Abstract
We study differential complexes of Kolmogorov–Alexander–Spanier type on metric measure spaces associated with unbounded non-local operators, such as operators of fractional Laplacian type. We define Hilbert complexes, observe invariance properties and obtain self-adjoint non-local analogues of Hodge Laplacians. For d-regular measures and operators of fractional Laplacian type we provide results on removable sets in terms of Hausdorff measures. We prove a Mayer–Vietoris principle and a Poincaré lemma and verify that in the compact Riemannian manifold case the deRham cohomology can be recovered.
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1 Introduction
In this article we study differential complexes of functions associated with unbounded non-local operators on metric measure spaces; particular examples are operators of fractional Laplacian type.
The classical deRham cohomology theory describes how topological features of a smooth manifold are detected by cochain complexes of differential forms [13, 30, 94]. There are numerous related cohomology theories, for instance those used for cell complexes and groups, [35, 71], and those based on commutative or non-commutative algebras [28, 45, 59]. Applications of related theories to data analysis, [20, 23, 36], sparked new interest in scaled differential complexes and cohomologies on metric spaces, [8, 42, 88]; ideas had been sketched earlier in [78, 79]. The complexes studied in these articles are of Kolmogorov–Alexander–Spanier type [3, 67, 68, 73]. They retrieve metric information and involve a parameter that determines the scale at which features are recognized. This is conceptually close to Vietoris–Rips complexes, [49, 51, 69, 93], and linked to uniform structures [15, Chap. II]. Approaches to homology and homotopy involving a metric scale parameter can be found in [6, 80, 81].
If the given metric space is endowed with a suitable measure, one can define Kolmogorov–Alexander–Spanier complexes based on \(L^2\)-spaces of (classes of) functions; this was done in [8, 42, 88]. Using the terminology of [16] in a slightly wider sense, they may be seen as examples of Hilbert complexes. This point of view is in line with the classical variational approach to Hodge theory, [30, 31, 40, 60, 66], and it emphasizes the link to elliptic partial differential equations.
At order zero—that is, acting on scalar functions—the exterior derivative on a Riemannian manifold M, seen as a closed unbounded operator between \(L^2\)-spaces, defines an unbounded local Dirichlet form [14, 39, 46]. Its infinitesimal generator is the self-adjoint Dirichlet Laplacian on M, clearly unbounded. The \(L^2\)-complexes on metric measure spaces X considered in [8, 42, 88] involve bounded coboundary operators, and at order zero one obtains a bounded (purely) non-local Dirichlet form. Its generator is a bounded non-local operator.
From the perspective of partial differential equations it is more natural to consider unbounded non-local operators and their Dirichlet forms, and there is rich literature on such operators on Euclidean spaces, including fractional Laplacians. See for instance [17, 33, 41] for basic concepts and applications, [19, 32, 34, 50, 64, 75, 77, 83, 87] for a number of well-known results and [1, 18, 21, 29, 41, 76, 91] for connections to geometric analysis and recent developments. Moreover, there is a well-established theory of unbounded purely non-local Dirichlet forms on metric measure spaces \((X,\varrho ,\mu )\); prototype examples are quadratic forms
associated with non-local operators of fractional Laplace type, where \(\mu \) is a d-regular measure on X and \(\alpha \in (0,2)\). See for instance [7, 25, 26, 47, 48, 92] and the references therein.
We propose an approach to higher order “differential forms” based on unbounded non-local operators on metric measure spaces and robust enough to include the complex induced by (1). To our knowledge, this has not been discussed anywhere else. For “differential forms” of order one some aspects of a more general theory based on non-local Dirichlet forms have been investigated, see for instance [27, 53, 56, 84, 85], but higher orders have not been studied.
We set ourselves two objectives:
-
(a)
To define “non-local” Hilbert complexes of Kolmogorov-Alexander-Spanier type, involving a metric scale and based on purely non-local Dirichlet forms with unbounded integral kernels.
-
(b)
To prove that in the case of compact Riemannian manifolds the cohomologies of suitable non-local complexes are isomorphic to the deRham cohomologies.
The passage from bounded kernels as in [8] to unbounded kernels such as \(j(x,y)=\varrho (x,y)^{-d-\alpha }\) in (1) must be carried out carefully. To meet objective (a), one has to introduce spaces of sufficiently regular multivariate functions that are compatible with applications of the coboundary operator. For (b) one needs partitions of unity, and this requires the boundedness of certain multipliers.
Objective (a) gives differential complexes connected to unbounded non-local operators in a similar way as the deRham complex is connected to the Laplace-Beltrami operator. One motivation for (a) is a close connection to partial differential equations. For instance, Poisson regularity was listed as an open problem in [8], and one may expect this problem to have a more familiar flavor in the context of unbounded non-local operators, cf. [87]. A second motivation for (a) is that non-local Dirichlet forms (1) and related operators can readily be defined on a wide class of metric measure spaces, including fractal spaces for which local Dirichlet forms or Laplacians are not known to exist or definitely do not exist. In other words, even if no theory of local complexes is available, such non-local complexes and their cohomologies can be studied. A third motivation for (a) is the possibility to tune the sensitivity of the complex towards the removal of small closed sets. If the starting point is (1), then there is the “new” (in comparison to [8, 42, 88]) parameter \(\alpha \in (0,2)\) that can be varied. We obtain Hilbert complexes with a metric scale parameter \(\varepsilon \) and the parameter \(\alpha \); finer details of X are noted by the complex if \(\varepsilon \) is decreased or \(\alpha \) is increased. Removable sets have also been studied for local complexes, [16, Theorem 4.4.], [44, Theorem 5.1], but as for Sobolev spaces, [2, 74], the possibility to parametrize the effect (while keeping the order of integrability fixed) is a feature of the fractional case. Apart from these motivations, objective (a) should also be seen as a key step towards a general theory of “differential forms” based on Dirichlet forms. In view of known results for first order forms, [27, 53, 56,57,58, 84, 85, 95] and related results for higher order forms under somewhat different hypotheses, [43], such a theory seems desirable.
Objective (b) is set to indicate that our approach is not detached from classical theories. The recovery of the deRham cohomologies on compact manifolds by non-local complexes may be seen as an analog of similar results proved in [8] and [42] for bounded kernels. A new obstacle is that a priori we have to distinguish between the cohomologies based on ’smooth functions or forms’ and those based on domains of closed operators. In the deRham case additional smoothing arguments are used to show that these cohomologies are isomorphic [16, 24]. Such arguments seem rather out of bounds for the non-local complexes we consider. However, in the non-local case the Poincaré lemma involves a homotopy operator that combines well with operator closures, and this can be used to obtain a similar isomorphy.
Our perspectives and results are as follows: Our starting point is an algebra of suitable bounded functions of a single variable; eventually they will be continuous, of bounded differentials and energy finite. We consider finite linear combinations of \((p+1)\)-fold antisymmetric tensor products of such functions, and we refer to them as elementary p-functions. The basic idea is standard, see for instance [12, 28] or [82, Sect. II.4]. By a slight change of notation one can express the tensor products in a more “simplicial” or “affine” manner, Proposition 3.2; for cochains of higher order the single difference in (1) is then replaced by a determinant of differences, Example 3.2. This perspective shows that if the individual factors are sufficiently regular, the integrability of elementary p-functions with respect to unbounded kernels is guaranteed, cf. Lemma 4.1 and Example 4.1. The dominating geometric idea is that of simple vectors, but with “infinitesimal displacements” replaced by differences (“jumps”) according to the kernel. For each given order p we view the elementary p-functions as a core for the coboundary operators and related energy forms, a perspective in line with the classical Beurling–Deny theory [4, 10, 11, 39]. To meet objective (a) we then assume that the kernels are absolutely continuous, as for example in (1), and pass to a Hilbert complex by taking closures of the coboundary operators, Theorem 5.1. We address questions of invariance, Theorem 5.2, introduce coderivations, Corollary 5.4, observe weak Hodge decompositions, Corollary 5.5, and obtain self-adjoint non-local analogs of Hodge Laplacians, Proposition 5.4. We discuss removable sets, Theorem 5.3, with results in terms of Hausdorff measures for the special case related to (1), Corollary 5.7. Objective (b) we discuss for compact spaces, following the classical path, [13, 97]. We prove a Mayer–Vietoris principle for finite open covers, Proposition 6.1. For suitable covers by open balls and under an assumption which is a variant of [8, Hypothesis (\(*\)), p. 34] we establish a Poincaré lemma; see Assumption 6.2, Proposition 6.2 and Corollary 6.2. The arguments are sufficiently robust to work for both the cores and the domains of the closures, and they show that the respective cohomologies are isomorphic to the Čech cohomology of the cover, Corollary 6.3. For compact Riemannian manifolds specific finite covers by sufficiently small balls satisfy all assumptions. In this case both the cohomologies based on elementary functions and those of the Hilbert complex are isomorphic to the respective deRham cohomologies.
We point out that under fairly general assumptions non-local complexes as studied here can be used to approximate local complexes in a natural way; this will be the subject of a follow-up article, [54].
Section 2 contains preliminaries and basic notation. Sections 3 and 4 discuss elementary p-functions and their integrability properties. Operator closures and Hilbert complexes are studied in Sect. 5. Open covers and the link to deRham cohomology in the compact manifold case are the subject of Sect. 6, along with simple examples in Sect. 7.
2 Preliminaries
Let X be a nonempty set and \(p\ge 1\) and integer. Let \({\mathcal {S}}_p\) denote the symmetric group of order p. A function \(F:X^p\rightarrow {\mathbb {R}}\) is called symmetric if \(F(x_{\sigma (1)}, \dotsc , x_{\sigma (p)}) = F(x_1, \dotsc , x_p)\) for all \(\sigma \in {\mathcal {S}}_p\), \(x_1, \dotsc , x_p\in X\), and antisymmetric if \(F(x_{\sigma (1)}, \dotsc , x_{\sigma (p)}) = ({{\,\textrm{sgn}\,}}\sigma ) F(x_1, \dotsc , x_p)\) for all \(\sigma \in {\mathcal {S}}_p\), \(x_1, \dotsc ,x_p\in X\); here \({{\,\textrm{sgn}\,}}\sigma \) denotes the sign of the permutation \(\sigma \). The symmetrizer \({{\,\textrm{Sym}\,}}_p\) of order p is defined by
it takes a given function \(F: X^p\rightarrow {\mathbb {R}}\) into a symmetric function \({{\,\textrm{Sym}\,}}_p(F): X^p\rightarrow {\mathbb {R}}\). Clearly \({{\,\textrm{Sym}\,}}_1=\textrm{id}\), \({{\,\textrm{Sym}\,}}_p(F) = F\) if F is symmetric, and \({{\,\textrm{Sym}\,}}_p^2 = {{\,\textrm{Sym}\,}}_p\). The pointwise product GF of a symmetric function G on \(X^p\) and a function F on \(X^p\) satisfies
if in addition F is symmetric, then FG is symmetric and equals (3). The antisymmetrizer \({{\,\textrm{Alt}\,}}_p\) of order p is defined by
it takes a given function \(F: X^p\rightarrow {\mathbb {R}}\) into an antisymmetric function \({{\,\textrm{Alt}\,}}_p(F): X^p\rightarrow {\mathbb {R}}\). Similarly as before, \({{\,\textrm{Alt}\,}}_1=\textrm{id}\), \({{\,\textrm{Alt}\,}}_p(F) = F\) if F is antisymmetric, and \({{\,\textrm{Alt}\,}}_p^2 = {{\,\textrm{Alt}\,}}_p\). The pointwise product GF of a symmetric function G on \(X^p\) and a function F on \(X^p\) satisfies
if in addition F is antisymmetric, then FG is again antisymmetric and equals (5).
Given a function \(F: X^p\rightarrow {\mathbb {R}}\), we consider the function \(\delta _{p-1} F: X^{p+1}\rightarrow {\mathbb {R}}\), defined by
where, as usual, \({\hat{x}}_i\) means that \(x_i\) is omitted. Obviously \(\delta _0{\textbf{1}}=0\), and it is easy to check that
The operators \(\delta _p\) are the (Kolmogorov–Alexander–Spanier) coboundary operators, [3, 73, 89, 90]. For functions of the form \(F = f_1\otimes \cdots \otimes f_p\) with \(f_i:X\rightarrow {\mathbb {R}}\) (cf. [12, 28]), we obtain
It is easily seen that for any integer \(p\ge 1\) we have
Identities (9) and (8) together imply that
This can be rewritten as
3 Complexes of elementary functions
Let \((X,\varrho )\) be a metric space and \({\mathcal {C}}\) an algebra of real valued functions on X. We now consider complexes of antisymmetric tensor products of elements of \({\mathcal {C}}\).
Let \({\mathcal {C}}^{a,0}:={\mathcal {C}}\), and for \(p\ge 1\) define
To the elements of \({\mathcal {C}}^{a,p}\) we refer as elementary p-functions. To save notation we will write \({\mathcal {C}}^p:={\mathcal {C}}^{a,p}\), except where we wish to point out the antisymmetry explicitly.
We call a family \(N_*=(N_p)_{p\ge 0}\) a system of diagonal neighborhoods for X if
-
(i)
the \(N_p\), \(p\ge 0\), are neighborhoods of the diagonal \({{\,\textrm{diag}\,}}_p:=\{(x_0,\ldots ,x_0): x_0\in X\}\) in \(X^{p+1}\), respectively, and either all open or all closed,
-
(ii)
the \(N_p\) are symmetric in the sense that for any \(\pi \in {\mathcal {S}}_{p+1}\) and any \((x_{0},\ldots ,x_p)\in N_p\) we have \((x_{\pi (0)},\ldots ,x_{\pi (p)})\in N_p\),
-
(iii)
for any \(p\ge 1\), any \((x_0,\ldots ,x_p)\in N_p\) and any \(0\le i\le p\) we have \((x_0,\ldots ,{\hat{x}}_i,\ldots ,x_p)\in N_{p-1}\).
Note that \(N_0=X\). The choice of a system of diagonal neighborhoods determines a metric scale. Given systems \(N_*=(N_p)_{p\ge 0}\) and \(N'_*=(N'_p)_{p\ge 0}\) we write \(N_*\prec N'_*\) if \(N_p\subset N'_p\) for all \(p\ge 0\). We use the symbol \(\subset \) in the non-strict sense; equality is permitted.
Example 3.1
-
(i)
Setting \(N_p:=X^{p+1}\), \(p\ge 0\), we obtain a system of diagonal neighborhoods; it is the largest possible in the sense of \(\prec \).
-
(ii)
Given \(\varepsilon >0\), the sets
$$\begin{aligned} N_p(\varepsilon ):=\{(x_0,\ldots ,x_p)\in X^{p+1}: \max _{0\le i<j\le p}\varrho (x_i,x_j)<\varepsilon \} \end{aligned}$$(13)form a system \(N_*(\varepsilon )\) of diagonal neighborhoods for X. The strict inequality in (13) could be replaced by \(\le \).
-
(iii)
For \(p\ge 0\) we write \(\varrho _p((x_0,\ldots ,x_p),(y_0,\ldots ,y_p)):=\max _{0\le i\le p}\varrho (x_i,y_i)\) and denote the resulting distance between points and sets by \({{\,\textrm{dist}\,}}_p\). Given \(\varepsilon >0\) also the sets
$$\begin{aligned} {\hat{N}}_p(\varepsilon ):=\{(x_0,\ldots ,x_p)\in X^{p+1}: {{\,\textrm{dist}\,}}_p((x_0,\ldots ,x_p),{{\,\textrm{diag}\,}}_p)\le \varepsilon \} \end{aligned}$$(14)form a system \({\hat{N}}_*(\varepsilon )\) of diagonal neighborhoods for X. These sets were used in [8, Sect. 7]. It is easily seen that \(N_*(\varepsilon )\prec {\hat{N}}_*(\varepsilon )\prec N_p(3\varepsilon )\), cf. [42, p. 17].
-
(iii)
Given an open cover \({\mathcal {V}}=\{V_\alpha \}_{\alpha \in I}\) of X, the sets
$$\begin{aligned} N_p({\mathcal {V}}):=\bigcup _{\alpha \in I} V_\alpha ^{p+1} \end{aligned}$$(15)form a system \(N_*({\mathcal {V}})\) of diagonal neighborhoods, cf. [28, p. 346].
