On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality

TL;DR

This paper explores spectral properties of Witten-Laplacians and their relation to covariance formulas, leading to a generalized Brascamp-Lieb inequality with applications in statistical mechanics.

Contribution

It establishes a spectral characterization of covariance formulas via Witten-Laplacians and derives a generalized Brascamp-Lieb inequality using advanced spectral analysis techniques.

Findings

Spectral properties of Witten-Laplacians are linked to covariance formulas.

A generalized Brascamp-Lieb inequality is derived.

Explicit criteria for measure properties in statistical mechanics are provided.

Abstract

A study is made of an integral identity of Helffer and Sj{\"o}strand, which for some class of probability measures yields a formula for the covariance of two functions (of a stochastic variable). In comparison with the Brascamp--Lieb inequality, this formula is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using a fine version of the Closed Range Theorem, the identity's validity is shown to be equivalent to some explicitly given spectral properties of Witten-Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp--Lieb's inequality is obtained. For a certain class of measures occurring in statistical mechanics, explicit criteria for the means are found from the Persson--Agmon formula, from compactness of embeddings and from the Weyl calculus of pseudo-differential operators, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.

Equations

$$\int_{{{\mathbb{R}}}^{n}}|f(x)-\langle f\rangle|^{2}e^{-\Phi(x)}\,dx\leq\int_{{{\mathbb{R}}}^{n}}\bigl{(}\nabla f(x)^{T}\Phi^{\prime\prime}(x)^{-1}\overline{\nabla f(x)}\bigr{)}e^{-\Phi(x)}\,dx;$$

$$(g_1-⟨g_1⟩|g_2-⟨g_2⟩{)}_{\mu }=(A_1^{-1}\nabla g_1|\nabla g_2{)}_{\mu }.$$

$$A_0=-\Delta +{\Phi }^{\prime }\cdot \nabla ,A_1=A_0\otimes I+{\Phi }^{\prime \prime }.$$

$${\Delta }_{\Phi }^{(0)}=-\Delta +\frac{1}{4}|{\Phi }^{\prime }{|}^2-\frac{1}{2}\Delta \Phi ,{\Delta }_{\Phi }^{(1)}={\Delta }_{\Phi }^{(0)}\otimes I+{\Phi }^{\prime \prime };$$

$${\int }_{{\mathbb{R}}^n}{e}^{-\Phi (x)}dx=1\text{and}\Phi \in C^2({\mathbb{R}}^n,\mathbb{R}).$$

$$\Vert \sum _{j_1<\cdots <j_k}{f}_{j_1\dots j_k}d{z}^{j_1}\wedge \cdots \wedge d{z}^{j_k}{\Vert }_{\mu }=({\int }_{{\mathbb{R}}^n}(\sum |f_{j_1\dots j_k}{|}^2)d\mu {)}^{1/2}.$$

$$A_k^{(H)}=d_k^{\ast }{d}_k+d_{k-1}{d}_{k-1}^{\ast },D(A_k^{(H)})=D(d_k^{\ast }{d}_k)\cap D(d_{k-1}{d}_{k-1}^{\ast }),$$

$$(0)⟶L^2(\mu )\stackrel{d_0}{⟶}{L}^2(\mu ,{\wedge }^1{\mathbb{C}}^n)\stackrel{d_1}{⟶}\dots \stackrel{d_{k-1}}{⟶}{L}^2(\mu ,{\wedge }^k{\mathbb{C}}^n)\stackrel{d_k}{⟶}\dots .$$

$$X_{k+1}:=\overline{R(d_k)}$$

$$(P{g}_1|P{g}_2{)}_{\mu }=(A_1^{-1}{d}_0{g}_1|d_0{g}_2{)}_{\mu }.$$

$$A_1{|}_X=(d_0{d}_0^{\ast }){|}_X.$$

$$(A_1^{-1}){|}_X=(d_0{d}_0^{\ast }{|}_X{)}^{-1};$$

$$g_1$$

$$⟺$$

$$((d_0{d}_0^{\ast }{)}^{-1}{d}_{0}f|d_{0}g{)}_{\mu }=(Pf|g{)}_{\mu }\text{holds for all}f,g\in H^1(\mu );$$

$$d_0^{\ast }(d_0{d}_0^{\ast }{|}_X{)}^{-1}{d}_{0}f=d_0^{\ast }\cdot Iv=f-⟨f⟩,$$

$$\forall f\in H^1(\mu ):f-⟨f⟩\in D(d_0)\cap R(d_0^{\ast })\text{?}$$

$$H^1(\mu )\ominus \mathbb{C}\subset R(d_0^{\ast }).$$

$$R(d_0^{\ast })=L^2(\mu )\ominus \mathbb{C};$$

$$P_k=d_k^{\ast }(A_{k+1}{|}_{X_{k+1}}{)}^{-1}{d}_k\text{on all of}D(d_k).$$

$$A_k{|}_{Z(d_k{)}^{\perp }}=U^{\ast }{A}_{k+1}{|}_{X_{k+1}}U$$

$$\sigma (A_k{|}_{Z(d_k{)}^{\perp }})=\sigma (A_{k+1}{|}_{X_{k+1}})$$

$$Pu=d_0^{\ast }{A}_1^{-1}{d}_{0}u$$

$$\sum {\Phi }_{jk}^{\prime \prime }(x)z_j\overline{z_k}>0\text{for all}x\in {\mathbb{R}}^n,\text{all }z\in {\mathbb{C}}^n\setminus \{0\}.$$

$$A_1\ge {\Phi }^{\prime \prime }\text{and therefore}{A}_1^{-1}\le ({\Phi }^{\prime \prime }{)}^{-1}\le \frac{1}{c_0}.$$

$$({\Phi }^{\prime }{)}^{\beta }{D}^{\alpha }u\in L^2({\mathbb{R}}^n,\mu )\text{for all α and β such that}|\alpha +\beta |\le 2.$$

$$m(T)=\inf\bigl{\{}\,\operatorname{Re}(\,Tx\,|\,x\,)\bigm{|}\|x\|=1\,\bigr{\}}$$

$$\mathrm{Re}s(u,u)\ge c\Vert u{\Vert }_V^2-k\Vert u{\Vert }_H^2\text{for all}u\in V,$$

$$f={\sum }_{|J|=k}^{\prime }{f}_J(x)d{z}^J;$$

$$F=\overline{R(T^{\ast })},R(T)\subset F_1.$$

$$S=T^{\ast }T{|}_F,S_1=T{T}^{\ast }{|}_{F_1}$$

Full text

∗Supported by TMR grant FMRX-CT960001 of the European Commision, ‘PDE and QM’ at Université de Paris-Sud, France; partly by the Danish Natural Sciences Research Council, grant 9700987.

Appeared in Integral equations and operator theory, 36 (2000), 288--324

1. Introduction and Main Results

1.1. Background

In 1976, H. J. Brascamp and E. H. Lieb [BL76] proved the following inequality for an arbitrary function $f$ in $C^{1}({{\mathbb{R}}}^{n})\cap L^{2}(\mu)$, when the given measure $d\mu=e^{-\Phi}\,dx$ has a real-valued, strictly convex $C^{2}$ ‘potential’ $\Phi$ with Hessian $\Phi^{\prime\prime}=(\partial^{2}_{jk}\Phi)_{j,k}$:

[TABLE]

the measure $\mu$ is finite and may be normalised to $\int d\mu=1$ without loss of generality (by adding $\log\int_{{{\mathbb{R}}}^{n}}d\mu$ to $\Phi$) which is done tacitly throughout, so $\langle f\rangle:=\int fe^{-\Phi}\,dx$ equals $f$’s mean.

Since then this inequality has been used in physics, where the strict convexity assumption on $\Phi$ in some contexts is a serious restriction; e.g. this is the case for the analysis of asymptotics of correlations in statistical mechanics.

As another technique for such problems, B. Helffer and J. Sjöstrand have recently introduced an exact formula [HS94, Sjö96, Hel98, Hel97a] for the covariance of two functions $g_{1}$, $g_{2}$ in $L^{2}(\mu)$, i.e. for $\operatorname{cov}(g_{1},g_{2}):=\int_{{{\mathbb{R}}}^{n}}(g_{1}-\langle g_{1}\rangle)\overline{(g_{2}-\langle g_{2}\rangle)}e^{-\Phi}\,dx$ (in comparison the variance of $f$ enters in (1.1)). Denoting the inner product of both $L^{2}({{\mathbb{R}}}^{n},\mu)$ and $L^{2}({{\mathbb{R}}}^{n},\mu,{{\mathbb{C}}}^{n})$ by $(\,\cdot\,|\,\cdot\,)_{\mu}$ for simplicity’s sake, their identity may be written as follows:

[TABLE]

This uses two elliptic differential operators $A_{0}\geq 0$ and $A_{1}\geq 0$ on ${{\mathbb{R}}}^{n}$ (although $A_{0}$ does not appear explicitly in (1.2)); these are equivalent, as observed in [Sjö96], to Witten’s Laplacians [Wit82], and they have the expressions

[TABLE]

In the proofs of Helffer and Sjöstrand, $\Phi$ has had a rather specific nature, e.g. with $\Phi^{\prime\prime}_{jk}$ bounded and $x\cdot\Phi^{\prime}\geq C|x|^{1+\omega}$ in [Sjö96]; the $\Phi$ treated in [Hel98] is essentially $|x|^{4}-|x|^{2}$, see Example 7.3 below. Formula (1.2) has also been used by A. Naddaf and T. Spencer [NS97], V. Bach, T. Jecko and J. Sjöstrand [BJS98], B. Helffer [Hel97b] and others. Indirectly $A_{0}$, $A_{1}$ appeared earlier in [Sjö93, HS94, Hel95].

