A simple O-dipole is a simplex cr ~ A simple 1-dipole is a simplicial chain of the
form 0.1 ---_ frO _
TvlO.O
where vl 6 R n, Ivll _ 1, and a ~ is a simplex disjoint from Tvla ~ We inductively define simple j-dipoles. Given a vector v j, ]vj[ < 1 and a simple (j - 1)-dipole crj-l, disjoint from T~ja j-l, define the simple j-dipole crJ as the simplicial chain a J = (rJ - 1 _ T v j f f J -1 .
Thus, (7j is generated by vectors Vl . . . . . vj with norm < 1, and a simplex a ~ where all translations of a ~ through sums of vectors vi are disjoint. Define (70 -1,c~
=
0-0
-1,a
and for j > 1
~J j-l,~ = ~o o l V l l ' " ] v J l
/
~
9
/ FIGURE 3
A 1-dipole.
A linear combination of simple j-dipoles DJ = z k i = l aia/with real coefficients ai is called a
j-dipole. Define DJ j - l , a = Z l a i l
a/ j-l,.
i
for j >_ 0. This is a semi-norm on the vector space of j-dipoles. Example. Any four oriented, paralM edges of a three-dimensional cube Q form a simple 2-dipote o~2 with llcr2112 = IQI0. This is called a quadrupole in physics.
777
Continuity of the Integral as a Function of the Domain
Cubical dipoles. An important example of a j-dipole QJ is one generated by a cube Q0 = ~ i ~ and vectors Vl . . . . . vj. Then QJ is called a cubical j-dipole. Its boundary 0 QJ is a cubical (j + 1)-dipole generated by the (p + 1) pairs of sides of 0 Q0 and vl . . . . . vj. Furthermore, OQ j j,c = ( p + l )
QJ j_l,c = ( p +
l)S Q~
,
(2.1)
for all j >_ 0. Every simple dipole a j has a 0-decomposition O"j = [~e<~i=l QJ ]0 into cubical dipoles.
,A r'a norms.
Let P be a polyhedral chain and (r, o~) > 0. Define
Ielr'== inf { ~ 'lDslls Dr+l r,ol -t-[Clr-l,~}
(2.2)
where the infimum is taken over all dipole decompositions r+l P=2[DS]+OC. s=0
In Section 4 we prove that [Plr,~ is a norm on the space of polyhedral chains. The Banach F,13/ space of polyhedral p-chains obtained on completion is denoted ,Ar,a = Ap . It is straightforward to verify that IP Ir,= is a semi-norm on the space of polyhedral chains. Given polyhedral chains Pi and ai E ]~, denote
ai Pi t_i=i
:
-tr, c~
lim ~ ai Pi k--~oc ~=i "="
where the limit is taken with respect to the .Ar'c~ norm with r > - 1 .
2.1. Examples In practice, it is usually straightforward to show that a domain is a chainlet. The general technique involves choosing a canonical series or sequence of "candidate" approximators. One then proves the Cauchy criterion by finding an upper bound for the norm of the Cauchy differences and showing this upper bound tends to zero. There is an abundance of examples of chainiets and the author is preparing a paper devoted to examples. Here are four. .
Van Koch Snowflake. One may write the snowflake arc S as a series of simplicial chains [Ek=0e~ Sk]~, log~l~ _ 1 < ot < 1 where for each k _> 1, Sk is the boundary of the' sum ak of 4 k triangles each of side length 3 -k. We show this series converges w.r.t, to the 1-norm: The partial sums satisfy S~ + 9 99 + Sj = O(a~ + .. 9 + aj). Thus, ISk + ' " +
Sj]I < lakl0 + " "
+ laj[o < 4 k / 3 2k 9
Since the r.h.s, tends to 0 as k, j ---> c~, we know the infinite sum S is a well defined chainlet. Results of this paper allow us to conclude that the snowflake supports a current and we may integrate Lipschitz differential forms over it. We leave the proof of convergence w.r.t to the a-norm as an exercise.
778
Jenny Harrison
)"
V
So
V 4x
, ~ ' ~ I~' ~ ' ~
~
FIGURE 4
2.
~]'~
'~ ~
~
$2
I0+S1+$2
The snowflake as a s u m of simplexes.
Dirac delta function and its derivatives. We work in dimension one for simplicity of notation, but the construction can be extended to any dimension. Fix p 6 ~1. For each k >_ 0, let Qk be a positively oriented interval with length 2 -k and centered at p. Let Dk = 2 k Qk. Notice that the mass of each polyhedral chain D/c is one. We estimate [Dk+l - Dk[1 by showing the difference Dk+l -- Dk is a one-dimensional 1-dipole, a sum of four simple 1-dipoles with real coefficients. Divide D~ into two intervals Qk with disjoint interiors, of length 2 -(k+l) and weighted by 2k. Now Dk+l can also be written as the sum of two intervals Pk of length 2 -(k+l) and weight 2 k, but the line segments are identical to each other. Since the distance between the line segments of P~ and those of Q~ is less than 2 -k, we deduce IDk+l -- Dkll < 22-k2k2 -(k+l) = 2 -k 9 We conclude that the series of dipoles ~ - - 1 Dk+l - Dk converges in the 1-norm. Let D = Do + ~ Dk+l -- Dk. The chainlet D is canonically associated to the Dirac delta function. First of all its support is the point p. Since D e AI, we may integrate smooth 1-forms Cdx over it. Hence, L 4)dx = lim f Odx = ~(p) = 3(4)). k~oo JDk
The derivative of the Dirac delta function can also be realized geometrically, but as a chainlet B 6 A 2. One considers the quadmpoles (or 2-dipoles) formed by small oppositely oriented intervals centered at the endpoints of the Dk. It is left as an exercise to show that
f~ q~dx --- 3'(4)). 3.