Given a system of diagonal neighborhoods \(N_*=(N_p)_{p\ge 0}\) and an integer \(p\ge 0\), we use the notation
Clearly \({\mathcal {C}}^p(X^{p+1})={\mathcal {C}}^p\).
By (7) and (10) the spaces \({\mathcal {C}}^p(N_p)\) are seen to form differential complexes; these complexes are of Kolmogorov–Alexander–Spanier type.
Proposition 1.1
Let \(N_*\) be a system of diagonal neighborhoods. Then the sequence
is a cochain complex.
We write \(({\mathcal {C}}^*(N_*), \delta _*)\) to denote this complex, and we refer to it as the elementary complex. Here and in the following we agree to set \(\delta _{-1}:=0\). For any integer \(p\ge 0\) the p-th cohomology of (17) is
Remark 3.1
It is well-known that whether the cohomologies for \(p\ge 1\) can be nontrivial or not depends on the choice of \(N_*\), see Sect. 7 or [8, 42].
We rewrite the spaces \({\mathcal {C}}^p(N_p)\) in a convenient way.
Proposition 1.2
For any integer \(p\ge 1\) we have
where \({\overline{g}}\) denotes the average
Moreover, for any \(g\in {\mathcal {C}}\oplus {\mathbb {R}}\) and \(f_1,\ldots , f_p \in {\mathcal {C}}\) the identity
holds.
Proof
Identity (19) is immediate from the fact that for any \(g\in {\mathcal {C}}\oplus {\mathbb {R}}\) and \(f_1,\dots , f_p, \in {\mathcal {C}}\) we have, by (10) and (5),
By (7) an application of \(\delta _p\) yields (21). \(\square \)
Example 3.2
-
(i)
For \(p=0\) we have
$$\begin{aligned} \delta _0 f(x_0,x_1)=f(x_1)-f(x_0). \end{aligned}$$ -
(ii)
For \(p=1\) we have
$$\begin{aligned} \delta _1 ({\overline{f}}_1\delta _0 f_2)(x_0,x_1,x_2)= & {} \delta _1{{\,\textrm{Alt}\,}}_2(f_1\otimes f_2)(x_0,x_1,x_2)\\= & {} \frac{1}{2}\det [((f_i(x_j)-f_i(x_0))_{i,j=1}^2]. \end{aligned}$$
Remark 3.2
The averaging (20) of the coefficients amounts to an antisymmetrization of the product: Given \(g\in {\mathcal {C}}\oplus {\mathbb {R}}\) and \(F:X^{p+1}\rightarrow {\mathbb {R}}\), we define a function \(g\cup F:X^{p+1}\rightarrow {\mathbb {R}}\) by \( g\cup F(x_0,\ldots ,x_p):=g(x_0)F(x_0,\ldots ,x_p)\). If F is antisymmetric, then
From (2) and (4) it is easily seen that for any integer \(p\ge 1\), any \(f_0,\ldots ,f_p,g_0,\ldots ,g_p\in {\mathcal {C}}\oplus {\mathbb {R}}\) we have
and
Let \({\mathcal {C}}^{s,p}\) be the vector space spanned by \(\{ {{\,\textrm{Sym}\,}}_{p+1}(f_0\otimes f_1\otimes \dots \otimes f_p): f_0, f_1,\dots , f_p \in {\mathcal {C}}\oplus {\mathbb {R}}\}\) and \({\mathcal {C}}^{s,p}(N_p):={\mathcal {C}}^{s,p}|_{N_p}\). From (23) it follows that \({\mathcal {C}}^{s,p}(N_p)\) with pointwise multiplication is a subring of the ring of real valued functions on \(N_p\). Using (24) it is seen that for any \(F\in {\mathcal {C}}^{a,p}(N_p)\) and \(\chi \in {\mathcal {C}}^{s,p}(N_p)\) we have \(\chi F\in {\mathcal {C}}^{a,p}(N_p)\), and that this multiplicative action of \({\mathcal {C}}^{s,p}(N_p)\) makes \({\mathcal {C}}^{a,p}(N_p)\) a module.
For later use we record an observation about differentials of functions of a special form. Given \(\chi \in {\mathcal {C}}\oplus {\mathbb {R}}\) and integer \(p\ge 0\) we write
where \(\chi \) appears \(p+1\) times on the right-hand side.
Lemma 1.1
Let \(p\ge 0\) be an integer and \(f_0,\ldots ,f_p,\chi \in {\mathcal {C}}\oplus {\mathbb {R}}\). Then
and
Proof
The first statement is obvious. To see the second, note that the evaluation of
at \((x_0,\ldots ,x_{p+1})\in X^{p+2}\) gives
\(\square \)
Suppose that \({\tilde{X}}\) is another metric space and \(\varphi :{\tilde{X}}\rightarrow X\) is a map. Given \(p\ge 0\) integer and a function \(F:X^{p+1}\rightarrow {\mathbb {R}}\) we define a function \(\varphi ^*F:{\tilde{X}}^{p+1}\rightarrow {\mathbb {R}}\) by
We write \(\varphi ^*{\mathcal {C}}:=\{\varphi ^*f: f\in {\mathcal {C}}\}\), and given a system \(N_*=(N_p)_{p\ge 0}\) of diagonal neighborhoods on X, we set \(\varphi ^*N_p:=\{({\tilde{x}}_0,\ldots ,{\tilde{x}}_p)\in {\tilde{X}}^{p+1}: (\varphi ({\tilde{x}}_0),\ldots ,\varphi ({\tilde{x}}_p))\in N_p\}\). The following is easily seen.
Proposition 1.3
Suppose that \(\tilde{{\mathcal {C}}}\) is an algebra of real valued functions on \({\tilde{X}}\) and \(\varphi ^*{\mathcal {C}}\subset \tilde{{\mathcal {C}}}\).
-
(i)
The map \(\varphi ^*:{\mathcal {C}}\rightarrow \tilde{{\mathcal {C}}}\) is an algebra homomorphism and \(\varphi ^*{\mathcal {C}}\) is a subalgebra of \(\tilde{{\mathcal {C}}}\). For each integer \(p\ge 0\) we have \((\varphi ^*{\mathcal {C}})^{a,p}=\varphi ^*{\mathcal {C}}^{a,p}\subset \tilde{{\mathcal {C}}}^{a,p}\) and \((\varphi ^*{\mathcal {C}})^{s,p}=\varphi ^*{\mathcal {C}}^{s,p}\subset \tilde{{\mathcal {C}}}^{s,p}\), and \(\varphi ^*\) is a module homomorphism. If \(\varphi \) is bijective and \(\varphi ^*{\mathcal {C}}= \tilde{{\mathcal {C}}}\), then the maps \(\varphi ^*\) are (algebra, module) isomorphisms.
-
(ii)
If \(\varphi \) is a homeomorphism from \({\tilde{X}}\rightarrow X\), then \(\varphi ^*N_*:=(\varphi ^*N_p)_{p\ge 0}\) is a system of diagonal neighborhoods on \({\tilde{X}}\) and \(\varphi ^*\) is a cochain map from the complex \(({\mathcal {C}}^*(N_*),\delta _*)\) to the complex \((\tilde{{\mathcal {C}}}^*(\varphi ^*N_*),\delta _*)\). In the case that \(\varphi ^*{\mathcal {C}}= \tilde{{\mathcal {C}}}\) the map \(\varphi ^*\) is an isomorphism of cochain complexes.
4 Kernels and measures
We introduce kernels and measures and related conditions on the algebra \({\mathcal {C}}\). Let \((X,\varrho )\) be a locally compact metric space. Let \({\mathcal {B}}(X)\) denote the Borel \(\sigma \)-algebra on X.
Assumption 1.1
-
(i)
We assume that \(j:X\times {\mathcal {B}}(X)\rightarrow [0,+\infty ]\) is a kernel in the sense that for any \(x\in X\) the map \(A\mapsto j(x,A)\) is a Borel measure on X, locally finite on \(X\setminus \{x\}\), and for any \(A\in {\mathcal {B}}(X)\) the function \(x\mapsto j(x,A)\) is Borel measurable.
-
(ii)
We assume that \({\mathcal {C}}\) is an algebra of bounded real valued Borel functions on X such that
$$\begin{aligned} \sup _{x\in X}\int \limits _{X}(f(x)-f(y))^2 j(x,dy)<+\infty ,\quad f\in {\mathcal {C}}. \end{aligned}$$(27)
If a volume measure \(\mu \) is given and satisfies the following assumption, then pointwise statements can be complemented by integrated versions.
Assumption 1.2
We assume that \(\mu \) is a nonnegative Radon measure on X with full support (that is, \({{\,\textrm{supp}\,}}\mu =X\)), \({\mathcal {C}}\subset L^2(X,\mu )\), and
Remark 4.1
If in the presence of a measure \(\mu \) as in Assumption 4.2 the supremum in (27) is replaced by an essential supremum, pointwise statements below remain true in the \(\mu \)-a.e. sense.
Given a system \(N_*=(N_p)_{p\ge 0}\) of diagonal neighborhoods and \(x_0\in X\), we write
We set \(D_0:=\emptyset \),
and write
Suppose that Assumption 4.1 is satisfied. We define kernels
from X to the p-fold product of \({\mathcal {B}}(X)\). Clearly \(j_1=j\). For each fixed \(x_0\) and p the measures \(j_p(x_0,\cdot )\) are Radon on \((X\setminus \{x_0\})^p\) and symmetric in the variables \(x_1,\ldots ,x_p\). Below we consider the Hilbert spaces
If in addition Assumption 4.2 is satisfied, we write \(J_0:=\mu \) and define measures
The measures \(J_p\) are Radon on \(X^{p+1}\setminus D_p\), respectively, and symmetric in the variables \(x_0,\ldots ,x_p\). Below we consider the Hilbert spaces
Remark 4.2
Rewriting (30) in terms of (29), we observe that
A somewhat similar point of view has been pursued in [8, Sect. 2], but with (bounded) kernels implemented into the definitions of operators, not spaces.
Remark 4.3
If \(N_*\) and \(N_*'\) are systems of diagonal neighborhoods and \(N_*'\prec N_*\), then \(\Vert \cdot \Vert _{L^2(N_p'{\setminus } D_p,J_p)}\le \Vert \cdot \Vert _{L^2(N_p{\setminus } D_p,J_p)}\) and in particular, \(L^2(N_p{\setminus } D_p,J_p)\subset L^2(N_p'{\setminus } D_p,J_p)\). Similarly for the spaces involving \(x_0\).
The maps \({{\,\textrm{Alt}\,}}_{p}\) and \({{\,\textrm{Alt}\,}}_{p+1}\) are linear and bounded on the spaces \(L^2(N_{p,x_0}\setminus D_{p,x_0},j_p(x_0,\cdot ))\) and \(L^2(N_p\setminus D_p,J_p)\), respectively. Since by the mentioned symmetries they are self-adjoint, they act as orthogonal projections. We write
and
for the closed subspaces obtained as their images.
Given \(x_0\in X\) we write \({\mathcal {C}}^p(N_p,x_0)\) for the space of functions \((x_1,\ldots ,x_p)\mapsto F(x_0,x_1,\ldots ,x_p)\) on \(N_{p,x_0}\) with \(F\in {\mathcal {C}}^p(N_p)\). To keep notation short we will denote the elements of \({\mathcal {C}}^p(N_p,x_0)\) also by F instead of the correct \(F(x_0,\cdot )\).
Lemma 1.2
Suppose that Assumption 4.1 is satisfied and let \(N_*\) be a system of diagonal neighborhoods.
-
(i)
For any \(p\ge 1\) and \(x_0\in X\) we have \({\mathcal {C}}^p(N_p,x_0) \subset L^2(N_{p,x_0}{\setminus } D_{p,x_0},j_p(x_0,\cdot ))\). For any \(g\in {\mathcal {C}}\oplus {\mathbb {R}}\) and \(f_1,\ldots ,f_p\in {\mathcal {C}}\) the inequality
$$\begin{aligned}{} & {} \left\| {\overline{g}}\delta _{p-1}{{\,\textrm{Alt}\,}}_p(f_1\otimes \dots \otimes f_p)\right\| _{L^2(N_{p,x_0}\setminus D_{p,x_0},j_p(x_0,\cdot ))}\nonumber \\{} & {} \quad \le \left\| g\right\| _{\sup }\prod _{i=1}^p\left( \int \limits _{N_{1,x_0}\setminus D_{1,x_0}} (f_i(y)-f_i(x_0))^2j(x_0,dy)\right) ^{1/2} \end{aligned}$$(33)holds. The functional \(\left\| \cdot \right\| _{L^2(N_{p,x_0}{\setminus } D_{p,x_0},j_p(x_0,\cdot ))}\) is a Hilbert seminorm on \({\mathcal {C}}^p(N_p,x_0)\); it is a norm if \({\mathcal {C}}\subset C(X)\) and \(j(x_0,\cdot )\) has full support on \(N_{1,x_0}\).
-
(ii)
Suppose that also Assumption 4.2 is satisfied. Then for any integer \(p\ge 1\) we have \({\mathcal {C}}^p(N_p) \subset L_a^2(N_p\setminus D_p,J_p)\). For any \(g\in {\mathcal {C}}\oplus {\mathbb {R}}\) and \(f_1,\ldots ,f_p\in {\mathcal {C}}\) and any \(i=1,\ldots ,p\) the inequality
$$\begin{aligned}{} & {} \left\| {\overline{g}}\delta _{p-1}{{\,\textrm{Alt}\,}}_p(f_1\otimes \dots \otimes f_p)\right\| _{L^2(N_p\setminus D_p,J_p)}\nonumber \\{} & {} \quad \le \left\| g\right\| _{\sup }\prod _{k\ne i}\left( \sup _{x\in X}\int \limits _{N_{1,x}\setminus D_{1,x}} (f_k(y)-f_k(x))^2j(x,dy)\right) ^{1/2}\nonumber \\{} & {} \qquad \times \left( \int \limits _X\int \limits _{N_{1,x}\setminus D_{1,x}} (f_i(y)-f_i(x))^2j(x,dy)\mu (dx)\right) ^{1/2} \end{aligned}$$(34)holds. The functional \(\left\| \cdot \right\| _{L^2(N_p{\setminus } D_p,J_p)}\) is a Hilbert seminorm on \({\mathcal {C}}^p(N_p)\); it is a norm if \({\mathcal {C}}\subset C(X)\) and for \(\mu \)-a.e. \(x_0\in X\) the measure \(j(x_0, \cdot )\) has full support on \(N_{1,x_0}\).
We write B(x, r) to denote the open ball of radius \(r>0\) centered at \(x\in X\).
Proof
Using (11),
By Cauchy–Schwarz with respect to the measure \(j_p(x_0,\cdot )\), its symmetry in \(x_1,\ldots ,x_p\) and Fubini this is seen to be bounded by
This is (33), which by (27) is finite, so (i) follows. Using (28) and (30) the inclusion in (ii) and estimate (34) are seen similarly.
The seminorm properties are clear. To see the last statement in (i) it suffices to note that if \(j_1(x_0,\cdot )\) has full support on \(N_{1,x_0}\), then \(j_p(x_0,\cdot )\) has full support on \(N_{p,x_0}\setminus D_{p,x_0}\): Given \(U\subset N_{p,x_0}{\setminus } D_{p,x_0}\) nonempty and open, we can find \((x_0,\ldots ,x_p)\in N_p{\setminus } D_p\) such that \(\varepsilon :=\frac{1}{2}\min _{0\le i<k\le p}\varrho (x_i,x_k)>0\) and \(B(x_1,\varepsilon )\times \cdots \times B(x_p,\varepsilon )\) is contained in U. By hypothesis this product of balls has positive \(j_p(x_0,\cdot )\)-measure, and therefore \(j_p(x_0,U)>0\). The last statement in (ii) follows similarly. \(\square \)
Example 4.1
Recall Examples 3.2.