Concerning formula (1.2), it should be noted that $g_{j}$ only enters in the covariance through $Pg_{j}:=g_{j}-\int g_{j}\,d\mu$, which is the orthogonal projection onto $L^{2}(\mu)\ominus{{\mathbb{C}}}$, i.e. onto the complement of the constant functions. Because the gradient provides another means to remove the part of $g_{j}$ lying in ${{\mathbb{C}}}$, it is natural to have $\nabla g_{1}$ and $\nabla g_{2}$ on the right hand side of (1.2) and to have an inverse of a second order differential operator, like $A_{1}^{-1}$, to counteract the gradients.

Viewed thus, (1.2) may seem plausible, and this article presents a study of it, resulting first of all in more general sufficient conditions for the formula; secondly, conditions that are equivalent to (1.2) are given at an abstract level (although these two improvements are formally substantial, the consequences for statistical mechanics are to be investigated). Thirdly a more systematic and streamlined approach to (1.2) is presented.

Remark 1.1*.*

For the general importance of formula (1.2), recall that for a stochastic variable $X\colon\Omega\to{{\mathbb{R}}}^{n}$ with distribution $\mu$ on ${{\mathbb{R}}}^{n}$, the left hand side of (1.2) equals $\operatorname{cov}(g_{1}(X),g_{2}(X))$. So when $\mu$ is of the type treated here, then (1.2) provides a formula also for such covariances. However, this is clear, and hence this consequence shall not be treated below.

1.2. Summary

In this paper various — abstract and explicit — conditions are deduced for (1.2). To give an application of these, (1.1) is derived in the general strictly convex case from (1.2) and moreover extended to the case $f\in H^{1}_{\operatorname{loc}}({{\mathbb{R}}}^{n})\cap L^{2}(\mu)$; this supplements an explanation of B. Helffer of how (1.2) implies the validity of (1.1) for certain $f$ when $\Phi$ is uniformly strictly convex [Hel98].

In the applications estimates like $A_{1}\geq c_{0}>0$ would clearly give a bounded inverse $A_{1}^{-1}$, so that (1.2) would yield control of the covariance on the left hand side there. However, it turns out that already (1.2) itself follows from such a strict positivity.

This fact is observed here (while further properties of the $A_{j}$ entered in [Sjö96, Hel98]) and proved for rather more general $\Phi$ than those considered hitherto. Actually, by means of Hilbert space methods (the Closed Range Theorem), even weaker sufficient conditions on $A_{0}$ or $A_{1}$ (such as closed range of $A_{0}$) are proved to be equivalent to (1.2); this analysis is furthermore valid for arbitrary probability measures $\mu$ on ${{\mathbb{R}}}^{n}$.

These techniques are also applied to the associated $d$-complex in $\mu$-weighted spaces and to the associated $A_{k}$ on forms of higher degrees, see (1.8) and (1.7) below. This is done because these objects may be of interest in statistical mechanics, and because the cases $k=0$ and $1$ have to be treated anyway in order to settle the relations between the various conditions found for $A_{0}$ and $A_{1}$.

Explicit criteria in terms of $\Phi$ are given partly by means of compact embeddings or bounds of essential spectra; partly by a pseudo-differential treatment of the Witten–Laplacians on functions and $1$-forms on ${{\mathbb{R}}}^{n}$, i.e. of

[TABLE]

these are unitarily equivalent to $A_{0}$ and $A_{1}$. This approach consists in a study of the $\operatorname{\Delta}^{(k)}_{\Phi}$ by means of the Weyl calculus of Hörmander [Hör85, 18.4–6], and it circumvents the difficulty that pseudo-differential techniques do not play well together with the weighted Sobolev spaces in which the $A_{k}$ act. In this analysis, sufficient conditions for closed range of $A_{0}$, strict positivity of $A_{1}$ and essential self-adjointness as well as characterisations of the maximal domains of $A_{0}$ and $A_{1}$ are established (through similar results for the $\operatorname{\Delta}^{(k)}_{\Phi}$).

When combined, the ps.d. and Hilbert space analyses show formula (1.2) for a class of $\Phi$ containing e.g. all polynomials of even degree $r\geq 2$, which have positive definite part of degree $r$ (a change of the constant term will renormalise to $\int d\mu=1$); in comparison the polynomials belonging to the class of [Sjö96] have $r=2$ (a rather simpler case because the term $\Phi^{\prime\prime}$ in (1.4) is bounded on $L^{2}({{\mathbb{R}}}^{n},{{\mathbb{C}}}^{n})$), while [Hel98] covered cases with $r=4$.

1.3. The main results

In formula (1.2) the probability measure $\mu$ will be arbitrary to begin with, while $g_{j}$ will be considered either in the weighted Sobolev space $H^{1}(\mu)$ equal to $\{\,u\in L^{2}(\mu)\mid\forall j=1,\dots,n\colon\partial_{j}u\in L^{2}(\mu)\,\}$, whereby $L^{2}(\mu):=L^{2}({{\mathbb{R}}}^{n},\mu,{{\mathbb{C}}})$, or in $L^{2}(\mu)$ when this is justified. When the measure $\mu$ is such that $d\mu=e^{-\Phi(x)}\,dx$, then it will throughout be assumed that

[TABLE]

To simplify notation (cf. (1.7) ff.  below), differentials will hereafter be used instead of gradients, so $A_{1}$ will act on suitable 1-forms in $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{1}{{\mathbb{C}}}^{n})$. To explicate the operators that appear as $d_{0}$ and $d^{*}_{0}$ in (1.8) below and onwards, note that when 1-forms are identified with vector functions $v$, then $d_{0}f$, $d_{0}^{*}v$ identify with $\nabla f$ and $(\Phi^{\prime}-\nabla)\cdot v$, respectively. Also for simplicity, $(\,\cdot\,|\,\cdot\,)_{\mu}$ will for any $k$ refer to the scalar product in $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{k}{{\mathbb{C}}}^{n})$, i.e. the space of $k$-forms with coefficients in $L^{2}(\mu)$; on this space the following norm is used:

[TABLE]

It is of course a central question how the operators $A_{0}$ and $A_{1}$ are defined precisely, but it will be equally important to make sense of the inverse $A_{1}^{-1}$. For this reason, it is worthwhile to define $A_{0}$ and $A_{1}$ as ‘Hodge Laplacians’ to begin with; this is not only a most general approach (it works for arbitrary probability measures $\mu$), but it also leads in a very natural way to a fruitful discussion, by simple operator theoretical methods, of the invertibility of $A_{1}$. This will be explained in the following, before the theorems are presented.

But first of all it should be emphasised that the present article focuses on the following problem for the identity (1.2):

Which probability measures $\mu$ have the property that formula (1.2) holds for all functions $f$, $g$ in $H^{1}(\mu)$?

Of course other problems would be equally meaningful (such as fixing two functions $f$ and $g$ and then search for the $\mu$ for which (1.2) would be true); but for simplicity’s sake the discussion will here be restricted to the above-mentioned problem.

Secondly, as the point of departure it is useful to adopt the following definitions of $A_{0}$ and $A_{1}$ as the Hodge Laplacians111For $d\mu=e^{-\Phi}\,dx$, an interpretation of $e^{-\Phi}$ as $\sqrt{\det(g_{ij})}$ would for any Riemannian metric $(g_{ij})$ lead to a Laplace–Beltrami operator which resembles $A_{0}$, $A_{1}$,…but differs; indeed, in this way already (1.6) would be unconventional in the study of the de Rham complex; see e.g. [Jos98, 2.1].

[TABLE]

associated to the complex (where $d_{-1}\equiv 0$)

[TABLE]

Here $d_{k}$ denotes the exterior differential of $k$-forms; this is in general a first order differential operator acting in the distribution sense, but in the context above, $d_{k}$ is equipped with its maximal domain as an unbounded, closed, densely defined operator from $L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ to $L^{2}(\mu,\wedge^{k+1}{{\mathbb{C}}}^{n})$; clearly $D(d_{0})=H^{1}(\mu)$. This is an example of a Hilbert complex in the sense of J. Brüning and M. Lesch [BL92], where such complexes are described in a clear way (by comparison the present article focuses on the conditions implying that (1.8) is a Fredholm complex rather than the conclusions that would follow from this property).

For $d_{k}$ the closure of the range is denoted by $X_{k+1}$,

[TABLE]

and the kernel by $Z(d_{k})$; as a convention $X_{0}:={{\mathbb{C}}}$. The Hilbert space adjoint, $d_{k}^{*}$, also enters in the Hodge Laplacians, cf. (1.7); for simplicity the superscript ‘(H)’ is suppressed in the sequel, for the definition in (1.7) will be in effect until Section 4, unless otherwise is explicitly stated.

Thirdly, a series of small remarks will clarify the situation: using the orthogonal projection $P$ onto $L^{2}(\mu)\ominus{{\mathbb{C}}}$, formula (1.2) may be written as

[TABLE]

Moreover, $d_{0}g_{1}$ and $d_{0}g_{2}$ belong to $X:=X_{1}$, and since $d\circ d\equiv 0$ implies $X_{k}\subset Z(d_{k+1})$, the subspace $X$ of $L^{2}(\mu,\wedge^{1}{{\mathbb{C}}}^{n})$ is invariant under $A_{1}$ by (1.7); in fact

[TABLE]

Hence one has the following identity of unbounded operators in $X$:

[TABLE]

in particular, $D(A_{1}^{-1})\cap X$ equals the range $R(d_{0}d_{0}^{*}|_{X})=d_{0}d_{0}^{*}(X)$.

