Metric currents and polylipschitz forms
aa r X i v : . [ m a t h . M G ] O c t METRIC CURRENTS AND POLYLIPSCHITZ FORMS
PEKKA PANKKA AND ELEFTERIOS SOULTANIS
Abstract.
We construct, for a locally compact metric space X , a space ofpolylipschitz forms Γ ∗ c ( X ), which is a pre-dual for the space of metric currentsof D ∗ ( X ) Ambrosio and Kirchheim. These polylipschitz forms may be seen asan analog of differential forms in the metric setting. Introduction
In [1], Ambrosio and Kirchheim extended the Federer–Fleming theory of currentsto general metric spaces by substituting the differential structure on the domain forcarefully chosen conditions on the functionals: a metric k -current T ∈ D k ( X ) on ametric space X is a ( k + 1)-linear map T : D k ( X ) := LIP c ( X ) × LIP ∞ ( X ) k → R satisfying continuity and locality conditions.In this article we construct a pre-dual for the space of metric currents. Ourstrategy is to pass from ( k + 1)-tuples of Lipschitz functions to linearized andlocalized objects we call polylipschitz forms . Linearization of multilinear functionalsnaturally involves tensor products, and we use sheaf theoretic methods to carry outthe localization.Williams [17] and Schioppa [12] have given different constructions for pre-dualsof metric currents. Their constructions are based on representation of currents offinite mass by duality using Cheeger differentiation and Alberti representations,respectively. Our motivation to consider polylipschitz forms stems from an appli-cation of metric currents to geometric mapping theory – polylipschitz forms inducea natural local pull-back for metric currents of finite mass under BLD-mappings.We discuss this application briefly in the end of the introduction and in more detailin [9]. Polylipschitz forms and sections.
Polylipschitz forms are introduced in threesteps: first polylipschitz functions and polylipschitz sections, then homogeneouspolylipschitz functions, and finally polylipschitz forms. Before stating our results,we discuss the motivation for this hierarchy of spaces.The space
Poly k ( X ) of k -polylipschitz functions on X is the projective tensorproduct of ( k + 1) copies of LIP ∞ ( X ). The collection { Poly k ( U ) } U , where U rangesover open sets in X , forms a presheaf and gives rise to the ´etal´e space of germsof polylipschitz functions. We denote Γ k ( X ) the space of continuous sections overthis ´etal´e space and call its elements polylipschitz sections. Date : October 11, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Metric currents, differential forms on metric spaces, polylipschitzsheaves.P.P. was supported in part by the Academy of Finland project
Although metric k -currents on X act naturally on compactly supported polylip-schitz sections Γ kc ( X ) (see Theorem 9.1), these sections are not the natural counter-part for differential forms on X , since the dual Γ kc ( X ) ∗ contains functionals, whichdo not satisfy the locality condition for metric currents. Recall that in Euclideanspaces the ( k + 1)-tuple ( π , . . . , π k ) ∈ D k ( X ) corresponds to the measurable dif-ferential form π dπ ∧ · · · ∧ dπ k . The locality condition of Ambrosio and Kirchheimfor metric currents states that T ( π , . . . , π k ) = 0 if π l is constant on spt π for some l >
0, while the polylipschitz section corresponding to ( π , . . . , π k ) need not bezero, cf. (7.1).For this reason, we introduce homogeneous polylipschitz functions . These areelements of the projective tensor product Poly k ( X ) of LIP ∞ ( X ) and k copies ofLIP ∞ ( X ), the space of bounded Lipschitz functions modulo constants. A polylips-chitz form is a continuous section over the ´etal´e space Poly k ( X ) associated to thepresheaf { Poly k ( U ) } U and we denote the space of polylipschitz forms by Γ k ( X ).The locality property of functionals in Γ k ( X ) ∗ , which motivated the homogeneousspaces, is discussed in Lemma 7.5.We consider polylipschitz forms as differential forms in the metric setting, al-though we do not impose antisymmetry on polylipschitz forms. Note that, byan observation of Ambrosio and Kirchheim, the other properties imply the corre-sponding antisymmetry property for metric currents. In Section 9 we discuss howantisymmetry of polylipschitz forms can imposed a posteriori.The space Γ kc ( X ) of compactly supported polylipschitz forms may be equippedwith a notion of sequential convergence. There is a natural, sequentially continuousexterior derivative ¯ d : Γ kc ( X ) → Γ k +1 c ( X ), a pointwise norm k · k x , for x ∈ X ,corresponding to the comass of a differential form, and a natural, sequentiallycontinuous map ¯ ι : D k ( X ) → Γ kc ( X ) , cf. (7.1).Our first main result states that the space of metric k -currents D k ( X ) on X embeds bijectively into the sequentially continuous dual Γ kc ( X ) ∗ of Γ kc ( X ). Theorem 1.1.
Let X be a locally compact metric space and k ∈ N . For each T ∈ D k ( X ) there exists a unique b T ∈ Γ kc ( X ) ∗ for which the diagram (1.1) D k ( X ) R Γ kc ( X ) ¯ ι T b T commutes. The map T b T : D k ( X ) → Γ kc ( X ) ∗ is a bijective and sequentiallycontinuous linear map.Moreover, for each T ∈ D k ( X ) and ω ∈ Γ k − c ( X ) , we have (1.2) c ∂T ( ω ) = b T ( ¯ dω ) . The sequential continuity of the map T b T is defined as follows: Supposethat a sequence ( T i ) in D k ( X ) weakly converges to T ∈ D k ( X ) , that is, for each ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 3 ( π , . . . , π k ) ∈ D k ( X ) , we have that lim i →∞ T i ( π , . . . , π k ) = T ( π , . . . , π k ) . Then,for each ω ∈ Γ kc ( X ) , we have that lim i →∞ b T i ( ω ) = b T ( ω ) . Remark 1.2.
A version of Theorem 1.1 for polylipschitz sections, shows that thereis a natural sequentially continuous embedding D k ( X ) → Γ kc ( X ) ∗ ; see Theorem 9.1.As already discussed, this embedding is, however, not a surjection. The space Γ kc ( X ) is a pre-dual to D k ( X ) in the sense of Theorem 1.1. In theother direction, we remark that De Pauw, Hardt and Pfeffer consider in [3] the dualof normal currents, whose elements are termed charges . We do not consider chargeshere and merely note that the bidual Γ kc ( X ) ∗∗ does not coincide with the space ofcharges. Currents of locally finite mass and partition continuous polylipschitzforms.
The extension of a current of locally finite mass, provided by Theorem 1.1,satisfies the following natural estimate.
Theorem 1.3.
For each T ∈ M k, loc ( X ) we have (1.3) | b T ( ω ) | ≤ Z X k ω k d k T k for every ω ∈ Γ kc ( X ) . We refer to Definition 8.1 for the pointwise norm of a polylipschitz form. Currentsof locally finite mass may further be extended, in the spirit of [5, Theorem 4.4], tothe space Γ k pc ,c ( X ) of partition continuous polylipschitz forms, that is, the spaceof partition continuous sections of the sheaf Poly k ( X ); we refer to Section 6 andDefinition 8.2 for definitions and discussion. Theorem 1.4.
Let T ∈ M k, loc ( X ) be a metric k -current of locally finite mass.Then there exists a unique sequentially continuous linear functional b T : Γ k pc ,c ( X ) → R satisfying b T ◦ ι = T . Furthermore, if T ∈ N k, loc ( X ) , then (1.4) c ∂T ( ω ) = b T ( ¯ dω ) for each ω ∈ Γ k − ,c ( X ) . Theorem 1.4 follows directly from Proposition 8.6 and Corollary 8.8, while The-orem 1.3 is implied by the more technical statement in Proposition 8.4. Note that,in Theorem 1.4, we do not claim that Γ k pc ,c ( X ) is a pre-dual of M k,loc ( X ). Motivation: Pull-back of metric currents by BLD-maps.
In [9] we applythe duality theory developed in this paper to a problem in geometric mappingtheory. To avoid the added layer of abstraction involved in polylipschitz formswe formulate the results in [9] for polylipschitz sections, which are sufficient forour purposes. For this reason, in Section 9 we briefly discuss duality theory inconnection with polylipschitz sections.In the Ambrosio–Kirchheim theory a Lipschitz map f : X → Y induces a naturalpush-forward f : M k ( X ) → M k ( Y ). In the classical setting of Euclidean spaces, PEKKA PANKKA AND ELEFTERIOS SOULTANIS this push-forward is associated to the pull-back of differential forms under the map-ping f . In [9], we consider BLD-mappings f : X → Y between metric generalized n -manifolds. A mapping f : X → Y is a mapping of bounded length distortion (or BLD for short) if f is a discrete and open mapping for which there exists a constant L ≥ L ℓ ( γ ) ≤ ℓ ( f ◦ γ ) ≤ Lℓ ( γ )for all paths γ in X , where ℓ ( · ) is the length of a path. We refer to Martio–V¨ais¨al¨a[6] and Heinonen–Rickman [4] for detailed discussions on BLD-mappings betweenEuclidean and metric spaces, respectively.For a BLD-mapping f : X → Y between locally compact spaces, the polylipschitzsections admit a push-forward f : Γ kc ( X ) → Γ k pc , c ( Y ), which in turn induces anatural pull-back f ∗ : M k, loc ( Y ) → M k, loc ( X ) for metric currents. We refer to [9]for detailed statements and further applications. Acknowledgments
We thank Rami Luisto and Stefan Wenger for discussions onthe topics of the manuscript.2.
Spaces of Lipschitz functions
We write A . C B if there is a constant c > C , for which A ≤ cB . We write A ≃ C B if A . C B . C A .Let X be a metric space. We denote by B r ( x ) ⊂ X the open ball of radius r > x ∈ X . The closed ball of radius r > x ∈ X is denoted by ¯ B r ( x ).2.1. The spaces
LIP c and LIP ∞ . Given a Lipschitz map f : X → Y betweenmetric spaces ( X, d ) and (
Y, d ′ ), we denote byLip( f ) = sup x = y d ′ ( f ( x ) , f ( y )) d ( x, y )the Lipschitz constant of f . Further, for each x ∈ X , we denoteLip f ( x ) = lim sup r → sup y,z ∈ B ( x,r ) d ′ ( f ( y ) , f ( z )) d ( z, y ) . the asymptotic Lipschitz constant of f at x .For Lipschitz functions f : X → R , we introduce the norms k f k ∞ = sup x ∈ X | f ( x ) | and L ( f ) = max { Lip( f ) , k f k ∞ } . In what follows, we denote by LIP ∞ ( X ) the space of all bounded Lipschitz func-tions on X . Note that (LIP ∞ ( X ) , L ( · )) is a Banach space [15]. Given a compact set K ⊂ X we denote by LIP K ( X ) the subspace of functions f ∈ LIP ∞ ( X ) satisfyingspt π ⊂ K .The subspace LIP c ( X ) ⊂ LIP( X ) consisting of compactly supported Lipschitzfunctions on X is the unionLIP c ( X ) = [ { LIP K ( X ) : K ⊂ X compact } . For each k ∈ N , we also denote by D k ( X ) the product space D k ( X ) = LIP c ( X ) × LIP ∞ ( X ) k . ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 5
Homogeneous Lipschitz space.
We introduce now the homogeneous Lips-chitz space LIP ∞ ( X ). The term homogeneous is taken from the theory of Sobolevspaces, where analogous homogeneous Sobolev spaces are defined.Let ∼ be the equivalence relation in LIP ∞ ( X ) for which f ∼ f ′ if f − f ′ is aconstant function. We denote the equivalence class of f ∈ LIP ∞ ( X ) by ¯ f and givethe quotient space LIP ∞ ( X ) := LIP ∞ ( X ) / ∼ the quotient norm¯ L ( ¯ f ) := inf { L ( f − c ) : c ∈ R } . The natural projection map q = q X : LIP ∞ ( X ) → LIP ∞ ( X ) , f ¯ f , is an open surjection satisfying¯ L ( ¯ f ) = inf { L ( g ) : q ( g ) = ¯ f } . Note that, given a subset E ⊂ X , the restriction map r E,X : LIP ∞ ( X ) → LIP ∞ ( E ) , f f | E , descends to a quotient map(2.1) ¯ r E,X : LIP ∞ ( X ) → LIP ∞ ( E )satisfying(2.2) q E ◦ r E,X = ¯ r E,X ◦ q X . This remark will be used later in Section 5.2.2.3.
Sequential convergence.
Following Lang [5], we give the spaces LIP c ( X )and LIP ∞ ( X ) with the topology of weak converge. We recall the notion of conver-gence of sequences in LIP c ( X ) and refer to [5] for the definition of the correspondingtopology; see also Ambrosio–Kirchheim [1]. Definition 2.1.
