Metric currents and polylipschitz forms

Abstract

We construct, for a locally compact metric space X, a space of polylipschitz forms ${\overline{\Gamma }}^*_c(X)$, which is a pre-dual for the space of metric currents ${\mathscr {D}}_*(X)$ of Ambrosio and Kirchheim. These polylipschitz forms may be seen as an analog of differential forms in the metric setting.

Appendix A. Cohomorphisms and their associated linear maps

In this appendix, we define cohomomorphisms between presheaves and describe a general construction yielding a linear map associated to a given cohomomorphism.

Let $f:X\rightarrow Y$ be a continuous map between paracompact Hausdorff spaces and let $A=\{A(U);\rho ^A_{U,V}\}_U$ and $B=\{B(U);\rho ^B_{U,V}\}_U$ be presheaves on X and Y, respectively. A collection

$$\begin{aligned} \{\varphi _U:B(U)\rightarrow A(f^{-1}U)\}_U \end{aligned}$$

of linear maps for each open $U\subset Y$, satisfying

$$\begin{aligned} \varphi _U\circ \rho ^B_{U,V}=\rho ^A_{f^{-1}U,f^{-1}V}\circ \varphi _V\quad \text { whenever }U\subset V, \end{aligned}$$

(A.1)

is called an f-cohomomorphism of presheaves; cf. [2, Chapter I.4]. For $f=\mathrm {id}:X\rightarrow X$, condition (A.1) becomes (5.2) and thus $\mathrm {id}$-cohomomorphisms are simply presheaf homomorphisms.

An f-cohomomorphism $\varphi :B\rightarrow A$ between presheaves induces a natural linear map

$$\begin{aligned} \varphi ^*:{\mathscr {G}}({\mathcal {B}}(Y))\rightarrow {\mathscr {G}}({\mathcal {A}}(X)), \end{aligned}$$

the linear map (on sections) associated to$\varphi$. Given a global section $\omega :Y\rightarrow {\mathcal {B}}(Y)$, the section $\varphi ^*(\omega ):X\rightarrow {\mathcal {A}}(X)$ is defined as follows: for $x\in X$,

$$\begin{aligned} \varphi ^*(\omega )(x){:}{=}[\varphi _U(g_U)]_x, \end{aligned}$$

(A.2)

where U is a neighborhood of f(x) and $g_U\in B(U)$ satisfies $\omega (f(x))=[g_U]_{f(x)}$.

To see that $\varphi ^*(\omega )(x)$ is well-defined, suppose that $\omega (f(x))=[g_U]_{f(x)}=[g'_V]_{f(x)}$, i.e., that there is a neighborhood $D\subset U\cap V$ of f(x) for which

$$\begin{aligned} \rho ^B_{D,U}(g_U)=\rho ^B_{D,V}(g'_V). \end{aligned}$$

By (A.1) we have that

$$\begin{aligned} \rho ^A_{f^{-1}D,f^{-1}U}(\varphi _U(g_U))=\varphi _D(\rho ^B_{D,U}(g_U))=\varphi _D(\rho ^B_{D,V}(g'_V))=\rho ^A_{f^{-1}D,f^{-1}V}(\varphi _V(g'_V)); \end{aligned}$$

in particular $[\varphi _U(g_U)]_x=[\varphi _V(g'_V)]_x$.

Remark A.1

Let $\omega \in {\mathscr {G}}({\mathcal {B}}(Y))$ be compatible with $\{ g_U \}_{\mathcal {U}}$. Suppose $U_V$ satisfies $V=f^{-1}U_V$ for each $V\in f^{-1}{\mathcal {U}}$. Then, by (A.1) and the fact that $\{g_U\}_{\mathcal {U}}$ is compatible $\omega$, we have that the collection $\{ \varphi _V(g_{U_V}) \}_{f^{-1}{\mathcal {U}}}$ is compatible with $\varphi ^*\omega$.

If $\omega \in \Gamma ({\mathcal {B}}(Y))$ and $\{g_U \}_{\mathcal {U}}$ represents $\omega$, then $\{ \varphi _V(g_{U_V}) \}_{f^{-1}{\mathcal {U}}}$ represents $\omega$, and if $\{g_U \}_{\mathcal {U}}$ satisfies the overlap condition (5.5) then $\{ \varphi _V(g_{U_V}) \}_{f^{-1}{\mathcal {U}}}$ also satisfies the overlap condition.

We collect some fundamental properties of linear maps associated to cohomorphisms in the next proposition.