It is now straightforward to verify the implications

[TABLE]

Indeed, these properties are, by (1.12) and the self-adjointness of $P$, both equivalent to

[TABLE]

e.g. it is found when (1.14) holds that $(d_{0}d_{0}^{*})^{-1}d_{0}f$ is an element of $D(d_{0}^{*})$, whence (1.15) follows from the definition of the adjoint. The other implications follow in a similar manner.

Therefore the formulated problem for (1.10) has been reduced to the just given property in (1.14) for $P$, and this rewriting as a ‘linear’ problem makes the analysis more straightforward, as we shall see immediately.

Lemma 1.2**.**

$Pf=d_{0}^{*}(d_{0}d_{0}^{*})^{-1}d_{0}f$* holds whenever $f-\langle f\rangle$ belongs to $D(d_{0})\cap R(d_{0}^{*})$.*

Proof.

When $f-\langle f\rangle$ is in $H^{1}(\mu)\cap R(d_{0}^{*})$, then $d_{0}f=d_{0}(d_{0}^{*}v)$ holds for some $v\in X\cap D(d_{0}^{*})$, because $X^{\perp}=Z(d_{0}^{*})$. Hence $d_{0}f$ belongs to $D(d_{0}d_{0}^{*}|_{X}^{-1})$, and so

[TABLE]

since $d_{0}d_{0}^{*}|_{X}^{-1}$ maps into $D(d_{0}d_{0}^{*})\subset D(d_{0}^{*})$. ∎

The usefulness of the lemma in connection with the proof of (1.2) was independently discovered by V. Bach, T. Jecko and J. Sjöstrand [BJS98, (II.15)].

Conversely (1.14) implies that $f-\langle f\rangle=Pf$ belongs to $D(d_{0})\cap R(d_{0}^{*})$ for any $f\in H^{1}(\mu)$. However, it is more useful to ask the following question: which probability mesures $\mu$ have the property that

[TABLE]

Since $H^{1}(\mu)=D(d_{0})$, it is clear that $\mu$ has this property if and only if

[TABLE]

(Here and throughout $F\ominus{{\mathbb{C}}}$ stands for the elements of a given subspace $F\subset L^{2}$ which are orthogonal to the constant functions.) The inclusion (1.18) would obviously be true if $\mu$ is such that

[TABLE]

since $\overline{R(d_{0}^{*})}=Z(d_{0})^{\perp}={{\mathbb{C}}}^{\perp}$, condition (1.19) is equivalent to closedness of the range $R(d_{0}^{*})$.

Because (1.19) also expresses that $R(d_{0}^{*})$ should be the maximal possible subspace of $L^{2}(\mu)$, one could in view of (1.18) envisage that (1.19) is sufficient but unnecessary.

However, this is not the case, for a fine version of the Closed Range Theorem yields that (1.10) is equivalent to the closedness of $d_{0}^{*}$’s range, hence to (1.19). This is explained for arbitrary unbounded operators in general Hilbert spaces in Section 3 below, but the consequences for the $A_{k}$ are summed up in the following theorem. In particular, the problem formulated for (1.2) above has been rephrased by means of a set of equivalent conditions on $A_{0}$, $A_{1}$, $d_{0}$ and $d_{0}^{*}$, that one finds for $k=0$ in

Theorem 1.3**.**

Let $k$ be a fixed index in the complex (1.8). Then closedness of each of the spaces $R(A_{k})$, $R(d_{k})$ and $R(d^{*}_{k})$ as well as strict positivity of $A_{k+1}|_{X_{k+1}}$ are equivalent properties, and if $P_{k}$ is the projection from $L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ onto $R(A_{k})$, these properties are also equivalent to the validity of

[TABLE]

In the affirmative case, $A_{k}|_{Z(d_{k})^{\perp}}=d^{*}_{k}d_{k}$ and has closed range $R(A_{k})=R(d^{*}_{k})$ and $A_{k+1}|_{X_{k+1}}=d_{k}d^{*}_{k}$. Moreover, there is an equivalence

[TABLE]

with $U$ equal to the unitary $d_{k}(d_{k}^{*}d_{k}|_{X_{k}})^{-1/2}$ from $L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})\ominus Z(d_{k})$ to $X_{k+1}$, and consequently

[TABLE]

with the analogous relation for the essential spectra.

The point of the proof of this result is to combine the usual estimates from below of the adjoint (here $d^{*}_{0}$) with the fact that $T^{*}T$ is self-adjoint for any densely defined, closed operator $T$. In the abstract set-up, the range projection of $T^{*}T$ has the form $P=T^{*}(TT^{*}|_{R(T)})^{-1}T$; when applied to $A_{0}$ and $A_{1}$, this yields (1.14) and thus (1.2) at least for $u$, $g_{1}$ and $g_{2}\in H^{1}(\mu)$.

However, neither closedness of $R(A_{0})$, $R(d_{0})$, $R(d_{0}^{*})$ nor positivity of $A_{1}|_{X}$ are easy to analyse when the $A_{k}$ are defined from (1.7) for an arbitrary probability measure $\mu$. From Section 4 below and onwards, we shall therefore work under the assumption that $\mu$ has a density $e^{-\Phi(x)}$ with respect to Lebesgue measure for some $\Phi$ fulfilling (1.5).

Using this assumption, an alternative variational definition of the $A_{k}$ (i.e. by means of sesqui-linear forms, or Lax–Milgram’s lemma) is introduced in Section 4 below; thereafter it is seen that this yields the Friedrichs extension from $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$, and it is then shown, for the variationally defined operators, that (1.7) holds both in the distribution sense and as a formula for unbounded operators. (Concerning essential self-adjointness of the $A_{k}$, see the remarks in Section 8 below.)

Using the definition by Lax–Milgram’s lemma, it is proved in Section 5 below that the regularity assumption on the $g_{1}$, $g_{2}$ and $f$ above may be relaxed from $H^{1}(\mu)$ to $L^{2}(\mu)$:

Theorem 1.4**.**

Let $\Phi$ satisfy (1.5), and suppose that $A_{0}$ as an unbounded operator in $L^{2}(\mu)$ has closed range, $R(A_{0})=\overline{R(A_{0})}$. Then (1.2) holds for all $g_{1}$ and $g_{2}$ in $L^{2}(\mu)$.

Moreover, it then holds that $L^{2}(\mu)=R(A_{0})\oplus{{\mathbb{C}}}$, and for every $u\in L^{2}(\mu)$,

[TABLE]

when $Pu=u-\int u\,d\mu$ denotes the orthogonal projection onto $L^{2}(\mu)\ominus{{\mathbb{C}}}$.

Note also that when $g_{2}$ is in $L^{2}(\mu)\setminus H^{1}(\mu)$, it is understood in Theorem 1.4 that the right hand side of (1.2) should be read as a duality $\langle\,\tilde{\cal A}_{1}^{-1}dg_{1},\,d\overline{g_{2}}\,\rangle_{\tilde{V}\times\tilde{V}^{\prime}}$ for a certain Hilbert space $\tilde{V}$ with isomorphism $\tilde{\cal A}_{1}\colon\tilde{V}\to\tilde{V}^{\prime}$ onto its dual. See Section 5.3 for details.

Although the identification of $A_{1}$, or rather $A_{1}|_{X}$, with a restriction of $\cal A_{1}$ is a well-known procedure for operators defined by the Lax–Milgram lemma, it is really the validity of a certain Poincaré inequality for $\tilde{V}$ which makes this possible. Because $\tilde{V}$ is the form domain of $A_{1}|_{X}$, one may of course ask directly whether such Poincaré inequalities for the form domain of $A_{1}$ or $A_{1}|_{X}$ would imply (1.2); moreover, also the exactness of the complex (1.8) may be considered. However, that the Poincaré inequalities are equivalent to e.g. positivity of $A_{0}$ and $A_{1}|_{X}$, respectively, while exactness is an ‘intermediate’ condition, is proved in Theorem 5.1 below, where the full interrelationship among the various criteria is settled.

By exploiting the possibility in Theorem 1.4 of taking the $g_{j}\in L^{2}(\mu)$, one finds as an application that (1.2) implies a generalisation of the inequality in (1.1) from $C^{1}$ to $H^{1}_{\operatorname{loc}}(\mu)$:

Corollary 1.5**.**

Let $\Phi\in C^{2}({{\mathbb{R}}}^{n},{{\mathbb{R}}})$ satisfy $\int e^{-\Phi}\,dx=1$ and be strictly convex, i.e.

[TABLE]

Then the inequality (1.1) holds for all $f\in L^{2}({{\mathbb{R}}}^{n},\mu)\cap H^{1}_{\operatorname{loc}}({{\mathbb{R}}}^{n})$.

This generalises [Hel98], where B. Helffer sketched how (1.2) implies (1.1) for uniformly strictly convex $\Phi$, when in addition $\nabla f$ is bounded. The general case was left open there, although with indications that (1.2), because it is an exact formula, is more powerful. (The reference to (1.2) refines an explanation from 1993, see [Hel95], where $A_{0}$ and $A_{1}$ were introduced for (1.1) without (1.2).)

For the sake of the proof, it should be recalled from [Hel98] that in the uniformly strictly convex case, say $\Phi^{\prime\prime}(x)\geq c_{0}>0$ on ${{\mathbb{R}}}^{n}$ for every $x$, the idea behind (1.2)$\implies$(1.1) is to infer from the formal expression in (1.3) that

[TABLE]

This actually requires justification, for inclusions between the domains $D(A_{1})$ and $D(\Phi^{\prime\prime})$ need not exist under the assumption (1.5). Nevertheless one may envisage that $A_{1}\geq c_{0}>0$ then holds — so that $A_{1}^{-1}$ would be well defined — and this is verified in Section 4.5 below. However, the proof of Corollary 1.5 is first completed in Section 5.3, after that of Theorem 1.4.