A sequence ( f n ) in LIP c ( X ) converges weakly to a function f : X → R in LIP c ( X ) , denoted f n → f in LIP c ( X ) , if (1) sup n Lip( f n ) < ∞ , (2) the set S n spt f n is pre-compact, and (3) f n → f uniformly as n → ∞ . In [5] Lang defines a topology on a larger space LIP loc ( X ), the space of locallyLipschitz functions, containing LIP ∞ ( X ). The weak convergence induced by thistopology for sequences could be used for functions in LIP ∞ ( X ) as well. HoweverLIP ∞ ( X ) ⊂ LIP loc ( X ) is not a closed subspace in this topology and we find it ismore convenient to modify the notion of convergence to suit the space LIP ∞ ( X )better. This does not cause any significant issues in the subsequent discussions.The weak convergence of sequences in LIP ∞ ( X ) is defined as follows. Definition 2.2.
A sequence ( f n ) in LIP ∞ ( X ) converges to a function f : X → R in LIP ∞ ( X ) , denoted f n → f in LIP ∞ ( X ) , if (1) sup n Lip( f n ) < ∞ , and (2) f n | K → f | K uniformly as n → ∞ , for every compact set K ⊂ X . PEKKA PANKKA AND ELEFTERIOS SOULTANIS
This notion of convergence for sequences arises from a topology in a similarmanner as in [5]. Another description of this topology is given in [15, Theorems2.1.5 and 1.7.2] in terms of the weak* topology with respect to the Arens–Eellsspace, which is a predual of LIP ∞ ( X ).We equip the product space D k ( X ) = LIP c ( X ) × LIP ∞ ( X ) k with the sequen-tial convergence arising from the product topology of the factors LIP c ( X ) andLIP ∞ ( X ).A sequence ( π n ) of ( k + 1)-tuples π n = ( π n , . . . , π nk ) ∈ D k ( X ) converges to a( k + 1)-tuple π = ( π , . . . , π k ) in D k ( X ) if and only if(1) π n → π in LIP c ( X ) and(2) π nl → π l in LIP ∞ ( X ), for each l = 1 , . . . , k as n → ∞ .We finish this section by defining the sequential continuity of multilinear func-tionals on D k ( X ). Definition 2.3.
A multilinear functional T : D k ( X ) → R is sequentially continu-ous if, for each sequence ( π n ) converging to π in D k ( X ) , lim n →∞ T ( π n ) = T ( π ) . Metric currents
Let X be a locally compact metric space. A sequentially continuous multilinearfunctional T : D k ( X ) → R is a metric k -current if it satisfies the following localitycondition:for each π ∈ LIP c ( X ) and π , . . . , π k ∈ LIP ∞ ( X ) having the propertythat one of the functions π i is constant in a neighborhood of spt π , wehave T ( π , . . . , π k ) = 0.By [5, (2.5)] the assertion in the locality condition holds if one of the π i ’s, i = 1 , . . . , k , is constant on spt π , i.e. no neighborhood is needed in the local-ity condition. We denote by D k ( X ) the vector space of k -currents on X . Remark 3.1. In [5] metric currents are defined as weakly continuous ( k + 1) -linearfunctionals on LIP c ( X ) × LIP loc ( X ) k . The present notion however coincides withthis class, see [5, Lemma 2.2] . Definition 3.2.
A sequence of k -currents T j : D k ( X ) → R on X converges weaklyto a k -current T : D k ( X ) → R in D k ( X ) , denoted T j → T in D k ( X ) , if, for each π ∈ D k ( X ) , lim j →∞ T j ( π ) = T ( π ) . The locality condition implies that the value T ( π , . . . , π k ) of a current T ∈ D k ( X ) at ( π , . . . , π k ) ∈ D k ( X ) only depends on the restriction π | K ∈ LIP ∞ ( K )and the equivalence classes of the restrictions π | K , . . . , π k | K ∈ LIP ∞ ( K ), where K = spt π .Given a compact set K ⊂ X , we define T : LIP K ( X ) × LIP ∞ ( K ) k → R as T ( π , . . . , π k ) := T ( π , f π , . . . , f π k )for any Lipschitz extensions e π i ∈ LIP ∞ ( X ) of π i ∈ LIP ∞ ( K ), for i = 1 , . . . , k . Thisyields a ( k + 1)-linear sequentially continuous functional. More precisely, there ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 7 exists
C > | T ( π , . . . , π k ) | ≤ CL ( π | K ) L ( π ) · · · L ( π k )for all π ∈ LIP K ( X ), π , . . . , π k ∈ LIP ∞ ( K ).3.1. Mass of a current. A k -current T : D k ( X ) → R has locally finite mass ifthere is a Radon measure µ on X satisfying(3.2) | T ( π , . . . , π k ) | ≤ Lip( π ) · · · Lip( π k ) Z X | π | d µ for each π = ( π , . . . , π k ) ∈ D k ( X ).For a k -current T of locally finite mass, there exists a measure k T k of minimaltotal variation satisfying (3.2); see Lang [5, Theorem 4.3]. The measure k T k is the mass measure of T . If k T k ( X ) < ∞ , we say T has finite mass . We denote by M k, loc ( X ) and M k ( X ) the spaces of k -currents of locally finite mass and of finitemass, respectively.The map D k ( X ) → M ( X ), T
7→ k T k , is lower semicontinuous with respect toweak convergence, that is, if the sequence ( T j ) in M k, loc ( X ) weakly converges to T ∈ M k, loc ( X ) then k T k ( U ) ≤ lim inf j →∞ k T j k ( U )for each open set U ⊂ X . Note that,spt T = spt k T k . We refer to Lang [5] for these results.A current T : D k ( X ) → R of locally finite mass admits a weakly continuousextension T : B ∞ c ( X ) × LIP ∞ ( X ) k → R satisfying (3.2); see Lang [5, Theorem 4.4]. Here B ∞ c ( X ) denotes the space ofcompactly supported and bounded Borel functions on X ; here the convergence offunctions in B ∞ c ( X ) is the pointwise convergence. Inequality (3.2) holds also forthis extension. We record this as a lemma. Lemma 3.3 ([5, Theorem 4.4]) . Let T ∈ M k, loc ( X ) be a k -current of locally finitemass. Then | T ( π , . . . , π k ) | ≤ Lip( π | E ) · · · Lip( π k | E ) Z E | π | d k T k for each ( k +1) -tuple ( π , . . . , π k ) ∈ B ∞ c ( X ) × LIP( X ) k and Borel set E ⊃ { π = 0 } Normal currents.
For the definition of a normal current, we first define theboundary operators ∂ = ∂ k : D k ( X ) → D k − ( X )for each k ∈ Z . For this reason, we set D k ( X ) = 0 for k < ∂ k = 0 for k ≤ k ≥
1, the boundary ∂T : D k − ( X ) → R of T is the ( k − ∂T ( π , . . . , π k − ) = T ( f, π , . . . , π k − ) , for ( π , . . . , π k ) ∈ D k ( X ), where f ∈ LIP c ( X ) is any Lipschitz function with com-pact support satisfying f | spt π ≡ . The current ∂T is well-defined, see [5, Definition 3.4]. PEKKA PANKKA AND ELEFTERIOS SOULTANIS
As a consequence of the locality of currents, we have that ∂ k − ◦ ∂ k ≡ D k ( X ) → D k − ( X )for each k ∈ Z , see the discussion following [5, Definition 3.4]. Definition 3.4. A k -current T ∈ M k, loc ( X ) is locally normal if ∂T ∈ M k − , loc ( X ) .A k -current T ∈ M k ( X ) is normal if ∂T ∈ M k − ( X ) . We denote by N k, loc ( X ) ⊂ M k, loc ( X ) and N k ( X ) ⊂ M k ( X ) the subspaces oflocally normal k -currents and normal k -currents on X , respectively. Note that thespace N ( X ) of normal 0-currents coincides with the space M ( X ) of all finite signedRadon measures on X . Remark 3.5.
By the lower semicontinuity of mass, if a bounded sequence ( T j ) in N k ( X ) weakly converges to a k -current T ∈ M k, loc ( X ) , then T ∈ N k ( X ) . Polylipschitz functions and their homogeneous counterparts
In this section we develop the notion polylipschitz functions and their homoge-neous counterparts. In the sequel we occasionally refer to (homogeneous) polylips-chitz forms and functions defined on Borel subsets of a locally compact space. Sincethese subsets are not necessarily locally compact, and since the treatment remainsessentially the same, we formulate all notions in this section for arbitrary metricspaces.
Algebraic and projective tensor product.
For the material on the tensor prod-uct, projective norm and projective tensor product, we refer to [10, Sections 1 and2]. Let ( V , k · k ) , . . . , ( V k , k · k k ) be Banach spaces. We denote the algebraic tensorproduct of V , . . . , V k by V ⊗ · · · ⊗ V k . There is a natural k -linear map : V × · · · × V k → V ⊗ · · · ⊗ V k , ( v , . . . , v k ) → v ⊗ · · · ⊗ v k . The projective norm of v ∈ V ⊗ · · · ⊗ V k is(4.1) π ( v ) = inf n X j k v j k · · · k v jk k k : v = n X j v j ⊗ · · · ⊗ v jk . The projective norm is a cross norm [10, Proposition 2.1], that is, for v l ∈ V l , l = 1 , . . . , k , we have(4.2) π ( v ⊗ · · · ⊗ v k ) = k v k · · · k v k k k . Note that by (4.2) the canonical k -linear map is continuous; see [10, Theorem2.9].The normed vector space ( V ⊗ · · · ⊗ V k , π ) is typically not complete. Its com-pletion V ˆ ⊗ π · · · ˆ ⊗ π V k is called the projective tensor product . We denote by b π : V ˆ ⊗ π · · · ˆ ⊗ π V k → [0 , ∞ )the norm on the completion V ˆ ⊗ π · · · ˆ ⊗ π V k of V ⊗ · · · ⊗ V k .It should be noted that the projective norm is one of many possible norms onthe algebraic tensor product, each giving rise to a completion. In general thesecompletions are not isomorphic and there is no canonical completion. However theprojective tensor product has the following universal property which characterizes it ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 9 up to isometric isomorphism in the category of Banach spaces: Let B be a Banachspace and A : V × · · · × V k → B a continuous k -linear map. Then there exists a unique continuous linear map A : V ˆ ⊗ π · · · ˆ ⊗ π V k → B for which the diagram(4.3) V × · · · × V k BV ˆ ⊗ π · · · ˆ ⊗ π V k AA commutes.Heuristically, the elements of V ˆ ⊗ π · · · ˆ ⊗ π V k can be viewed as series or as sum-mable sequences. More precisely, we have the following result. Theorem 4.1. [10, Proposition 2.8]
Let V , . . . , V k be Banach spaces, and let v ∈ V ˆ ⊗ π · · · ˆ ⊗ π V k . Then there is a sequence ( v j , . . . , v jk ) j in V × · · · × V k for which (4.4) ∞ X j k v j k · · · k v jk k k < ∞ and (4.5) lim n →∞ b π ( v − n X j v j ⊗ · · · ⊗ v jk ) = 0 . Polylipschitz functions.
We define in this section polylipschitz functionsand consider their representations. The counterpart of this discussion for homoge-neous polylipschitz functions is postponed to Section 4.2.
Definition 4.2.
Let X be a metric space and k ∈ N . A k -polylipschitz function on X is an element in the ( k + 1) -fold projective tensor product Poly k ( X ) := LIP ∞ ( X ) ˆ ⊗ π · · · ˆ ⊗ π LIP ∞ ( X ) . We denote by L k ( · ) the projective tensor norm on Poly k ( X ) . Given π , . . . π k ∈ LIP ∞ ( X ) the tensor product π ⊗ · · · ⊗ π k ∈ Poly k ( X ) maybe identified with the function in LIP ∞ ( X k +1 ) given by( x , . . . , x k ) π ( x ) · · · π k ( x k ) . Indeed, if Φ X : LIP ∞ ( X ) k +1 → LIP ∞ ( X k +1 ) is the continuous ( k + 1)-linear mapgiven byΦ X ( π , . . . , π k )( x , . . . , x k ) := π ( x ) · · · π k ( x k ) , ( x , . . . , x k ) ∈ X k +1 , the algebraic tensor product LIP ∞ ( X ) ⊗ · · · ⊗ LIP ∞ ( X ) may be identified withthe linear span of Φ X (LIP ∞ ( X ) k +1 ) and the unique continuous linear map Φ X : Poly k ( X ) → LIP ∞ ( X k +1 ) making the diagram (4.3) commute is injective.Thus, we may regard a polylipschitz function π ∈ Poly k ( X ) as a function π : X k +1 → R for which there exists a sequence ( π j , . . . , π jk ) j in LIP ∞ ( X ) k +1 satisfying(4.6) ∞ X j L ( π j ) · · · L ( π jk ) < ∞ and(4.7) π = ∞ X j π j ⊗ · · · ⊗ π jk pointwise. That is, we may identify Poly k ( X ) with Φ X ( Poly k ( X )) ⊂ LIP ∞ ( X k +1 )as sets. Definition 4.3.
For π ∈ Poly k ( X ) , any sequence ( π j , . . . π jk ) j in LIP ∞ ( X ) k +1 satisfying (4.6) and (4.7) – or, equivalently (4.4) and (4.5) – is said to represent π . We denote the collection of such sequences by
Rep( π ) . Conversely, if a sequence ( π j , . . . , π jk ) in LIP ∞ ( X ) k +1 satisfies (4.6) it representsa polylipschitz function.We denote by(4.8) = kX : LIP ∞ ( X ) k +1 → Poly k ( X ) , ( π , . . . , π k ) π ⊗ · · · ⊗ π k the natural ( k + 1)-linear bounded map, cf. (4.3).For metric spaces the standard McShane extension for Lipschitz functions yieldsimmediately an extension also for polylipschitz functions. We record this as alemma. Lemma 4.4.
