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DIFFERENTIAL FORMS FOR FRACTAL SUBSPACES AND FINITE ENERGY COORDINATES DANIEL J. KELLEHER 7 1 Abstract. This paper introduces a notion of differential forms on closed, potentially 0 fractal, subsets of the Rm by defining pointwise cotangent spaces using the restriction 2 of C1 functions to this set. Aspects of cohomology are developed: it is shown that the n differential forms are a Banach algebra and it is possible to integrate these forms along a rectifiable paths. These definitions are connected to the theory of differential forms on J Dirichlet spaces by considering fractals with finite energycoordinates. In this situation, 0 theC1differentialformsprojectontothespaceofDirichletdifferentialforms. Further,it 1 is shownthat ifthe intrinsicmetric ofa Dirichletformis a lengthspace, thenthe image ] of any rectifiable path through a finite energy coordinate sequence is also rectifiable. G The example of the harmonic Sierpinski gasket is worked out in detail. M . h t a 1. Introduction m Considering a closed, potentially fractal, subset of Rm, we define cotangent spaces as [ the quotients of C1 functions on the space. We refer to these as C1 differentials. While 1 this is a standard construction when considering subvarieties with algebraic, smooth or v 4 analytic functions, the current work makes use of functions which have lower regularity: 8 only assuming one continuous derivative, and we take an arbitrary closed subset rather 6 2 than an affine variety. Theorem 2.9 establishes the equivalence of three definitions of 0 these cotangent spaces. In section 3, C1 differential forms are defined. The main result of . 1 this section is that the 1-dimensional differential forms are a Banach algebra when added 0 to the scalar forms (functions). 7 1 In the classical setting, it is natural to define differential 1-forms as field which can be : v integrated along curves. This is a particular advantage of the present work — Section 4 i proves that it is possible to integrate C1 differential forms along rectifiable curves which X remain in the closed subset. [CGIS13] discusses the possibility of defining integrals of r a differential forms on the Sierpinski gasket along curves. Another method, which considers integration on fractal curves is [Har99]. In section 5, if the subset has a suitable measure, it is shown that the direct integral of the cotangent bundles can be taken to define L2 differential forms on the space which containtheC1 forms. Theorem4.1usesintegrationalongcurves toprovethattheexterior derivative is a closed operator with respect to this direct integral structure on these L2 forms. Recently there have emerged many points of view and techniques aimed at understand- ing differential structures on metric measure spaces. Works including [CS07, IRT12] de- velope differential forms on fractals from energy measures on Dirichlet spaces, which we shall refer to asDirichlet differential forms. This constrcution allows for thestudy of more Research supported in part by NSF grant DMS-0505622. 