-
(i)
For \(p=0\) we have
$$\begin{aligned} \big \Vert {\overline{g}}\delta _0 f\big \Vert _{L^2(N_1\setminus D_1,J_1)}^2&=\int \limits _{N_1\setminus D_1}{\overline{g}}(x_0,x_1)^2 (f(x_1)-f(x_0))^2 J_1(d(x_0,x_1))\\&=\frac{1}{2} \int \limits _X\int \limits _X {\overline{g}}(x_0,x_1)^2 (f(x_1)-f(x_0))^2{\textbf{1}}_{N_1}(x_0,x_1)\times \\&\quad \times (j(x_0,dx_1)\mu (dx_0)+j(x_1,dx_0)\mu (dx_1)). \end{aligned}$$ -
(ii)
For \(p=1\) we have
$$\begin{aligned}&\big \Vert {\overline{g}}\delta _1{{\,\textrm{Alt}\,}}_2(f_1 \otimes f_2)\big \Vert _{L^2(N_2\setminus D_2,J_2)}^2\\&\quad =\frac{1}{3}\int \limits _{N_2\setminus D_2}{\overline{g}}(x_0,x_1,x_2)^2\det [(f_i(x_j)-f_i(x_0))_{i,j=1}^2]^2 J_2(d(x_0,x_1,x_2)). \end{aligned}$$
A class of typical examples arises from Dirichlet forms, [39, 48, 72].
Example 4.2
Suppose that \(\mu \) is a nonnegative Radon measure on X with full support,
where j is as in Assumption 4.1 (i) and satisfies \(j(x,dy)\mu (dx)=j(y,dx)\mu (dy)\). Let
Then \(({\mathcal {E}},{\mathcal {D}}({\mathcal {E}}))\) is a Dirichlet form on X. If in addition all balls have finite measure and there is some \(\varepsilon >0\) such that
then the algebra \({{\,\textrm{Lip}\,}}_c(X)\) of compactly supported Lipschitz functions on X satisfies Assumptions 4.1 and 4.2 in place of \({\mathcal {C}}\); note that conditions (37) ensure (27). Given \(f\in {{\,\textrm{Lip}\,}}_c(X)\), let B be an open ball containing \(K:={{\,\textrm{supp}\,}}f\) and large enough to have \({{\,\textrm{dist}\,}}(K,B^c)>\varepsilon \). Then
By (27) and since \(\mu (B)<+\infty \) the first summand is finite. The second is bounded by \(\Vert f\Vert _{\sup }^2\mu (K)\sup _{y\in K}\int _{B^c} j(y,dx)\), which is finite by (37). This, together with \({{\,\textrm{Lip}\,}}_c(X)\subset L^2(X,\mu )\), implies that \({{\,\textrm{Lip}\,}}_c(X)\subset {\mathcal {D}}({\mathcal {E}})\).
Recall Lemma 3.1. We consider the linear extension of (26) and record a related norm estimate for derivatives.
Lemma 1.3
Suppose that Assumptions 4.1 and 4.2 are satisfied and let \(N_*\) be a system of diagonal neighborhoods. Let \(p\ge 0\) be an integer, \(F\in {\mathcal {C}}^{p}(N_{p})\) and \(\chi \in {\mathcal {C}}\oplus {\mathbb {R}}\). Then
Note that by Lemma 4.1 (ii) the norms of F and \(\delta _pF\) on the right-hand side are finite.
Proof
Since \((\delta _{p}(\chi ^{\otimes (p+1)}F)(x_0,\ldots ,x_{p+1}))^2\) is symmetric in \(x_0,\ldots ,x_{p+1}\), we have
similarly for \(\delta _p F\) or F in place of \(\delta _{p}(\chi ^{\otimes (p+1)}F)\). Since an element F of \({\mathcal {C}}^{p}(N_{p})\) is of the form \(F=\sum _{i=1}^m{{\,\textrm{Alt}\,}}_{p+1}(f_0^{(i)}\otimes \cdots \otimes f_{p}^{(i)})\), we can employ Lemma 3.1 to bound (38) by
and each of the multiple integrals in brackets in the second summand can be estimated by
\(\square \)
5 Non-local Hilbert complexes
5.1 Regularity and density
Let \((X,\varrho )\) be a locally compact metric space. We use the notation introduced in Sects. 3 and 4 and consider the following density assumption.
Assumption 1.3
The algebra \({\mathcal {C}}\cap C_c(X)\) is a uniformly dense subalgebra of \(C_c(X)\).
A generalization of [53, Lemma 3.1] shows that the elementary forms are dense in the Hilbert spaces \(L^2_a(N_p\setminus D_p,J_p)\).
Proposition 1.4
Let Assumptions 4.1, 4.2 and 5.1 be satisfied, let \(N_*\) be a system of diagonal neighborhoods and \(p\ge 0\) an integer.
-
(i)
The space \({\mathcal {C}}^p(N_p)\) is dense in \(L^2_a(N_p\setminus D_p,J_p)\).
-
(ii)
The operators \((\delta _p,{\mathcal {C}}^p(N_p))\) are densely defined as operators from \(L^2_a(N_p\setminus D_p,J_p)\) into \(L^2_a(N_{p+1}\setminus D_{p+1},J_{p+1})\).
Proof
It suffices to show that if \(F\in L^2_a(N_p\setminus D_p, J_p)\) is such that
for all \(f_1,\ldots ,f_p\in {\mathcal {C}}\) and \(f_0\in {\mathcal {C}}\oplus {\mathbb {R}}\), then we have
Suppose that \(f_0,f_1,\ldots ,f_p\) are as specified and that (39) holds. Since \({{\,\textrm{Alt}\,}}_{p+1}\) acts as an orthogonal projection in \(L^2(N_p{\setminus } D_p,J_p)\) and \({{\,\textrm{Alt}\,}}_{p+1}F=F\), it follows that
By Assumption 5.1 and Stone–Weierstrass the algebra generated by products \(f_0\otimes f_1\otimes \cdots \otimes f_p\) with \(f_0,\ldots ,f_p\in {\mathcal {C}}\cap C_c(X)\) is dense in \(C_c(X^{p+1}\setminus D_p)\) and therefore also in \(L^2(N_p{\setminus } D_p,J_p)\). This, together with (41), shows that
and this gives (40). \(\square \)
5.2 Closed extensions
We proceed under the following absolute continuity condition.
Assumption 1.4
The kernel j as in Assumption 4.1 is of the form
with a Borel function \(j:X^2\rightarrow (0,+\infty ]\), locally bounded on \(X^2\setminus D_1\).
Given \(p\ge 1\), let \(\mu ^p\) denote the p-fold product of \(\mu \) on \(X^p\). By Assumption 5.2 we have
and
Under Assumption 5.2 we can extend the derivations \(\delta _p\) to closed operators.
Theorem 1.1
Let Assumptions 4.1, 4.2 and 5.2 be satisfied. Let \(N_*\) be a system of diagonal neighborhoods and let \(p\ge 0\) be an integer.
-
(i)
The operators \((\delta _p,{\mathcal {C}}^p(N_p))\) extend to closed linear operators \((\delta _p,{\mathcal {D}}(\delta _p,N_p))\) from \(L^2_a(N_p{\setminus } D_p,J_p)\) into \(L^2_a(N_{p+1}\setminus D_{p+1},J_{p+1})\). Given \(F\in {\mathcal {D}}(\delta _p,N_p)\), we have
$$\begin{aligned} \delta _{p} F(x_0, \dotsc , x_{p+1})= \sum _{i=0}^{p+1} (-1)^i F(x_0, \dotsc , {\hat{x}}_i, \dotsc , x_{p+1}) \end{aligned}$$(43)in the \(J_{p+1}\)-a.e. sense. If in addition Assumption 5.1 holds, then the operators \((\delta _p,{\mathcal {D}}(\delta _p,N_p))\) are densely defined.
-
(ii)
We have
$$\begin{aligned} {{\,\mathrm{\textrm{im}}\,}}\delta _{p}|_{{\mathcal {D}}(\delta _{p},N_{p})} \subset {\mathcal {D}}(\delta _{p+1},N_{p+1})\quad \text {and}\quad \delta _{p+1}\circ \delta _p=0. \end{aligned}$$(44)The sequence
$$\begin{aligned} 0\longrightarrow {\mathcal {D}}(\delta _0,N_0){\mathop {\longrightarrow }\limits ^{\delta _0}} {\mathcal {D}}(\delta _1,N_1){\mathop {\longrightarrow }\limits ^{\delta _1}} {\mathcal {D}}(\delta _2,N_2) {\mathop {\longrightarrow }\limits ^{\delta _2}}\cdots \end{aligned}$$(45)is a cochain complex.
Similarly as in [16] one could call the complex \(({\mathcal {D}}(\delta _*,N_*),\delta _*)\) in (45) a Hilbert complex. It is “non-local” in the sense that the operators \(\delta _p\) are non-local.
For any integer \(p\ge 0\) the p-th cohomology of (45) is
Remark 5.1
-
(i)
In [16] the ambient Hilbert spaces of sufficiently high order p were assumed to be trivial; we do not make this assumption here.
-
(ii)
Obviously the complex (17) of elementary functions is a subcomplex of (45).
To prove Theorem 5.1 we consider quadratic forms associated with the operators \(\delta _p\). Note first that for each \(p\ge 0\) the application of \(\delta _p\) to an element F of \(L^2(N_{p}\setminus D_{p},J_{p})\) gives a well-defined \(\mu ^{p+2}\)-equivalence class \(\delta _pF\) of Borel functions on \(N_{p+1}\setminus D_{p+1}\); this is straightforward from (6) and Assumption 5.2. Therefore
defines a quadratic form
By Lemma 4.1 we have \(Q_{p}(F)<+\infty \) for any \(F\in {\mathcal {C}}^p(N_p)\).
Remark 5.2
Recall that if \(I:B\rightarrow [0,+\infty ]\) is a lower semicontinuous functional on a Banach space B and V is a subspace of B on which I is finite, then (I, V) is closable.
Proposition 1.5
Let Assumptions 4.1, 4.2 and 5.2 be satisfied. Let \(N_*\) be a system of diagonal neighborhoods and let \(p\ge 0\) be an integer.
Then the quadratic form \(Q_p\) defined in (47) is lower semicontinuous on \(L^2_a(N_p\setminus D_p, J_p)\). Its restriction \((Q_p,{\mathcal {C}}^p(N_p))\) to \({\mathcal {C}}^p(N_p)\) is closable.
Proof
Suppose that \((F_j)_j\subset L^2_a(N_p\setminus D_p, J_p)\) converges to F in \(L^2_a(N_p\setminus D_p, J_p)\) and let \((j_k)_k\) be such that
we may assume the right-hand side is finite. Passing to a further subsequence of \((F_{j_k})_k\) and relabelling, we may assume that
for all \((x_0,\ldots ,x_p)\in (N_p\setminus D_p)\setminus Z_{p}\), where \(Z_{p}\subset N_p\) is a set of measure zero for \(J_p\). Since by (42) the density of J with respect to \(\mu ^{p+1}\) is strictly positive, \(Z_{p}\subset N_p\) is also of zero measure for \(\mu ^{p+1}\). Now let
Then \(\mu ^{p+2}(Z_{p+1}')=0\), and for all \((x_0,\ldots ,x_{p+1})\in (N_{p+1}{\setminus } D_{p+1}){\setminus } Z_{p+1}'\) we have
Since \(J_{p+1}\ll \mu ^{p+2}\) by (42), Fatou’s lemma now implies that
and combining with (48), the lower semicontinuity of \(Q_p\) is observed. By Remark 5.2 the form \((Q_p,{\mathcal {C}}^p(N_p))\) is closable. \(\square \)
The preceding now yields a quick proof of Theorem 5.1.
Proof
Since the closability of \((\delta _p,{\mathcal {C}}^p(N_p))\) is equivalent to the closability of \((Q_p,{\mathcal {C}}^p(N_p))\), the first statement of Theorem 5.1 is immediate from Proposition 5.2.
To see (ii), let \(F\in {\mathcal {D}}(\delta _p,N_p)\) and let \((F_n)_n\subset {\mathcal {C}}^p(N_p)\) be such that \(\lim _{n\rightarrow \infty }F_n=F\) in \(L^2(N_p{\setminus } D_p,J_p)\) and \(\lim _{n\rightarrow \infty }\delta _pF_n=\delta _pF\) in \(L^2(N_{p+1}{\setminus } D_{p+1},J_{p+1})\). The lower semicontinuity of \(Q_{p+1}\) and (7) imply that
Consequently \(\delta _pF\in \ker \delta _{p+1}\subset {\mathcal {D}}(\delta _{p+1},N_p)\), and \(\delta _{p+1}\circ \delta _p=0\). \(\square \)
We write \(({\mathcal {Q}}_p, {\mathcal {D}}(\delta _{p},N_p))\) to denote the closure of \((Q_p,{\mathcal {C}}^p(N_p))\) in \(L^2_a(N_p\setminus D_p, J_p)\). By closedness the domain \({\mathcal {D}}(\delta _{p},N_p)\), endowed with the scalar product
is a Hilbert space. We write \(\left\| \cdot \right\| _{{\mathcal {D}}(\delta _{p},N_p)}\) for the associated Hilbert norm.
Remark 5.3
We point out that in general \({\mathcal {D}}(\delta _{p},N_p)\) may be smaller than the maximal domain \(\{F\in L^2_a(N_p{\setminus } D_p, J_p):\ Q_p(F)<+\infty \}\) of Q.
Corollary 1.1
Let Assumptions 4.1, 4.2, 5.1 and 5.2 be satisfied. Let \(N_*\) be a system of diagonal neighborhoods and let \(p\ge 0\) be an integer. Then \(({\mathcal {Q}}_p, {\mathcal {D}}(\delta _{p},N_p))\) is a densely defined closed quadratic form on \(L^2_a(N_p\setminus D_p, J_p)\), and
Remark 5.4
Let the hypotheses of Corollary 5.1 be satisfied, assume that \({\mathcal {C}}\subset C_c(X)\) and that X is separable. Then polarization of the quadratic form \(({\mathcal {Q}}_0, {\mathcal {D}}(\delta _{0},N_0))\) yields a regular Dirichlet form on \(X=N_0\) in the sense of [39], and this form is purely non-local. Its generator, that is, the unique nonpositive definite self-adjoint operator \(({\mathcal {L}}_0,{\mathcal {D}}({\mathcal {L}}_0))\) on \(L^2(X,\mu )\) satisfying
is of the form
understood in the \(\mu \)-a.e. sense.
Example 5.1
Suppose that j is symmetric and
with some fixed \(d>0\), \(\alpha \in (0,2)\) and \(c>1\). If \(\varepsilon >0\) and \(N_*=N_*(\varepsilon )\), then \({\mathcal {Q}}_0\) is comparable to the quadratic form
and \({\mathcal {L}}_0\) in (51) is comparable to a truncated fractional Laplacian of order \(\alpha /2\) on X.
Recall Lemmas 3.1 and 4.2. We observe that the linear extension of (26) becomes a multiplier on \({\mathcal {D}}(\delta _p,N_p)\).
Corollary 1.2
Let Assumptions 4.1, 4.2 and 5.2 be satisfied, let \(N_*\) be a system of diagonal neighborhoods and \(\chi \in {\mathcal {C}}\oplus {\mathbb {R}}\). Then for any \(p\ge 0\) the multiplication \(F\mapsto \chi ^{\otimes (p+1)}F\), \(F\in {\mathcal {C}}^p(N_p)\), extends to a bounded linear operator on \({\mathcal {D}}(\delta _p,N_p)\), and we have
where
Proof
By the trivial bound
and Lemma 4.2, estimate (53) is seen to hold for all \(F\in {\mathcal {C}}^p(N_p)\). Now let \(F\in {\mathcal {D}}(\delta _p,N_p)\) and let \((F_n)_n\subset {\mathcal {C}}^p(N_p)\) be such that \(\lim _{n\rightarrow \infty } \left\| F_n-F\right\| _{{\mathcal {D}}(\delta _p,N_p)}=0\). Clearly
By Lemma 4.2 the sequence \((\delta _p(\chi ^{\otimes (p+1)}F_n))_n\) is Cauchy in \(L^2(N_{p+1}{\setminus } D_{p+1},J_{p+1})\), so it has a limit G. Since \(\delta _p\) is closed, we have \(\chi ^{\otimes (p+1)}F\in {\mathcal {D}}(\delta _p,N_p)\) and \(\delta _p(\chi ^{\otimes (p+1)}F)=G\). Estimate (53) now follows easily. \(\square \)
5.3 Remarks on invariance
Choices of comparable kernels j in Assumption 5.2 lead to the same closures and therefore to the same complexes.