Let X be a metric space, E ⊂ X a subset, and let π ∈ Poly k ( E ) .Then there exists a k -polylipschitz function ˜ π ∈ Poly k ( X ) extending π and satisfying L k (˜ π ) = L k ( π ) .More precisely, if ( π j , . . . , π jk ) j is a representation of π and ˜ π jl ∈ LIP ∞ ( X ) isan extension of π jl satisfying L (˜ π jl ) = L ( π ji ) for each j ∈ N and l = 0 , . . . , k ,then the sequence (˜ π j , . . . , ˜ π jk ) j ⊂ LIP ∞ ( X ) k +1 represents a polylipschitz function ˜ π ∈ Poly k ( X ) for which L k (˜ π ) = L k ( π ) . Thus polylipschitz functions defined on a subset E ⊂ X can always be extendedto polylipschitz functions on X preserving the polylipschitz norm.4.2. Homogeneous polylipschitz functions.Definition 4.5. A homogeneous k -polylipschitz function is an element in Poly k ( X ) := LIP ∞ ( X ) ˆ ⊗ π LIP ∞ ( X ) ˆ ⊗ π · · · ˆ ⊗ π LIP ∞ ( X ) , where LIP ∞ ( X ) appears k times in the tensor product. We denote by L k ( · ) theprojective tensor norm on Poly k ( X ) . Denote by(4.9) ¯ = ¯ kX : LIP ∞ ( X ) k +1 → Poly k ( X ) , ¯ := ◦ Q the natural bounded ( k + 1)-linear map.The natural quotient map q : LIP ∞ ( X ) → LIP ∞ ( X ) induces a quotient map(4.10) Q = Q kX := id ⊗ π q ⊗ π · · · ⊗ π q : Poly k ( X ) → Poly k ( X ) , that is, q : Poly k ( X ) → Poly k ( X ) is an open surjection and¯ L k (¯ π ) = inf { L k ( π ) : Q ( π ) = ¯ π } for each ¯ π ∈ Poly k ( X ) , cf. [10, Proposition 2.5]. ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 11
Convergence of polylipschitz and homogeneous polylipschitz func-tions.
In this subsection, we define a notion of convergence of sequences on
Poly k ( X )and Poly k ( X ). These notions correspond to the weak − ∗ convergence in LIP ∞ ( X );see Section 2.1. We give the necessary notions of convergence in two separatedefinitions. Definition 4.6.
A sequence ( π n ) in Poly k ( X ) converges to π ∈ Poly k ( X ) if, for allcompact sets V := V × · · · × V k ⊂ X k +1 there exists representations ( π n,j , . . . , π j,nk ) ∈ Rep( π n − π ) for which (1) ∞ X j sup n L ( π j,n ) · · · L ( π j,nk ) < ∞ , and (2) lim n →∞ ∞ X j k π j,n | V k ∞ · · · k π j,nk | V k k ∞ = 0 . Definition 4.7.
A sequence (¯ π n ) in Poly k ( X ) converges to ¯ π ∈ Poly k ( X ) if thereare polylipschitz functions π ∈ Q − (¯ π ) and π n ∈ Q − (¯ π n ) for each n ∈ N , so that π n → π in Poly k ( X ) . Remark 4.8.
It follows immediately from Definitions 4.6 and 4.7 that the naturalmaps : LIP ∞ ( X ) k +1 → Poly k ( X ) and ¯ : LIP ∞ ( X ) k +1 → Poly k ( X ) in (4.8) and(4.9) are sequentially continuous. Before moving to polylipschitz forms, we record a notion of locality for linearmaps in LIP ∞ ( X ) k +1 and record some of its consequences. Definition 4.9. A ( k + 1) -linear map A : LIP ∞ ( X ) k +1 → V is local if, for a ( k + 1) -tuple ( π , . . . , π k ) ∈ LIP ∞ ( X ) k +1 , holds (4.11) A ( π , . . . , , π k ) = 0 whenever one of the functions π , . . . , π k is constant. Proposition 4.10.
Let V be a Banach space and let A : LIP ∞ ( X ) k +1 → V be abounded and local ( k + 1) -linear map. Then (1) A descends to a (unique) bounded ( k + 1) -linear map A ′ : LIP ∞ ( X ) × LIP ∞ ( X ) k → V , (2) the unique bounded linear maps A : Poly k ( X ) → V and A ′ : Poly k ( X ) → V satisfying A = A ◦ and A ′ = A ′ ◦ ( Q ◦ ) , respectively, satisfy A = A ′ ◦ Q ,and (3) if A is sequentially continuous (in the sense of Definition 4.6) then A ′ sequentially continuous (in the sense of Definition 4.7).Proof. It is clear that, since A is local, it descends to a unique bounded multilinearmap A ′ : LIP ∞ ( X ) × LIP ∞ ( X ) k → V satisfying A = A ′ ◦ (id × q X × · · · × q X ). Notethat Q = id × q X × · · · × q X . It follows from the uniqueness of the diagram (4.3) that A = A ′ ◦ (id × q X × · · · × q X ) = A ′ ◦ Q. Suppose A is sequentially continuous and let ¯ π n → Poly k ( X ). For each n ∈ N , we fix π n ∈ Poly k ( X ) so that Q ( π n ) = ¯ π n and that the sequence ( π n ) convergesto 0 ∈ Q − (0) in Poly k ( X ). Then A ( π n ) ∈ Q − ( A ′ (¯ π n )) and A ( π n ) →
0. Thus A ′ (¯ π n ) → Poly k ( X ). Since A ′ is linear this proves the sequential continuity of A ′ . (cid:3) Polylipschitz forms and sections
Since we consider presheaves of polylipschitz functions and homogeneous polylip-schitz functions and their ´etal´e spaces, we discuss the related terminology first inmore general. We refer to [14, Section 5.6] and [16, Chapter II] for a more detaileddiscussion.5.1.
Presheaves and ´etal´e spaces.
Let X be a paracompact Hausdorff space. A presheaf P on X is a collection { A ( U ) } U of vector spaces (over R ) for each open set U ⊂ X and, for each inclusion U ⊂ V , a linear map ρ U,V : A ( V ) → A ( U ) satisfying ρ U,U = id and(5.1) ρ U,V = ρ U,W ◦ ρ W,V whenever U ⊂ W ⊂ V .Given two presheaves { A ( U ) } and { B ( U ) } on X , a collection { ϕ U : B ( U ) → A ( U ) } of linear maps satisfying(5.2) ϕ U ◦ ρ BU,V = ρ AU,V ◦ ϕ V , U ⊂ V is called a presheaf homomorphism .Given an open set U ⊂ X , the support of f ∈ A ( U ), denoted spt( f ), is theintersection of all closed sets F ⊂ U with the property that ρ U \ F,U ( f ) = 0. Fine presheaves.
A presheaf { A ( U ) } on X is called fine if every open cover of X admits a locally finite open refinement U and, for each U ∈ U , there is a presheafhomomorphism { ( L U ) W : A ( W ) → A ( W ) } with the following properties:(a) spt( L U ) W ( f ) ⊂ U ∩ W for every f ∈ A ( W ) and U ∈ U ,(b) every point x ∈ X has a neighborhood D ⊂ X for which D ∩ U = ∅ foronly finitely many U ∈ U and X U ∈U ρ D,W ◦ ( L U ) W = ρ D,W . whenever D ⊂ W .Note that by (a) and the assumption on D , the sum in (b) has only finitely manynon-zero terms. ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 13
Space of germs and its sections.
Let x ∈ X and U and V be open neighborhoodsof x . Two elements f ∈ A ( U ) and g ∈ A ( V ) are equivalent if there exists an openneighborhood D ⊂ U ∩ V of x so that ρ D,U ( f ) = ρ D,V ( g ) . This defines an equivalence relation on the disjoint union G U A ( U ). We denote by A ( X ) the set of equivalence classes and say that A ( X ) is the space of germs for thepresheaf P .Given x ∈ X , an open neighborhood U of x and f ∈ A ( U ) we denote by [ f ] x ∈A ( X ) the equivalence class of f and call it the germ of f at x . There is a naturalprojection map(5.3) p : A ( X ) → X, [ f ] x x and the fibers p − ( x ) =: A x ( X ) are called stalks of A ( X ) over x . The stalk A x ( X )has a natural addition and scalar multiplication, making it a vector space (see [14,Section 5.6]).If U ⊂ X is an open set, a map ω : U
7→ A ( U ) satisfying p ◦ ω = idis called a section of A ( U ) over U and the space of all sections of A ( U ) over U is denoted by G ( U ; A ( U )) . Note that G ( U ; A ( U )) has a natural vector spacestructure given by pointwise addition and scalar multiplication. We abbreviate G ( A ( X )) = G ( X ; A ( X )) and call elements of G ( A ( X )) global sections of A ( X ). ´Etal´e space. There is a natural ´etal´e topology on A ( X ) so that the projection map(5.3) is a local homeomorphism.The ´etal´e topology has a basis of open sets of the form O U,f = { [ f ] x : x ∈ U } for U ⊂ X open and f ∈ A ( U ), cf. [14, Section 5.6]. We call A ( X ) equipped withthis topology the ´etal´e space associated to the presheaf P .If ω ∈ G ( A ( X )) and U is an open cover of X , a collection { f U ∈ A ( U ) } U is called compatible with ω if, for every x ∈ X , there exists U ∈ U so that ω ( x ) = [ f U ] x .Note that ω − O U,f U = { x ∈ U : ω ( x ) = [ f U ] x } ⊂ U forms a cover of X which is a refinement of U . When ω is continuous, the opencover V = { ω − O U,f U } U and the collection g V := ρ V,U ( f U ), where V = ω − O U,f U ,is compatible with ω and furthermore(5.4) [ g V ] x = ω ( x ) = [ g W ] x whenever V, W ∈ V and x ∈ V ∩ W .We say that a collection { f U } U represents a continuous section ω ∈ G ( A ( X )), if itsatisfies (5.4).Fine presheaves with a mild additional assumption admit a stronger form of(5.4), called the overlap condition , for collections representing continuous sections,which we record as the following lemma. This will be used in Section 7 to definethe action of a current on polylipschitz forms. Lemma 5.1.
Let { A ( U ) } is a fine presheaf and suppose that the linear maps ρ U,X are onto for each open U ⊂ X .If ω ∈ G ( A ( X )) is continuous and the collection { f U ∈ A ( U ) } U is compatiblewith ω , there exists a locally finite refinement V of U and a collection { g V ∈ A ( V ) } V satisfying the overlap condition(5.5) ρ V ∩ W,V ( f V ) = ρ V ∩ W,W ( f W ) , whenever V, W ∈ V and V ∩ W = ∅ . Proof.
The sets ω − O U,f U ⊂ X ( U ∈ U ) are open and, since { f U } U is compatiblewith ω , cover X . Let W be a locally finite refinement of { ω − O U,f U } U ∈U and { ( L W ) U : A ( U ) → A ( U ) } U ( W ∈ W ) be as in the definition of fine sheaves. Wedenote L W := ( L W ) X . For every W ∈ W choose U ∈ U such that W ⊂ ω − O U,f U ⊂ U and let h W ∈ A ( X ) be such that ρ W,X ( h W ) = ρ W,U ( f U ) . For each x ∈ X , let D x be a neighborhood of x satisfying (b) in the samedefinition. Set g x := X W ∈W ρ D x ,X ◦ L W ( h W ) ∈ A ( D x ) . The collection { g x ∈ A ( D x ) } x ∈ X now satisfies (5.5). Indeed, ρ D x ∩ D y ,D x ( g x ) = X W ∈W ρ D x ∩ D y ,X ( L W ( h W ))= X W ∈W ρ D x ∩ D y ,D y ( ρ D y ,X ◦ L W ( h W ))= ρ D x ∩ D y ,D y ( g y ) . We pass to a locally finite refinement V of { D x } x ∈ X and set, for any V ∈ V , g V = ρ V,D x ( g x )whenever V ⊂ D x . Clearly ρ U ∩ V,U ( g U ) = ρ U ∩ V,D x ∩ D y ( ρ D x ∩ D y ,D x ( g x ))= ρ U ∩ V,D x ∩ D y ( ρ D x ∩ D y ,D y ( g y )) = ρ U ∩ V,V ( g V )whenever U, V ∈ V and U ⊂ D x , V ⊂ D y .It remains to show that { g V } V is compatible with ω . Indeed, for x ∈ X and V ∈ V a neighborhood of x , we have V ⊂ D y ⊂ ω − O U,f U ⊂ U for some D y and U ∈ U . Since g V = X W ∈W ρ V,X ( L W ( h W )) = X W ∈W ( L W ) V ( ρ V,X ( h W ))= X W ∈W ( L W ) V ( ρ V,U ( f U ))) = ρ V,U ( f U ) , it follows that [ g V ] x = [ ρ V,U ( f U )] x = [ f U ] x = ω ( x ). (cid:3) Definition 5.2.