The identity (1.2) is established independently of compactness of $H^{1}(\mu)\hookrightarrow L^{2}(\mu)$, so in particular the essential spectra of $A_{0}$ and $A_{1}$ need not be empty. In this connection some simple sufficient conditions are given in Section 6, exploiting the Persson–Agmon formula and results due to P. Bolley, M. Dauge and B. Helffer [BDH89].

However, slightly stronger conditions allow $A_{0}$ and $A_{1}$ to be analysed by means of the Weyl calculus [Hör85, 18.4–6], leading to closed range, essential self-adjointness and positivity. With $v^{\alpha}:=v_{1}^{\alpha_{1}}\cdot\dots\cdot v_{n}^{\alpha_{n}}$ for $\alpha\in{\mathbb{N}}_{0}^{n}$ and $v\in{{\mathbb{C}}}^{n}$, or $v=-i\nabla=:D$, the following results are restatements of Theorems 7.3, 7.4 and 7.5 below:

Theorem 1.6**.**

Let $\Phi(x)\in C^{\infty}({{\mathbb{R}}}^{n},{{\mathbb{R}}})$ satisfy $\int e^{-\Phi(x)}\,dx=1$; suppose also that $|\Phi^{\prime}(x)|\to\infty$ for $|x|\to\infty$, that $|D^{\beta}\Phi(x)|\leq C_{\beta}(1+|\Phi^{\prime}(x)|^{2})^{\frac{1}{2}}$ for every multiindex $|\beta|\geq 1$ and that, for some $M$, any $D^{\beta}\Phi$ with $|\beta|=M$ is bounded on ${{\mathbb{R}}}^{n}$.

Then $A_{0}$ has closed range in $L^{2}({{\mathbb{R}}}^{n},\mu,{{\mathbb{C}}})$, so that the conclusions of Theorem 1.4 are valid, and $A_{0}$ is essentially self-adjoint (when considered) on $C^{\infty}_{0}({{\mathbb{R}}}^{n})$. The operator $A_{1}$ is essentially self-adjoint on $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{1}{{\mathbb{C}}}^{n})$ and has closed range.

Moreover, if there exist $\omega>0$, $C>0$ such that $x\cdot\Phi^{\prime}(x)\geq|x|^{1+\omega}/C$ holds for all $|x|\geq C$, then $A_{1}>0$ on $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{1}{{\mathbb{C}}}^{n})$.

The considered class of $\Phi$ is larger than those in [Sjö96, Hel98]; e.g. any polynomial $\Phi$ of even degree $\geq 2$ satisfies the assumptions when the part of highest degree is positive definite; this includes the condition for strict positivity of $A_{1}$ (seen as in Example 7.3 below).

Theorem 1.7**.**

Under the hypothesis for $\Phi$ made in the beginning of Theorem 1.6, a function $u$ belongs to $D(A_{0})$ precisely when it has the property

[TABLE]

A form $\sum_{l=1}^{n}v_{l}\,dx^{l}$ is in $D(A_{1})$ if and only if each $v_{l}$ satisfies (1.26).

Consequently $A_{0}$ and $A_{1}$ have compact resolvents, hence their spectra are discrete.

When the condition for $A_{1}>0$ is fulfilled for some $\omega\geq 1$, then $D(A_{0})$ contains the set $B_{2}({{\mathbb{R}}}^{n},\mu)$, defined as those $u$ for which $x^{\beta}D^{\alpha}u\in L^{2}(\mu)$ when $|\alpha+\beta|\leq 2$.

Theorem 1.7 may be used for example in distortion arguments for $A_{1}^{-1}$ in the correlation estimates; see [Hel98, Thm. 4.1]. Recently Wei-Min Wang [Wan99] has applied Theorem 1.6, or rather the corresponding facts on $\operatorname{\Delta}^{(1)}_{\Phi}$ in (the proofs of) Theorems 7.4–7.5 below; indeed the explicit conditions on $\Phi$ implying closed range and injectivity was used in [Wan99, Sect. 3–4]. In addition the unitary equivalence in (1.21) has been used by V. Bach, T. Jecko and J. Sjöstrand [BJS98, Prop. V1].

Remark 1.8*.*

For general probability measures $\mu$ it is questionable which expressions the associated operators $A_{0}$, $A_{1}$ can have: already for $\mu$ absolutely continuous with respect to Lebesgue measure, $\Phi$ equals $+\infty$ in ${{\mathbb{R}}}^{n}\setminus\operatorname{supp}\tilde{\mu}$, and this Borel set may be so irregular that when one attempts a calculation of the $d_{0}^{*}$ in (1.7), then $\partial_{j}e^{-\Phi}$ is unequal to $-\Phi^{\prime}_{j}e^{-\Phi}$ (and the latter may even be undefined in $\cal D^{\prime}$).

Acknowledgement

I am very grateful to B. Helffer for several discussions on the subject and for letting me carry further some ideas from [Hel98] with this work, and I thank V. Bach and T. Jecko for fruitful conversations and for explaining me about the applications to statistical mechanics.

2. Notation and Preliminairies

For an operator $T$ in a Hilbert space with scalar product $(\,\cdot\,|\,\cdot\,)$ and norm $\|\cdot\|$, the domain, range and kernel is written $D(T)$, $R(T)$ and $Z(T)$, respectively, while $\rho(T)$ and $\sigma(T)$ stand for the resolvent and spectrum of $T$; the space of bounded operators is denoted by ${{\mathbb{B}}}(H)$. The essential spectrum $\sigma_{\operatorname{ess}}(T)=\sigma(T)\setminus\sigma_{\operatorname{disc}}(T)$ consists in the self-adjoint case of those $\lambda\in{{\mathbb{C}}}$ for which there is a sequence with $\|x_{n}\|=1$ and $(T-\lambda)x_{n}\to 0$ while $x_{n}\rightarrow 0$ weakly; it is the complement of the $\lambda$ for which $T-\lambda$ is Fredholm $D(T)\to H$ in the graph topology, i.e. has $Z(T-\lambda)$ of finite dimension and $R(T-\lambda)$ closed (using [Hör85, Prop. 19.3]). Moreover,

[TABLE]

is the lower bound of $T$. Occasionally the norm $\|\cdot\|_{X}$ in a space $X$ is written $\mathinner{\|}\cdot\,|X\|$, to avoid unnecessary subscripts.

Given a triple $(H,V,s)$ consisting of two Hilbert spaces $V\hookrightarrow H$ with bounded, dense injection and a bounded sesqui-linear form $s(\cdot,\cdot)$ on $V$, then coerciveness — i.e. existence of $c>0$ and $k\in{{\mathbb{R}}}$ such that

[TABLE]

gives the following for the operator $S$ defined on \bigl{\{}\,u\in V\bigm{|}\exists f\in H\forall v\in V:s(u,v)=(\,f\,|\,v\,)_{H}\,\bigr{\}} by the formula $Su=f$:

Lemma 2.1**.**

$1^{\circ}$* $S$ is closed in $H$ with $D(S)$ dense in $V$. The adjoint $S^{*}$ is the analogous operator defined from $s^{*}(v,w):=\overline{s(w,v)}$.*

$2^{\circ}$* When $s(\cdot,\cdot)$ is positive definite on $V$ (i.e. $s(v,v)>0$ for $v\neq 0$, e.g. for $k=0$), then $S$ extends to the isometry $\cal A$ of $(V,s(\cdot,\cdot))$ onto $V^{*}$, its antidual, and $D(S)=\cal A^{-1}(H)$.*

This result, known as Lax–Milgram’s lemma, may be proved straightforwardly in the fashion of J.–L. Lions and E. Magenes [LM68, Sect. 9.1].

$C^{\infty}_{0}({{\mathbb{R}}}^{n})$ denotes the space of infinitely differentiable functions with compact support, and $\cal D^{\prime}({{\mathbb{R}}}^{n})$ its dual; $\cal D^{\prime}{}^{k}$ the subspace of distributions of order $k$; $\langle\,u,\,\varphi\,\rangle=u(\varphi)$ for all $u\in\cal D^{\prime}$ and $\varphi\in C^{\infty}_{0}$. For $L^{2}({{\mathbb{R}}}^{n},\mu)$ the scalar product and norm is written $(\,\cdot\,|\,\cdot\,)_{\mu}$ and $\|\cdot\|_{\mu}$ respectively, although with $\mu$ omitted in case of the Lebesgue measure. Similar notation is adopted for the space of $k$-forms $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{k}{{\mathbb{C}}})$; in general, for a space $F$ of functions ${{\mathbb{R}}}^{n}\to{{\mathbb{C}}}$, the set $F({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ consists of the differential forms with coefficients therein.

A differential form of degree $k$ with complex $C^{\infty}$-coefficients has the form

[TABLE]

here $\sum^{\prime}$ indicates summation over increasing $k$-tuples $J=(j_{1},\dots,j_{k})$, i.e. strictly increasing maps $\{\,1,\dots,k\,\}\to\{\,1,\dots,n\,\}$; and $dz^{J}:=dz^{j_{1}}\wedge\dots\wedge dz^{j_{k}}$ stands for the usual $k$-linear anti-symmetric map $({{\mathbb{C}}}^{n})^{k}\to{{\mathbb{C}}}$ derived from the Cartesian coordinates in ${{\mathbb{C}}}^{n}$. By anti-symmetry in the indices, $f_{jL}=f_{J}\varepsilon^{jL}_{J}$ where $\varepsilon^{jL}_{J}=0$ unless $J$ has the same elements as $jL:=(j,l_{1},\dots,l_{k-1})$, in which case $\varepsilon^{jL}_{J}$ is the sign of the permutation $\left(\begin{smallmatrix}jL\\ J\end{smallmatrix}\right)$.