Proposition 1.6
Suppose that both j and \(j'\) are kernels such that Assumptions 4.1, 4.2 and 5.2 are satisfied and let \(N_*\) be a system of diagonal neighborhoods. If there is a constant \(c>1\) such that \(c^{-1}j(x,y)\le j'(x,y)\le c\,j(x,y)\), \((x,y)\in N_1\setminus D_1\), then the resulting complexes \(({\mathcal {D}}(\delta _*,N_*),\delta _*)\) and \(({\mathcal {D}}'(\delta _*,N_*),\delta _*)\) coincide.
Recall Proposition 3.3 and the notation used there. Suppose that \(({\tilde{X}},{\tilde{\varrho }})\) is another (locally compact) metric space with associated data \(\tilde{{\mathcal {C}}}\), \({\tilde{j}}\), \({\tilde{\mu }}\) and that \(\varphi :{\tilde{X}}\rightarrow X\) is a given Borel map. Under suitable conditions the resulting Hilbert complexes on X and \({\tilde{X}}\) are isomorphic. We write \(\varphi ^*j({\tilde{x}},{\tilde{y}}):=j (\varphi ({\tilde{x}}),\varphi ({\tilde{y}}))\) and \(\varphi _*{\tilde{\mu }}:={\tilde{\mu }}\circ \varphi ^{-1}\), and we denote the Hilbert spaces obtained from \(\tilde{{\mathcal {C}}}^p(\varphi ^*N_p)\) by taking closures as in Theorem 5.1 by \(\tilde{{\mathcal {D}}}(\delta _p,\varphi ^*N_p)\).
Theorem 1.2
Suppose that Assumptions 4.1, 4.2 and 5.2 are satisfied for X, \({\mathcal {C}}\), j, \(\mu \) and also for \({\tilde{X}}\), \(\tilde{{\mathcal {C}}}\), \({\tilde{j}}\), \({\tilde{\mu }}\). Let \(N_*\) be a system of diagonal neighborhoods for X. Assume that \(\varphi \) is a homeomorphism from \({\tilde{X}}\) onto X,
and
with constants \(c_j>1\), \(c_\mu >1\), and that \(\varphi ^*{\mathcal {C}}\subset \tilde{{\mathcal {C}}}\) is dense in \(\tilde{{\mathcal {C}}}\) with respect to \(\left\| \cdot \right\| _{\tilde{{\mathcal {D}}}(\delta _0,\varphi ^*N_0)}\). Then for any integer \(p\ge 0\) the map \(\varphi ^*:{\mathcal {D}}(\delta _p,N_p)\rightarrow \tilde{{\mathcal {D}}}(\delta _p,\varphi ^*N_p)\) is an isomorphism of equivalently normed spaces, and
with a constant \(c_p>1\). Moreover, \(\varphi ^*:{\mathcal {D}}(\delta _*,N_*)\rightarrow \tilde{{\mathcal {D}}}(\delta _*,\varphi ^*N_*)\) is an isomorphism of cochain complexes.
Proof
It is quickly seen that \({\tilde{D}}_p\), defined similarly as \(D_p\), equals \(\varphi ^*D_p:=\{({\tilde{x}}_0,\ldots ,{\tilde{x}}_p)\in {\tilde{X}}^{p+1}: (\varphi ({\tilde{x}}_0),\ldots ,\varphi ({\tilde{x}}_p))\in D_p\}\). Using (54), change of variables and (55) we obtain
and an analogous lower bound with the reciprocal constant. Since \(\delta _p\varphi ^*F=\varphi ^*\delta _p F\), similar estimates are seen to hold with \(\delta _p \varphi ^*F\) and \(\delta _p F\) in place of \(\varphi ^*F\) and F, respectively. As in (38) these estimates carry over to the \(L^2\)-norms with respect to \(J_p\) and \(J_{p+1}\) by the symmetry of the integrands. Combining, we arrive at (56), and this implies that \(\varphi ^*\) preserves the property to be a Cauchy sequence and therefore induces isomorphisms as stated. \(\square \)
Let \(d>0\). Recall that a nonnegative Radon measure \(\mu \) on X is said to be (Ahlfors) d-regular, [52, p. 62], if there is a constant \(c>1\) such that
If in this situation \({{\,\textrm{supp}\,}}\mu =X\), then d equals the Hausdorff dimension of X.
A special case of Theorem 5.2 gives the following Lipschitz invariance property.
Corollary 1.3
Suppose that \(\varphi \) is a bi-Lipschitz map from \({\tilde{X}}\) onto X and that both \(\mu \) and \({\tilde{\mu }}\) are d-regular and of full support. Let \(\alpha \in (0,2)\) and assume that j is as in (52) and \({\tilde{j}}\), too, but with \({\tilde{\varrho }}\) in place of \(\varrho \).
-
(i)
If \({\mathcal {C}}\) satisfies Assumptions 4.1 (ii) and 4.2, then so does \(\varphi ^*{\mathcal {C}}\).
-
(ii)
If \(\tilde{{\mathcal {C}}}\) satisfies Assumptions 4.1 (ii), 4.2 and 5.2, \(N_*\) is a system of diagonal neighborhoods for X and \(\varphi ^*{\mathcal {C}}\) is dense in \(\tilde{{\mathcal {C}}}\) with respect to \(\left\| \cdot \right\| _{\tilde{{\mathcal {D}}}(\delta _0,\varphi ^*N_0)}\), then the norm estimates (56) hold and the cochain complexes \({\mathcal {D}}(\delta _*,N_*)\) and \(\tilde{{\mathcal {D}}}(\delta _*,\varphi ^*N_*)\) are isomorphic.
Proof
To verify (55) one can use the fact that \(\mu \) is comparable to the d-dimensional Hausdorff measure, [52, Exercise 8.11], the other hypotheses of Theorem 5.2 are easily seen. \(\square \)
5.4 Non-local Hodge Laplacians
The next result on associated coderivations \(\delta _p^*\) follows from general theory, [96, Theorem 5.3].
Corollary 1.4
Let Assumptions 4.1, 4.2, 5.1 and 5.2 be satisfied. Let \(N_*\) be a system of diagonal neighborhoods and let \(p\ge 0\) be an integer. The adjoint \((\delta _p^*,{\mathcal {D}}(\delta _p^*,N_{p+1}))\) of \((\delta _p,{\mathcal {D}}(\delta _p,N_p))\) is a densely defined closed linear operator from \(L^2_a(N_{p+1}\setminus D_{p+1},J_{p+1})\) into \(L^2_a(N_{p}\setminus D_p,J_p)\). It is characterized by the identity
and satisfies
Moreover, \(({\mathcal {Q}}_p^*, {\mathcal {D}}(\delta _{p}^*,N_{p+1}))\), defined by
is a densely defined closed quadratic form on \(L^2(N_{p+1}{\setminus } D_{p+1},J_{p+1})\).
The complex
is the dual complex of (45).
Example 5.2
Recall (51). Given \(f\in {\mathcal {D}}(\delta _0,N_0)\) we have \(\delta _0 f\in {\mathcal {D}}(\delta _0^*,N_1)\) if and only if \(f\in {\mathcal {D}}({\mathcal {L}}_0)\). In this case, we have \({\mathcal {L}}_0f=-\delta _0^*\delta _0 f\).
As usual we set \(\delta _{-1}:=0\); then also \(\delta _{-1}^*=0\). For any integer \(p\ge 0\) let
The following weak Hodge decomposition, [40, 66], into orthogonal closed subspaces is straightforward, see [16, Lemma 2.1].
Corollary 1.5
Let Assumptions 4.1, 4.2, 5.1 and 5.2 be satisfied. Let \(N_*\) be a system of diagonal neighborhoods. For any integer \(p\ge 0\) the space \(L^2_a(N_{p}\setminus D_p,J_p)\) admits the orthogonal decomposition
here the closures are taken with respect to the norm in \(L^2(N_{p}\setminus D_p,J_p)\).
The preceding can be used to introduce non-local analogs of Hodge Laplacians. Recall (50) and (59). For any integer \(p\ge 0\) consider the subspace
of \(L^2_a(N_{p}\setminus D_p,J_p)\) and set
The quadratic forms in (61) may be viewed as non-local analogs of the Dirichlet integrals for differential forms, [40]. The following is a consequence of Corollaries 5.1 and 5.4.
Proposition 1.7
Let Assumptions 4.1, 4.2, 5.1 and 5.2 be satisfied and let \(N_*\) be a system of diagonal neighborhoods. The quadratic form \(({\mathbb {D}}_p,{\mathcal {D}}({\mathbb {D}}_p,N_p))\) is closed and densely defined.
By polarization \({\mathbb {D}}_p\) may be seen as a closed and densely defined nonnegative definite symmetric bilinear form on \(L^2_a(N_p\setminus D_p,J_p)\). For any \(p\ge 0\) let \(({\mathbb {L}}_p,{\mathcal {D}}({\mathbb {L}}_p))\) be the unique nonnegative definite self-adjoint operator associated with \(({\mathbb {D}}_p,{\mathcal {D}}({\mathbb {D}}_p))\) in the sense that
The operator \(({\mathbb {L}}_p,{\mathcal {D}}({\mathbb {L}}_p))\) may be viewed as a non-local analog of the Hodge Laplacian, [40, 66]. Formally, it satisfies the identity
Note that for \(p=0\) we have \({\mathbb {L}}_0=-{\mathcal {L}}_0\) with \({\mathcal {L}}_0\) as defined in (51).
Proof of Proposition 5.4
Note first that by Proposition 5.2 and Corollary 5.4 the forms \(({\mathcal {Q}}_p, {\mathcal {D}}(\delta _{p},N_p))\) and \(({\mathcal {Q}}_{p-1}^*, {\mathcal {D}}(\delta _{p-1}^*,N_p))\) are densely defined closed quadratic forms. Their closedness implies that also \(({\mathbb {D}}_p,{\mathcal {D}}({\mathbb {D}}_p,N_p))\) must be closed.
It remains to show that \({\mathcal {D}}({\mathbb {D}}_p,N_p)\) is dense in \(L^2_a(N_p\setminus D_p,J_p)\). To see this, note first that the nonnegative definite self-adjoint operators uniquely associated with \({\mathcal {Q}}_p^*\) and \({\mathcal {Q}}_{p-1}\) are
respectively. In particular, \({\mathcal {D}}(\delta _{p}\delta _{p}^*,N_{p+1})\) is dense in the Hilbert space \({\mathcal {D}}(\delta _{p}^*,N_{p+1})\) and \({\mathcal {D}}(\delta _{p-1}^*\delta _{p-1},N_{p-1})\) is dense in the Hilbert space \({\mathcal {D}}(\delta _{p-1},N_{p-1})\). This implies that
with respect to the norm in \(L^2(N_p\setminus D_p,J_p)\).
Given \(G\in {\mathcal {D}}(\delta _{p}^*,N_{p+1})\), the co-exact function \(\delta _{p}^*G\) is in \({\mathcal {D}}(\delta _{p},N_p)\) if and only if G is in \({\mathcal {D}}(\delta _{p}\delta _{p}^*,N_{p+1})\). Given \(G\in {\mathcal {D}}(\delta _{p-1},N_{p-1})\), the exact function \(\delta _{p-1} G\) is in \({\mathcal {D}}(\delta _{p-1}^*,N_p)\) if and only if G is in \({\mathcal {D}}(\delta _{p-1}^*\delta _{p-1},N_{p-1})\). Together with (44) and (58) this implies that
Now suppose that \(F\in L^2_a(N_p\setminus D_p,J_p)\) is such that
Since obviously \({\mathbb {H}}_p\subset {\mathcal {D}}({\mathbb {D}}_p,N_p)\), we have
By (63) also
and
By (62) this means that
The combination of (60), (64) and (65) shows that \(F=0\) in \(L^2_a(N_p{\setminus } D_p,J_p)\). \(\square \)
5.5 Approximation results
The following estimate complements (34).
Lemma 1.4
Let Assumptions 4.1, 4.2 and 5.2 be satisfied and let \(N_*\) be a system of diagonal neighborhoods. Then for any integer \(p\ge 1\) and any \(g\in {\mathcal {C}}\oplus {\mathbb {R}}\), \(f_1,\ldots ,f_p\in {\mathcal {C}}\) we have
Proof
As in (38) we can use the symmetry of \(({\overline{g}}\delta _{p-1}{{\,\textrm{Alt}\,}}_p(f_1\otimes \cdots \otimes f_p))^2\) to see that the left-hand side of (66) equals
the estimate uses (11) and the triangle inequality. Similarly as in Lemma 4.1 it follows that the summand for \(k=0\) does not exceed
For fixed \(k\ne 0\) the corresponding summand is bounded by
The first of these two terms admits the estimate (67) as before. The square of the second is bounded by
Using Cauchy–Schwarz with respect to the variables \(x_1,\ldots ,{\hat{x}}_k,\ldots ,x_p\) in the inner integral and Fubini, we can bound the preceding by
\(\square \)
For certain applications it can be convenient to restrict attention to smaller cores. We have the following approximation result for the norms induced by (49).
Corollary 1.6
Let Assumptions 4.1, 4.2 and 5.2 be satisfied and let \(N_*\) be a system of diagonal neighborhoods. Let \({\mathcal {C}}'\) be a subalgebra of \({\mathcal {C}}\), dense in \({\mathcal {C}}\) with respect to \(\Vert \cdot \Vert _{{\mathcal {D}}(\delta _0,N_0)}\). Then for any integer \(p\ge 0\) the space \({\mathcal {C}}'^p(N_p)\) is dense in \({\mathcal {C}}^p(N_p)\) with respect to \(\Vert \cdot \Vert _{{\mathcal {D}}(\delta _p,N_p)}\).
Proof
For \(p=0\) the result is immediate. Suppose that \(p\ge 1\) and let
be an element of \({\mathcal {C}}^p(N_p)\). Let \((g_n^{(i)})_n\) and \((f_{j,n_j}^{(i)})_{n_j}\) be sequences from \({\mathcal {C}}'\) such that for all \(i=1,\ldots ,m\) and \(j=1,\ldots ,p\) we have \(\lim _n \Vert g_n^{(i)}-g^{(i)}\Vert _{L^2(X,\mu )}=0\) and \(\lim _{n_j}\Vert f_{j,n_j}^{(i)}-f_j^{(i)}\Vert _{{\mathcal {D}}(\delta _0,N_0)}=0\). Consider the elementary p-functions \(F_{n,n_1,\ldots ,n_p}\in {\mathcal {C}}'^p(N_p)\) defined by
We claim that
For fixed i we have
The first summand goes to zero as \(n\rightarrow \infty \) by Lemma 5.1. For fixed n the second summand goes to zero as \(n_1\rightarrow \infty \) by Lemma 4.1 (ii), and so on. Similar estimates hold for
and using the triangle inequality we arrive at (68). \(\square \)
5.6 Removable sets
In this subsection we assume that \((X,\varrho )\) is locally compact and separable.
The Hilbert complex \(({\mathcal {D}}(\delta _*,N_*),\delta _*)\) in (45) was obtained from the elementary complex \(({\mathcal {C}}^*(N_*),\delta _*)\) in (17) by taking closures. Now suppose that \(\Sigma \subset X\) is a closed set and let \(\mathring{X}:=X\setminus \Sigma \). Let \(\mathring{{\mathcal {C}}}\) denote the ideal in \({\mathcal {C}}\) consisting of all elements whose support is contained in the open set \(\mathring{X}\) and let \(\mathring{{\mathcal {C}}}^p(N_p)\) be defined as in (19) but with \(\mathring{{\mathcal {C}}}\) in place of \({\mathcal {C}}\). Proposition 3.1 remains true and \((\mathring{{\mathcal {C}}}^*(N_*),\delta _*)\) is a subcomplex of \(({\mathcal {C}}^*(N_*),\delta _*)\). If \(\Sigma \) is of \(\mu \)-measure zero, then \(\mu ^{p+1}\) in (42) can be replaced by \((\mu |_{\mathring{X}})^{p+1}\) without changing \(J_p\), and taking the closures of the operators \((\delta _p,\mathring{{\mathcal {C}}}^p(N_p))\) in the spaces \(L^2(N_p\setminus D_p,J_p)\) gives a Hilbert complex \((\mathring{{\mathcal {D}}}(\delta _*,N_*),\delta _*)\). Clearly it is a subcomplex of \(({\mathcal {D}}(\delta _*,N_*),\delta _*)\). One expects that for small enough \(\Sigma \) the entire complexes coincide,
in the sense that for all integers \(p\ge 0\) we have
For too large \(\Sigma \) they should differ. This can be discussed in terms of capacities.