1 general differential equations on fractal spaces. For example, one can construct magnetic Schro¨dinger operators, as in [HT14, HR16, HKM+16], allowing for the rigorous mathe- matical study of physical objects from [DABK83, Bel92]. For one-dimensional fractals Hodge theory is defined in [HT14], and [HKT15] defines a Dirac operator and spectral triples with these fractals. Navier-Stokes equations are also studied in [HT14, HRT13]. Section 6 discusses the relationship between the C1 differential forms and those defined for Dirichlet Spaces which have finite energy coordinates. One of the main result of this section is to prove that there is a closed projection from the C1 differential forms defined to the Dirichlet differential forms. Subsection 6.1 gives the details of this relationship in the special case of harmonic coordinates on the Sierpinski Gasket as defined in [Kaj12, Kaj13, Kig93, Kig08]. The work in [Gig15] constructs first and second order differential structures for metric spaces which satisfy Ricci curvatures lower bounds. The works [BSSS12, SS12, Sma15], construct abstract versions of Hodge–DeRham and Alexander–Spanier cohomologies for use on metric spaces. A large part of the motivation is to understand data sets by there global structure which is determined by these cohomologies. Thereisastrongrelationshipbetween geometryofametricmeasurespaceandDirichlet forms on the space. This is discussed at length in [Stu94, Sto10, HKT12], where intrinsic metrics induced by the Dirichlet space are proven to be geodesic metrics in the sense that the distance is given by the length of the shortest path between two points. Theorem 6.1 of the current work proves that rectifiable curves on our fractal (with respect to the intrinsic metric) have rectifiable (with respect to Euclidean distance) images through our coordinates. Areas of interest for further research would be extending these results to infinite dimensional spaces. This would allow for the study of Dirichlet forms with infinite coordinate sequences, as was considered in previous works [Hin10]. Further, one could consider sub-Riemannian spaces, as considered in [GL14]. 2. Definitions of Cotangent spaces for closed subsets Consider U be an open subset of Rm, C (U) shall denote the set of continuous func- 0 tions on the closure of U vanishing at infinity, and C1(U) will denote the continuous functions which have continuous first-order partial derivatives. Define the following norm on C1(U) := C1(U)∩C (U) 0 0 m ∂u kuk := kuk + C1 ∞ ∂xi i=1 (cid:13) (cid:13)∞ X(cid:13) (cid:13) (cid:13) (cid:13) where (cid:13) (cid:13) kuk = sup|u(x)| ∞ x∈U and {xi}m are the coordinates of Rm. It is elementary to prove i=1 Proposition 2.1. C1(U) is a commutative Banach algebra with pointwise addition, mul- 0 tiplication, and the norm k.k . If U has compact closure, then C1(U) has multiplicative C1 0 identity 1 . U 2 For a subset K ⊂ U define I = u ∈ C1(U) | u(p) = 0 for all p ∈ K K 0 If K is a relatively closed sub(cid:8)set of U, then IK is a closed ideal o(cid:9)f C01(U). We shall take I := I for a point p ∈ U. p {p} Proposition 2.2. For a given closed K ⊂ U, The space C1(K) := C1(U)/I , is a 0 0 K Banach algebra with the norm d kuk := kuk + inf k∂ vk K ∞,K i ∞,K v|K=u|K i=1 X where kvk = sup |v(p)|. ∞,K p∈K Proof. I is a closed ideal of C1(U) with respect to the norm above and this norm is the K 0 quotient norm of C1(U)/I . (cid:3) 0 K Remark 2.3. We interpret this to mean that every function in an equivalence class of C1(K)takesthesamevaluesonK,sowetendtothinkofelementsinC1(K)asrestrictions 0 0 of elements in C1(U). 