Let ω ∈ G ( A ( X )) be continuous. If U is a locally finite opencover, { f U } U is compatible with ω and satisfies the overlap condition (5.5), we saythat { f U } U is overlap-compatible with ω . ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 15
Remark 5.3.
Representations { f U } U of continuous sections ω ∈ G ( A ( X )) arestable under passing to refinements. Indeed, if D is a refinement of U and we set f ′ V = ρ V,W ( f W ) , for V ∈ D and V ⊂ W ∈ U , the collection { f ′ V } V ∈D again represents ω . The sameholds true for the overlap condition (5.5).Thus we may always assume that the underlying cover in a representation of ω is locally finite consists of precompact sets if X is locally compact. The vector space of continuous sections over A ( X ) is denoted Γ( A ( X )). Weremark that there is a canonical linear map(5.6) γ : A ( X ) → Γ( A ( X )) , f ( x [ f ] x ) . Support.
Let { A ( U ); ρ U,V } U be a presheaf on X and A ( X ) the associated ´etal´espace. For ω ∈ G ( A ( X )), we definespt ω = { x ∈ X : ω ( x ) = 0 } . We say that the section ω ∈ G ( A ( X )) has compact support if spt ω is compact. Wedenote by G c ( A ( X )) the vector space of compactly supported (global) sections of A ( X ) and Γ c ( A ( X )) = Γ( A ( X )) ∩ G c ( A ( X )).5.2. Polylipschitz forms and sections.
We move now the discussion from ab-stract presheaves to presheaves of polylipschitz and homogeneous polylipschitz func-tions. Let X be a locally compact metric space and k ∈ N . Recall the notationintroduced in Section 2.2. We consider two presheaves, namely the collections { Poly k ( U ) } U and { Poly k ( U ) } U together with the restriction maps ρ U,V : Poly k ( V ) → Poly k ( U ) , ρ U,V = r ⊗ π ( k +1) U,V , ¯ ρ V,U : Poly k ( V ) → Poly k ( U ) , ¯ ρ U,V := r U,V ⊗ π ¯ r ⊗ π kU,V for inclusions U ⊂ V . Note that under the identification described in Section 4.1the map ρ U,V is simply the restriction map π π | U k +1 . It is not difficult to see(using the corresponding facts for r U,V and ¯ r U,V ) that ρ U,V and ¯ ρ U,V satisfy (5.1)for U ⊂ W ⊂ V . For the purposes of Section 6, we note that this property remainstrue for the quotient maps ρ U,V and ¯ ρ U,V for any sets U ⊂ V ⊂ X , in particularalso for sets which are not open.The overlap condition (5.5) for polylipschitz forms and sections is crucial fordefining the action of currents on them. The next proposition establishes this byshowing that the presheaves { Poly k ( U ) } and { Poly k ( U ) } are fine. Proposition 5.4.
The presheaves { Poly k ( U ) } and { Poly k ( U ) } are fine, and themaps ρ U,X and ¯ ρ U,X are onto.Proof.
The last claim is immediate since ρ U,X and ¯ ρ U,X are quotient maps. Since X is locally compact, any open cover of X admits a locally finite precompactrefinement U . Let { ϕ U } U be a Lipschitz partition of unity subordinate to U . Foreach U ∈ U and W ⊂ X open, consider the bounded ( k + 1)-linear maps( L U ) W : LIP ∞ ( W ) k +1 → Poly k ( W ) , ( π , . . . , π k ) W ( ϕ U π , π , . . . , π k ) . The bounded linear maps ( L U ) W : Poly k ( W ) → Poly k ( W ) making the diagram(4.3) commute form a presheaf homomorphism of { Poly k ( W ) } . Since spt ϕ U ⊂ U it follows that spt ( L U ) W ( π ) ⊂ U ∩ W for any π ∈ Poly k ( W ). This shows (a) in thedefinition of fine presheaves.Let x ∈ X and D be a neighborhood of x meeting only finitely many of the setsin U . The fact that { ϕ U } is a partition of unity implies that for any ( π , . . . , π k ) ∈ LIP ∞ ( W ) k +1 X U ∈U ( L U ) D ( π , . . . , π k ) = D X U ∈U ϕ U ! π , π , . . . , π k ! = D ( π , . . . , π k ) . This implies (b) in the definition of fine presheaves.Since the bounded ( k + 1)-linear maps Q W ◦ ( L U ) W : LIP ∞ ( W ) k +1 → Poly k ( W )satisfy (4.11) we obtain maps ( L U ) ′ W : Poly k ( W ) → Poly k ( W ) by Proposition 4.10,for each open W ⊂ X , that form a presheaf homomorphism. Condition (a) nowfollows from the corresponding statement for Poly k ( W ) and (5.8). Condition (b)follows as above. (cid:3) We denote by
Poly k ( X ) and Poly kx ( X ) (respectively, Poly k ( X ) , Poly kx ( X )) the´etal´e space and stalk at x associated to { Poly k ( U ) } U (respectively for { Poly k ( U ) } U ).We further denote the various spaces of sections associated to Poly k ( X ) and Poly k ( X )by G k ( X ) := G ( Poly k ( X )) , Γ k ( X ) := Γ( Poly k ( X )) , G kc ( X ) := G c ( Poly k ( X )) , Γ kc ( X ) := Γ k ( X ) ∩ G kc ( X ) G k ( X ) := G ( Poly k ( X )) , Γ k ( X ) := Γ( Poly k ( X )) , G kc ( X ) := G c ( Poly k ( X )) , Γ kc ( X ) := Γ k ( X ) ∩ G kc ( X ) Definition 5.5.
A continuous section in Γ k ( X ) is a polylipschitz k -form on X . Acontinuous section of Γ k ( X ) is called a k -polylipschitz section . We denote by γ = γ kX : Poly k ( X ) → Γ k ( X ) , π ( x [ π ] x ) , (5.7) ¯ γ = ¯ γ kX : Poly k ( X ) → Γ k ( X ) , ¯ π ( x [¯ π ] x ) , the natural linear maps in (5.6) associated to the presheaves { Poly k ( U ) } U and { Poly k ( U ) } U , respectively.5.3. Relationship of polylipschitz forms and polylipschitz sections.
Webriefly describe the relationship between Γ k ( X ) and Γ k ( X ). The natural operatorsin this section arise as linear maps associated to presheaf cohomomorphisms . Wegive a general sheaf theoretic construction in Appendix A, and establish some of itsbasic properties there; see Proposition A.2. Here we apply the results in AppendixA without further mention. We assume throughout this section that X and Y arelocally compact metric spaces and k, m ∈ N are possibly distinct natural numbers.Using (2.2) and the uniqueness in diagram (4.3), we see that the quotient map(4.10) satisfies Q U ◦ ρ U,V = ¯ ρ U,V ◦ Q V . (5.8) ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 17
Thus the collection { Q U : Poly k ( U ) → Poly k ( U ) } U is a presheaf homomorphism.Let Q be the associated linear map(5.9) Q = Q k : G k ( X ) → G k ( X ) . The next proposition shows that cohomomorphisms { Poly k ( U ) } → { Poly m ( V ) } satisfying the locality condition (4.11) descend to cohomomorphisms { Poly k ( U ) } →{ Poly m ( V ) } . Proposition 5.6.
Let f : X → Y be a continuous map and ϕ = { ϕ U : Poly k ( U ) → Poly m ( f − U ) } an f -cohomomorphism, where each ϕ U is bounded. Assume ϕ U ◦ U satisfies (4.11) and let ϕ U : Poly k ( U ) → Poly m ( f − U ) be the unique bounded linear map satisfying ϕ U = ¯ ϕ U ◦ Q kU , for each open U ⊂ Y . Then ϕ = { ϕ U : Poly k ( U ) → Poly m ( f − U ) } is an f -cohomomorphism. The linear maps ϕ ∗ and ϕ ∗ associated to ϕ and ϕ satisfy ϕ ∗ = ϕ ∗ ◦ Q . Proof.
The existence and uniqueness of ϕ U follows from Proposition 4.10. For opensets U ⊂ V ⊂ Y , ϕ U ◦ ρ U,V ◦ U and ρ f − U,f − V ◦ ϕ V ◦ V satisfy (4.11). Since ϕ isan f -cohomomorphism, (5.8) and the uniquenenss in diagram (4.3) implies that ϕ U ◦ ¯ ρ U,V = ¯ ρ f − U,f − V ◦ ϕ V . The identity ϕ ∗ = ϕ ∗ ◦ Q follows from the fact that ϕ U = ¯ ϕ U ◦ Q U for each open U ⊂ Y . (cid:3) Sequential convergence on Γ kc ( X ) and Γ kc ( X ) . To study sequential con-tinuity of linear maps between polylipschitz forms and sections, we introduce anotion of sequential convergence on Γ kc ( X ) and Γ kc ( X ). Recall that, by Proposi-tion 5.4 and Lemma 5.1, polylipschitz forms and sections admit overlap-compatiblerepresentations indexed by a locally finite precompact open cover; see also Remark5.3. Definition 5.7.
We say a sequence ( ω n ) in Γ kc ( X ) convergences to ω ∈ Γ kc ( X ) ,denoted ω n → ω in Γ kc ( X ) , if there exists a compact set K ⊂ X , a locally finiteprecompact open cover U of X , and, for each n ∈ N , a collection { π nU } U ∈U overlap-compatible with ω n − ω having the following properties: (1) spt( ω n − ω ) ⊂ K for each n ∈ N , and (2) π nU → in Poly k ( U ) for each U ∈ U .Convergence of a sequence (¯ ω n ) in Γ kc ( X ) to ¯ ω ∈ Γ kc ( X ) is defined analogously. For metric spaces X and Y , let A = { A ( U ) } U and B = { B ( V ) } V denote eitherof the presheaves { Poly k ( U ) } U or { Poly m ( U ) } U on X and Y , respectively, and let L : Γ c ( B ( Y )) → Γ c ( A ( X ))be a linear map. We say that L is sequentially continuous if L ( ω n ) → L ( ω ) in Γ c ( A ( X )) whenever ω n → ω in Γ c ( B ( X )) . The natural quotient map from polylipschitz sections to polylipschitz forms issequentially continuous.
Proposition 5.8.
The map Q : Γ kc ( X ) → Γ kc ( X ) is sequentially continuous. This follows immediately from an abstract result on sequential continuity oflinear maps associated to cohomomorphisms.
Proposition 5.9.
Suppose f : X → Y is a proper continuous map, and ϕ = { ϕ U : B ( U ) → A ( f − ( U )) } an f -cohomomorphism, where A = { A ( U ) } U and B = { B ( V ) } V denote either ofthe presheaves { Poly k ( U ) } U or { Poly m ( U ) } U on X and Y , respectively.If ϕ U is bounded and sequentially continuous for each open U ⊂ Y , then theassociated linear map ϕ ∗ : Γ c ( B ( Y )) → Γ c ( A ( X )) is sequentially continuous.If A = { Poly m ( U ) } U , B = { Poly k ( U ) } U , and ϕ U ◦ U satisfies (4.11) for eachopen U ⊂ Y , then the linear map ϕ ∗ : Γ kc ( Y ) → Γ mc ( X ) in Proposition 5.6 is sequentially continuous.Proof. We prove the first claim in case A = { Poly m ( U ) } U and B = { Poly k ( U ) } .The other cases are analogous.Since ϕ ∗ is linear it suffices to prove sequential continuity at the origin. Let ω n → kc ( Y ), and let K ⊂ X , U and { π nU } U be as in Definition 5.7. For each V ∈ f − U choose U V ∈ U such that V = f − U . Then the collection { ϕ U V ( π nU V ) } f − U is overlap-compatible with ϕ ∗ ω n for each n ∈ N ; cf. proof of Proposition A.2(2).Since spt ϕ ∗ ω n ⊂ f − K and ϕ U V is sequentially continuous for each n ∈ N and V ∈ f − U , we have that ϕ ∗ ω n → mc ( X ).To prove the last claim assume that A = { Poly m ( U ) } U , B = { Poly k ( U ) } U , andthat ϕ U ◦ U satisfies (4.11) for each U ⊂ Y . By Proposition 4.10 the unique map ϕ U satisfying ϕ U = ϕ U ◦ Q U is sequentially continuous. Thus, the associated linearmap ϕ ∗ : Γ kc ( Y ) → Γ mc ( X ) is sequentially continuous. (cid:3) Remark 5.10.
Propositions 5.9 and 5.6 have natural bilinear analogues in thesituation of Remark A.3. The proofs are similar and we omit the details. Exterior derivative, pull-back, and cup-product of polylipschitzforms
In this section we introduce the exterior derivative, pull-back and cup product onpolylipschitz forms and sections. We prove that they are sequentially continuouswith respect to a natural notion of sequential convergence on Γ kc ( X ) and Γ kc ( X );cf. Definition 5.7. The results in this section are important for applications tocurrents, and will be used extensively in [9]. ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 19
Pull-back.