Toral solenoid. Let T be the 2-torus in R 3 and f : T -+ T a smooth hyperbolic mapping that contracts the toms in one direction, expands it in the other, and then wraps the toms
Continuity of the Integral as a Function of the Domain
779
around inside itself twice. The solenoid is defined as the intersection ('~n=l~176 f n T . This set of points supports a chainlet. Let Q be the solid toms positively oriented and P0 = Q/L Q 10. For k _> 0, let Pk+l = f ( P D / I f ( P D I o . Since the mass stays constant, the analysis here is similar to that for the Dirac delta function and on e can use dipoles to show that Pk converges to a nonzero chainlet in A~ with support of the solenoid. One can also find chainlets in AI with support of the solenoid as follows. Let B0 be the oriented core circle in the toms that is not null homotopic. For k _> 0, let B~+I = f(B~)/lf(B~)lO. Then B/: forms a Cauchy sequence in A~ and thus converges to a 1chainlet B. It is also possible to find chainlets in .A01 with support of the solenoid by choosing a countable dense subset and forming a Dirac mass at each of these points so that their total mass is finite.
FIGURE 5 .
Toralsolenoid.
G r a p h s of L 1 f u n c t i o n s . A nonnegative L 1 function f over the unit interval is the a.e. limit in the L 1 norm of step functions Sn. Let F0 = [0, 1] and Fn be the graph of Sn, n >_ 1, all positively oriented. For each pair n, m the simplicial chain Fn - P m is equivalent to a 1-dipole with dipole mass the same as the area between the graphs. Of course, this area tends to 0 as n, m -+ c~. Thus, the graph of f supports an .41 chain
lPf = F0 -I-
Fn - Fn-1
1
We later examine the boundary of the graph
OFf and the relation
of ]lPf]l to If ]L1.
The preceding example shows that dipoles naturally arise and how the L 1 and dipole norms are closely related. Theflat norm of a p-dimensional polyhedral chain P is defined as IPI ~ = i n f { I B I o +
IClo}
where the infimum is taken over all decompositions P --- B + OC where B and C are p-dimensional and (p + 1)-dimensional polyhedral chains, respectively [14]. When we consider dipoles D 1 in the
Jenny Harrison
780
decomposition of P as well, we recover the A 1 norm:
IPIl=inf{ [Bl~176
D1 11
where the infimum is taken over all decompositions P = B + OC + [D1]. In this way the ,A 1-norm can be seen as a combination of the fiat and L 1 norms.
Relations between spaces of chainlets Lemma 2.3.
I f - 1 < (r, or) < (s, fl), then
IPIs,~ _< [elr,~ for all polyhedral chains P, P r o o f . I f 0 < a < fl _< 1 the weighted dipole masses satisfy [[D j+l Ilj,~ _< [[D j+l [[j,c~. Itremains to see that ]Plr,~ <_ IP]r. For r - 0, we know [PI~ _< IP[0 since P itself is a 0-dipole. For r >_ 1, the inequality follows directly from the definitions since the left-hand side is found by infimizing over (r + D-dipole decompositions and the right-hand side is found by infimizing over r-dipole decompositions. [] Let - 1 < (s, fi) < (r, o0. For any A s'~ chain A we have
where the Pi are polyhedral. Set |
= [Y~i Pi]r,~" It follows from L e m m a 2.3 that | is a well
defined linear mapping of A s'~ into .4r, cL
Theorem 2.4.
| 9 ,,4s,~ ~ A r,~ is a linear mapping with |
~'~) dense in A ",~ in the norm
I Ir,~P r o o f . The set of polyhedral chains is dense in A ~,~ . Since | (A '~'g) contains all polyhedral chains, it is also dense in , A r'c~ . Linearity is immediate from the definitions. []
Separability of spaces of chainlets Theorem 2.5.
The space A r,~ is separable for each (r, or) > O.
P r o o f . The set S of polyhedral chains represented by simplicial chains of the form y~ aiffi where the ai are rational and the vertices of the o-i are rational is countable. We shall show S is dense in .Ar,~. It is sufficient to show it is dense in the set of polyhedral chains. This will follow if we show that for each simplex o- and e > 0, there is a simplex a / with rational vertices so that la - cr1], < e. This, in turn, reduces to showing that for each vertex q of a simplex cr and e > 0, there is a simplex a I with the same vertices as a , except that q has been replaced by a rational vertex qr and [or - o"11~ < e. We are assuming cr is p-dimensional. Suppose q, ql . . . . . qp are its vertices. Then qt, q, ql . . . . . qp determine a (p + 1)-dimensional simplex r. If q and q~ are sufficiently close together, then its weighted mass satisfies Iv[-1,~ < e/2. The boundary of z consists of cr - a ~ and simplexes ~ ai
Continuity of the Integral as a Function of the Domain
781
with total p-dimensional volume < e/2. Since each o-i includes both vertices q and ql, we have [o'-o-'1~ ~ ~ D i l 0 +
Irl-l,= < e .