For the distribution space $\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$, see Appendix A.

3. An Operator Approach

It is shown in this section how Hilbert space methods can provide detailed information about (1.2), using the rewriting given in (1.14). The basic observation is that a similar projection appears in the Closed Range Theorem, at least in the version established below where the self-adjointness of $T^{*}T$ and $TT^{*}$ is incorporated for this purpose.

Let in the sequel $T\colon H\to H_{1}$ be a densely defined, closed operator between Hilbert spaces $H$ and $H_{1}$, and let $F\subset H$ and $F_{1}\subset H_{1}$ denote two closed subspaces such that

[TABLE]

Here the possibility of taking $F_{1}$ different from $\overline{R(T)}$ is adopted from Hörmander’s treatment of the $\bar{\partial}$-complex [Hör66, Ch. 4]; in an analogous way this is useful for the below study of the exterior derivative $d$ in $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ spaces, where $F_{1}=Z(d_{k+1})$ is a natural choice when $T=d_{k}$.

The closedness of $T$’s range is closely connected to the properties of the operators

[TABLE]

and to the orthogonal projection $P$ onto $F$. In fact one has the next result, which might be folklore, but nevertheless is formulated as a theorem in view of the clarification it gives for (1.2):

Theorem 3.1** (Closed Range Theorem).**

When $T$ is an operator as above, and the set-up in (3.1)–(3.2) is used, then the following properties are equivalent:

• (i)

$R(T)$* is closed and equal to $F_{1}$.*

• (ii)

There exists $c>0$ such that $\mathinner{\|}y\,|H_{1}\|\leq c\mathinner{\|}T^{*}y\,|H\|$ for $y\in D(T^{*})\cap F_{1}$.

• (iii)

$R(T^{*})$* is closed and $F_{1}^{\perp}=Z(T^{*})$.*

• (iv)

$S_{1}$* is injective and has closed range.*

• (v)

$S$* is injective and has closed range.*

• (vi)

$S_{1}$* is injective and*

[TABLE]

In the affirmative case, $S$ and $S_{1}$ are unitarily equivalent, that is

[TABLE]

holds for $U=TS^{-\frac{1}{2}}$, which is an isometry $U\in{{\mathbb{B}}}(F,F_{1})$. Consequently $\sigma(S)=\sigma(S_{1})$ and $\sigma_{\operatorname{ess}}(S)=\sigma_{\operatorname{ess}}(S_{1})$.

Proof.

Note first that $F_{1}^{\perp}\subset Z(T^{*})$ because of the assumption on $F_{1}$, and that $D(T^{*})$ is invariant under projection onto $F_{1}$ and $F_{1}^{\perp}$; indeed, if $D(T^{*})\ni y=f+f^{\perp}$ where $f\in F_{1}$ and $f^{\perp}\in F_{1}^{\perp}$, then $f^{\perp}\in Z(T^{*})$ and $f\in D(T^{*})$. Moreover, $S_{1}$ is densely defined in $F_{1}$: if $D(TT^{*})\ni y_{k}\to y\in F_{1}$, one may split $y_{k}=f_{k}+f^{\perp}_{k}$, where $f_{k}\in F_{1}\cap D(TT^{*})=D(S_{1})$ while $\|y-f_{k}\|\leq\|y-y_{k}\|\searrow 0$; the self-adjointness of $TT^{*}$ then carries over to $S_{1}$. Because the roles of $T$ and $T^{*}$ may be interchanged, also $S$ is self-adjoint.

(i)$\iff$(ii) is proved in [Hör66, Lem. 4.1.1]. When (ii) holds, then $Z(T^{*})\ni z=f+f^{\perp}$ (with $f^{(\perp)}$ as above) implies $T^{*}f=0$, hence $f=0$, so $Z(T^{*})=F_{1}^{\perp}$; hence $R(T^{*})$ equals $R(T^{*}|_{F_{1}\cap D(T^{*})})$, and the latter is closed by (ii) so (iii) is obtained. To deduce (ii) from (iii), it suffices to consider $T^{*}\colon F_{1}\cap D(T^{*})\to H$ as a bounded operator in the graph norm and apply the Open Mapping Theorem. Because (i) and (iii) are equivalent, so would (v) and (iv) be once (i)$\iff$(iv) is proved.

From (iii) injectivity follows since $TT^{*}z=0$ yields $z\in Z(T^{*})=F_{1}^{\perp}$; clearly $R(S_{1})\subset R(T)$, but any $y\in R(T)$ equals $Tx$ for some $x\in D(T)\ominus Z(T)\subset Z(T)^{\perp}=R(T^{*})$; thus $R(S_{1})$ equals $R(T)$, hence is closed. This shows (iv).

When (iv) holds, $S_{1}^{-1}$ is defined on $F_{1}$ (because $R(S_{1})=Z(S_{1})=\{0\}$), and $S_{1}^{-1}\in{{\mathbb{B}}}(F_{1})$ since $S_{1}$ is closed. $S_{1}^{-1}$ maps into $D(TT^{*})$, so for every $x\in D(T)$ it is obvious that $x\in D(T^{*}S_{1}^{-1}T)$ and that

[TABLE]

since $1-P$ is the projection onto $Z(T)$, this entails (3.3), for

[TABLE]

Finally, (vi)$\implies$(i), for injectivity of $S_{1}$ yields $F_{1}=\overline{R(T)}$, while (3.3) shows that

[TABLE]

hence $R(T)=R(TT^{*})$, and since this implies that $R(T)$ is closed, (i) is obtained.

However, for completeness’ sake an elementary proof of the just mentioned implication shall be given. When $R(T)=R(TT^{*})$, then one can pass to a domain consideration for the operator $\bar{T}=(T|_{F})^{-1}$ and use that

[TABLE]

Since $\bar{T}^{*}\bar{T}=(TT^{*}|_{F})^{-1}$, it is self-adjoint $\geq 0$ in $F_{1}$ with a square root $Q=Q^{*}=(\bar{T}^{*}\bar{T})^{1/2}\geq 0$ fulfilling

[TABLE]

Indeed, $D(Q^{2})=D((TT^{*})^{-1})=R(T)=D(\bar{T})$ so it remains to be shown that $D(Q)=D(\bar{T})$. But if $(x_{k})$ is a sequence in $D(Q^{2})=D(\bar{T})$,

[TABLE]

whence the closures of this set in the graph topologies on $D(Q)$ and $D(\bar{T})$ coincide.

Combining the boundedness of the resolvent with (3.9) it follows that

[TABLE]

and it is seen from the first of these lines and (3.9) that $\bar{T}(I+\bar{T}^{*}\bar{T})^{-1}$ belongs to ${{\mathbb{B}}}(F_{1},F)$; then the third line gives that $\bar{T}\bar{T}^{*}\bar{T}(I+\bar{T}^{*}\bar{T})^{-1}\in{{\mathbb{B}}}(F_{1},F)$, and (3.8) implies that $R(T)=D(\bar{T})=F_{1}$, which is closed.

Given that (i)–(vi) hold, then (v) gives both that $S^{\frac{1}{2}}$ is injective and that $S^{-\frac{1}{2}}\in{{\mathbb{B}}}(F)$, because it is closed and everywhere defined, and similarly $U:=TS^{-\frac{1}{2}}\in{{\mathbb{B}}}(F,F_{1})$.

For any $x\in D(S)$ it holds that $S^{-\frac{1}{2}}x\in D(S)\subset D(T^{*}T)$, so

[TABLE]

This extends to all $x\in F$, and $TS^{-\frac{1}{2}}$ maps onto $F_{1}$ by (i), for the fact that $D(S^{\frac{1}{2}})=D(T|_{F})$ (shown analogously to (3.9)) shows that $S^{-\frac{1}{2}}$ maps onto $D(T)\ominus Z(T)$. Hence $U$ is unitary.

The identity (3.4) holds on vectors $x\in D(S^{2})$, for $S^{2}x=(T^{*}T)^{2}x=T^{*}S_{1}Tx$, so that

[TABLE]

note that $S^{-\frac{1}{2}}T^{*}\subset U^{*}$. Now any $x\in D(S)$ may be approximated in the graph norm by $x_{k}\in D(S^{2})$: then $Sx_{k}\to Sx$ while

[TABLE]

therefore $Ux\in D(S_{1})$ with $S_{1}Ux=USx$, from where (3.4) follows for $x$ by application of $U^{*}$. Conversely, note that $U^{*}S_{1}U$ acts as $U^{*}TS^{\frac{1}{2}}$, hence has its domain contained in $D(TS^{\frac{1}{2}})=D(S)$ (since $U^{*}\in{{\mathbb{B}}}(F_{1},F)$); therefore $U^{*}S_{1}Ux$ can only be defined when $Sx$ is so, and then we have already seen that $U^{*}S_{1}Ux=Sx$.

Finally, for $\lambda\in\sigma(S)$ there is $x_{k}\in D(S)$ with $\|x_{k}\|=1$ and $(S-\lambda)x_{k}\to 0$, and $y_{k}=Ux_{k}$ is also normalised while

[TABLE]

moreover, $y_{k}\to 0$ weakly if the $x_{k}$ do so, hence also $\sigma_{\operatorname{ess}}(S)\subset\sigma_{\operatorname{ess}}(S_{1})$, and the opposite inclusions are equally easy. ∎

The requirement in (iv) is equivalent to [math] belonging to the resolvent set of $S_{1}$, and by the minimax principle this may, of course, be replaced by strict positivity of $S_{1}$. Applied to the complex (1.8) this yields, because of (1.12):

Corollary 3.2**.**

The conclusions of Theorem 1.3 are valid.