Let f : X → Y be a Lipschitz map. Consider the f -cohomomorphism ϕ := { f U : Poly k ( U ) → Poly k ( f − U ) } , where f U is given by π π ◦ ( f | f − U × · · · × f | f − U ) . The maps ϕ ′ := { Q f − U ◦ f U : Poly k ( U ) → Poly k ( f − U ) } form an f -cohomomorphismand Q f − U ◦ f U ◦ U satisfies (4.11) for each U ⊂ Y . By Proposition 5.6 the linearmaps f : G k ( Y ) → G k ( X ) and ¯ f : G k ( Y ) → G k ( X ) associated to ϕ and ϕ ′ ,respectively, satisfy Q ◦ f = ¯ f ◦ Q . We refer to the linear maps f and ¯ f as pull-backs . If f is proper, Proposition5.9 implies that f and ¯ f are sequentially continuous.If E ⊂ X , the pull-backs ι E and ¯ ι E given by the construction above for theinclusion map ι E : E ֒ → X are called restrictions to E . We denote ω | E := ι E ( ω ) and ¯ ω | E := ¯ ι E (¯ ω )for ω ∈ G k ( X ) and ¯ ω ∈ G k ( X ). Note that, when E ⊂ X is closed, the inclusion ι E is a proper map. Thus the restriction operator to closed sets is sequentiallycontinuous.6.2. Exterior derivative.
As in Alexander-Spanier cohomology (see [14, Section5.26]) we define a linear map d = d kX : LIP ∞ ( X k +1 ) → LIP ∞ ( X k +1 )by(6.1) d kX π ( x , . . . , x k +1 ) = k +1 X j =0 ( − j π ( x , . . . , ˆ x j , . . . x k +1 )for π ∈ LIP ∞ ( X k +1 ) and x , . . . , x k ∈ X . It is a standard exercise to show that(6.2) d k +1 X ◦ d kX = 0 Lemma 6.1.
For each π ∈ Poly k ( X ) and each open set V ⊂ X , we have (6.3) L k +1 ( dπ ; V ) ≤ ( k + 2) L k ( π ; V ) . Thus d defines a bounded linear map d : Poly k ( X ) → Poly k +1 ( X ) which, moreover,is sequentially continuous. It follows from Lemma 6.1 that { d kU : Poly k ( U ) → Poly k +1 ( U ) } , and consequently { Q k +1 U ◦ d kU : Poly k ( U ) → Poly k +1 ( U ) } , are presheaf homomorphisms, and Q k +1 U ◦ d kU ◦ U satisfies (4.11) for each open U ⊂ X . By Propositions 5.6 and 5.9 we obtainsequentially continuous associated linear maps¯ d := ¯ d kX : G k ( X ) → G k +1 ( X )and d := d kX : G k ( X ) → G k +1 ( X ) , called the exterior derivative of polylipschitz forms and sections, respectively. Remark 6.2.
In fact the identity (6.4) ¯ d kA (¯ ρ A,B (¯ π )) = ¯ ρ A,B ( ¯ d kB ¯ π ) for all ¯ π ∈ Poly k ( B ) . holds for any sets A, B ⊂ X . Thus, if E ⊂ X and ¯ ω ∈ G k ( X ) we have ¯ d ( ¯ d ¯ ω ) = 0 and ( ¯ d ¯ ω ) | E = ¯ d (¯ ω | E ) . The first identity follows from (6.2) and Proposition A.2(4),while the second is implied by (6.4).The same identities hold for restrictions and the exterior derivative of polylips-chitz sections.
These properties of the exterior derivative and restriction are used in the sequelwithout further mention.We conclude this subsection with the proof of Lemma 6.1. For the proof, thefollowing expression, for ( π , . . . , π k ) ∈ LIP ∞ ( X ) k +1 and π = π ⊗ · · · ⊗ π k , will beuseful.(6.5) d kX ( π ⊗ · · · ⊗ π k ) = k +1 X l =0 ( − l π ⊗ · · · ⊗ π l − ⊗ ⊗ π l ⊗ · · · ⊗ π k Proof of Lemma 6.1.
Let π , . . . , π k ∈ LIP ∞ ( X ) and π = π ⊗ · · · ⊗ π k ∈ Poly k ( X ).By (6.5) we have L k ( d kX π ; V ) ≤ k +1 X l =0 L ( π | V ) · · · L ( π k | V ) = ( k + 2) L k ( π ; V ) . Thus, by the subadditivity of L k ( · ; V ), we have, for each π ∈ Poly k ( X ) and( π j , . . . , π jk ) j ∈ Rep( π ), the estimate L k +1 ( d kX π ; V ) ≤ X j ( k + 2) L ( π j | V ) · · · L ( π jk | V ) . Taking infimum over all such representatives yields (6.3).To show that d kX is sequentially continuous suppose π n → Poly k ( X ) and let V = V × · · · × V k be compact. For each n ∈ N , let π n = ∞ X j π j,n ⊗ · · · ⊗ π j,nk be a representation of π n satisfying (1) and (2) in Definition 4.6. Then(6.6) d X π n = k +1 X l =0 ∞ X j ( − l π j,n ⊗ · · · ⊗ π j,nl − ⊗ ⊗ π j,nl · · · ⊗ π j,nk is a representation of d kX π n .To show that condition (1) in Definition 4.6 is satisfied it suffices to observe that,for every j ∈ N and l ∈ { , . . . , k + 1 } , we havesup n L (( − l π j,n ) L ( π j,n ) · · · L ( π j,nk ) = sup n L ( π j,n ) · · · L ( π j,nk ) . Moreover, for each compact set V × · · · × V k ⊂ X k +1 , we have k +1 X l =0 ∞ X j k ( − l π j,n | V k ∞ · · · k π j,nk | V k k ∞ ≤ ( k + 2) ∞ X j k π j,n | V k ∞ · · · k π j,nk | V k k ∞ . ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 21
Thus condition (2) in Definition 4.6 is satisfied by the representation (6.6). Itfollows that d X π n → Poly k +1 ( X ). (cid:3) Cup product.
Given polylipschitz functions π ∈ Poly k ( X ) and σ ∈ Poly m ( X ),their cup product is the function π ⌣ σ : X k + m +1 → R ,( x , x , . . . x k + m ) π ( x , x , . . . , x k ) σ ( x , x k +1 , . . . , x m + k ) . If ( π j , . . . , π jk ) ∈ Rep( π ) and ( σ j , . . . , σ jk ) ∈ Rep( σ ) are representations of polylip-schitz functions π ∈ Poly k ( X ) and σ ∈ Poly m ( X ), respectively, we observe that( π j σ i , π j , . . . , π jk , σ i , . . . , σ im ) i,j is a representation of π ⌣ σ and that π ⌣ σ ∈ Poly k + m ( X ). The proof of the next Lemma follows from Definition 4.6 and straight-forward calculations and estimates. We omit the details. Lemma 6.3.
The cup-product · ⌣ · : Poly k ( X ) × Poly m ( X ) → Poly k + m ( X ) is asequentially continuous bounded bilinear map. The collection { ⌣ : Poly k ( U ) × Poly m ( U ) → Poly k + m ( U ) } is a bilinear presheafhomomorphism, and we note that Q k + mU ◦ ⌣ ◦ ( kU × mU ) : LIP ∞ ( U ) k +1 × LIP ∞ ( U ) m +1 → Poly k + m ( U )satisfies the bi-linear analogue of (4.11) for each open U ⊂ X . By Lemma 6.3 andRemark 5.10 (see also Remark A.3) we obtain bilinear maps ⌣ : G k ( X ) × G m ( X ) → G k + m ( X )and ⌣ : G k ( X ) × G m ( X ) → G k + m ( X ) , called the cup product of polylipschitz forms and sections, respectively. Note thatspt(¯ ω ⌣ ¯ σ ) ⊂ spt ¯ ω ∩ spt ¯ σ, Γ k ( X ) ⌣ Γ m ( X ) ⊂ Γ k + m ( X ) , see Remark A.3. We record the following standard identities for cup products,pull-backs and the exterior derivative; cf. [8, 7]. Lemma 6.4.
Let X and Y be metric spaces, f : X → Y a Lipschitz map. Let ¯ α ∈ G k ( Y ) and ¯ β ∈ G m ( Y ) . Then (a) ¯ f (¯ α ⌣ ¯ β ) = ( ¯ f ¯ α ) ⌣ ( ¯ f ¯ β ) , and (b) ¯ d (¯ α ⌣ ¯ β ) = ¯ d ¯ α ⌣ ¯ β + ( − k ¯ α ⌣ ¯ d ¯ β. The same identities hold for α ∈ G k ( Y ) and β ∈ G m ( Y ) . Metric currents as the dual of polylipschitz forms
In this section X is a locally compact metric space and k ∈ N . We prove thatmetric currents act sequentially continuously on the space of polylipschitz forms.Recall the natural maps (4.8), (4.9) and (5.7) and denote(7.1) ι = ι kX : D k ( X ) → Γ kc ( X ) , ι := γ ◦ | D k ( X ) , and(7.2) ¯ ι = ¯ ι kX : D k ( X ) → Γ kc ( X ) , ¯ ι := ¯ γ ◦ ¯ | D k ( X ) , It follows from the respective definitions of sequential convergence that ι and ¯ ι aresequentially continuous. Theorem 7.1.
For each T ∈ D k ( X ) , there exists a unique sequentially continuouslinear map b T : Γ kc ( X ) → R satisfying T = b T ◦ ¯ ι . We use an auxiliary result for the proof of Theorem 7.1. For the next lemma,let T ∈ D k ( X ) be a metric current, ϕ ∈ LIP c ( X ), and U an open set containing K := spt ϕ . We define the ( k + 1)-linear map(7.3) T Uϕ : LIP ∞ ( U ) k +1 → R , ( π , . . . , π k ) T ( ϕπ , e π , . . . , e π k )for any extension e π l ∈ LIP ∞ ( X ) of π l . By (3.1) and the discussion preceding it wehave that the map T U is well-defined and bounded, with the bound(7.4) | T Uϕ ( π , . . . , π k ) | ≤ CL ( ϕ ) L ( π | K ) · · · L ( π k | K )for ( π , . . . , π k ) ∈ LIP ∞ ( U ) k +1 . Lemma 7.2.
The bounded ( k + 1) -linear map T Uϕ : LIP ∞ ( U ) k +1 → R in (7.3)descends to a unique sequentially continuous bounded linear map T Uϕ : Poly k ( U ) → R satisfying T Uϕ ◦ ¯ = T Uϕ .Proof. By the locality properties of currents, we have that T Uϕ ( π , . . . , , . . . , π k ) = 0 , ( π , . . . , , . . . , π k ) ∈ LIP ∞ ( U ) k +1 if one of the functions π l is the constant one for l = 1 , . . . , k ; cf. Definition 4.9.By Proposition 4.10 T Uϕ descends to a bounded linear map T Uϕ : Poly k ( U ) → R ,satisfying T Uϕ = T Uϕ ◦ ¯ , and the sequential continuity of T Uϕ is implied by thesequential continuity of the bounded linear map A : Poly k ( U ) → R for which T Uϕ = A ◦ . To prove sequential continuity of A , suppose that the sequence ( π n ) n convergesto zero in Poly k ( U ). It suffices to prove that each subsequence of { A ( π n ) } n has afurther subsequence converging to zero.Since π n → Poly k ( U ) there are representatives ( π j,n , . . . , π j,nk ) j ∈ Rep( π n )satisfying ∞ X j sup n L ( π j,n ) · · · L ( π j,nk ) < ∞ and lim n →∞ ∞ X j k π j,n | K k ∞ · · · k π j,nk | K k ∞ = 0 . For each j ∈ N , we denote t j := sup n L ( π j,n ) · · · L ( π j,nk ). We may assume L ( π j,nl ) = 0for all l = 0 , . . . , k and j, n ∈ N . For each j, n ∈ N and l = 0 , . . . , k , let σ j,nl = a j,nl π j,nl , where a j,nl = (cid:0) L ( π j,n ) · · · L ( π j,nk ) (cid:1) / ( k +1) L ( π j,nl ) . Then σ j,n ⊗ · · · ⊗ σ j,nk = π j,n ⊗ · · · ⊗ π j,nk ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 23 which implies T ( ϕπ j,n , π j,n , . . . , π j,nk ) = A ( π j,n ⊗ · · · ⊗ π j,nk ) = A ( σ j,n ⊗ · · · ⊗ σ j,nk )= T ( ϕσ j,n , σ j,n , . . . , σ j,nk ) . (7.5)For each j, n ∈ N and l = 0 , . . . , k , we have L ( σ j,nl ) = [ L ( π j,n ) · · · L ( π j,nk )] / ( k +1) ≤ t / ( k +1) j . By the Arzela-Ascoli theorem and a diagonal argument, there exists a subsequenceand σ jl ∈ LIP ∞ ( U ) for which σ j,nl → σ jl in LIP ∞ ( U ) as n → ∞ , for each j ∈ N and l = 0 , . . . , k . Thus(7.6) lim n →∞ T ( ϕσ j,n , σ j,n , . . . , σ j,nk ) = T ( ϕσ j , σ j , . . . , σ jk ) . Moreover, for fixed j ∈ N , we have thatlim n →∞ k σ j,n | K k ∞ · · · k σ j,nk | K k ∞ = lim n →∞ k π j,n | K k ∞ · · · k π j,nk | K k ∞ = 0and k σ j,nl | K k ∞ = a j,nl k π j,nl | K k ∞ ≤ t / ( k +1) j for all n and l = 0 , . . . , k. It follows that, up to passing to a further subsequence, there is, for each j ∈ N , anindex l = 0 , . . . , k for which lim n →∞ k σ j,nl | K k ∞ = 0. Consequently, for each j ∈ N ,there exists l = 0 , . . . , k for which(7.7) σ jl | K = 0 . The locality of T together with (7.5),(7.6), and (7.7) now implies that(7.8) lim n →∞ T ( ϕπ j,n , π j,n , . . . , π j,nk ) = 0for each j ∈ N . The estimate (7.4) yields | T ( ϕπ j,n , π j,n , . . . , π j,nk ) | ≤ Ct j for each j ∈ N and for some constant C >
0. The dominated convergence theoremnow implies thatlim n →∞ A ( π n ) = lim n →∞ ∞ X j T ( π j,n , . . . , π j,nk ) = ∞ X j lim n →∞ T ( π j,n , . . . , π j,nk ) = 0 . This concludes the proof. (cid:3)
Remark 7.3.