[]
i
In [7] we show that the dual space of cochains (,Ar'~) * is isomorphic to the space of differential forms with bounded C r,~ norm and is thus not separable. It follows that ,A r''~ is not reflexive.
Continuity of the boundary operator. We know the boundary operator is continuous directly from Definition (2.2).
Theorem 2.6.
For C apolyhedral cha/n and (r, a) > 0, IOCIr,~ ~ ICIr--l,~ -
The boundary operator Ar-l,e~ r,~ 0 :~p+l --+ ,Ap
is continuous and therefore defined for all (r, or) > 0. Continuity also gives us the important relation OoO=O. Of course, the boundary operator is not continuous in the mass norm, so this theorem does not extend to (r, ct) = O.
E x a m p l e s . (i) Let U c R n be oriented, bounded, and open. Then U = U Qi, an infinite union of oriented cubes Qi with disjoint interiors and sides parallel to the coordinate hyperplanes. Then a = [ ~ 1 Q i ] o E A ~ By Theorem 2.6 o a = [ ~ 1 0 Q i ] l ~ -4~1-19 We conclude that the boundary of an oriented, bounded, open subset naturally supports an infinite series of 1-dipoles. (ii) Let f be a nonnegative L 1 function on [0, 1]. Recall that the graph of f supports a chainlet of the following form: I ' f = I'0 q-
Fn -- I ' n - 1 Ln=l
1
Its boundary can be written
OI'f=aI'O+IOOnS__.,=lO(I'n--I'n--1)l 2 The boundary of the graph of a step function 0 I'n is a dipole. Thus, the differences O(i'n - I'm) are 2-dipoles. By Theorem 2.4 0F0 is a 0-chainlet in A 2 and so OI'f is a 0-chainlet in ~4~. We conclude that the boundary of a nonnegative L 1 function minus the boundary of its domain naturally supports an infinite series of 2-dipoles. The chain 0 I'f identifies the points of"true discontinuity" of f . For example, if f = X[1/3,2/3] + XQ then f is discontinuous at all points in [0, 1], but 0I'f is the 0-chain (2/3, 1) - (1/3, 1).
782
Jenny Harrison Characterization
o f t h e . A r'~ n o r m
Theorem 2.7.
For (r, ~) > O, the .Ar,~ norm is the largest semi-norm I I~ in the space of semi-norms defined for polyhedral chains such that (i) ]as[ ~ _< HaSlls forevery simple s-dipole crs, 0 < s < r + 1, and (ii) [(0C)]' _< [CI~-I,~ for every (p + 1)-dimensional polyhedral chain C.
Proofi
First observe that the A r," norm satisfies (i) and (ii) by Definition (2.2). Now let I any semi-norm satisfying (i) and (ii). Suppose D s = ~ i ai~
Use (i) and the fact that I
[/be
[' is a semi-norm to deduce
ID=I' <- ~ lai i I+ ~I' -< ~ lai ill<+ I1., 9 i
i
Hence,
Iosl' ~
[Ioslls,
for all0 < s < r + 1. Let P be a polyhedral chain and e > O. There exists a decomposition P = ~-'r+lros Z_~s=0t 1, + OC so that
Inlr,~ > k i i D S i l s + Dr+l r,o-[-lClr-l.c~ - 8 . s=0
Applying (ii) we have r+l
r+l
IPI' -< ~ IDOl' + laCl' <- E IIDSlIs+ lclr-'.~ s =0 r
s=0
_< Elloslls+
D r+l r, + lClr_l,a
s=0
-< Since this holds for all e, we conclude IP
I' ~
[elr,~+e.
[]
[P Ir,~-
3. Cochains A cochain X is a linear functional on polyhedral chains. We denote by X 9 P, the action of X on the polyhedral chain P. The (r, ot)-norm [Xlr,~ is, by definition,
IXlr,,~
=
sup IX" el IPIr,==l
=
Ix.el
sup IPIr,~#0 [P[r.c+
(3.1)
the supremum taken over all polyhedral chains P. By L e m m a 2.3, we know for any cochain X,
ISls,~ <_ IX[r,c~
(3.2)
783
Continuity o f the Integral as a Function of the Domain
Given a cochain X define its coboundary d X by dX.P=X.OP.
Proposition 3.4.
(3.3)
I f X is a p-coehain, then d X is (p + 1)-cochain with [dXl~-l,~ ~ [Xlr,~.
Proof.
By Theorem 2.6 []
IdX . Cl = I X . aCl <~ IXlr,~laClr,~ <_ IXlr,~lClr-l,~ 9 Since O o O = 0 we deduce d ( d X ) = O.