For $k=0$ this almost gives the main part of Theorem 1.4, for clearly $A_{0}=d^{*}_{0}d_{0}$ has kernel ${{\mathbb{C}}}$ so that $P_{0}$ must equal $u-\int u\,d\mu$; whence (1.23).

The goal is not yet attained, however. First of all we shall in Section 4 below give a definition of $A_{0}$ and $A_{1}$ using sesqui-linear forms, and then verify in (4.13) and (4.21) below that this coincides with the $A_{k}$ in (1.7) above and gives a meaning to (1.3). Secondly, the formula for $P_{k}$ in Theorem 1.3 is obtained for $H^{1}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ only, whereas for Theorem 1.4 it is necessary to make sense of the right hand side of (1.23) when the $u$ there is arbitrary in $L^{2}(\mu)$. This is based on the Lax–Milgram definition in Section 4, and is carried out in Section 5.

Remark 3.3*.*

Corollary 3.2 and the remark following it entails

[TABLE]

Earlier Sjöstrand [Sjö96] obtained that the gap between the first two eigenvalues of $A_{0}$ is larger than $A_{1}$’s first eigenvalue. This also follows immediately from the above formula when the assumptions on $\mu$, or $\Phi$, imply that the spectra are discrete, as in [Sjö96].

4. The $d$-complex in Weighted $L^{2}$ Spaces

From this section and onwards, the probability measure is assumed to have the form $d\mu=e^{-\Phi(x)}\,dx$ in order to derive more explicit conditions.

4.1. The Operators

To avoid cumbersome justification of integration by parts, it is worthwhile to define $A_{0}$, $A_{1}$, …  variationally, i.e. by Lax–Milgram’s lemma (in contrast with [Hel95, Sjö96, Hel98] were the Friedrichs extension was used). This is based on the weighted space $L^{2}(\mu):=L^{2}({{\mathbb{R}}}^{n},\mu,{{\mathbb{C}}})$ with measure $\mu:=e^{-\Phi(x)}dx$ and scalar product

[TABLE]

and its analogue for $k$-forms $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{k}{{\mathbb{C}}}^{n})$, where $(\,v\,|\,w\,)_{\mu}$ is defined instead by integration of $\sum{}^{\prime}v_{J}(x)\overline{w_{J}(x)}$, with prime denoting summation over increasing $k$-tuples $J$, see Section 2.

For each $k$ the operator $A_{k}$ is defined in $H_{k}:=L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ by means of the triple $(H_{k},V_{k},a_{k})$, where

[TABLE]

with $a_{k}(\cdot,\cdot)$ defined on $V_{k}$ and $d_{k}$ equal to the exterior differential from $D(d_{k})\subset H_{k}$ to $H_{k+1}$ (with derivatives calculated in $\cal D^{\prime}({{\mathbb{R}}}^{n})$), while $d^{*}_{k}$ denotes the Hilbert space adjoint with respect to $(\,\cdot\,|\,\cdot\,)_{\mu}$, see (4.11) below for the expression. Recall that in this way $A_{k}$ is defined as follows:

[TABLE]

To substantiate this, note that $V_{k}$ in (4.3), in view of $d_{k}^{*}$’s closedness and differentiation’s continuity in $\cal D^{\prime}({{\mathbb{R}}}^{n})$, is a Hilbert space with

[TABLE]

the sesqui-linear form is clearly bounded

[TABLE]

Since $a_{k}(\cdot,\cdot)$ is symmetric and (4.6) yields

[TABLE]

$A_{k}$ is well defined, self-adjoint with spectrum in $[0,\infty[\,$ and with domain $D(A_{k})$ dense in $V_{k}$. When obtaining this from Lemma 2.1, it is important to have density of $V_{k}\subset H_{k}$, but more than that holds in the present set-up:

Lemma 4.1**.**

The space $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ is dense in each of the domains $D(d_{k})$, $D(d^{*}_{k-1})$ and $D(d_{k})\cap D(d^{*}_{k-1})=V_{k}$ with respect to their graph topologies (for $V_{k}$ this is given by (4.6)).

This may be proved by a cut-off and convolution procedure, as in [Hör66, Lem. 4.1.3] mutatis mutandem. (Specifically one should let $\Omega={{\mathbb{R}}}^{n}$, $S=d_{k}$ and $T^{*}=d^{*}_{k-1}$ there, while $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$ should equal our $\Phi$; the required inequality (4.1.6) there is redundant for we may take $\eta_{\nu}(x)=\eta(\nu^{-1}x)$ on ${{\mathbb{R}}}^{n}$.)

As a second application of Lemma 4.1 we have a characterisation of $A_{k}$:

Lemma 4.2**.**

$A_{k}$* equals the Friedrichs extension from $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$.*

Proof.

Let $S$ denote $A_{k}$’s restriction to $C^{\infty}_{0}$ and let $T$ be the Friedrichs extension (using that $A_{k}\geq 0$). The completion of $C^{\infty}_{0}$ with respect to

[TABLE]

is a Hilbert space $V_{S}\subset L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})$. If $\varphi\in C^{\infty}_{0}$ and $c=1+m(S)_{-}$,

[TABLE]

so by Lemma 4.1 it is found that $D(A_{k})\subset V_{k}\subset V_{S}$. Because $T$ is the only self-adjoint extension of $S$ with domain contained in $V_{S}$, this yields $A_{k}=T$. ∎

As a part of the above-suggested proof of Lemma 4.1 it is found that $d^{*}_{k-1}$ has the following action on forms $f=\sum^{\prime}_{|J|=k}f_{J}\,dz^{J}$, cf. Section 2, when derivatives are calculated in $\cal D^{\prime}$:

[TABLE]

For later reference the argument is recalled: if $f\in H_{k}$ and $w\in C^{1}_{0}({{\mathbb{R}}}^{n},\wedge^{k-1}{{\mathbb{C}}}^{n})$,

[TABLE]

This should be justified since $w$ is not $C^{\infty}$, but by reading $\langle\,\cdot,\,\cdot\,\rangle$ as the duality of $\cal D^{\prime}{}^{1}$ and $C^{1}_{0}$, the $f_{J}$ may be approximated from $C^{\infty}_{0}({{\mathbb{R}}}^{n})$ so that Leibniz’ rule may be applied together with the continuity of $\partial_{j}\colon\cal D^{\prime}{}^{0}\to\cal D^{\prime}{}^{1}$ in the last line.

Taking $f\in D(d^{*}_{k-1})$ and $w$ such that $w_{L}=\delta_{LK}\varphi$ for arbitrary $K$ and $\varphi\in C^{\infty}_{0}({{\mathbb{R}}}^{n})$, the left hand side equals $\langle\,(d^{*}_{k-1}f)_{K},\,e^{-\Phi}\overline{\varphi}\,\rangle$ so that (4.11) results.

Using Lemma 4.1 once more, we now observe (as a first step in the verification of (1.7)) two different identifications for the domain of $d^{*}_{k-1}$:

Lemma 4.3**.**

For the exterior differential $d$ going from $H_{k-1}$ to $H_{k}$, whereby $H_{j}$ stands for the space $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{j}{{\mathbb{C}}}^{n})$, and its formal adjoint $d^{*}$ given by (4.11), the minimal and maximal realisation coincide, i.e. $d_{\operatorname{min}}=d_{\operatorname{max}}$ and $(d^{*})_{\operatorname{min}}=(d^{*})_{\operatorname{max}}$.

Indeed, for $d$ itself this is just a reformulation of Lemma 4.1, so by duality $(d^{*})_{\operatorname{max}}=(d^{*})_{\operatorname{min}}=d^{*}_{k-1}$. For this reason the true adjoint $d_{k}^{*}$ may be abbreviated as $d^{*}$.

When it is understood that $d$ and $d^{*}$ act in the distribution sense (as opposed to their maximal realisations in $H_{k}$, viz. $d_{k}$ and $d^{*}_{k-1}$, which have subscripts), it is now easy to infer that $A_{k}$ acts according to the formula

[TABLE]

For this it is advantageous to test against $w=e^{\Phi}\varphi$ for $\varphi\in C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$:

[TABLE]

if $v\in D(A_{k})$, while taking $f=d_{k}v\in H_{k+1}$ in (4.12) yields

[TABLE]

Strictly speaking the right hand side should be read as a sum (over $|J|=k$) of distributions acting on $\varphi_{J}$, cf. (4.12), for the dual of $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ is not considered here. Using the compact support of $w$ it follows analogously to (4.12) that, since $d^{*}v$ is in $\cal D^{\prime}{}^{0}$ (or rather has coefficients there),

[TABLE]

From the definition of $a_{k}$ this shows (4.13).

Combining (4.13) with (4.11) a calculation now yields an explicit formula for $A_{k}$’s action. The details of this will be given partly to verify the expressions for $A_{0}$ and $A_{1}$ in the introduction, and partly because such formulae may be of interest in their own right.

Since $dv=\sum^{\prime}_{K}\sum_{m,M}\partial_{m}v_{M}\varepsilon^{mM}_{K}\,dz^{K}$, where $|K|=|M|+1$,

[TABLE]

while the other contribution becomes, with $|L|=|J|-1$,

[TABLE]

Taking $j=m$ in (4.17) for fixed $J$, only $j\notin J=M$ gives a non-trivial term, viz. $(\Phi^{\prime}_{j}-\partial_{j})\partial_{j}v_{J}$; and for $j\neq m$ there are contributions when $j\in M$ and $m\in J$, in which case deletion of $j$ from $M$ and of $m$ from $J$ gives the same tuple, say $L$, so that

[TABLE]

and hence $\partial_{m}v_{M}\varepsilon^{mM}_{jJ}=-\partial_{m}v_{jL}\varepsilon^{mL}_{J}$. Therefore

[TABLE]

When $j=l$ in (4.18), then $\varepsilon^{jL}_{J}=0$ unless $j$ belongs to $J$, so $\sum_{L}^{\prime}$ has only one term $\neq 0$; hence $\underset{J\ni j}{\sum^{\prime}\sum}(\Phi^{\prime}_{j}-\partial_{j})\partial_{j}v_{J}\,dz^{J}$ is the contribution. For $j\neq l$ there appears a term present in (4.20) with opposite sign, plus one involving $\Phi^{\prime\prime}$.