By the multilinearity of currents and the uniqueness in Lemma 7.2,we observe the following functorial property: If ϕ, ψ ∈ LIP c ( X ) , U ⊂ V are opensets, and spt ϕ ∪ spt ψ ⊂ U , we have that T Uϕ + ψ = T Uϕ + T Uψ and T Vϕ = T Uϕ ◦ ¯ ρ U,V for each T ∈ D k ( X ) .Proof of Theorem 7.1. Let T ∈ D k ( X ). We define b T : Γ kc ( X ) → V as follows. Let ω ∈ Γ kc ( X ) be overlap-compatible with { ¯ π U ∈ Poly k ( U ) } U ∈U for a locally finiteprecompact open cover; cf. Remark 5.3. Let { ϕ U } U be a Lipschitz partition ofunity subordinate to U and set(7.9) b T ( ω ) := X U ∈U T Uϕ U (¯ π U ) , where T Uϕ U is as in Lemma 7.2. Note that, since spt ω is compact, only finitely many π U are nonzero and the sum has only finitely many nonzero terms.We prove that b T is well-defined. Let { ¯ σ V } V ∈V be overlap-compatible with ω ,and { ψ V } V a Lipschitz partition of unity subordinate to V . Let W be a locallyfinite refinement of U ∩ V = { U ∩ V : U ∈ U , V ∈ V} with the property that,whenever U ∈ U , V ∈ V , W ∈ W , and W ⊂ U ∩ V , we have that¯ ρ W,U (¯ π U ) = ¯ ρ W,V (¯ σ V );cf. Remark 5.3 and the discussion after it. In particular(7.10) ¯ ρ U ∩ V ∩ W,U (¯ π U ) = ¯ ρ U ∩ V ∩ W,V (¯ σ V )for all U ∈ U , V ∈ V and W ∈ W .Let { θ W } W be a Lipschitz partition of unity subordinate to W . By the functorialproperties in Remark 7.3, and (7.10), we have that X U ∈U T Uϕ U ( π U ) = X U ∈U X V ∈V X W ∈W T Uϕ U ψ V θ W (¯ π U )= X U ∈U X V ∈V X W ∈W T U ∩ V ∩ Wϕ U ψ V θ W (¯ ρ U ∩ V ∩ W,U (¯ π U ))= X U ∈U X V ∈V X W ∈W T U ∩ V ∩ Wϕ U ψ V θ W (¯ ρ U ∩ V ∩ W,V (¯ σ V ))= X V ∈V T Vϕ V (¯ σ V ) . Note that all the sums above have only finitely many non-zero summands. Thisshows that b T is well-defined.To prove that b T : Γ kc ( X ) → R is sequentially continuous, let ω n → kc ( X ).By Definition 5.7 there is a compact set K ⊂ X , a locally finite precompact opencover U and { ¯ π nU } U overlap-compatible with ω n , for each n ∈ N , so that spt ω n ⊂ K for each n ∈ N and ¯ π nU → Poly k ( U )as n → ∞ , for each U ∈ U .Since K is compact and U is locally finite, the collection U K := { U ∈ U : U ∩ K = ∅ } is finite. Moreover, for each n ∈ N and U / ∈ U K , we have that ¯ π nU = 0, sinceotherwise ω n ( x ) = [¯ π nU ] x = 0 for some x / ∈ K . It follows by Lemma 7.2 thatlim n →∞ b T ( ω n ) = X U ∈U K lim n →∞ T Uϕ U (¯ π nU ) = 0 . To prove the factorization, suppose ( π , . . . , π k ) ∈ D k ( X ) and { ϕ U } U is a Lip-schitz partition of unity subordinate to a locally finite precompact open cover U .Then { ¯ π U := π | U ⊗ π | U ⊗ · · · ⊗ π k | U } U is overlap-compatible with ¯ ι ( π , . . . , π k ).Proposition 7.2 and (7.3) now imply that b T (¯ ı ( π , . . . , π k )) = X U ∈U T Uϕ U (¯ π U ) = X U ∈U T ( ϕ U π , π , . . . , π k ) = T ( π , . . . , π k ) . For uniqueness, let A : Γ kc ( X ) → R be linear and sequentially continuous, and A ◦ ¯ ι = T . Let ω ∈ Γ kc ( X ) be overlap-compatible with { ¯ π U } U , and let { ϕ U } U be a ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 25
Lipschitz partition of unity subordinate to U . We may assume that ¯ π U ∈ Poly k ( X )for each U ∈ U by Lemma 4.4 and the surjectivity of 4.10.Note that ϕ U ⌣ ω = ¯ γ ( ϕ U ⌣ ¯ π U ) and, by linearity, A ( ω ) = X U ∈U A ( ϕ U ⌣ ω ) = X U ∈U A (¯ γ ( ϕ U ⌣ ¯ π U )) . For each U ∈ U , the linear map A U : Poly k ( U ) → R , A U = A (¯ γ ( ϕ U ⌣ · ))is bounded and satisfies A U ◦ ¯ ( π , . . . , π k ) = A (¯ γ (( ϕ U π ) ⊗ ¯ π ⊗ · · · ⊗ ¯ π k ))= A (¯ γ ◦ ¯ ( ϕ U π , π , . . . , π k )) = T ( ϕ U π , π , . . . , π k )for ( π , . . . , π k ) ∈ D k ( X ); cf. (7.1). The uniqueness in Lemma 7.2 implies that A U = T Uϕ U . Hence A ( ω ) = X U ∈U A (¯ γ ( ϕ U ⌣ ¯ π U )) = X U ∈U T Uϕ U (¯ π U ) = b T ( ω ) . The proof is complete. (cid:3)
The next proposition establishes (1.2).
Proposition 7.4.
Let T ∈ D k ( X ) be a k -current on X . Let b T : Γ kc ( X ) → R and c ∂T : Γ k − c ( X ) → R be extensions of T and ∂T , respectively. Then we have c ∂T ( ω ) = b T ( ¯ dω ) for each ω ∈ Γ k − c ( X ) .Proof. Since ¯ d is sequentially continuous it follows that b T ◦ ¯ d defines an element inΓ k − c ( X ) ∗ . Since the extension in Theorem 7.1 is unique, it suffices to show that b T ( ¯ d (¯ ι ( π , . . . , π k − )) = ∂T ( π , . . . , π k − )for ( π , . . . , π k − ) ∈ D k − ( X ).The expression (6.5) shows that, for each open U ⊂ X , the l th term in the sum(6.5) is constant the x l -variable, where l = 1 , . . . , k , and thus belongs to ker Q kU .Therefore Q kU ◦ d ( U ( π | U , . . . , π k − | U )) = Q k +1 U ◦ U (1 , π , . . . , π k − ) . It follows that ¯ d U (¯ ( π , . . . , π k − )) = ¯ U (1 , π , . . . , π k − ). Since U is arbitrary wehave ¯ d (¯ ι ( π , . . . , π k − )) = ¯ ι (1 , π , . . . , π k − ) = ¯ ι ( ϕ, π , . . . , π k − )for any ϕ ∈ LIP c ( X ) which is 1 on a neighborhood of spt π . This implies that b T ◦ ¯ d (¯ ι ( π , . . . , π k )) = b T (¯ ι ( ϕ, π , . . . , π k )) = T ( ϕ, π , . . . , π k ) = ∂T ( π , . . . , π k ) . The claim follows. (cid:3)
Pre-dual for metric currents – Proof of Theorem 1.1.
Before provingTheorem 1.1, we record he following locality property of ¯ ι . Note that the claim ofLemma 7.5 is not true for ι , and thus Γ kc ( X ) is not a pre-dual for metric currents. Lemma 7.5.
Let ( π , . . . , π k ) ∈ D k ( X ) . If π l is constant on a neighborhood of spt π for some l = 1 , . . . , k then ¯ ι ( π , . . . , π k ) = 0 .Proof. If x ∈ spt π and U is a neighborhood of x so that π l is constant on U , then Poly k ( U ) ∋ π | U ⊗ π | U ⊗· · ·⊗ π k | U = 0. Thus ¯ ι ( π , . . . , π k ) = [ π ⊗ ¯ π ⊗· · ·⊗ ¯ π k ] x = 0If x / ∈ spt π , then there is a neighborhood V of x on which π vanishes, so that Poly k ( V ) ∋ π | V ⊗ · · · ⊗ π k | V = 0. (cid:3) Proof of Theorem 1.1.
By Theorem 7.1 we have a linear mapΞ : D k ( X ) → Γ kc ( X ) ∗ , T b T .
The uniqueness of the extension implies injectivity Ξ. Indeed, the only current T ∈ D k ( X ) for which b T = Ξ( T ) = 0 is the zero current T = 0.If b T ∈ Γ kc ( X ) ∗ , then T := b T ◦ ¯ ı : D k ( X ) → R defines a metric k -current. Indeed,( k +1)-linearity and sequential continuity are clear, and locality follows from Lemma7.5. Clearly Ξ( T ) = b T and thus we have shown the surjectivity of Ξ.The identity (1.2) is proven in Proposition 7.4. It remains to prove the sequentialcontinuity of Ξ. By linearity, it suffices to show that, if T i → D k ( X ), then b T i → kc ( X ) ∗ .Let ω ∈ Γ kc ( X ) and suppose { ¯ π U } U ∈U is overlap-compatible with ω . Let { ϕ U } U ∈U be a Lipschitz partition of unity subordinate to U . Since ω is compactly supported,there is a compact set K ⊂ X containing every set U ∈ U for which ¯ π U = 0. Thecollection of these elements of U is a finite set and we denote it by U K .For each U ∈ U K let π U ∈ Q − U (¯ π U ) and fix a representation ( π j,U , . . . , π j,Uk ) j ∈ Rep( π U ). We have b T i ( ω ) = X U ∈U K ( T i ) Uϕ U (¯ π U ) = X U ∈U K ∞ X j T i ( ϕ U π j,U , π j,U , . . . , π j,Uk ) . Since lim i →∞ ( T i ) Uϕ U ( π , . . . , π k ) = 0 for every ( π , . . . , π k ) ∈ LIP ∞ ( U ) k +1 thebound (7.4) and the multilinear uniform boundedness principle, cf. [11, Theorem1] and [13, Theorem 1], implies that there is a constant C > | T i ( ϕ U π , π , . . . , π k ) | ≤ CL ( π ) · · · L ( π k )for all i ∈ N and every ( π , . . . , π k ) ∈ LIP ∞ ( U ) k +1 . By the dominated convergencetheorem, we have thatlim i →∞ b T i ( ω ) = X U ∈V ∞ X j lim i →∞ T i ( ϕ U π j,U , π j,U , . . . , π j,Uk ) = 0 . The proof is complete. (cid:3) Currents of locally finite mass: extension topartition-continuous polylipschitz forms
In this section we show that currents of locally finite mass admit a further ex-tension to partition-continuous sections. We introduce the following notation: for
ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 27 a subset E ⊂ X and ¯ π ∈ Poly k ( X ), set L k (¯ π ; E ) := L k ( ρ E,X (¯ π ))andLip k (¯ π ; E ) = inf sup x ∈ E ∞ X j | π j ( x ) | Lip( π j | E ) · · · Lip( π j | E ) : ( π j , . . . , π jk ) ∈ Rep( π ) for any π ∈ Q − (¯ π ). This is clearly independent of the choice of π and the estimateLip k (¯ π ; E ) ≤ L k (¯ π ; E )holds. Definition 8.1.
For ω ∈ G k ( X ) and x ∈ X , the pointwise norm k ω k x of ω at x is k ω k x = lim r → Lip k (¯ π ; B r ( x )) , where ¯ π ∈ Poly k ( U ) satisfies [¯ π ] x = ω ( x ) and U is a neighborhood of x . A simple argument using a representation of ω shows that, for each ω ∈ Γ k ( X ),the function x
7→ k ω k x is upper semicontinuous.We have, for π = ( π , . . . , π k ) ∈ D k ( X ), that k ¯ ι ( π ) k x ≤ | π ( x ) | Lip π ( x ) · · · Lip π k ( x )for all x ∈ X .8.1. Partition continuous polylipschitz forms and their convergence.