Proposition A.2

Let $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ be continuous maps between paracompact 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

$$\begin{aligned} \varphi =\{\varphi _U:B(U)\rightarrow A(f^{-1}U)\}_U,\quad \varphi '=\{\varphi '_U:B(U)\rightarrow A(f^{-1}U) \}_U \end{aligned}$$

are f-cohomomorphisms, and

$$\begin{aligned} \psi =\{\psi _U:C(U)\rightarrow B(g^{-1}U)\}_U \end{aligned}$$

is an g-cohomomorphism.

• (1) For $\omega \in {\mathscr {G}}({\mathcal {B}}(Y))$ we have

  $$\begin{aligned} {\text {spt}}(\varphi ^*\omega )\subset f^{-1}({\text {spt}}(\omega )). \end{aligned}$$

• (2) The associated linear map $\varphi ^*:{\mathscr {G}}({\mathcal {B}}(Y))\rightarrow {\mathscr {G}}({\mathcal {A}}(X))$ satisfies

  $$\begin{aligned} \varphi ^*(\Gamma ({\mathcal {B}}(Y)))\subset \Gamma ({\mathcal {A}}(X)). \end{aligned}$$

• (3) The collection $\{ \varphi _U+\varphi '_U:B(U)\rightarrow A(f^{-1}U) \}_U$ is an f-cohomomorphism and

  $$\begin{aligned} (\varphi +\varphi ')^*=\varphi ^*+\varphi '^*:{\mathscr {G}}({\mathcal {B}}(Y))\rightarrow {\mathscr {G}}({\mathcal {A}}(X)). \end{aligned}$$

• (4) The collection $\{\varphi _{g^{-1}U}\circ \psi _U:C(U)\rightarrow A((g\circ f)^{-1}U)\}_U$ is an $(g\circ f)$-cohomomorphism and

  $$\begin{aligned} (\varphi \circ \psi )^*=\varphi ^*\circ \psi ^*:{\mathscr {G}}({\mathcal {C}}(Z))\rightarrow {\mathscr {G}}({\mathcal {A}}(X)). \end{aligned}$$

Remark A.3

Given presheaves

$$\begin{aligned} A=\{ A(U) \}_U\text { on { X} and }B_1=\{B_1(U) \},\ B_2=\{ B_2(U) \}\text { on }Y, \end{aligned}$$

and bilinear maps $\{ \varphi _U;B_1(U)\times B_2(U)\rightarrow A(f^{-1}U) \}$ an analogous construction gives an associated bilinear map

$$\begin{aligned} \varphi ^*:{\mathscr {G}}({\mathcal {B}}_1(Y))\times {\mathscr {G}}({\mathcal {B}}_2(Y))\rightarrow {\mathscr {G}}({\mathcal {A}}(X)). \end{aligned}$$

The induced bi-linear map satisfies (2) and (3) and also

• (1’) For each $(\omega ,\sigma ) \in {\mathscr {G}}({\mathcal {B}}_1(Y))\times {\mathscr {G}}({\mathcal {B}}_2(Y))$, we have

  $$\begin{aligned} {\text {spt}}(\varphi ^*(\omega ,\sigma ))\subset f^{-1}({\text {spt}}\omega \cap {\text {spt}}\sigma ). \end{aligned}$$

We will need this only for the case $\mathrm {id}:X\rightarrow X$ 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 sketch them.

If $\varphi ^*\omega (x)\ne 0$ then, since $\varphi _U$ is linear, (A.2) implies that $\omega (f(x))=[g_U]_{f(x)}\ne 0$, 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 $\varphi ':B\rightarrow A$ is another f-cohomorphism between presheaves, then $\varphi +\varphi '$ is an f-cohomomorphism and we have

$$\begin{aligned} (\varphi +\varphi ')^*=\varphi ^*+\varphi '^{*}. \end{aligned}$$

To prove (4), note that condition (A.1) follows for $\phi \circ \psi$ from the fact that it holds for $\varphi$ and $\psi$. Using (A.2) (and the same notation) we see that

$$\begin{aligned} (\varphi \circ \psi )^*\omega (x)=[\varphi _{g^{-1}U}(\psi _U(g_U))]_x=\varphi ^*(\psi ^*\omega )(x). \end{aligned}$$

$\square$

Proposition A.2 has the following immediate corollary.

Corollary A.4

If $f:X\rightarrow Y$ is a proper continuous map and $\varphi :B\rightarrow A$ an f-cohomomorphism between presheaves B on Y and A on X, then

$$\begin{aligned} \varphi ^*({\mathscr {G}}_c({\mathcal {B}}(Y)))\subset {\mathscr {G}}_c({\mathcal {A}}(X))\text { and }\varphi ^*(\Gamma _c({\mathcal {B}}(Y)))\subset \Gamma _c({\mathcal {A}}(X)). \end{aligned}$$

In particular presheaf homomorphisms always have this property.