Therefore, when $v=\sum^{\prime}_{J}v_{J}\,dz^{J}$ the action of $d^{*}d+dd^{*}$, and hence of $A_{k}$, is altogether given by the relatively simple formula

[TABLE]

where $J_{j\to l}$ means $J$ with $j$ replaced by $l$. Note that the degree $k$ of $v$ really only enters in the determination of how $\Phi^{\prime\prime}$ acts on $v$. (This formula was also given by Sjöstrand [Sjö96] for the Witten Laplacians ensuing after the transformation in Section 4.2 below.)

In particular, if $1$-forms are identified with vector functions,

[TABLE]

as claimed in the introduction. Note that for $k=0$ or $n$ the action of $A_{k}$ is given by (4.17) or (4.18), respectively, that is

[TABLE]

when $f$ in $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{n}{{\mathbb{C}}})$ is (considered as) a function in $D(A_{n})$.

4.2. Unitary Transformation

The operators $A_{k}$ are easily transformed into the Witten-Laplacians denoted by $\operatorname{\Delta}^{(k)}_{\Phi}$ in [Sjö96]. E.g. multiplication by $e^{-\Phi/2}$ defines a unitary $U\colon L^{2}(\mu)\to L^{2}({{\mathbb{R}}}^{n})$ transforming $A_{0}$ into (a realisation of) $B_{0}=-\!\operatorname{\Delta}+\frac{1}{4}|\Phi^{\prime}|^{2}-\frac{1}{2}\operatorname{\Delta}\Phi$.

For later reference, this is done consisely here. Writing $d=\sum\partial_{j}\,dz_{j}\wedge$ for the differential on $k$-forms $v=\sum^{\prime}_{J}v_{J}\,dz^{J}$, where $v_{J}$ is in $\cal D^{\prime}({{\mathbb{R}}}^{n})$ in general, the formal adjoint with respect to $(\,\cdot\,|\,\cdot\,)_{\mu}$ on $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ is

[TABLE]

whereby $dz_{j}\rfloor$ either removes $dz_{j}$ when present (and anti-commuted to the left) or gives zero. When denoting (with subscript $k$ if necessary)

[TABLE]

then $d_{\Phi}^{*}$ is the formal adjoint of $d_{\Phi}$ on $L^{2}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ and (on $\cal D^{\prime}{}^{1}$ forms)

[TABLE]

Using this, $v\in D(A_{k})$ with $v_{1}=A_{k}v$ if and only if for all $w\in V_{k}$,

[TABLE]

Hence $UA_{k}U^{*}$ equals the operator $B_{k}$ for the triple $(L^{2}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n}),V_{k,\Phi},b_{k})$ where $b_{k}(\cdot,\cdot)$ equals $(\,d_{\Phi}\cdot\,|\,d_{\Phi}\cdot\,)+(\,d_{\Phi}^{*}\cdot\,|\,d_{\Phi}^{*}\cdot\,)$ while $V_{k,\Phi}:=UV_{k}$ equals $D(d_{k,\Phi})\cap D(d_{k-1,\Phi}^{*})$ as subspaces of $L^{2}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$; cf. Lemma 2.1.

Using (4.28) it follows that the $B_{k}$ acts as $\operatorname{\Delta}^{(k)}_{\Phi}$.

4.3. Identification with the Hodge Laplacian

Denoting by $X_{k+1}$ the closure of $d_{k}$’s range in $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{k+1}{{\mathbb{C}}}^{n})$, that is $X_{k+1}=\overline{R(d_{k})}$ and $X_{0}={{\mathbb{C}}}$ as in (1.9),

[TABLE]

It is now elementary to see that $A_{k}$ commutes with the orthogonal projections onto the summands, and exploiting Lemma 4.1 once more it also follows that (4.13) may be read with $d$ and $d^{*}$ as the respective unbounded operators associated with the complex (1.8):

Lemma 4.4**.**

If $P_{k}$ is the orthogonal projection onto $X_{k}$,

[TABLE]

Furthermore, the restriction $A_{k}|_{X_{k}}$ is injective, and $A_{k}=d^{*}_{k}d_{k}+d_{k-1}d^{*}_{k-1}$ holds as a formula for unbounded operators, i.e. with $D(A_{k})=D(d^{*}_{k}d_{k})\cap D(d_{k-1}d_{k-1}^{*})$.

Proof.

Omitting some $k$’s for simplicity, it follows from (4.30) that $PV\subset V$, since $V=D(d^{*})\cap D(d)$. Using this one finds: if $v\in D(A_{k})$ and $w\in V$, then $d^{*}(1-P)\equiv 0$ and $dP\equiv 0$ so that

[TABLE]

consequently $PA_{k}\subset A_{k}P$. If $v\in X\cap Z(A_{k})$, then $0=(\,A_{k}v\,|\,v\,)_{\mu}=\|d^{*}v\|_{\mu}^{2}$, so $v$ is also in $Z(d^{*}_{k-1})=X^{\perp}$, whence $v=0$.

If $v\in D(A_{k})$ and $A_{k}v=v_{1}$ then $(\,Pv_{1}\,|\,w\,)_{\mu}=(\,d^{*}Pv\,|\,d^{*}w\,)_{\mu}$ and $(\,(1-P)v_{1}\,|\,w\,)=(\,d(1-P)v\,|\,dw\,)_{\mu}$ for all $w$ in $V$. Because $C^{\infty}_{0}$ is dense with respect to the graph norms in $D(d)$ and $D(d^{*})$, this gives by closure that $d^{*}Pv\in D(d)$ with $dd^{*}Pv=Pv_{1}$ and that $d(1-P)v\in D(d^{*})$ with $d^{*}d(1-P)v=(1-P)v_{1}$; hence that $v\in D(d^{*}_{1}d_{1})\cap D(d_{0}d_{0}^{*})$ with $dd^{*}v=Pv_{1}$ and $d^{*}dv=(1-P)v_{1}$. Conversely $d^{*}_{k}d_{k}+d_{k-1}d_{k-1}^{*}\subset A_{k}$ follows easily from (4.2)–(4.3). ∎

Note that by (4.31) the subspaces $X_{k}$ and $Z(d^{*}_{k-1})$ are invariant under $A_{k}$, and that the terms $d^{*}_{k}d_{k}$ and $d_{k-1}d^{*}_{k-1}$ vanish there, respectively.

Also both $d_{k}A_{k}$ and $A_{k+1}d_{k}$ are defined on $D(d_{k+1}d^{*}_{k}d_{k})$, so in this way we have the intertwining properties

[TABLE]

on $D(d_{k}d^{*}_{k}d_{k})$ and $D(d^{*}_{k-1}d_{k-1}d^{*}_{k-1})$, respectively, for the unbounded operators, as well as in general in the distribution sense.

Since Lemma 4.4 shows that the $A_{k}$ of this section coincide with (1.7) above, it is clear that Theorem 1.3 holds for the operators given in (4.2)–(4.5) and (4.21).

4.4. A direct $\boldsymbol{H^{1}}$-proof

The injectiveness of $A_{1}|_{X}$ shown in Lemma 4.4 may be used for a short proof of Theorem 1.4’s essential parts. This is done in the spirit of [Hel98, Sjö96], but now for our general $\Phi$ and with much sharper assumptions:

Proposition 4.5**.**

Suppose (1.5) holds and that $A_{0}$ defined above satisfies:

[TABLE]

Then it holds true for all $g_{1}$, $g_{2}\in H^{1}(\mu)$ that

[TABLE]

Proof.

Since $A_{0}=A_{0}^{*}$, with closed range by (4.33) and $Z(A_{0})={{\mathbb{C}}}$ by (4.4)–(4.5), there is a decomposition $L^{2}(\mu)=R(A_{0})\oplus{{\mathbb{C}}}$; hence $g_{1}-\langle g_{1}\rangle=A_{0}f$ for some $f\in D(A_{0})$, so according to (4.4),

[TABLE]

In the distribution sense $d^{*}df=A_{0}f$, since $f$ is picked in $D(A_{0})$; therefore we moreover have for $w\in C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{1}{{\mathbb{C}}}^{n})$,

[TABLE]

when $\varphi_{k}\in C^{\infty}_{0}$ tends to $e^{-\Phi}w$ in the $V_{1}$-topology.

By completion (4.36) also holds for every $w\in V_{1}$, cf. the density in Lemma 4.1 and (4.7), so it follows that $df\in D(A_{1})$ with $A_{1}^{-1}dg_{1}=df$ (using the injectivity of Lemma 4.4). This and (4.35) yields the proof. ∎

In addition to the above, we may observe that using (4.34), partial integration gives for each test function $\varphi$

[TABLE]

which shows (1.23) when $u\in H^{1}(\mu)$. Since $P$ is bounded in $L^{2}(\mu)$, we can extend $d^{*}A_{1}^{-1}d$ by continuity such that (1.23) holds by definition on $L^{2}(\mu)$.

A closer analysis given in Section 5.3 below will show that each of the individual factors in $d^{*}A_{1}^{-1}du$ are well defined too, when $u\in L^{2}(\mu)$.

4.5. Brascamp–Lieb’s inequality

When $\Phi$ is strictly convex, Corollary 1.5 may now be proved for $f\in H^{1}(\mu)$, i.e.