0 For f ∈ C1(U) denote the classical gradient ∇f : U → Rm as ∇f := ∂f ,..., ∂f for ∂x1 ∂xm all f ∈ C1(U). (cid:0) (cid:1) Proposition 2.4. If f is a C1(U) with f(p) = 0 and ∇f(p) = 0, then there is g ∈ 0 1 C(U) and g (p) ∈ I such that f(x) = g (x)g (x) + g (x) where g (p) = g (p) = 0, 2 p 1 2 0 1 2 g ∈ C1(U \{p}), and g is constant in a neighborhood of p. If U is compact, we can take 1 0 g ≡ 0. 0 Proof. By a partition of unity argument, it follows that f is the sum of a function compactly supported in a neighborhood around p and a function which is 0 in a neighborhood of p. Thus we assume that f is compactly supported around p without losing generality. Further, we may assume that p = 0. Becauselim |f(x)|/|x| = 0, wecanfindanincreasing C1((0,N))∩C([0,N))function x→p h(t) > sup |f(x)|/|x| and h(0) = 0. To construct this function we define h(1/n) = |x|≤t sup |f(x)|/|x| and interpolate with a C1 function. With this g (x) = (h(|x|))1/2 x≤1/(n−1) 1 and f(x)/(h(|x|))1/2 if x 6= p g (x) = 2 0 if x = p, ( which is continuous because f(x) lim ≤ lim(|x|f(x))1/2 = 0 x→p (h(|x|))1/2 x→p Clearly g is in C(U)∩C1(U \{p}), and 1 1/2 |f(x)| |f(x)| lim ≤ lim = 0. x→p |x|(h(|x|))1/2 x→p |x| (cid:18) (cid:19) so g is in C1(U) and vanishes at p. (cid:3) 2 3 We define I := clos(I2,k.k ), 2,p p C1 that is, the closure of the square of the ideal I2 with respect to the norm k.k . p C1 Corollary 2.5. The ideal I consists the set of functions of the form g g + g where 2,p 1 2 0 g ∈ C is equivalently 0 in a neighborhood of p, and g (p) = g (p) = 0 with g ∈ 0 1 1 2 1 C(U)∩C1(U \{p})and g ∈ C1(U). 2 0 Definition 2.6. Define T∗U := I /I , and K the span of a compactly supported p p 2,p p smooth function φ which is constant 1 in a neighborhood of p. Then C1(U) = K ⊕ 0 p T∗U ⊕I , and thus T∗U is an m-dimensional vector space. The natural projection from p 2,p p d : C1(U) → T∗U as the exterior derivative. Note that d does not depend on our choice p p p of φ, because if ψ was another such function, then φ−ψ ∈ I . 2,p There is a unique decomponsition of f into a sum of elements from K , T∗U and I p p 2,p given by the Taylor expansion f(x) = f(p)φ(x)+(x−p)·∇f(p)+o(|x−p|2). Proposition 2.7. d (fg) = f(p)d g +g(p)d f and p p p ∂f d f = d xi. p ∂xi p Remark 2.8. This is classical, of course, but the following proof exhibits thinking which is useful herein. Proof. Assuming, without loss of generality, p = 0, the direct product decomposition (Taylor’s theorem) implies that, if [f] and [g] are the equivalence classes of f,g ∈ C1(U) 0 mod I , 2,p [f(x)] = f(p)φ(x)+x·∇f(p)+I and [g(x)] = g(p)φ(x)+x·∇g(p)+I 2,p 2,p where φ is the bump function mentioned above. The multiplying we discover [g(x)f(x)] = f(p)g(p)φ(x)+φ(x)(g(p)(x·∇f(p))+f(p)(x·∇g(p)))+I 2,p noting that φ2 = φ modulo I and that φ is equivalently 1 in a neighborhood of p. (cid:3) 2,p Now, fixing a closed K ⊂ U, we define