Given \(K\subset X\) compact, let
with the convention that \(\inf \emptyset :=+\infty \). For general \(E\subset X\) we set
The following is related to [16, Theorem 4.4]. For the order \(p=0\) the result is classical, [74], and by Corollary 5.6 the order \(p=0\) determines whether (69) holds or not.
Theorem 1.3
Suppose that \({\mathcal {C}}\subset C_c(X)\), Assumptions 4.1, 4.2 and 5.2 are satisfied and \(\Sigma \subset X\) is closed. Then \({{\,\textrm{cap}\,}}_{N_0}(\Sigma )=0\) if and only if \(\mu (\Sigma )=0\) and (69) holds.
The proof follows a standard pattern, we provide it for convenience.
Proof
We write \({{\,\textrm{cap}\,}}:={{\,\textrm{cap}\,}}_{N_0}\). Suppose that \(\mu (\Sigma )=0\) and (69) holds. By (72) we can find a sequence \((\Sigma _i)_i\) of compact sets \(\Sigma _i\subset \Sigma \) such that
Let i be fixed. By (70) with \(p=0\) any \(u\in {\mathcal {C}}\) with \(u=1\) on a neighborhood of \(\Sigma _i\) can be approximated in \(\Vert \cdot \Vert _{{\mathcal {D}}(\delta _0,N_0)}\) by a sequence \((u_n)_n\subset \mathring{{\mathcal {C}}}\), and this implies that \({{\,\textrm{cap}\,}}(\Sigma _i)\le \lim _n \Vert u-u_n\Vert _{{\mathcal {D}}(\delta _0,N_0)}^2=0\). Now \({{\,\textrm{cap}\,}}(\Sigma )=0\) follows using (73).
If \({{\,\textrm{cap}\,}}(\Sigma )=0\), then by the inner regularity of \(\mu \), (71) and since \(\Vert \cdot \Vert _{L^2(X,\mu )}\le \Vert \cdot \Vert _{{\mathcal {D}}(\delta _0, N_0)}\) we must have \(\mu (\Sigma )=0\). Clearly \(\mathring{{\mathcal {D}}}(\delta _p,N_p)\subset {\mathcal {D}}(\delta _p,N_p)\). We claim that \(\mathring{{\mathcal {C}}}\) is dense in \({\mathcal {C}}\) with respect to \(\Vert \cdot \Vert _{{\mathcal {D}}(\delta _0, N_0)}\). If so, then Corollary 5.6 implies that for any integer \(p\ge 0\) the space \(\mathring{{\mathcal {C}}}^p(N_p)\) is dense in \({\mathcal {D}}(\delta _p,N_p)\), and this proves (70). To see the claim, let \(f\in {\mathcal {C}}\). Then \(\Sigma _f:=\Sigma \cap {{\,\textrm{supp}\,}}f\) is compact and \({{\,\textrm{cap}\,}}(\Sigma _f)\le {{\,\textrm{cap}\,}}(\Sigma )=0\). By (71) there is a sequence \((u_n)_n\subset {\mathcal {C}}\) such that \(\lim _n\Vert u_n\Vert _{{\mathcal {D}}(\delta _0,N_0)}=0\) and \(u_n=1\) on a neighborhood of \(\Sigma _f\), respectively. The functions \(f_n:=(1-u_n)f\) are in \(\mathring{{\mathcal {C}}}\), and \(\lim _n \Vert f-f_n\Vert _{{\mathcal {D}}(\delta _0,N_0)}=\lim _n\Vert f u_n\Vert _{{\mathcal {D}}(\delta _0,N_0)}=0\) since \(\Vert u_n f\Vert _{L^2(X,\mu )}\le \Vert f\Vert _{\sup }\Vert u_n\Vert _{L^2(X,\mu )}\) and
\(\square \)
In applications one can start from a non-local Dirichlet form as in Example 4.2. Recall that if \(({\mathcal {E}},{\mathcal {D}}({\mathcal {E}}))\) is a regular Dirichlet form on \(L^2(X,\mu )\), [39], then the associated capacity \({{\,\textrm{Cap}\,}}^{\mathcal {E}}\) is defined by
with \(\inf \emptyset :=+\infty \) for \(A\subset X\) open and by
for general \(E\subset X\). Here \(u\mapsto \left\| u\right\| _{{\mathcal {D}}({\mathcal {E}})}:=({\mathcal {E}}(u)+\left\| u\right\| _{L^2(X,\mu )}^2)^{1/2}\) denotes the norm in the Hilbert space \({\mathcal {D}}({\mathcal {E}})\). See [39, Sect. 2.1].
Lemma 1.5
Let \(({\mathcal {E}},{\mathcal {D}}({\mathcal {E}}))\) be a purely non-local regular Dirichlet form on \(L^2(X,\mu )\) with representation
for all \(f\in {\mathcal {D}}({\mathcal {E}})\), where j is a symmetric density. Assume that (37) holds and that \({\mathcal {C}}:={{\,\textrm{Lip}\,}}_c(X)\) is dense in \({\mathcal {D}}({\mathcal {E}})\). Let \(\varepsilon >0\) and let \(N_*=N_*(\varepsilon )\) be as in (13).
-
(i)
We have \({\mathcal {D}}(\delta _0,N_0)={\mathcal {D}}({\mathcal {E}})\) and
$$\begin{aligned} {\mathcal {Q}}_0(f)\le {\mathcal {E}}(f)\le {\mathcal {Q}}_0(f)+4\left\| f\right\| _{L^2(X,\mu )}^2\sup _{x\in X}\int \limits _{B(x,\varepsilon )^c}j(x,y)\mu (dy), \quad f\in {\mathcal {D}}({\mathcal {E}}). \end{aligned}$$ -
(ii)
There is a constant \(c(\varepsilon )>1\) such that for any Borel set \(E\subset X\) we have
$$\begin{aligned} {{\,\textrm{cap}\,}}_{N_0(\varepsilon )}(E)\le {{\,\textrm{Cap}\,}}^{\mathcal {E}}(E)\le c(\varepsilon )\,{{\,\textrm{cap}\,}}_{N_0(\varepsilon )}(E). \end{aligned}$$(76)
Let
Remark 5.5
To \(T_1\) one refers as the unit contraction, [4, 10, 11, 39, 70]. Clearly \(u\in {{\,\textrm{Lip}\,}}_c(X)\) implies \(T_1(u)\in {{\,\textrm{Lip}\,}}_c(X)\).
Proof
Item (i) follows from [48, Proposition 4.1]. For (ii), note that since \({{\,\textrm{Cap}\,}}^{\mathcal {E}}\) is inner regular, [39, Theorem 2.1.1], it suffices to verify (76) for compact K. Let \(K\subset X\) be compact and \(\eta >0\). By (71) we can find \(u\in {\mathcal {C}}\) such that \(u=1\) on a neighborhood of K and \(\left\| u\right\| _{{\mathcal {D}}(\delta _0,N_0)}^2<{{\,\textrm{cap}\,}}_{N_0}(K)+\eta \). By (i) there is a constant \(c(\varepsilon )>0\) such that \(\left\| u\right\| _{{\mathcal {D}}({\mathcal {E}})}^2\le c(\varepsilon )\,\left\| u\right\| _{{\mathcal {D}}(\delta _0,N_0)}^2\). Therefore \({{\,\textrm{Cap}\,}}^{\mathcal {E}}(K)\le \left\| u\right\| _{{\mathcal {D}}({\mathcal {E}})}^2\le c(\varepsilon )\,({{\,\textrm{cap}\,}}_{N_0}(K)+\eta )\), and letting \(\eta \rightarrow 0\) gives the inequality on the right-hand side of (76) for K in place of E. To see the inequality on the left-hand side, let K again be compact. Since \({\mathcal {C}}={{\,\textrm{Lip}\,}}_c(X)\) is a special standard core, [39, Lemma 2.2.7 (ii)] gives
and we may replace \(\ge \) by >. For any \(\eta >0\) we can therefore find some \(u\in {\mathcal {C}}\) with \(u>1\) on a neighborhood U of K and \(\left\| u\right\| _{{\mathcal {D}}({\mathcal {E}})}^2<{{\,\textrm{Cap}\,}}^{\mathcal {E}}(K)+\eta \). Clearly \(T_1(u)=1\) on U, \(T_1(u)\in {\mathcal {C}}\) and \(\left\| T_1(u)\right\| _{{\mathcal {D}}({\mathcal {E}})}\le \left\| u\right\| _{{\mathcal {D}}({\mathcal {E}})}\), and by (i) therefore
Letting \(\eta \rightarrow 0\) gives the left part of (76). \(\square \)
Remark 5.6
The expression (75) makes sense for all \(f\in L^2(X,\mu )\). The norm \(\Vert \cdot \Vert _{{\mathcal {D}}({\mathcal {E}})}\) extends to the maximal domain \(\{f\in L^2(X,\mu ): {\mathcal {E}}(f)<+\infty \}\) of \({\mathcal {E}}\) and makes it a Hilbert space.
Recall that the 1-resolvent of a Dirichlet form \(({\mathcal {E}},{\mathcal {D}}({\mathcal {E}}))\) is the bounded linear operator \(G_1^{\mathcal {E}}\) on \(L^2(X,\mu )\) uniquely determined by
We say that \(G_1^{\mathcal {E}}\) has a pointwise defined density \(g_1^{\mathcal {E}}\) (with respect to \(\mu \)) if \(g_1^{\mathcal {E}}\) is a Borel function from \(X^2\) to \((0,+\infty ]\) and for all \(f\in L^2(X,\mu )\) we have
If metric estimates for \(g_1^{\mathcal {E}}\) are available, the critical size of \(\Sigma \) can be characterized in terms of Hausdorff measures. Given \(s\ge 0\), we write \({\mathcal {H}}^s\) to denote the s-dimensional Hausdorff measure, and given a Hausdorff function h, we write \({\mathcal {H}}^h\) for the corresponding generalized Hausdorff measure, see [2, Sect. 5.1], [38, Sect. 2.10] or [74, Sect. 7.2.3].
Corollary 1.7
Assume that X is complete. Let \(\mu \) be a nonnegative Radon measure on X with full support and d-regular as in (57). Let \(\alpha \in (0,2)\) and let j be a symmetric kernel satisfying (52). Let \({\mathcal {E}}\) be as in (75), and let \({\mathcal {D}}({\mathcal {E}})\) be the closure of \({{\,\textrm{Lip}\,}}_c(X)\) in the maximal domain of \({\mathcal {E}}\). Then \(({\mathcal {E}},{\mathcal {D}}({\mathcal {E}}))\) is a purely non-local regular Dirichlet form and its 1-resolvent admits a pointwise defined density \(g_1^{\mathcal {E}}(x,y)\). If in addition \(\varepsilon >0\) and \(N_*=N_*(\varepsilon )\) is as in (13), then we observe the following for a closed set \(\Sigma \subset X\) of measure zero:
-
(i)
For \(d<\alpha \) the density \(g_1^{\mathcal {E}}(x,y)\) is bounded on X. In this case (69) holds if and only if \(\Sigma =\emptyset \).
-
(ii)
For \(d>\alpha \) the density \(g_1^{\mathcal {E}}(x,y)\) is bounded by \(c\varrho (x,y)^{\alpha -d}\) on X and lower bounded by \(c^{-1}\varrho (x,y)^{\alpha -d}\) on \(\{\varrho (x,y)<1\}\), where \(c>1\) is a universal constant. In this case \({\mathcal {H}}^{d-\alpha }(\Sigma )<+\infty \) implies (69); and if (69) holds, we have \({\mathcal {H}}^{d-\alpha +\eta }(\Sigma )=0\) for any \(\eta >0\).
-
(iii)
For \(d=\alpha \) the density \(g_1^{\mathcal {E}}(x,y)\) is comparable to \((1+(-\log \varrho (x,y))_+)\) on \(\{\varrho (x,y)<1\}\). In this case \({\mathcal {H}}^{h}(\Sigma )<+\infty \) with \(h(r)=(1+(-\log r)_+)^{-1}\) implies (69); and if X is bounded and (69) holds, we have \({\mathcal {H}}^{k}(\Sigma )=0\) for any Hausdorff function k satisfying \(\int _0^1k(r)\frac{dr}{r}<+\infty \).
Proof
By [26, Proposition 2.2] the Dirichlet form \(({\mathcal {E}},{\mathcal {D}}({\mathcal {E}}))\) is regular. By [47, Theorem 1.12] and the remarks following it \(({\mathcal {E}},{\mathcal {D}}({\mathcal {E}}))\) admits a heat kernel that obeys typical two-sided estimates. Standard calculations and well-known potential theoretic arguments then imply (i), (ii) and (iii), see [2, 39, 74]. \(\square \)
Remark 5.7
Since X is assumed to be complete, we may apply Frostman’s lemma to go from zero capacity to zero Hausdorff measure in Corollary 5.7. The other implication (finite Hausdorff measure giving zero capacity) does not need completeness.
Remark 5.8
Lemma 5.2 and Corollary 5.7 can also be adapted to \({\hat{N}}_*={\hat{N}}_*(\varepsilon )\) as in (14).
Example 5.3
Corollary 5.7 applies in particular if \(0<d\le n\) and X is a compact d-set in \({\mathbb {R}}^n\), [25].
Another class of examples can be obtained by subordination, [9, 41, 63]. Suppose that \((P_t)_{t>0}\) is a strongly continuous Markov semigroup on \(L^2(X,\mu )\), symmetric in the sense that
for all \(t>0\) and \(f,g\in L^2(X,\mu )\), [14, 39]. We say that \((P_t)_{t>0}\) has a heat kernel \(p_t(x,y)\) (with respect to \(\mu \)) if \((t,x,y)\mapsto p_t(x,y)\) is a real valued function on \((0,+\infty )\times X^2\), Borel measurable in (x, y) for each \(t>0\) and such that for all \(t>0\) and \(f\in L^2(X,\mu )\) we have
Example 5.4
Suppose that \(\mu \) is d-regular and \((P_t)_{t>0}\) is a symmetric strongly continuous Markov semigroup on \(L^2(X,\mu )\) with heat kernel \(p_t(x,y)\) admitting two-sided Gaussian estimates of the form
here \(c_1>1\) and \(c_2>1\) are fixed constants. Given \(\alpha \in (0,2)\) let
denote the Lévy measure of the strictly \(\alpha /2\)-stable subordinator on \((0,\infty )\), see [9, Sect. 3.1] or [63, Example 3.9.16]; here 
denotes the Euler Gamma function. The Lévy jump density
satisfies (52) and in particular, fits Assumption 4.1 (i) and Assumption 5.2. Let \({\mathcal {E}}\) be the form having representation (75) with this density j and endowed with the natural domain \({\mathcal {D}}({\mathcal {E}})\) defined using \((P_t)_{t>0}\) and spectral theory. The heat kernel of \(({\mathcal {E}},{\mathcal {D}}({\mathcal {E}}))\) then is
where \(q_t^{\alpha /2}\) denotes the law of the strictly \(\alpha /2\)-stable subordinator at time t, and the behavior of \(g_1^{\mathcal {E}}\) can be read off from
using the estimates (78) for \(p_t(x,y)\). If in addition \({{\,\textrm{Lip}\,}}_c(X)\) is dense in \({\mathcal {D}}({\mathcal {E}})\), then Corollary 5.7 applies. This is, for instance, the case if \(X=M\) is a complete smooth Riemannian manifold of dimension d or X is an \(\textrm{RCD}^*(0,d)\) space, [5, 37, 43].
6 Covers and cohomology
Throughout this section \((X,\varrho )\) is a compact metric space and \({\mathcal {C}}\) is a subalgebra of C(X).