A cochain X is an element of the dual space (Ar'~) * of chainlets A r'~. (Until we prove that [ [r,~ is a norm, we must consider (,Ar'~) * as a semi-conjugate space.) Cochains with bounded (r, a)-norm extend uniquely to cochains in (Ar'c~) *.
Support of ehainlets and cochains. The support spt(A) of a chainiet A c r ~ is the set of points q such that for each s > 0 there is a cochain X e (~A~) * such that X . A # 0, X 9a = 0 for all simplexes a outside Be(q) 9 Say A is compact if spt(a) is compact. The support spt(X) of a cochain X c ( A ~ ) * is the set of points q such that for each s > 0 there is a simplex cr such that X.a
Derivatives of cochains.
# O , cr C BE(q).
The following norm measures, in some sense, the
S th
derivative
of a cochain X. Define
IIXIl-l,~ = I S l - l , ~ 9 For (r, a) > 0, define X 9D r§
l[X IIr,~ = sup IIDr+l
IIr,,
where the supremum is taken over all nonzero (r + 1)-dipoles D r+l . If X is a cochain
IlXllr,~ ~<
IXIr,~
since [[Dr+l]lr, a < lIDr+lllr,~. L e m m a 3.6.
K X is a cochain, X 9 O" r + l
[[X[[r,~ = sup [[ar+l [["r,~ where the supremum is taken over all nonzero simple (r + 1) -dipoles a r + 1, (r, or) > - 1.
(3,5)
784
Jenny Harrison
Proof.
First assume IlXllr,+ < ~ . We are reduced to showing
X 9 o.r+l
IlXllr,,+ < We m a y thus assume I]Xl[r,~ that
#
Ho.r+a
sup
.
(3.7)
O. Given e > O, choose a nonzero dipole D r+l = ~-~4aiar+l such X 9 D r+l
_> IlXllr,~ -- ~ / 2 .
Dr+l
(3.8)
r, Ol
If we can show that for some i
X .o.7 +1
llXllr,+
-
0-7+1 r,+ then (3.7) follows. Suppose that for every i,
S.o.r+l
<
O.r+X r,~ (llXllr,~ -- 0 "
Then
~-~ilai] X'O.7 +1
IIXIIr,~--~/2 < I X ' D r + l ]
- I~n-i[[~,~
-<
IIDr+~llr,+ ~ i lail o-2+1 r,a
_< (llXIIr,~ -- 0
=
[ior+~ lit, ~
IlXllr,~--e,
a contradiction. The p r o o f is similar if [IXl[r,~ = ~x~. Given a constant k > 0, there exists D r+l = ,_..,azo. 3-' 9 ir+l such that X . D r+l >k.
Assume
x.o.r+l < k o./~+1 r,+ for all o_/,-+1. Then
k<
Ix 9o*+~ I
IfVr+ llr,
Z , la~I X. ~,."+~
--
Ilvr+lk~ Eilai[
<
k
=
k.
~
][Dr4-1[[r, ~
r,a
Continuity of the Integral as a Function of the Domain
785
We conclude for all k, there exists a simple dipole 0-[ +1 such that
x, ~/,-+1[ >k err+ 1
--
,
F,lY
[]
completing the proof. L e m m a 3.9.
I f X E ( A - l , u ) * for 0 < ol < 1, then X . Q0
IIx II-1,~ = sup IIOOII-1,a where the supremum is taken over all nonzero cubes QO. / f X 9 (A-1'r
* f o r s o m e 0 < fl < 1 and (r, or) > O, then
X. Q~+I IIX IIr,~ = sup IIo r+l IIr,c~ where the supremum is taken over all nonzero cubical (r + 1)-dipoles Qr + 1.
Proof.
By Leinma 3.6 this reduces to proving that if O-r+l is a simple (r + 1)-dipole and e > 0, there exists a cubical (r + 1)-dipole Qr+l such that
Ix. Or+l I ..r+l I ]~r~ill,.~ -> (1- ~')l[;r+lllr, 9 Suppose ~rr+l is generated by a simplex rr ~ and vectors vl . . . . . Vr+l. For a cube QO c cr~ let Q~+I denote the cubical dipole generated by Qo and vl . . . . . Vr+l. Set r = - 1 . For e > 0 there exists a 0-decomposition rr ~ = [~i~=1 Q/~ such that oo
O-0 -1,c~ =
Now [cr ~
~1
o.o
QO
i -1,~ (1 - s/2) .
- l , u -> E i=1
QO] = [~ioc__t:+] QO]o ' implying
O.0_ ~~Q? i=1
_< ~ i_l,ce
QOi
-1,o~
i=k+l
It follows that
i=1
.3 -1,a
Assuming X e (A-I'~) * we deduce
x.,~~ = E x . Qo. i=1
(3.1o)
78 6
Jenny Harrison
Suppose
Ix. o~ Ix. ~~ iiQOll_l,o ~ < (1 -- E)ii-~--~il--~,a for each i. Then sup I X ' Q ~
<
0 i 11a i II-1,a
(l-s)
IX" a~
II~-~ i'-~,c~ (l-e)
<
E i ~ i lX " O~ l
-
( l - s / 2 ) E~i Ila~
<
sup
Ix o~
i IIQ~
~
a contradiction. Now set r > 0. Let a ~ = [ ~ i Q~ be the Whitney decomposition of a ~ into nonoverlapping cubes. The number of cubes of sidelength 2 -k is bounded by a constant C (depending on a ~ times 2 ~(p-1). Summing over all cubes with sidelength < 2 -k we obtain -
Q~ "
< C
2i(p-1)2-i(p-l-fl)
< 2 C 2 -k~
i=k
.