C1(K) := C1(U)/I 0 0 K Further, we define I (K) = I /(I ∩I ), alternatively p p K p I /I if p ∈ K p K I (K) = p {0} if p ∈/ K. ( Note that this is a subspace of C1(K), and is a Banach algebra with the inherited norm. 0 We define ∼ I (K) := I /I ∩I = (I +I )/I , 2,p 2,p K 2,p 2,p K K noting that both are Banach algebras, because I and I are both closed, so their K 2,p intersection isalsoaclosedideal. ThisnotationallowsustodefineT∗K = I (K)/I (K). p p 2,p The homomorphism theorems for rings provides the following equivalent definitions. 4 Theorem 2.9. Using the notation above, T∗K := I (K)/I (K) ∼= I /(I +I ) ∼= T∗U/d (I ). p p 2,p p 2,p K p p K Thus C1(U) = K ⊕T∗K ⊕(I +I ) 0 p p 2,p K = K ⊕T∗K ⊕I (K)⊕I p p 2,p K and thus C1(K) = K ⊕T∗K ⊕I (K). 0 p p 2,p and thus we define the differential dK : C1(K) → T∗K by the natural projection asso- p 0 p ciated with the above decomposition. If we define ρ : T∗U → T∗K and σ : C1(U) → p p p 0 C1(K) to be the natural projections, then dK ◦ σ = ρ ◦ d , i.e. the following diagram 0 p p p commutes C1(U) −−d−p→ T∗U 0 p σ ρ  dK  C1(K) −−−p→ T∗K 0y py This commutative diagram implies the following. Proposition 2.10. dK(fg) = f(p)dKg +g(p)dKf. p p p 3. Definition of differential forms Definition 3.1. Assuming T∗U has the standard norm, for ω = m ω dxi, then kωk2 = p i=1 i p m ω2(p). Since T∗K is a quotient space of T∗U, then we define the quotient norm for i=1 i p p P any equivalence class [ω] ∈ T∗K by p P m m k[ω]k = inf η2, for η = η dxi. Tp∗K η∈[ω]v i i ui=1 i=1 uX X t Similarly, assumingthatT∗U hasthestandardinnerproducthη dxi,ω dxii = m η ω , p i i p i=1 i i we give T∗K the standard quotient inner product: h[η],[ω]i = hη˜,ω˜i, where η˜,ω˜ ∈ p K,p P (dI )⊥ are the unique representative of [η],[ω] respectively. K Definition 3.2. On the other hand if T U is the tangent space at a point p, define the p tangent space with respect to K at a point p T K = {X ∈ T U | Xf = 0 ∀f ∈ I }. p p K T K is a subspace of T U, and we shall define P : T U → T K to be the orthogonal p p p p p projection (with respect to the dot product). Since the definition is equivalent to being the set of X ∈ T U such that ωX = 0 for all ω ∈ dI , T K is the dual of T∗K and visa p K p p versa. 5 If we take ♯ : T∗U → T U to be the standard musical operator such that ♯( ω dxi) = p p i ω ∂ then we have that the following diagram commutes i∂xi P P T∗U −−−♯→ T U p p ρ P T∗K −−−♯→ TK py py i.e. P♯ = ♯ρ, and k[ω]k = kP♯ωk. T∗K p Define the cotangent bundle of U to be T∗U = T∗U and Ω1(U) to be the contin- p∈U p C uous sections of T∗U which have fiberwise bounded norms vanishing at infinity, i.e. maps from U → T∗U of the form p 7→ m g (p)d xi`for g ∈ C (U), which we will denote i=1 i p i 0 m g dxi. Elements of Ω1(U) can also be interpreted as continuous functions from U toiR=1m.i C P P Proposition 3.3. Ω1(U) is a Banach space with the norm C kωk = supkω k = sup ω2 +ω2 +···+ω2 , for ω = ω dxi. Ω1 p p 1 2 m i C p∈U p∈U q Similarly, define TU = T U to be the tangent bundle. We shall use ♯ to refer to p∈U p the musical isomorphism between T∗U and TU. ` We shall define the cotangent bundle of K to be T∗K = T∗K, and Ω1(K) to be p∈U p C the sections of T∗K which are of the form p 7→ ρ ω where ω ∈ Ω1(U). That is we shall p p ` C consider the maps p 7→ g (p)dKx˜i, where g ∈ C(K), and x˜i is the equivalence class of i i p i xi in C1(K). Two forms ω and η are equal, if ω −η are in dI . 