6.1 Partitions of unity
Let I be a finite ordered set of cardinality |I|. Suppose that \({\mathcal {U}}=\{U_\alpha \}_{\alpha \in I}\) is an open cover of X and \({\mathcal {V}}=\{V_\alpha \}_{\alpha \in I}\) is a shrinking of \({\mathcal {U}}\) in the sense that \({\mathcal {V}}\) is an open cover of X and for any \(\alpha \in I\) we have \({\overline{V}}_\alpha \subset U_\alpha \). Consider the system \(N_*({\mathcal {V}})=(N_p({\mathcal {V}}))_{\alpha \in I}\) of diagonal neighborhoods
as in (15) and suppose that \(N_*=(N_p)_{p\ge 0}\) is a system of diagonal neighborhoods such that \(N_*\prec N_*({\mathcal {V}})\). For each integer \(p\ge 0\) let
note that since \(N_0=X\), we have \(U_\alpha ^{(1)}=U_\alpha \), \(\alpha \in I\). The family
is a finite open cover of \(N_p\).
We make an assumption on the existence of bump functions in \({\mathcal {C}}\).
Assumption 1.5
For any open set \(U\subset X\) and any compact \(K\subset U\) we can find some \(f\in {\mathcal {C}}\) with \(0\le f\le 1\), \({{\,\textrm{supp}\,}}f\subset U\) and \(f=1\) on K.
Remark 6.1
By Stone–Weierstrass Assumption 6.1 implies Assumption 5.1. If \({\mathcal {C}}\) is stable under composition with the unit contraction \(T_1\) as defined in (77), then Assumption 5.1 also implies Assumption 6.1.
Under Assumption 6.1 one can find related symmetric partitions of unity.
Lemma 1.6
Suppose that Assumption 6.1 holds. Let \({\mathcal {V}}\), \({\mathcal {U}}\) and \(N_*\) be as above. Then for each integer \(p\ge 0\) there is a partition of unity \(\{\chi _{p,\alpha }\}_{\alpha \in I}\) on \(N_p\) subordinate to \({\mathcal {U}}^{(p+1)}\) and such that each \(\chi _{p,\alpha }\) is a finite linear combination of functions of form \(\chi ^{\otimes (p+1)}\) with \(\chi \in {\mathcal {C}}\) and, in particular, \(\chi _{p,\alpha }\in {\mathcal {C}}^{s,p}\).
Proof
We use a variant of standard arguments, [46, Theorem 2.2]: Assumption 6.1 ensures that for each \(\alpha \in I\) we can find \(\varphi _\alpha \in {\mathcal {C}}\) such that \(0\le \varphi _\alpha \le 1\), \({{\,\textrm{supp}\,}}\varphi _\alpha \subset U_\alpha \) and \(\varphi _\alpha \equiv 1\) on \(V_\alpha \). Given an integer \(p\ge 0\) and \(\alpha \in I\) we write
for the \((p+1)\)-fold tensor product of \(\varphi _\alpha \) with itself. Then
Writing \(I=\{\alpha _0,\ldots ,\alpha _{|I|-1}\}\), we consider the functions
on \(X^{p+1}\); note that \({\textbf{1}}^{\otimes (p+1)}\equiv 1\). All functions \(\chi _{p,\alpha }\) are finite linear combinations of \((p+1)\)-fold tensor powers of functions from \({\mathcal {C}}\), they satisfy \(0\le \chi _{p,\alpha }\le 1\) and \({{\,\textrm{supp}\,}}\chi _{p,\alpha }\subset U_{\alpha }^{p+1}\). We have
and by the last item in (81) the right-hand side of this equality is zero. It follows that \(\sum _{\alpha \in I}\chi _{p,\alpha }\equiv 1\). Seen as a function on \(N_p\), each \(\chi _{p,\alpha }\) has support in \(U_{\alpha }^{(p+1)}\). \(\square \)
6.2 Mayer–Vietoris sequences
As in the classical case of the deRham complex on a smooth manifold, [13, Chap. II, Sect. 8], one can obtain generalized Mayer–Vietoris sequences. Let \({\mathcal {V}}\), \({\mathcal {U}}\) and \(N_*\) be as above.
It is not difficult to see that for each \(\alpha \in I\), the family \((U_\alpha ^{(p+1)})_{p\ge 0}\) is a system of diagonal neighborhoods for \(U_\alpha \); this is a slight variation of Examples 3.1 (i). More generally, given an integer \(q\ge 0\) and distinct \(\alpha _0,\ldots ,\alpha _q\in I\), we write
note that \(U_{\alpha _0\cdots \alpha _q}^{(p+1)}=U_{\alpha _0\cdots \alpha _q}^{p+1}\cap N_{p}\) and in particular, \(U_{\alpha _0\cdots \alpha _q}^{(1)}=U_{\alpha _0\cdots \alpha _q}\). The family \((U_{\alpha _0\cdots \alpha _q}^{(p+1)})_{p\ge 0}\) is a system of diagonal neighborhoods for \(U_{\alpha _0\cdots \alpha _q}\). We continue to use the shortcut notation (16), that is, \({\mathcal {C}}^p(U_{\alpha _0\cdots \alpha _q}^{(p+1)}):={\mathcal {C}}^p|_{U_{\alpha _0\cdots \alpha _q}^{(p+1)}}\). By Proposition 3.1 the sequence
is a cochain complex.
Now suppose that also Assumptions 4.1, 4.2 and 5.2 hold. For any integer \(p\ge 0\) let \({\mathcal {D}}(\delta _p,U_{\alpha _0\cdots \alpha _q}^{(p+1)})\) be the domain of the closure of \((\delta _p, {\mathcal {C}}^p(U_{\alpha _0\cdots \alpha _q}^{(p+1)}))\) in \(L^2(U_{\alpha _0\cdots \alpha _q}^{(p+1)}\setminus D_p,J_p)\) as constructed in Theorem 5.1; the closability follows similarly as there. Then also the sequence
is a cochain complex.
Let r be the natural restriction that takes an element F of \({\mathcal {C}}^p(N_p)\) into the element \(rF=((rF)_\alpha )_{\alpha \in I}\) of the direct product \(\prod _{\alpha \in I} {\mathcal {C}}^p(U_\alpha ^{(p+1)})\) with components
note that \({\mathcal {C}}^p(N_p)|_{U_{\alpha }^{(p+1)}}\subset {\mathcal {C}}^p(U_{\alpha }^{(p+1)})\) by (16). Assume that < is a strict order on I. Given an ordered subset \(\alpha _0<\dots <\alpha _q\) of I and an element F of the direct product \(\prod _{\alpha _0<\dots <\alpha _q}{\mathcal {C}}^p(U_{\alpha _0\cdots \alpha _q}^{(p+1)})\) with components \(F_{\alpha _0\dots \alpha _q}\in {\mathcal {C}}^p(U_{\alpha _0\cdots \alpha _q}^{(p+1)})\), we define an element \({\check{\delta }}F=(({\check{\delta }}F)_{\alpha _0\dots \alpha _{q+1}}))_{\alpha _0<\dots <\alpha _{q+1}}\) of \(\prod _{\alpha _0<\dots <\alpha _{q+1}}{\mathcal {C}}^p(U_{\alpha _0\cdots \alpha _{q+1}}^{(p+1)})\) component-wise by
It is quickly seen that \({\check{\delta }}\circ {\check{\delta }}=0\). Under Assumptions 4.1, 4.2 and 5.2 we can apply the operators r and \({\check{\delta }}\) similarly to the spaces \({\mathcal {D}}(\delta _p,N_p)\) and \(\prod _{\alpha _0<\dots <\alpha _q}{\mathcal {D}}(\delta _p,U_{\alpha _0 \cdots \alpha _q}^{(p+1)})\). Note that by (16) and Theorem 5.1 we have \({\mathcal {D}}(\delta _p,N_p)|_{U_{\alpha }^{(p+1)}}\subset {\mathcal {D}}(\delta _p,U_{\alpha }^{(p+1)})\) and so on.
With the convention that the interchange of two indices provokes a change of sign,
we may drop the requirement that \(\alpha _0<\dots <\alpha _q\), see [13, Exercise 8.4].
Proposition 1.8
Let Assumption 6.1 be in force. Let \({\mathcal {V}}\), \({\mathcal {U}}\) and \(N_*\) be as above.
-
(i)
For any integer \(p\ge 0\) the sequence
$$\begin{aligned} 0\longrightarrow {\mathcal {C}}^p(N_p) {\mathop {\longrightarrow }\limits ^{r}}\prod _{\alpha _0} {\mathcal {C}}^p(U_{\alpha _0}^{(p+1)}) {\mathop {\longrightarrow }\limits ^{{\check{\delta }}}}\prod _{\alpha _0<\alpha _1} {\mathcal {C}}^p(U_{\alpha _0\alpha _1}^{(p+1)}) {\mathop {\longrightarrow }\limits ^{{\check{\delta }}}} \cdots \end{aligned}$$(85)is exact.
-
(ii)
Suppose that also Assumptions 4.1, 4.2 and 5.2 hold. Then for any integer \(p\ge 0\) the sequence
$$\begin{aligned} 0\longrightarrow {\mathcal {D}}(\delta _p,N_p){\mathop {\longrightarrow }\limits ^{r}}\prod _{\alpha _0} {\mathcal {D}}(\delta _p,U_{\alpha _0}^{(p+1)}){\mathop {\longrightarrow }\limits ^{{\check{\delta }}}}\prod _{\alpha _0<\alpha _1} {\mathcal {D}}(\delta _p,U_{\alpha _0\alpha _1}^{(p+1)}){\mathop {\longrightarrow }\limits ^{{\check{\delta }}}} \cdots \end{aligned}$$(86)is exact.
For convenience we provide a suitable variant of the well-known classical arguments.
Proof
The injectivity of r in (85) and (86) is clear. Let \(F=(F_\alpha )_{\alpha \in I}\in \prod _\alpha {\mathcal {C}}^p(U_\alpha ^{(p+1)})\) be such that \({\check{\delta }}F=0\). Then \(F_{\alpha _0}=F_{\alpha }\) on \(U_{\alpha \alpha _0}^{(p+1)}\) for any distinct \(\alpha ,\alpha _0 \in I\). Let \(\chi _{p,\alpha }\), \(\alpha \in I\), be as in Lemma 6.1 and set \(G:=\sum _{\alpha \in I}\chi _{p,\alpha } F_\alpha \). Since each \(F_\alpha \) is a restriction to the corresponding open set \(U_\alpha ^{(p+1)}\) of an element of \({\mathcal {C}}^p\) and \(\chi _{p,\alpha }\) is in \({\mathcal {C}}^{s,p}\) and supported in \(U_\alpha ^{(p+1)}\), we have \(\chi _{p,\alpha } F_\alpha \in {\mathcal {C}}^p(N_p)\) by (24) and the comments following it. Consequently \(G\in {\mathcal {C}}^p(N_p)\). It is quickly seen that \(rG=F\), and we can conclude that \({{\,\mathrm{\textrm{im}}\,}}r=\ker {\check{\delta }}\) for \({\check{\delta }}\) acting along the third arrow in (85). Under the hypotheses in (ii) the same argument applies to \(F=(F_\alpha )_{\alpha \in I}\in \prod _\alpha {\mathcal {D}}(\delta _p,U_\alpha ^{(p+1)})\), because for each \(\alpha \) we have \(\chi _{p,\alpha }F_\alpha \in {\mathcal {D}}(\delta _p,N_p)\) by Corollary 5.2. If \(q\ge 1\) and \(F=(F_{\alpha _0\cdots \alpha _q})_{\alpha _0< \cdots <\alpha _q}\) is an element of \(\prod _{\alpha _0< \cdots <\alpha _q}{\mathcal {C}}^p(U_{\alpha _0 \ldots \alpha _q}^{(p+1)})\) satisfying \({\check{\delta }}F=0\), then
for any distinct \(\alpha , \alpha _0, \ldots \alpha _q\in I\). Let \(G=(G_{\alpha _0\cdots \alpha _{q-1}})_{\alpha _0< \cdots <\alpha _{q-1}}\) be defined by
Again by (24) this defines an element G of \(\prod _{\alpha _0< \cdots <\alpha _{q-1}}{\mathcal {C}}^p(U_{\alpha _0 \ldots \alpha _{q-1}}^{(p+1)})\). Since
we see that \({\check{\delta }}G=F\). This shows \({{\,\mathrm{\textrm{im}}\,}}{\check{\delta }}=\ker {\check{\delta }}\) along the remaining arrows. As before the arguments remain valid for \(F\in \prod _{\alpha _0< \cdots <\alpha _q}{\mathcal {D}}(\delta _p,U_{\alpha _0 \ldots \alpha _q}^{(p+1)})\) if the hypotheses of (ii) are assumed; note that in this case \(\chi _{p,\alpha } F_{\alpha \alpha _0\cdots \alpha _{q-1}}\in {\mathcal {D}}(\delta _p,U_{\alpha _0\cdots \alpha _{q-1}}^{(p+1)})\) by Corollary 5.2 and therefore \(G\in \prod _{\alpha _0< \cdots <\alpha _{q-1}}{\mathcal {D}}(\delta _p,U_{\alpha _0\cdots \alpha _{q-1}}^{(p+1)})\). \(\square \)
The complexes (83) and (85), together with the operators \(\delta _*\) and r respectively \({\check{\delta }}\), define an augmented Čech–Alexander type bicomplex of the form
Given integers \(p,q\ge 0\), let \(K^{q, p}_{\mathcal {C}}({\mathcal {U}},N_*):= \prod _{\alpha _0< \dots <\alpha _q} {\mathcal {C}}^p(U_{\alpha _0\cdots \alpha _q}^{(p+1)})\). Defining
we obtain a cochain complex \(({\mathcal {K}}_{\mathcal {C}}^*({\mathcal {U}},N_*),D_*)\).
Analogously the complexes (84) and (86) define a bicomplex with \({\mathcal {D}}(\delta _p,U_{\alpha _0\cdots \alpha _q}^{(p+1)})\) in place of \({\mathcal {C}}^p(U_{\alpha _0\cdots \alpha _q}^{(p+1)})\). We write \(K^{q, p}_{\mathcal {D}}({\mathcal {U}},N_*)\) and \({\mathcal {K}}_{\mathcal {D}}^\ell ({\mathcal {U}},N_*)\) for the corresponding counterparts of the above spaces; this gives a complex \(({\mathcal {K}}_{\mathcal {D}}^*({\mathcal {U}},N_*),D_*)\).
We write
and \(H^\ell {\mathcal {K}}_{\mathcal {D}}^*({\mathcal {U}},N_*)\), defined similarly, to denote the respective \(\ell \)-th cohomologies.
Recall (18) and (46). By Proposition 6.1 and well-known abstract arguments, [13, Proposition 8.8], [8, Lemma 4], one can see that r is a cochain map from \({\mathcal {C}}^*(N_*)\) to \({\mathcal {K}}_{\mathcal {C}}^*({\mathcal {U}},N_*)\) respectively from \({\mathcal {D}}(\delta _*, N_*)\) to \({\mathcal {K}}_{\mathcal {D}}^*({\mathcal {U}},N_*)\) and induces an isomorphism in cohomology.
Corollary 1.8
Let Assumption 6.1 be in force. Let \({\mathcal {V}}\), \({\mathcal {U}}\) and \(N_*\) be as above.
-
(i)
For any integer \(\ell \ge 0\) the spaces \(H^\ell {\mathcal {C}}^*(N_*)\) and \(H^\ell {\mathcal {K}}_{\mathcal {C}}^*({\mathcal {U}},N_*)\) are isomorphic.
-
(ii)
Suppose that also Assumptions 4.1, 4.2 and 5.2 hold. Then for any integer \(\ell \ge 0\) the spaces \(H^\ell {\mathcal {D}}(\delta _*, N_*)\) and \(H^\ell {\mathcal {K}}_{\mathcal {D}}^*({\mathcal {U}},N_*)\) are isomorphic.
6.3 Poincaré lemma
We prove versions of Poincaré’s lemma for specific finite covers by open balls. We start with a small observation.