[ 0~ for any 0 < fl -<- 1. It follows that a ~ = ~ i = 1 Q~ Since X is in (A-l'~) *, we have
and hence a r+l
-~-
c~ ,qr+ll [~i=1 ~i 1-1,8"
OQ
X . a r+l = E X .
Q[+I.
(3.11)
i=1 Since the Q/O are nonoverlapping [a ~ [o = E ~
[Q/~lo, and thus
(X)
ur+l r,c~ ~---E
Q[+I r,~ for all (r, a) > 0.
i=1 Suppose
X. Qr+l]
X 9o-r+l I
Qr+l r,cr for each i. Then
IX 9fir+l]
IX" Q~+I I
i sup
Qir+l r,et E i ~ i X . ~ior+ll _<
( 1 - s ) E%i Q~+I
<
(1 - e) sup . . . . .
,.~
x. Q~+~ --
i
r+l Qi r,c~
.
Continuity of the Integral as a Function of the Domain
787
Since this holds for all e > 0, we deduce X . Q~+I sup i
x . Q~+I
< sup Q~+I r,~ i
, Q~+I r,~
[]
a contradiction.
Theorem 3.12.
I f X is a cochain, then IXl-l,~ = IlXll-l,~
and, for a/1 (r, or) > O,
IXl~,~ = sup {Iglr, IIXllr,~, [dXIr-l,c~} P r o o f . Suppose e > O. Let P be a polyhedral chain. r+l s ~ s = o [ D ] + OC with
tPlr,~
9
There exists a decomposition P =
~l[DSlls + Dr+l rot"]-IfIr-l,eL--C. s=O
Let K = sup {IXlr, IlXllr,~, IdXIr-l,,}. Then r+l
ix. PI
< ~IX.D'I+Ix.ocl s=O
<_
~lXls [DSls +
IlXlIr,~ D r+l r,a + IdXlr-l,~[C[~-l,~
s=O
<_ g (lPlr,~ + e) . Since this holds for all e > 0 and all polyhedral P, we deduce ]Sir,c, < sup
{IXIr, IlXllr,~,
]dXlr-l,~}
9
The opposite inequality follows directly from (3.2), (3.5), and Proposition 3.4.
[]
Expand the terms of Theorem 3.12, and apply the fact that [1Yl]- 1,,~ = [Y [- 1,c~for all cochains Y to conclude the following.
Theorem 3.13.
I f X is a cochain, then [X]~ = sup {llXIIo, ]lXl[~, IldXl[-l,~}
and for all (r, a) > 1, [XIr, c~ = sup {[[XIIo, [IXlI1 . . . . , IlXllr, IlXll~,~,
IIdXllo . . . . . IldXllr-l,c~} 9
Jenny Harrison
788
This theorem is useful for establishing whether specific cochains are elements of (~4r'a) * and hence correspond to smooth differential forms. If it is known that a cochain X 6 (A ])*, then we do not need information about d X to conclude X ~ (At,") *.
Theorem 3.14.
I f X ~ ( A " ) * for 0 < ot <_ 1, then
IX[~ ~ sup (llXII0, (p + 1)llXll~} ~ (p + l)lXl~, I f X E ( A ~ ) * for s o m e 0 < fl < 1, then IXlr,~ < sup {llXll0, (p + 1)]lXl]l . . . . . (p + 1)llXllr,~} _< (p + 1)lXlr,~, for all (r, or) > O.
Proof.
For the first part, it suffices to prove that
IIdXIl-l,~ ~ (p + 1)llXll~ 9 By Lemma 3.9
IIdXll-t,~
=
X . OQ ~ sup ]l QO]l_],~
il0Q~ IIx lid sup IIo ~ =
( p + 1)llXll~.
Since X 6 (A~) *, then d X ~ (.A-I'~) *. It follows from L e m m a 3.9 that for all 1 < (s, V) < (r, o0 [X.OQSl IldXJls-l,y
--
sup Q,#0 IIeSlls-l,y
IIOQSll,.y _< IlXll~w sup as~0 IIQS I1~-i,• =
(p + 1)llXIIs,y.
[]
The next example shows this theorem is false if X r (J[~)*. Examples. Let X be the cochain on R 2 defined by X - a = [a Io if a is a positively oriented 1-simplex, parallel to either of the coordinate axes. Otherwise, set X 9a = 0. Then [[XI[o = 1 and ]1X [1~ = 0. Let r be an oriented equilateral triangle with side length e and one edge along the x-axis. Since [0r[~ < Ir[-1,. _< el+~/2, we conclude
IX.Orl IXl~ >_ - -
>-2/e
~.