0 P p p p Define ρ : Ω1(U) → Ω1(K) to be defined fibre-wise as the projection ρ : T∗U → T∗K. C C p p p Since convergence with respect to the sup norm on Ω1(U) implies pointwise convergence, C it follows that that kerρ is a closed subspace of Ω1(U). C We can define the norm on Ω1(K) by C kωk = inf kηk . Ω1(K) Ω1 (U) C η∼ω C1 Proposition 3.4. Ω1(K) is a Banach space with the norm k.k . C Ω1(K) C Proof. First, we claim that the this is a well defined norm on Ω1(K), and in particular C that kωk = 0 if and only if ω ∈ dI for all p ∈ K. If kωk = 0 this implies Ω1(K) p p,2 Ω1(K) C C that ω ∼ 0 and thus ω ∈ dI for all p ∈ K. p p,2 On the other hand, let ω = ω (p)dKxi is such that ω ∈ dI . Because for ω ∈ i p p p,2 i C (K) it can be extended to a function in ω˜i ∈ C (U), and defining ω˜ = ω˜i(p)dxi is 0 0 equivalent to 0. Hence kωk P= 0. (cid:3) Ω1(K) C P On the other hand define ΩC(K) = PX | X ∈ ΩC(U) 1 1 where ΩC(U) is the set of continuous(cid:8)vector fields on U(cid:9), i.e. X = Xi ∂ where 1 ∂xi Xi ∈ C1(U), and (PX) = P X . We can consider ♯ as a map from Ω1(K) to ΩC(K), 0 p p p CP 1 6 and this allows for the characterization kωk = supkP♯ωk. Ω1 C p Proposition 3.5. With fiberwise multiplication, C (K) acts on Ω1(K) by bounded op- 0 C erators. In particular kfωk ≤ kfk kωk . Similarly, if f ∈ C1(K), then defining Ω1 ∞ Ω1 0 C C fω = f˜ω to be fiberwise multiplication by any representative f˜ ∈ C1(U) of f, then 0 kfωk ≤ kfk kωk . Ω1 C1 Ω1 C C Now, if we define Ω(U) = C1(U)⊕Ω1(U) then we have can define a multiplication 0 C (f ,ω )(f ,ω ) = (f f ,f ω +f ω ). 1 1 2 2 1 2 1 2 2 1 Theorem 3.6. Ω(U) is a Banach algebra with the norm k(f,ω)k = kfk +kωk . Ω C1 Ω1 C Proof. Ω(U) is a Banach space because C1(U) and Ω1(U) are both Banach spaces with 0 C their norms. To see the algebra condition k(f ,ω )(f ,ω )k = kf f k +kf ω +f ω k 1 1 2 2 Ω 1 2 C1 1 1 2 1 Ω1 C ≤ kf k kf k +kf k kω k +kf k kω k 1 C1 2 C1 1 C1 1 Ω1 2 C1 1 Ω1 C C ≤ k(f ,ω )k k(f ,ω )k . 1 1 Ω 2 2 Ω (cid:3) 4. Homology and Cohomology It is important to note that if γ = (γ ,...,γ ) : [a,b] → U is a curve which is 1 m differentiable at t , then γ induces an element of γ˙(t ) ∈ T U, where γ(t ) = p in a 0 0 p 0 standard way d dγ dγ 1 m γ˙(t )f = (f ◦γ)(t ) = ,..., ·∇f. 0 0 dt dt dt (cid:18) (cid:19) Proposition 4.1. Say γ : [a,b] → K is a continuous curve which is differentiable at time t = t such that γ(t ) = p ∈ K, then γ˙(t ) ∈ T K, i.e. if f and g are elements of C1(K) 0 0 0 p 0 such that dKf = dKg, then p p d d f ◦γ(t ) = g ◦γ(t ). 0 0 dt dt Proof. Assumingf−g ∈ I (K),theforanyrepresentativesfromC1(U),f˜,g˜respectively, p,2 0 ˜ ˜ f −g˜ ∈ span(I +I ), that is f −g˜ = φ+ψ where φ ∈ I and ψ is constant on K, 2,p K 2,p ˜ thus ∇(f −g˜) = ∇ψ and d d ˜ (f −g)◦γ(t) = γ˙(t)(f −g˜) = γ˙(t)ψ = ψ ◦γ(t ) = 0. 