Lemma 1.7
Given \(f_0,\ldots ,f_p\in {\mathcal {C}}\oplus {\mathbb {R}}\), we have
Proof
The evaluation at \((x_0,\ldots ,x_p)\) of the left-hand side in (88) equals
Varying \(\sigma \) in the first summand, \((\sigma (1),\ldots ,\sigma (p))\) runs through all permutations of the tuple \((1,\ldots ,p)\), and if \((\sigma (1),\ldots ,\sigma (p))=(1,\ldots ,p)\), then clearly \({{\,\textrm{sgn}\,}}\sigma =1\). Varying \(\sigma \) in the second summand, \((\sigma (1),\ldots ,\sigma (p))\) runs through all permutations of \((0,2,\ldots ,p)\), and if \((\sigma (1),\ldots ,\sigma (p))=(0,2,\ldots ,p)\) (that is, if the natural order is preserved), we have \({{\,\textrm{sgn}\,}}\sigma =-1\). The other summands behave similarly. Therefore the preceding equals
here \({\mathcal {S}}_{p}(1,\ldots ,p)\) means that the permutations are applied to \((1,\ldots ,p)\), and so on. \(\square \)
Now let \(\varepsilon >0\) and let \({\hat{N}}_*(\varepsilon )=({\hat{N}}_p(\varepsilon ))_{p\ge 0}\) be as in (14). We follow [8, Sect. 9] and, given \(p\ge 1\) and \((x_0,\ldots ,x_{p-1})\in {\hat{N}}_{p-1}(\varepsilon )\), define the slice
The following assumption is a variant of [8, Hypothesis (\(*\)), p. 34]. It is a condition on X, \(\varepsilon \) and an integer \(K\ge 0\). It guarantees that for a suitable finite cover of X by open balls we can prove a Poincaré lemma up to order K, compatible with Proposition 6.1.
Assumption 1.6
There is a number \(\eta >0\) such that for any nonempty intersection \(B_{\alpha _0\cdots \alpha _q}=\bigcap _{k=0}^q B_{\alpha _k}\) of finitely many open balls \(B_{\alpha _k}\) of radius \(\varepsilon +2\eta \) we can find a nonempty open set \(W_{\alpha _0\cdots \alpha _q}\) satisfying
Here \(B_{\alpha _0\cdots \alpha _q}^{(p)}\) is defined as in (82).
Suppose that X, \(\varepsilon \) and K satisfy Assumption 6.2 and let \(\eta \) be as stated there. Let \(\{B(y_\alpha ,\eta )\}_{\alpha \in I}\) be a finite cover of X by open balls \(B(y_\alpha ,\eta )\) having centers \(y_\alpha \in X\) and common radius \(\eta \). Since X is assumed to be compact, we can always find such a cover. Then clearly also \({\mathcal {V}}:=\{B(y_\alpha ,\varepsilon +\eta )\}_{\alpha \in I}\) covers X, and it is easily seen that the system \({\hat{N}}_*(\varepsilon )\) as defined in (14) is dominated by the system \(N_*({\mathcal {V}})\) as defined in (15) respectively (79) in the sense that
We consider the finite open cover
Here we add one more \(\eta \) to enlarge the radius so that each \(B_\alpha \) contains the closure of the concentric smaller ball from \({\mathcal {V}}\). Note that Lemma 6.1 and Proposition 6.1 may be applied with these ball covers \({\mathcal {V}}\), \({\mathcal {U}}\) and with \({\hat{N}}_*(\varepsilon )\) in place of \(N_*\).
Suppose that \(\mu \) is a finite nonnegative Borel measure on X with full support and that Assumption 6.2 is in force. Given \(p,q\ge 0\) and a bounded Borel function \(F:B_{\alpha _0\cdots \alpha _q}^{(p+1)}\rightarrow {\mathbb {R}}\), where \(B_{\alpha _0\cdots \alpha _q}\) is a nonempty intersection, we can define a bounded Borel function
by
where \(W_{\alpha _0\cdots \alpha _q}\) is as in (89). For brevity we suppress p from notation, although \(\Psi _{B_{\alpha _0\cdots \alpha _q}}\) clearly depends on it. Note that \(\Psi _{B_{\alpha _0\cdots \alpha _q}}\) is well-defined, since by (89) we have
If instead F is a \(\mu ^{p+1}\)-class of functions on \(B_{\alpha _0\cdots \alpha _q}^{(p+1)}\), then—subject to suitable integrability conditions—also \(\Psi _{B_{\alpha _0\cdots \alpha _q}}F\) is well-defined as a \(\mu ^{p}\)-class of functions on \(B_{\alpha _0\cdots \alpha _q}^{(p)}\).
Proposition 1.9
Let \(\mu \) be a finite nonnegative Borel measure on X with full support. Let \(\varepsilon >0\) and \(K\ge 0\) be such that Assumption 6.2 holds, let \(\eta >0\) be as there and \({\mathcal {U}}\) as in (91). Suppose that \(B_{\alpha _0\cdots \alpha _q}\) is a nonempty intersection.
-
(i)
For any \(1\le p\le K+1\) identity (92) defines a linear map
$$\begin{aligned} \Psi _{B_{\alpha _0\cdots \alpha _q}}:{\mathcal {C}}^p(B_{\alpha _0\cdots \alpha _q}^{(p+1)})\rightarrow {\mathcal {C}}^{p-1}(B_{\alpha _0\cdots \alpha _q}^{(p)}). \end{aligned}$$Moreover, for any \(F\in {\mathcal {C}}^p(B_{\alpha _0\cdots \alpha _q}^{(p+1)})\) we have
$$\begin{aligned} (\Psi _{B_{\alpha _0\cdots \alpha _q}}\circ \delta _p+\delta _{p-1}\circ \Psi _{B_{\alpha _0\cdots \alpha _q}})F=F. \end{aligned}$$(94) -
(ii)
Suppose in addition that Assumptions 4.1, 4.2 and 5.2 are satisfied and that
$$\begin{aligned} \inf _{(x,y)\in {\hat{N}}_1(\varepsilon )\setminus D_1}j(x,y)\ge c \end{aligned}$$(95)with a constant \(c>0\). Then for any \(1\le p\le K+1\) identity (92) defines a bounded linear map
$$\begin{aligned} \Psi _{B_{\alpha _0\cdots \alpha _q}}:L^2_a(B_{\alpha _0\cdots \alpha _q}^{(p+1)}\setminus D_p,J_p)\rightarrow L^2_a(B_{\alpha _0\cdots \alpha _q}^{(p)}\setminus D_{p-1},J_{p-1}) \end{aligned}$$and there is a constant \(c>0\) such that
$$\begin{aligned}{} & {} \left\| \Psi _{B_{\alpha _0\cdots \alpha _q}}F\right\| _{L^2(B_{\alpha _0\cdots \alpha _q}^{(p)}\setminus D_{p-1},J_{p-1})}\nonumber \\{} & {} \quad \le c\,\left\| F\right\| _{L^2(B_{\alpha _0\cdots \alpha _q}^{(p+1)}\setminus D_{p},J_{p})},\quad F\in L^2_a(B_{\alpha _0\cdots \alpha _q}^{(p+1)}\setminus D_p,J_p). \end{aligned}$$(96)Moreover, for any \(F\in {\mathcal {D}}(\delta _p,B_{\alpha _0\cdots \alpha _q}^{(p+1)})\) we have \(\Psi _{B_{\alpha _0\cdots \alpha _q}}F\in {\mathcal {D}}(\delta _p,B_{\alpha _0\cdots \alpha _q}^{(p)})\) and (94) holds on \(B_{\alpha _0\cdots \alpha _q}^{(p+1)}\) in the \(J_p\)-a.e. sense.
Proof
By Assumption 6.2 the map \(\Psi _{B_{\alpha _0\cdots \alpha _q}}\) in (i) is well-defined. Since it is obviously linear, we may assume that \(F={{\,\textrm{Alt}\,}}_{p+1}(f_0\otimes \cdots \otimes f_p)\) with \(f_0\in {\mathcal {C}}\oplus {\mathbb {R}}\) and \(f_1,\ldots ,f_p\in {\mathcal {C}}\). Let \(W:=W_{\alpha _0\cdots \alpha _q}\) be as in (89) and (92). By Lemma 6.2
and consequently \(\Psi _{B_{\alpha _0\cdots \alpha _q}}F \in {\mathcal {C}}^{p-1}(B_{\alpha _0\cdots \alpha _q}^{(p)})\). Identity (94) is straightforward: Given \(F\in {\mathcal {C}}^p(B_{\alpha _0\cdots \alpha _q}^{(p+1)})\),
This completes the proof of (i).
To see the first part of (ii), let \(F\in L^2_a(B_{\alpha _0\cdots \alpha _q}^{(p+1)}{\setminus } D_p,J_p)\). By Jensen’s inequality we have
where c is as in (95). To see the second part of (ii), let \(F\in {\mathcal {D}}(\delta _p,B_{\alpha _0\cdots \alpha _q}^{(p+1)})\) and let \((F_k)_k\subset {\mathcal {C}}^p(B_{\alpha _0\cdots \alpha _q}^{(p+1)})\) be such that
and
By (94) we have
for all k. From (96) and (98) it follows that
and taking into account (97) and (99), we see that \((\delta _{p-1}\Psi _{B_{\alpha _0\cdots \alpha _q}}F_k)_k\) is a Cauchy sequence in \(L^2(B_{\alpha _0\cdots \alpha _q}^{(p+1)}{\setminus } D_p,J_p)\). Since by (96) and (97) we have
in \(L^2(B_{\alpha _0\cdots \alpha _q}^{(p+1)}\setminus D_p,J_p)\) and the operator \((\delta _p,{\mathcal {D}}(\delta _p,B_{\alpha _0\cdots \alpha _q}^{(p+1)}))\) is closed, the function \(\Psi _{B_{\alpha _0\cdots \alpha _q}}F\) is in \({\mathcal {D}}(\delta _p,B_{\alpha _0\cdots \alpha _q}^{(p+1)})\) and
Taking limits in (99) we obtain the desired variant of (94). \(\square \)
As a consequence we have the following variant of Poincaré’s lemma: Functions, closed on a given nonempty intersection, are also exact there.
Corollary 1.9
Let \(\mu \) be a finite nonnegative Borel measure on X with full support. Let \(\varepsilon >0\) and \(K\ge 0\) be such that Assumption 6.2 holds, let \(\eta >0\) be as there and \({\mathcal {U}}\) as in (91). Suppose that \(B_{\alpha _0\cdots \alpha _q}\) is a nonempty intersection and \(1\le p\le K+1\).
-
(i)
For any \(F\in {\mathcal {C}}^p(B_{\alpha _0\cdots \alpha _q}^{(p+1)})\) with \(\delta _pF=0\) on \(B_{\alpha _0\cdots \alpha _q}^{(p+2)}\) there is some \(G\in {\mathcal {C}}^{p-1}(B_{\alpha _0\cdots \alpha _q}^{(p)})\) such that \(\delta _{p-1}G=F\).
-
(ii)
Suppose in addition that Assumptions 4.1, 4.2 and 5.2 are satisfied and that (95) holds. Then for any \(F\in {\mathcal {D}}(\delta _p,B_{\alpha _0\cdots \alpha _q}^{(p+1)})\) with \(\delta _pF=0\) \(J_{p+1}\)-a.e. on \(B_{\alpha _0\cdots \alpha _q}^{(p+2)}\) there is some \(G\in {\mathcal {D}}(\delta _{p-1},B_{\alpha _0\cdots \alpha _q}^{(p)})\) such that \(\delta _{p-1}G=F\) on \(B_{\alpha _0\cdots \alpha _q}^{(p+1)}\) \(J_p\)-a.e.
For any integer \(q\ge 0\) let \({\check{C}}^q({\mathcal {U}},{\mathbb {R}})\) be the space of Čech cochains of order q with real coefficients associated with the ball cover \({\mathcal {U}}\) as defined in (91). We make the following additional assumption.
Assumption 1.7
For any distinct \(\alpha _0,\ldots ,\alpha _q\in I\) the intersection \(B_{\alpha _0\cdots \alpha _q}\) is connected or consists of finitely many connected components that have positive minimal distance.
If Assumptions 6.1 and 6.3 are satisfied, then for any distinct \(\alpha _0,\ldots ,\alpha _q\in I\) the locally constant functions on \(B_{\alpha _0\cdots \alpha _q}\) are contained in \({\mathcal {C}}^0(B_{\alpha _0\cdots \alpha _q})\). As a consequence, there are natural inclusions i taking elements of \({\check{C}}^q({\mathcal {U}},{\mathbb {R}})\) into elements of \(K^{q,0}_{{\mathcal {C}}}({\mathcal {U}},{\hat{N}}_*(\varepsilon )):=\prod _{\alpha _0<\dots <\alpha _q} {\mathcal {C}}^0(B_{\alpha _0\cdots \alpha _q})\). The complex (87) can be augmented further by a bottom line,
Note that a function \(f\in {\mathcal {C}}^0(B_{\alpha _0\cdots \alpha _q})\) with \(\delta _0f=0\) on \(B_{\alpha _0\cdots \alpha _q}^{(2)}\) is locally constant on \(B_{\alpha _0\cdots \alpha _q}\).
If in addition Assumptions 4.1, 4.2 and 5.2 are satisfied, then similar inclusions i take elements of \({\check{C}}^q({\mathcal {U}},{\mathbb {R}})\), interpreted in the sense of \(\mu \)-equivalence classes, into elements of \(K^{q,0}_{{\mathcal {D}}}({\mathcal {U}},{\hat{N}}_*(\varepsilon )):=\prod _{\alpha _0<\dots <\alpha _q} {\mathcal {D}}(\delta _0,B_{\alpha _0\cdots \alpha _q})\). This interpretation is unambiguous, because each \(\mu \)-equivalence class of locally constant functions contains exactly one representative locally constant in the strict (everywhere) sense. Any \(f\in {\mathcal {D}}(\delta _0, B_{\alpha _0\cdots \alpha _q})\) with \(\delta _0f=0\) \(J_1\)-a.e. is \(\mu \)-a.e. locally constant on \(B_{\alpha _0\cdots \alpha _q}\).
We write \(H^\ell {\check{C}}^*({\mathcal {U}},{\mathbb {R}})\) for the \(\ell \)-th Čech cohomology of the ball cover \({\mathcal {U}}\). Corollaries 6.1 and 6.2, together with similar abstract arguments as before, [13, Theorem 8.9], [8, Corollary 3], give the following.
Corollary 1.10
Let Assumption 6.1 be in force. Suppose that \(\varepsilon >0\) and \(K\ge 0\) are such that Assumption 6.2 holds, \(\eta >0\) is as there, \({\mathcal {U}}\) as in (91) and \(\mu \) as specified above. Suppose that also Assumption 6.3 holds.
-
(i)
For any integer \(0\le \ell \le K\) the spaces \(H^\ell {\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\), \(H^\ell {\mathcal {K}}_{\mathcal {C}}^*({\mathcal {U}},{\hat{N}}_*(\varepsilon ))\) and \(H^\ell {\check{C}}^*({\mathcal {U}},{\mathbb {R}})\) are isomorphic.
-
(ii)
Suppose that also Assumptions 4.1, 4.2 and 5.2 are satisfied and that (95) holds for any nonempty intersection of sets from \({\mathcal {U}}\). Then for any integer \(0\le \ell \le K\) the spaces \(H^\ell {\mathcal {D}}(\delta _*, {\hat{N}}_*(\varepsilon ))\), \(H^\ell {\mathcal {K}}_{\mathcal {D}}^*({\mathcal {U}},{\hat{N}}_*(\varepsilon ))\) and \(H^\ell {\check{C}}^*({\mathcal {U}},{\mathbb {R}})\) are isomorphic, and they are also isomorphic to \(H^\ell {\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\) and \(H^\ell {\mathcal {K}}_{\mathcal {C}}^*({\mathcal {U}},{\hat{N}}_*(\varepsilon ))\).
Remark 6.2
In the case of local complexes on manifolds one typically uses smoothing methods to show that the cohomologies defined in terms of cores and the cohomologies defined in terms of their closures are isomorphic, see for instance [16, Theorems 2.12 and 3.5] or [24, Sect. 8]. In the non-local case the implementation of similar smoothing arguments does not seem straightforward, but one can pass directly from the core to its closure as done in Propositions 6.1 and 6.2 to obtain \(H^\ell {\mathcal {D}}(\delta _*, {\hat{N}}_*(\varepsilon ))\cong H^\ell {\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\) as in Corollary 6.3 (ii).