Let E ⊂ X be a Borel set, and define G kE ( X ) := ρ − E (Γ k ( E )) . Given ω ∈ G kE ( X ), a collection { ¯ π U } U is said to be overlap-compatible with ω in E , if U is a locally finite precompact open cover of E and { ¯ ρ E ∩ U,U (¯ π U ) } U isoverlap-compatible with ω | E . Set G kE,c ( X ) = G kE ( X ) ∩ G kc ( X ) . Definition 8.2.
Let E be a countable Borel partition of X . A section ω ∈ G k ( X ) is called E -continuous if there is a locally finite precompact open cover U of X anda collection { ¯ π U ∈ Poly k ( U ) } U so that (1) { ¯ π U } U is overlap-compatible with ω in E , for every E ∈ E ; (2) sup E ∈E L k ( π U ; E ∩ U ) = C U < ∞ for every U ∈ U .Given E and { ¯ π U } U satisfying (1) and (2) above we say that { π U } U represents ω with respect to E .A section ω ∈ G k ( X ) is called partition-continuous if it is E -continuous forsome countable Borel partition of X . We also call a partition E , for which ω is E -continuous, an admissible partition for ω ∈ G k ( X ).We denote Γ k pc ( X ) the vector space of of partition-continuous forms, and setΓ k pc , c ( X ) := G kc ( X ) ∩ Γ k pc ( X ). Definition 8.3.
A sequence ( ω n ) in Γ k pc , c ( X ) converges to ω ∈ Γ k pc , c ( X ) , denoted ω n → ω in Γ k pc , c ( X ) , if there is a compact set K ⊂ X , a countable Borel partition E ,a locally finite precompact open cover U and collections { ¯ π nU } U representing ω n − ω with respect to E , n ∈ N , so that (1) spt( ω n − ω ) ⊂ K , for each n ∈ N , (2) ρ E ∩ U,U (¯ π n ) → in Poly k ( E ∩ U ) for all E ∈ E and U ∈ U , and (3) sup n,E L k (¯ π n ; E ∩ U ) = C K < ∞ for each U ∈ U K := { U ∈ U : U ∩ K = ∅ } . Note that the inclusion ı pc : Γ kc ( X ) ֒ → Γ k pc , c ( X ) is sequentially continuous. Wedenote by ι pc := ı pc ◦ ι : D k ( X ) → Γ k pc , c ( X )the natural inclusion.8.2. Mass bounds for restrictions of currents to Borel sets.
In this sectionwe prove Theorem 1.3. In fact the bound (1.3) in Theorem 1.3 follows directly fromthe more technical statement (8.3) in Proposition 8.4. We begin by discussing avariant of Lemma 7.2 for currents of locally finite mass and their restrictions toBorel sets.Let T ∈ M k, loc ( X ) and E ⊂ X be a Borel set. If ϕ ∈ LIP ∞ ( E ) is the re-striction to E of a compactly supported function (equivalently, if spt ϕ is a totallybounded set) and U ⊂ X is an open set with spt ϕ ⊂ U , consider the map ( T ⌊ E ) Uϕ in (7.3). By the locality properties of currents the value ( T ⌊ E ) Uϕ ( π , . . . , π k ) for( π , . . . , π k ) ∈ LIP ∞ ( U ) k +1 depends only on π | E ∩ U , . . . , π k | E ∩ U . Thus we get a( k + 1)-linear map T E ∩ Uϕ : LIP ∞ ( E ∩ U ) k +1 → R . By Lemma 3.3, the map T E ∩ Uϕ satisfies the bound(8.1) | T E ∩ Uϕ ( π , . . . , π k ) | ≤ Lip( π | E ∩ U ) · · · Lip( π k | E ∩ U ) Z E ∩ U | ϕ || π | d k T k . In particular it is bounded and satisfies (4.11). We denote by T E ∩ Uϕ : Poly k ( E ∩ U ) → R the sequentially continuous linear map for which T E ∩ Uϕ = T E ∩ Uϕ ◦ ¯ E ∩ U ; cf. Lemma7.2. Note that the statements in Remark 7.3 remain true for all Borel sets U ⊂ V ⊂ X .Given T ∈ M k, loc ( X ) and a Borel set E ⊂ X , we define b T E : G kE,c ( X ) → R by(8.2) b T E ( ω ) := X U ∈U T E ∩ Uϕ U (¯ ρ E ∩ U,U (¯ π U )) , whenever ω ∈ G kE,c ( X ) is represented by { π U } U in E , and { ϕ U } is a Lipschitzpartition of unity (in E ) subordinate to U . Proposition 8.4.
Let T ∈ M k, loc ( X ) and let E ⊂ X be a Borel set. Then b T E iswell-defined, linear and satisfies T ⌊ E = b T E ◦ ¯ ι . Moreover, (8.3) | b T E ( ω ) | ≤ Z E k ω | E k x d k T k ( x ) , for each ω ∈ G kE,c ( X ) . ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 29
Proof.
Arguing as in the proof of Theorem 7.1, we see that b T E is well-defined.Linearity is clear and the factorization follows from the identities T E ∩ Uϕ U = T E ∩ Uϕ U ◦ ¯ E ∩ U as in the proof of Theorem 7.1.Next we prove the bound in the claim. Let ϕ ∈ Lip ∞ ( E ) and spt ϕ ⊂ U aprecompact open set. For π ∈ Poly k ( X ), π ∈ Q − (¯ π ) and ( π j , . . . , π jk ) j ∈ Rep( π ),we have T E ∩ Uϕ (¯ π ) = ∞ X j T E ∩ Uϕ ( π j , . . . , π jk ) = ∞ X j T ( χ E ϕπ j , π k , . . . , π jk )and, by (8.1), the estimate | T E ∩ Uϕ (¯ π ) | ≤ ∞ X j Lip( π j | E ∩ spt ϕ ) · · · Lip( π jk | E ∩ spt ϕ ) Z E | ϕπ j | d k T k≤ sup x ∈ E ∩ spt ϕ ∞ X j Lip( π j | E ∩ spt ϕ ) · · · Lip( π jk | E ∩ spt ϕ ) | π j ( x ) | Z E | ϕ | d k T k , yielding(8.4) | T E ∩ Uϕ (¯ π ) | ≤ Lip k (¯ π ; E ∩ spt ϕ ) Z E | ϕ | d k T k . Let ω ∈ G kE,c ( X ). We extend ω | E : E → Poly k ( E ) as the zero map ω | E : X → Poly k ( E ) ⊔ Poly k ( X \ E )outside E , and thus k ω | E k x = 0 if x / ∈ E . Let K := spt ω and let g : X → R be asimple Borel function satisfying k ω | E k ≤ g. We may assume that g = m X l a l χ A l , where m ∈ N and { A , . . . , A m } is a Borel partition of K .Let ε >
0. Since k T k is a Radon measure there is, for each l , an open set U l ⊃ A l ,satisfying k T k ( U l ) < k T k ( A l ) + ε. We construct a collection overlap-compatible with ω in E as follows: for x ∈ K there exists a unique l = 0 , . . . , m for which x ∈ A l . Fix a radius r x > B x := B r x ( x ) ⊂ U l is precompact. Let π x ∈ Poly k ( B x ) satisfy[¯ ρ E ∩ B x ,B x ( π x )] x = ω | E ( x ) and Lip k (¯ π x ; E ∩ B x ) ≤ a l + ε, if x ∈ E , and π x = 0 if x ∈ K \ E . For x / ∈ K let B x ⊂ X \ K be a precompactball around x and set π x = 0.By Lemma 5.1, we may pass to a locally finite refinement U and a collection π U := ρ U,B x ( π x ) ∈ Poly k ( U ) , where U ⊂ B x so that { ¯ ρ E ∩ U,U (¯ π U ) } U is overlap-compatible with ω | E . Note that ρ E ∩ U,U (¯ π U ) = 0if U / ∈ U K := { U ∈ U : U ∩ K = ∅ } , since otherwise there would be x ∈ E \ K forwhich ω ( x ) = [ ρ E ∩ U,U (¯ π U )] x = 0. Let { ϕ U } be a Lipschitz partition of unity subordinate to U (in E ). By thedefinition of b T E and (8.4) we have | b T E ( ω ) | ≤ X U ∈U K | T E ∩ Uϕ U (¯ π U ) | ≤ X U ∈U K Lip k ( π U ; E ∩ U ) Z E ϕ U d k T k . We may express the collection U K as U K = m [ l =0 U l , where U l = { U ∈ U : U ⊂ B x with x ∈ A l } . Thus | b T E ( ω ) | ≤ m X l X U ∈U l Lip k ( π j ; E ∩ U ) Z E ϕ j d k T k≤ m X l ( a l + ε ) Z E X U ∈U l ϕ U ! d k T k≤ m X l ( a l + ε ) Z E χ U l d k T k ≤ m X l ( a l + ε ) k T k ( U l ∩ E ) . By the choice of the open sets U l , we have that k T k ( U l ∩ E ) + k T k ( U l \ E ) < k T k ( A l ∩ E ) + k T k ( A l \ E ) + ε< k T k ( A l ∩ E ) + k T k ( U l \ E ) + ε. Thus k T k ( U l ∩ E ) < k T k ( A l ∩ E ) + ε for each l ≤ m . Since ε > | b T E ( ω ) | ≤ m X l a l k T k ( A l ∩ E ) = Z E g d k T k . By taking infimum over all simple functions g satisfying k ω | E k ≤ g , we obtain theclaim. (cid:3) Proposition 8.5.
Let T ∈ D k ( X ) . The map b T E : G kE,c ( X ) → R in (8.2) is uniqueamong linear maps satisfying (8.3) .Proof. Let A : G kE,c ( X ) → R be a linear map such that A ◦ ¯ ι = T ⌊ E and A satisfies(8.3). We observe that, by (8.3), the value A ( ω ) depends only on ω | E .Let ω ∈ Γ kE,c ( X ) and suppose { ¯ π U } U is overlap-compatible with ω in E , and let { ϕ U } be a Lipschitz partition of unity in E subordinate to U . For each U ∈ U notethat ϕ U ⌣ ω | E = ¯ γ E ( ϕ U ⌣ ¯ ρ E ∩ U,U (¯ π U )). Consider the multilinear map A U : LIP ∞ ( U ∩ E ) k +1 → R , A U = A (¯ γ E ( ϕ U ⌣ E ∩ U ( · )))where ¯ γ E : Poly k ( E ) → Γ k ( E ) is the canonical map in (5.7). By (8.3), A U isbounded. As in the proof of Theorem 7.1 we see that A U = T E ∩ Uϕ U . Thus T E ∩ Uϕ U = A U = A (¯ γ E ( ϕ U ⌣ ¯ ρ E ∩ U,U ( · ))) : Poly k ( E ∩ U ) → R ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 31
It follows that A ( ω ) = X U ∈U A ( ϕ U ⌣ ω ) = X U ∈U A (¯ γ E ( ϕ U ⌣ ¯ ρ E ∩ U,U (¯ π U )))= X U ∈U T E ∩ Uϕ U (¯ π U ) = b T E ( ω ) . (cid:3) Extending currents of locally finite mass.
The remainder of this section isdevoted to the proof of Theorem 1.4. The existence and uniqueness of the extensionis proved in Proposition 8.6 below.
Proposition 8.6.
Let T ∈ M k, loc ( X ) and let b T : Γ k pc , c ( X ) → R be the linear map ω X E ∈E b T E ( ω ) whenever E is an admissible partition for ω . Then b T is well-defined. Moreover, b T is the unique sequentially continuous linear map Γ kpc,c ( X ) → R satisfying b T ◦ ¯ ι pc = T. Proof. If ϕ ∈ LIP c ( X ), spt ϕ ⊂ U is open, and E , E ⊂ X are disjoint Borel sets E = E ∪ E , the identity T ⌊ E = T ⌊ E + T ⌊ E implies that T E ∩ Uϕ = T E ∩ Uϕ ◦ ¯ ρ E ∩ U,E ∩ U + T E ∩ Uϕ ◦ ¯ ρ E ∩ U,E ∩ U . This and the estimate (8.4) can be used to show that, if E , E , . . . is a Borelpartition of E , we have b T E ( ω ) = ∞ X i b T E i ( ω ) , ω ∈ G kE,c ( X ) . Note that by (8.3) the sum above is absolutely convergent. The well-definedness of b T : Γ k pc , c ( X ) → R follows easily from this.The factorization b T ◦ ι pc = T follows immediately from the observation that b T | Γ kc ( X ) = e T , where e T denotes the extension given by Theorem 7.1. Uniquenessfollows from Proposition 8.5 and (8.3).Next we prove that b T is sequentially continuous. Suppose ω n → ω in Γ k pc , c ( X ),and let K ⊂ X , E and { ¯ π n } U be as in Definition 8.3. Denote U K := { U ∈ U : U ∩ K = ∅ } and set C K := sup n,E,U ∈U K ¯ L k ( π n ; E ∩ U ) < ∞ . Let { ϕ U } be a Lipschitz partition of unity subordinate to U . For each n ∈ N , wehave b T ( ω n ) − b T ( ω ) = b T ( ω n − ω ) = X E ∈E X U ∈U T E ∩ Uϕ U (¯ π n ) = X E ∈E X U ∈U K T E ∩ Uϕ U (¯ π n ) . Since | T E ∩ Uϕ U (¯ π n ) | ≤ Lip k (¯ π n ; E ∩ U ) Z E ϕ U d k T k ≤ C K Z E ϕ U d k T k we may apply the dominated convergence theorem to conclude thatlim n →∞ b T ( ω n − ω ) = X E ∈E X U ∈U K lim n →∞ T E ∩ Uϕ U (¯ π n ) = 0 . The claim follows. (cid:3)
Boundaries of normal currents.