0 dt dt (cid:3) This fact allows us to integrate differential forms in T∗K along paths which stay in K. 7 Theorem 4.1. For a rectifiable curve γ : [a,b] → K, the linear functional from Ω1(K) → C R ω 7→ ω := η Zγ Zγ where η ∈ Ω1(U) is fiber-wise a representative of ω, is well defined. C Proof. First, we know that we can chose such an η by the definition of Ω1(K). Say η and C η◦ are two representatives of ω, as proven in the proposition above γ˙(t)(η −η◦) = 0 for all t, thus b b η = γ˙(t)η dt = γ˙(t)η◦ dt = η◦. Zγ Za Za Zγ (cid:3) This implies a version of the fundamental theorem of line integrals for the restricted cotangent space. Theorem 4.2. For any rectifiable curve γ : [a,b] → K and any function f ∈ C1(K), 0 dKf = f(b)−f(a). γ TR heorem 4.3. If forevery two points in x,y ∈ K there is a rectifiable curve γ : [a,b] → K such that γ(a) = x and γ(b) = y, then dK : C1(K) → Ω1(K), is closed as an operator 0 C from C (K) → Ω1(K), where C (K) is the continuous functions vanishing at infinity 0 C 0 with the uniform norm. Proof. Say that f → f and ω → ω, for f ,f ∈ C1(K) andω ,ω ∈ Ω1(K) withdKf = ω . i i i 0 i C i i Pick x ∈ K, then, for each x ∈ K pick γ : [a,b] → K with γ (a) = x and γ (b) = x, 0 x x 0 x then we can define g(x) = f(x )+ ω. 0 Zγx Note that by the theorem 4.2, this function is independent of our choice of γ . It is also x clear that f (x) = f (x )+ ω i i 0 i Zγx by the theorem 4.2. Finally, because γ˙(t)ω → γ˙(t)ω uniformly, we have that f = g. (cid:3) i 5. If K is a metric measure space In this section, concepts from measurable spaces are introduced to help us understand theunderlying space fromanintrinsic viewpoint. Assume thatK isendowed with σ-finite Borel regular measure µ. The subset TK = T K is closed in TU because if (x ,X ) → (x,X) in TU p∈K p n n and X f = 0 for all f ∈ I , then Xf = 0. Consider h.,.i to be the quotient inner n ` K K,p product induced by the the standard metric on T∗U, and k.k be the associated norm. p K,p If P : T U → T K is taken to be the orthogonal projection onto the tangent space, then p p p if X = ∂ is the standard frame for TU, then kdxik = kP X k. i ∂xi K,p p i Lemma 5.1. The values dKxj,dKxi are measurable on K and hence T∗K is a K,p x measurable field. (cid:10) (cid:11) 8 Proof. If p → p, then dKxi ≥ limsup dKxi . This follows because, taking k p K,p k pk K,p T K to be the tangent space at p, P : T U → T K to be the orthogonal projection, p (cid:13) (cid:13) pk pk (cid:13) pk(cid:13) this means that dKxi (cid:13) = k(cid:13)P Xik. (cid:13) (cid:13) p K,p p Ifwerestrict toasubsequence suchthat dKxi converges tosomething greaterthan (cid:13) (cid:13) pk K,p (cid:13) (cid:13) 0 (the claim is trivial if such a subsequence does not exist). Then, because the sequence (cid:13) (cid:13) is eventually contained in a compact neig(cid:13)hborh(cid:13)ood of (p,0) ∈ TU, there is a further subsequence that Y = P X converges to Y ∈ T U. Since the norm is continuous, k pk i p hY ,X i hY,X i k i i Y = lim Y = lim Y k→∞ kY k2 k k→∞ kYk2 k but since Y is in T K this implies that p hY,X i kP X k ≥ i = lim dKxi , p i kYk k→∞ pk K,p (cid:13) (cid:13) (cid:13) (cid:13) (cid:3) Since, T K is a measurable field over K, it is possible to consider the direct integral of p this field, as follows. Definition 5.2. For the rest of this section we shall assume that µ is a Radon measure on U with support