It follows that ]XI~ = oo. Theorem 3.14 shows that if X c (,A~) * one may use cubes to estimate the cochain norms. This example demonstrates that it is advisable to use simplexes to define the norms, rather than cubes.
789
Continuity of the Integral as a Function of the Domain
Proposition 3.15. I f X is a O-dimensional cochain, then IdXl-l,~ = IlXll~ and IldXllr-~a = IlXllr,~ for a11 (r, a) > O.
Proof.
Let a be a 1-simplex with endpoints a, b. Letting v be the vector b - a, we have [dX.a[
IX.a~l
I~1-1,~
I~1-*,~
IX.a-
X.bl
IX.(a-
Ivl ~
where D 1 = a - Tva. Hence, [dXl-l,c, =
] X ' D 11
Tva)[
-[DI[~
Ivl ~
IIXl[~,
The other relations are similar.
[]
4. Integration of differential forms over chainlet domains Norms on differential forms and cochains. A function f is of class C r if its r thorder derivatives g exist and are continuous. The C r norm Ilfllc, is the supremum of the norms Ilgl[0 = SUpq [g(q)l. Suppose0 < o~ < 1. If there exists C > 0 suchthat I f ( x ) - f ( y ) [ < C l x - y l ~ for all Ix - Y l < 1, then f is said to satisfy an o~-H61der condition for oe < 1 and a Lipschitz condition if o~ = 1. Define "t
Ilfllc- = max
fllo,
sup
I f ( x ) - f(Y)l [ .
O
IX -- y[C~
/
Let r _> 0 be an integer. We say f is of class C r,~ if it is of class C r and its r th-Order partial derivatives satisfy ot-H61der conditions for ot < 1 and they satisfy a Lipschitz condition if a = 1.
Differential forms f i r , %
An L 1 differential p-form co determines a cochain qu(w) through
integration qJ (no) 9 A = JA (1)
for all polyhedral A. We next identify the maximal subspace of L 1 differential p-forms ,fr,~ = ff~,mapped linearly by qJ onto the space (,A~.~) * of cochains. For r > - 1 , define
]Lr +10' I IIoollr,~ =
sup
~r+l#0
[io_r+1
[[r,~
where the supremum is taken over all p-dimensional simple (r + 1)-dipoles a r+l. Define Ifarr+~ col lien rd,. = sup [[Tr+l
3#0
[hr,
where the supremum is taken over all (p + 1)-dimensional simple (r + 1)-dipoles r r+l. Note that the right-hand side has meaning, even if co has no exterior derivative. If den is defined and satisfies Stokes's theorem, then
Ildo, llr,~ =
co ~ .
790
Jenny Harrison
Define
Icol-l,~
=
IIo)ll-l,~,
I~1o -- sup {11~,0, ,~11o, II~,~ ~}, Icolr,~
=
sup
and
I1o)110,Ilcolll,'", [IcolIr, II~Ollr,~, I1~oI1~I..... Ilcollr-l,~
(4.1)
for ( r , a ) > 1. Write co E ffr,~ if co is a differential form with Icolr,~ < ~ and co ~ f f ~ if co ~ fir for all r _> 0. We prove below B r'~ C j r . ~ . It follows from Theorem 3.13 that: L e m m a 4.2.
I f o ) c ~fr,ee, then qJ(co) E (Ar'c~) * forall r > - 1 andlcolr,~ = IqJ(co)lr,~.
Observe that an L 1 form can be infinitely smooth in the ,]r,o~ norms and not even be continuous. This permits definition of "smooth" forms on nonsmooth manifolds and formulation therein of classical results of calculus such as Stokes's theorem. This definition ties in Whitney's flat theory, for
Icolb : sup {[[collo, ]lco[Iod } contrasted with
I~l~
sup(Ll~,0 ,~lll ,~LIo~}
I~l~ -- sup/,~,o H~, ll~,~ ,~llo~ ,~,~}, etc. so the theory of ffr,~ forms is a higher order theory of flat forms.
Example 4.3 ( *flat is not flat).
L e t co be the 1-form in R 2 that is identically zero for y >_ - x and co = dx + dy elsewhere. Then co is fiat but ,co = dy - dx is not fiat. (Consider small right triangles with diagonal satisfying y = - x . )
L e m m a 4.4.
For any L1 form co then
II~OIIr~ : ~0 d :
II~(~o)llr,~ IId~(~o)llr,~
and
ICOlr,~ : I~COlr,~ forall r > - 1
Proof.
This follows immediately on comparing Theorem 3.13 with Definition (4.1).
Proposition 4.5.
I f co C f f ~ for 0 < ot < 1, then fooo w
I1~o11~1,~ = sup
Qo~o 11Q~
[]
Continuity of the Integral as a Function of the Domain
791
I f co ~ ff~ for some 0 < fi <_ 1, then
IIcOIIr~
sup
:
LQr+] co
Qr+l IIQr+l [[r,~
for all (r, a) > O.
Proof.
By Lemma 4.4 co d = IldqJ(co)IIr,~. The proposition then follows from L e m m a 3.9, and Lemma 4.2. []
Integration over infinite series of simplexes.