6.4 Recovering deRham cohomology
Suppose that \(X=M\) is a compact smooth Riemannian manifold of dimension d. Let \({\mathcal {C}}:={{\,\textrm{Lip}\,}}(M)\). Then clearly Assumption 6.1 is satisfied. Let \(\textrm{r}_c(M)\) be the convexity radius of M, [22, Sect. IX.6]. Recall that if \(d\ge 2\), then \(\textrm{r}_c(M)\ge \min \{\frac{{{\,\mathrm{\textrm{inj}}\,}}(M)}{2},\frac{\pi }{2\sqrt{k}}\}\), where \({{\,\mathrm{\textrm{inj}}\,}}(M)>0\) denotes the injectivity radius of M and \(k>0\) is an upper bound on its sectional curvatures, [22, Theorem IX.6.1]. Assume that
Let \({\hat{N}}_*(\varepsilon )\) be as in (14). Then a combination of results from [8], classical theorems and Corollary 6.3 shows that the complexes \({\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\) and \({\mathcal {D}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) can be used to recover the deRham cohomology of M. By \(H^\ell _{dR}\Omega ^*(M)\) we denote the \(\ell \)-th deRham cohomology and by \(H^\ell {\check{C}}^*(M,{\mathbb {R}})\) the \(\ell \)-th Čech cohomology of M. To a measure \(\mu \) with strictly positive and smooth density (with respect to the Riemannian volume) we refer as a smooth measure.
Theorem 1.4
Let M be a compact smooth Riemannian manifold of dimension d and \(\mu \) a smooth measure on M. Suppose that \(\varepsilon \) is as in (101) and \(N_*(\varepsilon )\) as defined in (14).
-
(i)
For any integer \(\ell \ge 0\) the spaces \(H^\ell {\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\), \(H^\ell {\check{C}}^*(M,{\mathbb {R}})\) and \(H^\ell _{dR}\Omega ^*(M)\) are finite dimensional and isomorphic; they are trivial for \(\ell >d\).
-
(ii)
If j is a kernel such that Assumptions 4.1, 4.2 and 5.2 are satisfied and (95) holds, then for any integer \(\ell \ge 0\) also \(H^\ell {\mathcal {D}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) is isomorphic to the spaces in (i).
Example 6.1
If j is as in (52), then the hypotheses in (ii) are satisfied.
Proof
Let \(K\ge 0\) be an integer and let \(\varepsilon \) be as in (101). In the case \(d\ge 2\) [8, Theorem 11 and Propositions 20 and 21] show that Assumption 6.2 holds with \(\eta >0\) satisfying \(\varepsilon +2\eta <\min \{\textrm{r}_c(M), \frac{\pi }{2\sqrt{k}}\}\). The proof of [8, Proposition 21] uses a well-known consequence of the Rauch comparison theorem, [65, Theorem 2.7.6]. In the case \(d=1\) Assumption 6.2 holds with \(\eta >0\) such that \(\varepsilon +2\eta <\textrm{r}_c(M)\) by Lemma 6.3 below.
Let \({\mathcal {U}}=\{B_\alpha \}_{\alpha \in I}\) be a finite cover of M by open balls \(B_\alpha \) of radius \(\varepsilon +2\eta \). Since the \(B_\alpha \) have radii less or equal to \(\textrm{r}_c(M)\), \({\mathcal {U}}\) is a good cover, that is, for any \(q\ge 0\) and distinct \(\alpha _0,\ldots ,\alpha _q\in I\) the intersection \(B_{\alpha _0\cdots \alpha _q}\) is diffeomorphic to \({\mathbb {R}}^d\). Consequently for any \(\ell \ge 0\) the cohomologies \(H^\ell {\check{C}}^*({\mathcal {U}},{\mathbb {R}})\), \(H^\ell _{dR}\Omega ^*(M)\) and \(H^\ell {\check{C}}^*(M,{\mathbb {R}})\) are isomorphic, [13, Theorem 8.9 and Proposition 10.6]. The space \(H^\ell _{dR}\Omega ^*(M)\) is finite dimensional, and it is trivial for \(\ell >d\).
Clearly any good cover satisfies Assumption 6.3, so that Corollary 6.3 now yields statements (i) and (ii). \(\square \)
Remark 6.3
-
(i)
The space \(H^\ell {\check{C}}^*({\mathcal {U}},{\mathbb {R}})\) with \({\mathcal {U}}\) as in the proof, can be added to Theorem 6.1 (i).
-
(ii)
Theorem 6.1 result also holds with \({\mathcal {C}}=C^\infty (M)\).
Lemma 1.8
Let M be a smooth Riemannian manifold of dimension one. Suppose that \(\varepsilon \) is as in (101) and \(N_*(\varepsilon )\) as defined in (14). Then Assumption 6.2 holds.
Proof
Let \(0<2\eta <\min \{\varepsilon /8,\textrm{r}_c(M)-\varepsilon \}\). A nonempty finite intersection \(B_{\alpha _0\cdots \alpha _q}\) of balls of radius \(\varepsilon +2\eta \) is a geodesically convex arc with midpoint \(y_{\alpha _0\cdots \alpha _q}\), say.
Suppose first that \(q=0\). Given \((x_0,\ldots ,x_{p-1})\in B_{\alpha _0}^p\cap {\hat{N}}_{p-1}(\varepsilon )\), we can find a closed ball \(B_{\alpha _0}'\) of radius \(\varepsilon \) containing \(x_0,\ldots ,x_{p-1}\), and shifting it along M, we may assume that \(B_{\alpha _0}'\subset B_{\alpha _0}\). Then \(B(y_{\alpha _0},\varepsilon -4\eta )\subset B_{\alpha _0}'\). This means that for any \(t\in B(y_{\alpha _0},\varepsilon -4\eta )\) and any \((x_0,\ldots ,x_{p-1})\in B_{\alpha _0}^{(p)}\) we can identify a closed ball \(B_{\alpha _0}'\) of radius \(\varepsilon \) containing \(t,x_0,\ldots ,x_{p-1}\), in other words,
Now suppose that \(B_{\alpha _0\cdots \alpha _q}\) is a general nonempty intersection. If its length is less than or equal to \(2\varepsilon \), then
If not, then its length is \(2\varepsilon +2\varepsilon _{\alpha _0\cdots \alpha _q}\) with some \(\varepsilon _{\alpha _0\cdots \alpha _q}>0\); we may always assume that \(\varepsilon _{\alpha _0\cdots \alpha _q}<2\eta \) (otherwise the balls are identical). Given \((x_0,\ldots ,x_{p-1})\in B_{\alpha _0\cdots \alpha _q}^p\cap {\hat{N}}_{p-1}(\varepsilon )\), we can again find a closed ball \(B_{\alpha _0\cdots \alpha _q}'\) of radius \(\varepsilon \) containing \(x_0,\ldots ,x_{p-1}\), and we may again assume it is a subset of \( B_{\alpha _0\cdots \alpha _q}\). Similarly as before this gives \(B(y_{\alpha _0\cdots \alpha _q},\varepsilon -2\varepsilon _{\alpha _0\cdots \alpha _q})\subset B_{\alpha _0\cdots \alpha _q}'\), and therefore
\(\square \)
Remark 6.4
The proof shows that under the hypotheses of Lemma 6.3, \(H^\ell {\mathcal {C}}^*(B_{\alpha }^{(*+1)})=\{0\}\), \(\ell \ge 1\), for all \(\alpha \in I\). This follows using Corollary 6.2 (i). Under the additional assumptions in Corollary 6.2 (ii) also \(H^\ell {\mathcal {D}}(\delta _*, B_{\alpha }^{(*+1)})=\{0\}\), \(\ell \ge 1\), holds for all \(\alpha \in I\).
7 Some basic examples
We provide some very basic examples.
Example 7.1
Let \((X,\varrho )\) be a metric space and \({\mathcal {C}}={{\,\textrm{Lip}\,}}_b(X)\) the algebra of bounded Lipschitz functions on X. Let \(N_*(\varepsilon )\) be as in Example 3.1 (ii). If \(\varepsilon >{{\,\textrm{diam}\,}}(X)\), then \(H^0{\mathcal {C}}^*(N_*(\varepsilon ))\cong {\mathbb {R}}\) and \(H^\ell {\mathcal {C}}^*(N_*(\varepsilon ))=\{0\}\) for all \(\ell \ge 1\); note that if \(\ell \ge 1\) and \(F\in {\mathcal {C}}^\ell (N_\ell (\varepsilon ))\) is such that \(\delta _\ell F=0\), then for any \(y\in X\) the function
is in \({\mathcal {C}}^{\ell -1}(N_{\ell -1}(\varepsilon ))\) and satisfies \(\delta _{\ell -1}G=F\). If \((X,\varrho )\) is locally compact, \(\mu \) and j satisfy Assumptions 4.1, 4.2 and 5.2, then an averaged version of this argument gives again \(H^0{\mathcal {D}}(\delta _*, N_*(\varepsilon ))\cong {\mathbb {R}}\) and \(H^\ell {\mathcal {D}}(\delta _*, N_*(\varepsilon ))=\{0\}\), \(\ell \ge 1\).
Example 7.2
Suppose that the metric space \((X,\varrho )\) has two connected components \(X_1\) and \(X_2\), \({\mathcal {C}}={{\,\textrm{Lip}\,}}_b(X)\) and that \(N_*(\varepsilon )\) is as in Example 3.1 (ii). If \(\varepsilon >{{\,\textrm{dist}\,}}(X_1,X_2)\), then \(H^0{\mathcal {C}}^*(N_*(\varepsilon ))\cong {\mathbb {R}}\); if \(\varepsilon \le {{\,\textrm{dist}\,}}(X_1,X_2)\), then \(H^0{\mathcal {C}}^*(N_*(\varepsilon ))\cong {\mathbb {R}}^2\). If \((X,\varrho )\) is locally compact, both \(X_1\) and \(X_2\) have nonempty interior and \(\mu \) and j are such that Assumptions 4.1, 4.2 and 5.2 hold, then a similar observation is true for \(H^0{\mathcal {D}}(\delta _*, N_*(\varepsilon ))\).
Example 7.3
Let X be the unit circle \(S^1:=\{e^{i\theta }: 0\le \theta <2\pi \}\), let \({\mathcal {C}}={{\,\textrm{Lip}\,}}(S^1)\) be the algebra of Lipschitz functions on \(S^1\) and \(\mu \) the Riemannian volume (Haar measure) on \(S^1\). Assume that j satisfies (52) with \(d=1\) and some \(\alpha \in (0,2)\). An application of Theorem 6.1 to \(M=S^1\) with \(0<\varepsilon <\pi /2\) shows that \(H^0{\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\) and \(H^0{\mathcal {D}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) are both one-dimensional and also
The results for order zero could also be concluded directly from the fact that \(\ker \delta _0={\mathbb {R}}\). An alternative way to see (102) is to (repeatedly) inspect long exact sequences as in [13, Example 2.6]: We can patch single balls \(B_\alpha \) together to larger and larger open arcs, the fact that their first cohomologies are trivial (cf. Remark 6.4) propagates from the \(B_\alpha \) to the larger arcs by the exactness of the long sequence. Eventually we cover all of \(S^1\) by two arcs whose intersection has two connected components, and exactness gives (102). A generating element for \(H^1{\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\) can be constructed as in [13, Example 2.6] or using a “spiral with constant slope”.
Example 7.4
Let X be the unit interval [0, 1], let \({\mathcal {C}}={{\,\textrm{Lip}\,}}([0,1])\) and let \(\mu \) be the one-dimensional Lebesgue measure, restricted to [0, 1]. Assume that \(\varepsilon \ll 1\) and that j satisfies Assumptions 4.1, 4.2 and 5.2. Then \(H^0{\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\) and \(H^0{\mathcal {D}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) are both one-dimensional. We can cover [0, 1] by finitely many open balls of radius slightly larger than \(\varepsilon \), on each of these balls a closed 1-function is exact (Remark 6.4), and we can use long exact sequences to see that \(H^1{\mathcal {C}}^*({\hat{N}}_*(\varepsilon ))\) and \(H^1{\mathcal {D}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) are trivial.
Example 7.5
Let again \(X=[0,1]\). Let \(\mathring{X}:=[0,1]{\setminus } \{1/2\}\) and let \(\mathring{{\mathcal {C}}}\) be the ideal in \({{\,\textrm{Lip}\,}}([0,1])\) consisting of functions that vanish in a neighborhood of 1/2. Let \(\mu \) be the one-dimensional Lebesgue measure, restricted to [0, 1], \(\varepsilon \ll 1\) and j a kernel satisfying (52) with \(d=1\) and some \(\alpha \in (0,2)\). Let \(\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) be as explained before (69). If \(\alpha \le 1\), then \(H^0\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) is one-dimensional and \(H^1\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) is trivial by Example 7.4, Corollary 5.7, Remark 5.7 and Remark 5.8. If \(\alpha >1\), then by Corollary 5.7 and Remark 5.8 the complex \(\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) is known to be different from \({\mathcal {D}}(\delta _*,{\hat{N}}_*(\varepsilon ))\). In fact, we have \(H^0\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))=\{0\}\) since \(\mathring{{\mathcal {D}}}(\delta _0,{\hat{N}}_0(\varepsilon ))\) does not contain nonzero constants. We can cover \(\mathring{X}\) by finitely many open balls of radius slightly larger than \(\varepsilon \), and proceeding similarly as in Lemma 6.3, we can see that Assumption 6.2 holds. These balls have trivial first cohomology (Remark 6.4 resp. Corollary 6.2). A variant of [13, Proposition 2.3] remains true, and using long exact sequences we find that also \(H^1\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))=\{0\}\).
Example 7.6
Consider the unit circle \(X=S^1\) with \({\mathcal {C}}={{\,\textrm{Lip}\,}}(S^1)\), the Riemannian volume \(\mu \) and with j satisfying (52) with \(d=1\) and some \(\alpha \in (0,2)\). Let \(x\in S^1\) and \(\mathring{S}^1:=S^1{\setminus } \{x\}\). Let \(0<\varepsilon <\pi /2\). If \(\alpha \le 1\), then \(H^0\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))\cong {\mathbb {R}}\) and \(H^1\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))\cong {\mathbb {R}}\) by Example 7.3, Corollary 5.7, Remark 5.7 and Remark 5.8. If \(\alpha >1\), then \(\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))\ne {\mathcal {D}}(\delta _*,{\hat{N}}_*(\varepsilon ))\). Again \(H^0\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))=\{0\}\) since nonzero constants are lost. Similarly as in Example 7.5 a small enough open arc containing a neighborhood of x has trivial zero and first cohomologies. Patching together covering arcs eventually produces two arcs which cover \(\mathring{S}^1\). If any small neighborhood of x is contained in only one of these two arcs, their overlap has two connected components, both arcs have zero \(H^1\), one has trivial \(H^0\), the other one-dimensional \(H^0\). Using a long exact sequence as in Example 7.3 (resp. [13, Example 2.6]), \(H^1\mathring{{\mathcal {D}}}(\delta _*,{\hat{N}}_*(\varepsilon ))\) is seen to be one-dimensional. If a small neighborhood of x is contained in the overlap, a slightly different calculation gives the same result.
Example 7.7
Consider the unit sphere \(X=S^2\), let \({\mathcal {C}}={{\,\textrm{Lip}\,}}(S^2)\) and \(N_*=N_*(\varepsilon )\) with small \(\varepsilon >0\). Suppose that \(\mu \) is the Riemannian volume and that j satisfies (52) with \(d=2\) and some \(\alpha \in (0,2)\). Then \(H^0{\mathcal {D}}(\delta _*,N_*)\cong {\mathbb {R}}\). Now let \(\Sigma \subset S^2\) be a closed set of Hausdorff dimension \(0\le \beta < 2\) and consider \(\mathring{X}:=S^2{\setminus } \Sigma \). If \(\alpha < 2-\beta \), then \(H^0\mathring{D}(\delta _*,N_*)\cong {\mathbb {R}}\) by Corollary 5.7. This remains true if \(\alpha =2-\beta \) and \({\mathcal {H}}^\beta (\Sigma )<+\infty \), for instance if \(\beta =1\) and \(\Sigma \) is the equator. If \(\alpha >2-\beta \), then \(H^0\mathring{D}(\delta _*,N_*)=\{0\}\), because nonzero constants are no longer in \(\mathring{D}(\delta _0,N_0)\).
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Hinz, M., Kommer, J. A tensor product approach to non-local differential complexes. Math. Ann. 389, 2357–2409 (2024). https://doi.org/10.1007/s00208-023-02703-w
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DOI: https://doi.org/10.1007/s00208-023-02703-w