in K. We define measurable forms on K by the direct integral ⊕ Ω1 (K,µ) = T∗K dµ(x). L2 x ZU We shall write Ω1 (K) when the choice of measure is clear. This is also a Hilbert space L2 with hω,ηi = hω ,η i dµ(p). K p p K,p ZK We consider Ω1(K) ⊂ Ω1 (K) in the natural way. C L2 Theorem 5.3. Ω1(K) is a dense subspace of Ω1 (K). C L2 Proof. Every element in Ω1 (K) is the space of measurable sections m ω dKxi for L2 i=1 i ωi ∈ L2(K,µ). In this case P m kωk = ω ω dKxi,dKxj dµ ωL2(K,µ) i j i,j=1Z X (cid:10) (cid:11) By approximating each ω by continuous functions with respect to µ and noting that i | dKxi,dKxj | ≤ 1, one sees that ω can be approximated by a forms with these coeffi- (cid:3) cients. (cid:10) (cid:11) If ν is another Radon measure with support on K with ν ≪ µ, then there is a natural map Ω1 (K,ν) → Ω1 (K,µ) by ω 7→ (dµ/dν)ω. L2 L2 9 6. Two notions of 1-forms In this section we shall compare the differential forms from [IRT12, CS07, HRT13] to C1 differential forms. First we shall recall some basic definitions and concepts from the theoryofDirichlet forms, formoresee[BH91,FOT11]. Foralocallycompact metricspace K, consider a regular Dirichlet form (E,F) on L2(K,µ), where µ is a Radon measure. That is (E,F) satisfy (DF1) F ⊂ L2(K) is a dense subspace and E : F × F → R is a non-negative definite symmetric bilinear form. (DF2) Closed: F is a a Hilbert space with the inner product E (f,g) := E(f,g)+hf,gi . 1 L1(K,µ) ˆ ˆ ˆ (DF3) Markov property: f ∈ F implies that f = (0∨f)∧1 ∈ F and E(f,f) ≤ E(f,f). (DF4) Regular: The space C := C (K)∩F is uniformly dense in C (K) and dense in F c c with respect to the norm induced by E . 1 It shall also be assumed that (E,F) is strongly local: For u,v ∈ F, if u is constant in the support of v, then E(u,v) = 0. From [BH91, Chapter I 3.3], the space C from (DF4) is an algebra with respect to pointwise multiplication and addition, thus we shall refer to it as the Dirichlet algebra. For any f ∈ C define the energy measure 1 φdΓ(f) = E(φf,f)− E(φ,f2) when φ ∈ C. 2 Z Consider the space C ⊗C with the bilinear form ha⊗b,c⊗di = bd dΓ(a,c). H ZK As in [HRT13], define the differential forms on the space K to be the space H, which is attained from C ⊗C by factoring out the zero space of h.,.i and completing. H Here we shall assume, that there is a finite coordinate sequence Φ = (φi)m of finite i=1 energy functions φi : K → R, that is (CO1) for all i,j ∈ N, dΓ(φi,φj)/dµ ∈ L (K,µ)∩L (K,µ), 1 ∞ (CO2) the space of functions C1(Φ) = F(φ1,...,φm) | F ∈ C1(Rn) 0 is dense in C with respect to E (the inner product from (DF2)). (cid:8) 1 (cid:9) Notice, that the assumption that C1(Φ) is dense in C implies that it is dense in C (K) 0 which means that it is point separating, so that it is an embedding into the space. We shall refer to Φ as a function from K → Rm. We shall define K = Φ(K) and Φ µ˜ = Φ∗µ, where µ is a σ-finite Borel regular measure with full support. Because Φ is point separating, it is a homeomorphism between K and K . Φ The energy measures Γ satisfy the following chain rule from theorem 3.2.2 in [FOT11]: if for f,h,g ,...,g ∈ F and F ∈ C1(Rk), if f = F(g ,...,g ), then 1 k 1 k k ∂F Γ(f,h) = Γ(g ,h). ∂xi i i=1 X 10