We finish the proof of Theorem 1.4 byshowing the validity of (1.4) in Corollary 8.8.
Proposition 8.7.
The differentials d : G kc ( X ) → G k +1 c ( X ) and ¯ d : G kc ( X ) → G k +1 c ( X ) restrict to sequentially continuous linear maps d : Γ k pc , c ( X ) → Γ k +1pc ,c ( X ) and ¯ d : Γ k pc , c ( X ) → Γ k +1pc ,c ( X ) . Proof.
We prove the statement for ¯ d . The other case is similar. Let ω ∈ Γ k pc , c ( X )and let E , { ¯ π U } U , and C U be as in Definition 8.2. Since ¯ d ( ω | E ) = ( ¯ dω ) | E , RemarkA.1 implies that { ¯ d ¯ π U } U is overlap-compatible with ¯ dω in E for every E ∈ E .Moreover by Lemma 6.1 we have¯ L k +1 ( ¯ d ¯ π U ; E ∩ U ) ≤ C U ( k + 2)for every E ∈ E . Thus ¯ dω ∈ Γ k +1pc ,c ( X ).To see sequential continuity, let ω n → ω in Γ k pc , c ( X ), and let E , K , C K , and { ¯ π n } U be as in Definition 8.3. Since ρ E ∩ U,U (¯ π n ) → Poly k ( E ∩ U ) and ¯ d E ∩ U issequentially continuous it follows that ¯ d E ∩ U ( ρ E ∩ U,U (¯ π n )) → Poly k +1 ( E ∩ U ).Thus ¯ d ( ω n − ω ) → k +1pc ,c ( X ) and the proof of the proposition is complete. (cid:3) Proposition 8.7 and the uniqueness in Proposition 8.6 immediately yield thefollowing corollary.
Corollary 8.8.
Let T ∈ N k, loc ( X ) be a locally normal k -current on X . Let b T :Γ k pc , c ( X ) → R and c ∂T : Γ k − ,c ( X ) → R be extensions of T and ∂T , respectively.Then we have c ∂T ( ω ) = b T ( dω ) for each ω ∈ Γ k − ,c ( X ) . Remark 8.9.
Corollary 8.8 implies that ∂ b T : Γ k − ,c ( X ) → R , ω b T ( dω ) , coincideswith the extension c ∂T : Γ k − ,c ( X ) → R of ∂T : D k ( X ) → R to partition-continuouspolylipschitz forms. Thus the use of the symbol ∂ is unambiguous. Final remarks
We briefly discuss polylipschitz sections in connection with duality, and antisym-metrization on polylipschitz forms.
ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 33
Extending currents to polylipschitz sections.
Define the space Γ k pc , c ( X )and sequential convergence in Γ k pc , c ( X ) as in Definitions 8.2 and 8.3. It is straight-forward to check that the map Q : G k ( X ) → G k ( X ) in (5.9) restricts to a sequen-tially continuous linear map Q : Γ k pc , c ( X ) → Γ kpc,c ( X ) . We record the following theorem for extensions of currents to polylipschitz sections.The claims follow directly from Theorems 1.4 and 1.3 together with the fact that¯ d ◦ Q = Q ◦ d , cf. Proposition 5.6. Theorem 9.1.
Suppose T ∈ M k, loc ( X ) , let b T be the unique extension given byProposition 8.6, and e T := b T ◦ Q . Then e T : Γ k pc , c ( X ) → R linear, sequentiallycontinuous and satisfies e T ◦ ι = T .Moreover, for T ∈ M k, loc ( X ) and T ′ ∈ N k +1 , loc ( X ) , the identities g ∂T ′ ( ω ) = e T ′ ( dω ) and | e T ( ω ) | ≤ Z X k ω k x d k T k ( x ) hold for all ω ∈ Γ k pc , c ( X ) . Alternating polylipschitz forms and metric currents.
In [1], Ambrosioand Kirchheim point out that the other axioms of metric currents imply that ametric k -current T on space X is alternating in the sense that T ( π , π σ (1) , . . . , π σ ( k ) ) = sign( σ ) T ( π , π , . . . , π k )for all π = ( π , . . . , π k ) ∈ D k ( X ) and permutations σ of { , . . . , k } .Taking into account the particular role of the function π in the ( k + 1)-tuple( π , . . . , π k ) in the definition of a k -current, we use this property of metric currentsto define an antisymmetrization operator Alt = Alt X : Poly k ( X ) → Poly k ( X ) byAlt( π )( x , . . . , x k ) = 1 k ! X σ sign( σ ) π ( x , x σ (1) , . . . , x σ ( k ) )for π ∈ Poly k ( X ) and ( x , . . . , x k ) ∈ X k +1 . This map descends to a linear mapAlt : Poly k ( X ) → Poly k ( X ) satisfying Q ◦ Alt = Alt ◦ Q. We call the images Alt(
Poly k ( X )) and Alt( Poly k ( X )) alternating polylipschitz func-tions and alternating homogeneous polylipschitz functions , respectively.Continuous sections of the ´etal´e space associated to the corresponding presheavesgives rise to alternating polylipschitz sections and forms, Γ k Alt ( X ) and Γ k Alt ( X ), re-spectively. Since the exterior derivatives d and ¯ d preserve the property of beingalternating on (homogeneous) polylipschitz functions, they induce exterior deriva-tives d : Γ k Alt ( X ) → Γ k +1Alt ( X ) , ¯ d : Γ k Alt ( X ) → Γ k +1Alt ( X ) . Since classical differential k -forms on a manifold may be viewed either as sec-tions of the k th exterior bundle or as sections of the bundle of alternating k -linearfunctions, we observe that alternating polylipschitz forms on a metric space areanalogous to the latter. It is now straightforward to check, using the observation of Ambrosio and Kirch-heim, that for each metric k -current T , we have e T = e T ◦ Alt and b T = b T ◦ Alt , where Alt and Alt are the linear maps associated to the presheaf homomorphisms { Alt U } and { Alt U } . Appendix A. Cohomorphisms and their associated linear maps
In this appendix, we define cohomomorphisms between presheaves and describea general construction yielding a linear map associated to a given cohomomorphism.Let f : X → Y be a continuous map between paracompact Hausdorff spacesand let A = { A ( U ); ρ AU,V } U and B = { B ( U ); ρ BU,V } U be presheaves on X and Y ,respectively. A collection { ϕ U : B ( U ) → A ( f − U ) } U of linear maps for each open U ⊂ Y , satisfying(A.1) ϕ U ◦ ρ BU,V = ρ Af − U,f − V ◦ ϕ V whenever U ⊂ V, is called an f -cohomomorphism of presheaves ; cf. [2, Chapter I.4]. For f = id : X → X , condition (A.1) becomes (5.2) and thus id-cohomomorphisms are simplypresheaf homomorphisms.An f -cohomomorphism ϕ : B → A between presheaves induces a natural linearmap ϕ ∗ : G ( B ( Y )) → G ( A ( X )) , the linear map (on sections) associated to ϕ . Given a global section ω : Y → B ( Y ),the section ϕ ∗ ( ω ) : X → A ( X ) is defined as follows: for x ∈ X ,(A.2) ϕ ∗ ( ω )( x ) := [ ϕ U ( g U )] x , where U is a neighborhood of f ( x ) and g U ∈ B ( U ) satisfies ω ( f ( x )) = [ g U ] f ( x ) .To see that ϕ ∗ ( ω )( x ) is well-defined, suppose that ω ( f ( x )) = [ g U ] f ( x ) = [ g ′ V ] f ( x ) ,i.e., that there is a neighborhood D ⊂ U ∩ V of f ( x ) for which ρ BD,U ( g U ) = ρ BD,V ( g ′ V ) . By (A.1) we have that ρ Af − D,f − U ( ϕ U ( g U )) = ϕ D ( ρ BD,U ( g U )) = ϕ D ( ρ BD,V ( g ′ V )) = ρ Af − D,f − V ( ϕ V ( g ′ V ));in particular [ ϕ U ( g U )] x = [ ϕ V ( g ′ V )] x . Remark A.1.
Let ω ∈ G ( B ( Y )) be compatible with { g U } U . Suppose U V satisfies V = f − U V for each V ∈ f − U . Then, by (A.1) and the fact that { g U } U iscompatible ω , we have that the collection { ϕ V ( g U V ) } f − U is compatible with ϕ ∗ ω .If ω ∈ Γ( B ( Y )) and { g U } U represents ω , then { ϕ V ( g U V ) } f − U represents ω , andif { g U } U satisfies the overlap condition (5.5) then { ϕ V ( g U V ) } f − U also satisfies theoverlap condition. We collect some fundamental properties of linear maps associated to cohomor-phisms in the next proposition.
ETRIC CURRENTS AND POLYLIPSCHITZ FORMS 35
Proposition A.2.
Let f : X → Y and g : Y → Z be continuous maps betweenparacompact Hausdorff spaces and let A = { A ( U ) } U , B = { B ( U ) } U and C = { C ( U ) } U be presheaves on X, Y and Z respectively. Suppose ϕ = { ϕ U : B ( U ) → A ( f − U ) } U , ϕ ′ = { ϕ ′ U : B ( U ) → A ( f − U ) } U are f -cohomomorphisms, and ψ = { ψ U : C ( U ) → B ( g − U ) } U is an g -cohomomorphism. (1) For ω ∈ G ( B ( Y )) we have spt( ϕ ∗ ω ) ⊂ f − (spt( ω )) . (2) The associated linear map ϕ ∗ : G ( B ( Y )) → G ( A ( X )) satisfies ϕ ∗ (Γ( B ( Y ))) ⊂ Γ( A ( X )) . (3) The collection { ϕ U + ϕ ′ U : B ( U ) → A ( f − U ) } U is an f -cohomomorphismand ( ϕ + ϕ ′ ) ∗ = ϕ ∗ + ϕ ′∗ : G ( B ( Y )) → G ( A ( X )) . (4) The collection { ϕ g − U ◦ ψ U : C ( U ) → A (( g ◦ f ) − U ) } U is an ( g ◦ f ) -cohomomorphism and ( ϕ ◦ ψ ) ∗ = ϕ ∗ ◦ ψ ∗ : G ( C ( Z )) → G ( A ( X )) . Remark A.3.
Given presheaves A = { A ( U ) } U on X and B = { B ( U ) } , B = { B ( U ) } on Y, and bilinear maps { ϕ U ; B ( U ) × B ( U ) → A ( f − U ) } an analogous constructiongives an associated bilinear map ϕ ∗ : G ( B ( Y )) × G ( B ( Y )) → G ( A ( X )) . The induced bi-linear map satisfies (2) and (3) and also (1’)
For each ( ω, σ ) ∈ G ( B ( Y )) × G ( B ( Y )) , we have spt( ϕ ∗ ( ω, σ )) ⊂ f − (spt ω ∩ spt σ ) . We will need this only for the case id : X → X in the construction of cup products.The details are similar as above and we omit them.Proof of Proposition A.2. The proofs are straightforward and we merely sketchthem.If ϕ ∗ ω ( x ) = 0 then, since ϕ U is linear, (A.2) implies that ω ( f ( x )) = [ g U ] f ( x ) = 0,proving (1). Claim (2) follows directly from Remark A.1.To prove (3) we observe that from (A.2) it is easy to see that, if ϕ ′ : B → A isanother f -cohomorphism between presheaves, then ϕ + ϕ ′ is an f -cohomomorphismand we have ( ϕ + ϕ ′ ) ∗ = ϕ ∗ + ϕ ′∗ . To prove (4), note that condition (A.1) follows for φ ◦ ψ from the fact that itholds for ϕ and ψ . Using (A.2) (and the same notation) we see that( ϕ ◦ ψ ) ∗ ω ( x ) = [ ϕ g − U ( ψ U ( g U ))] x = ϕ ∗ ( ψ ∗ ω )( x ) . (cid:3) Proposition A.2 has the following immediate corollary.
Corollary A.4. If f : X → Y is a proper continuous map and ϕ : B → A an f -cohomomorphism between presheaves B on Y and A on X , then ϕ ∗ ( G c ( B ( Y ))) ⊂ G c ( A ( X )) and ϕ ∗ (Γ c ( B ( Y ))) ⊂ Γ c ( A ( X )) . In particular presheaf homomorphisms always have this property.
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