It follows immediately from L e m m a 4.4
that:
Lemma 4.6.
I f co is an L 1 form and P is a polyhedral chain, then
fp
co 5
ICOI~,~IPIr,~
for each r > - 1 .
For a chainlet A c A ~'" and co 6 ffr, u define
fACO:Zfpico i=1
where A -- [ ~ i ~ 1 Pi]r,~ and the Pi are polyhedral. By the preceding lemma, the integral is well defined. The next theorems show that Ico[r,c~ ~ forms may be used to estimate integrals.
Theorem 4.7.
[IwlIcr~ for all co s B ~'c~, so the standard C r'~ norm on
I f co c BOp, then
lco]o ~ Ilcollc0 9 I f co c B p for 0 < ~ ~ 1, then
Icol~ ~ (p + 1)llcollc~. I f o9 ~ B~, then
Icolr,~ ~ (p + 1)IlcolIcr,~ for all (r, ~) > 1. In particular, B ~''~ C ,j'r,~ for all (r, u) > O.
Proof.
o~ Suppose co E 13p. Since co is continuous
Icolo = Ilcoll0 _< Ilcollco _< Ilcollc~ 9 Let
a 1 =
(Y - -
T v a be a simple 1-dipole and fl the p-direction ofo-. Then
Jenny Harrison
792
Io-lo sup Ico(x; f ) - co (T~(x; ~))l <
[o.lollco[Ic~ Ivl ~
=
o- 1 ~llcollc~
Therefore, Ilcoll~ -- Ilcollc~. If Q is a p-dimensional cube and v ~ ~n, then
fQ-TvQ
co=fQco-Tyco < -
ial01vl ~lco-
T*~ ivl~
<_ Ialolvl'~llcollc~ It follows that for Q0 a (p + 1)-dimensional cube
Q0
co ~ ( p + l ) l l c o l l c =
Q0
-1,~ "
By Proposition 4.5,
Ilcolld~l,~ ~ (p + l)llcollc= 9 It follows from Definition (4.1) that IwIu _< (p + 1)llcollc,. We use induction to show for all ~ 6 / 3 1
9
Ilo[l~ ~ II~llc~
9
II~llsd_l ~ (p § 1)11,711cs-1,1
1,1
and
for all 1 < s < r and s = (r, or). The two estimates are shown to be valid for s = 1 by setting a = 1 above. (Here we use 0 6 131.) Assume the estimates hold for s = t - 1. We prove they hold for s = t < r. Given a simple t-dipole o. t generated by o., Wl . . . . , wt-1 and v, with [wi I, Ivl _ 1, let crt-1 be the simple t-dipole generated by o., Wl . . . . . wt-1. Let f d V be the component of 77 in the direction of o. and g(p) = f (p) - f (p + v). Note that gdV ~ /31p. Then by induction
~, 77
=
f~,-1 g(p)dV
5
o-t-1 t-I IlgdVllt-1 o't-1 t-1 IlgdVllct-2'l
< -
o-t-1 t-1 Ivl IIf - f o Tt.l[c,-i Ivl
-<
I[o.t[[ t tlf tlc,-11
_
II tll, Ilollc,-1,1
Thus,
11771i~~ I[~116~-1,1 9
Continuity of the Integral as a Function of the Domain
793
Therefore, fa
Qt-1
0
< =
OQ t-1 t ll~llc , 1,1 ( p -1- 1)
Qt-i
t-1 II0llct-l'l "
By Proposition 4.5 Ilolltd 1 ~ (p -t- 1)llrlllct-l,1. This establishes the estimates for all 1 < s < r. The proof for s = (r, or) is similar. One sets t = r + 1 and multiplies and divides by Ivl ~ instead of Ivl in the preceding proof. []
Finally apply to COE B~. We may now state the main integral inequality for chainlets.
Corollary 4.8.
Let A be a chainlet in A p . I f co ~ Bp, then
fA
CO ~ (P ~- 1)llCO[Ic=lAl~ 9
I f co E B;, then fACO < (P + 1)IICOIIcr,~IAIr,~
for all (r, ~) > 1.
Proof.
This follows immediately from Lemma 4.6 and Theorem 4.7.
[]
Since A r'~ is separable (Theorem 2.5), integration of differential forms over chainlets reduces to integration over simplexes or cubes having vertices wifla rational coordinates.
Corollary 4.9. Every chainlet A E A r'~ determines a unique boundedcurrent CA by CA(CO) = f A COwhere CO is a C ~ form. Proof.
CA is bounded (in the sense of Schwartz [13]) since ACOk ~
0
if IIcokllcr,~ ~ 0.
[]
The four examples given in this paper are all currents as they are chainlets in Alp. One may therefore integrate Lipschitz forms over them. L e m m a 4.10.
co E B ~ i f and o n l y i f COE ff~ and COis continuous.
Proof. The proof is similar to the preceding proof and we omit it. T h e o r e m 4.11.
Let co E 13r'~ and (r, ~) > O. Then
IIcoIIfr,~ ___ IICOIIr,~-----]COlr,c~-
[]
Jenny Harrison
794
P r o o f . We verify the first inequality since the second follows immediately from the definitions. Let r = 0 and 0 < ot < 1. Then co ~ J ~ and co is continuous. Let f l d X i be a component of co corresponding to a simple p-vector ei. Choose x # y ~ R n with v = y - x, Ivl _< 1. Let a be a p-simplex containing x in its interior and with p-direction the same as the p-direction of ei. Since fx is continuous =
Ifz(x) - fz(y)l Ix - y V
lim
fcr-T, cr f l d X l
I~10-+0
lal01vl"
=
lim f~-T,~ co I~1o--,o IcrloivV
<
sup
--
O"1
=
Ikoll~
f~co I1~'11~ -
It follows that Ilcollc. _ Ilcoll. 9
For r >__ 1 the problem reduces to showing that if g is an #h-order partial derivative of the component function f I , then IIg Ilca _< Ilcollr,~. Define cr to be a p-simplex in the direction of f l d x i and let o"~ be the r-dipole generated by a and vectors ul . . . . . ur in the directions of the successive derivatives of f / . Since g is continuous, and letting v = x - y , we have
]g(x) - g(Y)l Ix - yV
=
f~r-r,,.,
lim
co
I~10,I.zl-~0 Ikrrllr Ivl ~ <
fcrr+l co
_
sup ii r+
=-
Ilcoll~,~ 9
f~0 co Finally, if co is continuous then Ilcolico _< sup~o#o ~ = [[col[o. Corollary
4.12.
The ,4 r,a norm I
[]
Ir,~ isafullnormonpolyhedral p-chainsfora11 (r, a) > - 1 .
P r o o f . The semi-norm properties are easy to verify. Let P # 0 be a polyhedral p-chain. Now P is represented by a simplicial chain S = ~ i aiai where the cri are simplexes with disjoint interiors. Choose one of these, ao, say. Let q E int(a~ There exists an n-ball B = Be (q) that misses the interiors of the other simplexes of S. Let coo be a smooth differential form supported in B that is the p-volume form for Be/3(q) f7 a 0 and Ilco0llc0 -< 1. Then fp coo # 0 and Ilco0llcr,~ < oe. C l a i m . Icoolr,. is bounded for all (r, or) > - 1 . This follows for (r, ce) > 0 since ICO01r,~ ~< (p + 1)tlcoolicr,~ by Theorem 4.7. Note that lf~o cool -< la~ < CIIrr~ where C depends only on a ~ and el. The claim follows. By L e m m a 4.4 for (r, o~) > - 1, 0 # i * (coo)- PI ~ 1" (coo)[r,,~ IPb-,~ = tcoo[,.,~ IPlr,~ 9 We conclude IPIr,~ # O.
[]
795
Continuity of the Integral as a Function of the Domain
C o r o l l a r y 4.13. Proof.
Let A, B ~ A r'~. I f X . A = X . B forallcochains X ~ ( X ' ~ ) * then A = B.
I A - Blr,,~ = suplxl,,,#0 ]X'(A-B)I IXl,,~ - O. Since ]
Ir,~ is a full norm A - B = O.
C o r o l l a r y 4.14 ( G e n e r a l i z e d S t o k e s ' s T h e o r e m for C h a i n l e t D o m a i n s ) . differential form in Nr+l'a and A ~ A~ ~, then ~p-I
[]
I f o) is a
References [1] Almgren, F. Mass continuous coehains are differential forms, Proc., AMS, 16, 1291-1294, (1965). [2] Daniell, A general form of integral, Ann. Math., 19, 279-294, (1917/18). [3] Federer, H. Geometric Measure Theory, Springer-Verlag, New York, 1969. [4] Harrison, J. and Norton, A. Geometric integration on fractal curves in the plane, Ind. J., 40, 567-594, (1991). [5] Harrison, J. and Norton, A. The Gauss-Green theorem for fractal boundaries, Duke J. Math., 67, 575-588, (1992). [6] Harrison, J. Stokes's theorem on nonsmooth chains, Bull AMS, October (1993). [7] Harrison, J. Isomorphisms of differential forms and cochains, in J. Geom. Anal., preprint, to appear. [8] Harrison, J. Geometric realizations of distributions and currents, preprint. [9] [10] [11]
Harrison, J. Geometric dual to the Hodge * operator with applications to the theorems of Gauss and Green, preprint. Harrison, J. Geometric dual to the Hodge * operator with applications to Poincar6 duality, preprint in preparation. Kats, B.A. Gap problem and integral over non-rectifiable curves. Izvestija Vyssh. Uchebn. Zaved. Mathem., 5, 49-57, (1987). [12] Lebesgue, H. Lecons SurL'intdgration, 2nd ed., Chelsea, New York, 1928. [13] Schwartz, L. Thdorie des Distributions, Hermann, 1966. [14] Whitney, H. Geometric Integration Theory, Princeton University Press, Princeton, NJ, 1957. [15] Whitney, H. Algebraic topology and integration theory, Proc. Natl. Acad. Sci., 33, 1-6, (1947).
Department of Mathematics, University of California, Berkeley e-mail: hurrison @math.berkeley.edu