[TABLE]

To begin with it is first assumed that $\Phi^{\prime\prime}(x)\geq c_{0}>0$ in ${{\mathbb{C}}}^{n}$ for each $x\in{{\mathbb{R}}}^{n}$. Partial integration shows that on $C^{\infty}_{0}$ we have

[TABLE]

so $m(A_{1})\geq c_{0}>0$ follows by Lemma 4.2. Then Theorem 1.3 yields that $A_{0}$ restricted to ${{\mathbb{C}}}^{\perp}$ has [math] in the resolvent, hence that (4.33) and (4.34) hold.

Because $A_{1}^{-1}$ is symmetric $\geq 0$, Cauchy–Schwarz’ inequality (for such operators) applied to $(\,A_{1}^{-1}v\,|\,A_{1}w\,)_{\mu}=(\,v\,|\,w\,)_{\mu}$ for $v\in L^{2}(\mu,\wedge^{1}{{\mathbb{C}}})$ and $w\in D(A_{1})$ yields

[TABLE]

with equality if $v=A_{1}w$. Hence

[TABLE]

and analogously for $\Phi^{\prime\prime}$, so (4.39) and the density of $D(A_{1})$ and $C^{\infty}_{0}$ in $V$ give

[TABLE]

[regardless of whether $C^{\infty}_{0}$ is dense in $D(\Phi^{\prime\prime})$], which proves (4.38) in this case.

In general this applies for $0<\varepsilon<1$ to

[TABLE]

which is uniformly strictly convex with $\int d\mu_{\varepsilon}=1$; note that $C_{\varepsilon}\nearrow 1$ for $\varepsilon\searrow 0$. Clearly (4.38) holds with $\mu_{\varepsilon}$ instead of $\mu$; because $e^{-\varepsilon|x|^{2}-\log C_{\varepsilon}}\leq C_{1}^{-1}$,

[TABLE]

by majorised convergence for $\varepsilon\searrow 0$. Indeed, in this way $\int fe^{-\Phi_{\varepsilon}}\,dx$ tends to $\langle f\rangle$ and the whole left hand side is controlled by $(|f(x)|+\mathinner{\|}f\,|L^{1}(\mu)\|)^{2}e^{-\Phi(x)}$.

Being positive, $I_{\varepsilon}:=\nabla f^{T}(\Phi^{\prime\prime}_{\varepsilon})^{-1}\overline{\nabla f}$ always has an integral; if this is finite for $\varepsilon=0$ then (4.38) must be proved. But then $I_{0}$ itself may serve as a majorant, and because $\Phi^{\prime\prime}_{\varepsilon}(x)\geq\Phi^{\prime\prime}(x)$ in ${{\mathbb{B}}}(\ell^{2}(\{1,\dots,n\}))$ for all $x$,

[TABLE]

Pointwise convergence is clear from the norm continuity of inversion. This completes the proof for $f\in H^{1}(\mu)$.

5. Extensions to Integrable Functions

5.1. Sufficient Conditions Revisited

Instead of merely establishing estimates sufficient for the full proof of Theorem 1.4, their relation to the other conditions treated is given below, for they all fit so well together that a discussion should be of interest in its own right; the following ten conditions are elucidated (in and) after the proof. Note that subscripts are suppressed on $d$ and $d^{*}$ when the context makes it clear what the domain is.

Theorem 5.1**.**

For $A_{k}=d^{*}_{k}d_{k}+d_{k-1}d^{*}_{k-1}$ with $k>0$ it holds true that

[TABLE]

when the properties (1)–(10) are as follows:

1. (1)

$0<m(A_{k}):=\inf\{\,(\,A_{k}v\,|\,v\,)_{\mu}\mid v\in D(A_{k}),\ \|v\|_{\mu}=1,\}$; 2. (2)

the norms $a_{k}(\cdot,\cdot)^{1/2}$ and $\mathinner{\|}\cdot\,|V_{k}\|$ are equivalent on $V_{k}$**; 3. (3)

\|v\|_{\mu}^{2}\leq c^{2}(\|d^{*}v\|_{\mu}^{2}+\|dv\|_{\mu}^{2})\text{ for all k-forms }v\in D(d^{*})\cap D(d); 4. (4)

\|v\|_{\mu}^{2}\leq c^{2}(\|d^{*}v\|_{\mu}^{2}+\|dv\|_{\mu}^{2})\text{ for all k-forms }v\in D(d^{*})\cap Z(d); 5. (5)

\|v\|_{\mu}^{2}\leq c^{2}(\|d^{*}v\|_{\mu}^{2}+\|dv\|_{\mu}^{2})\text{ for all k-forms }v\in D(d^{*})\cap X_{k}; 6. (6)

\|v\|_{\mu}\leq c\|d^{*}v\|_{\mu}\text{ for all k-forms }v\in D(d^{*})\cap X_{k}; 7. (7)

the range of $d\colon D(d_{k-1})\to L^{2}(\mu,\wedge^{k}{{\mathbb{C}}})\text{ is closed, i.e.~{}}R(d_{k-1})=X_{k}$; 8. (8)

the range $R(A_{k-1}|_{Z(d)^{\perp}})$ is closed in $L^{2}(\mu,\wedge^{k-1})$; 9. (9)

the norms $\|d^{*}\cdot\|_{\mu}$ and $\mathinner{\|}\cdot\,|V_{k}\|$ are equivalent on $V_{k}\cap X_{k}$**; 10. (10)

$0<m(A_{k}|_{X_{k}}):=\inf\{\,(\,A_{k}v\,|\,v\,)_{\mu}\mid v\in D(A_{k})\cap X_{k},\ \|v\|_{\mu}=1,\}$.

In the affirmative case $m(A_{k})=m(A_{k}|_{X_{k}})\geq c^{-2}$, where $c$ is any of the constants in (3)–(6).

Moreover, the closed forms in $L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ belong to $R(d_{k-1})$, i.e. $Z(d_{k})=R(d_{k-1})$, when any of (1)–(4) holds.

Proof.

(1)$\iff$(2) because they are both equivalent to (3) in view of $D(A_{k})$’s density in $V_{k}$ and (4.2)–(4.6). In addition $m(A_{1})\geq c^{-2}$ is found.

Now (3)$\implies$(4)$\implies$(5)$\iff$(6) by shrinking of the set $F_{1}$ of $v$’s from $D(d)$ to $X_{k}$ (and since $d\equiv 0$ on $X$). That (6)$\iff$(7)$\iff$(8) follows from Theorem 3.1 with $F_{1}=X_{k}$; this moreover gives that $R(d_{k-1})=Z(d_{k})$ when (4) holds.

Finally (6) trivially gives (9), and (9)$\implies$(10) is clear. When (10) holds, the inequality in (6) is valid in the subspace $D(A_{k})\cap X_{k}$. Therefore, if $v_{0}\in V_{k}\cap X_{k}$ then $v_{m}\underset{m}{\rightarrow}v_{0}$ in $V_{k}$ for some $v_{m}\in D(A_{k})$ where $P_{k}v_{m}\to v_{0}$ in $L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ with $P_{k}v_{m}\in D(A_{k})\cap X_{k}$ by Lemma 4.4. Using (10) and $d^{*}(1-P)\equiv 0$,

[TABLE]

Consequently (6) holds. ∎

While (8) is the central point because of its implications for Theorem 1.4, (1) and (10) are probably the most convenient to verify for a given $\Phi(x)$; but if this can be done, then $A_{k}^{-1}$ and $A_{k}|_{X_{k}}^{-1}$ extend automatically, by the Poincaré inequalities (2), (9) and $2^{\circ}$ of Lemma 2.1, to bounded operators from certain spaces of distributions of order $1$ (or of forms with such coefficients).

Since (3)$\implies$(4)$\implies$(5), it is clear that (4) is an intermediate property in comparison with the two situations described by (1)–(3) on the one hand and (5)–(10) on the other.

Actually condition (4) is equivalent to exactness of the $d$-complex at $L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})$: $Z(d_{k})$ is closed in $L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ because $d$ has the maximal domain there and is continuous in $\cal D^{\prime}$, so Theorem 3.1 with $F_{1}=Z(d_{k})$ shows that (4) holds if and only if $R(d_{k-1})=F_{1}$.

Injectiveness of $A_{k}$ is furthermore a consequence of (4). For by the Lax–Milgram definition of $A_{k}$, cf. (4.2),

[TABLE]

so (4) implies $Z(A_{k})=\{0\}$. In addition, if (5) holds but (4) doesn’t, then there is some $v\in Z(d_{k})\setminus R(d_{k-1})$; writing this as $v=x+x^{\perp}$ with $x\in X_{k}$ and $x^{\perp}\in X_{k}^{\perp}$, then $x\in R(d_{k-1})$ since (5) implies (7), whence $0\neq x^{\perp}\in Z(d_{k})\cap Z(d^{*}_{k-1})$. So when $A_{k}$ is injective, then either (4), i.e. exactness, holds or $R(A_{k-1}|_{Z(d)^{\perp}})$ is unclosed.

5.2. Proof preparations

As mentioned, (2) and (9) imply the extendability of $A_{1}^{-1}$ and $A_{1}|_{X}^{-1}$ to larger spaces than just the $L^{2}$-forms, which is crucial for Theorem 1.4:

Corollary 5.2**.**

$1^{\circ}$* Let (2) in Theorem 5.1 hold. One has then $A_{k}\subset\cal A_{k}$, when $\cal A_{k}\colon V_{k}\overset{\sim}{\longrightarrow}V^{\prime}_{k}$ is the linear isometry from $(V_{k},a_{k}(\cdot,\cdot))$ onto its dual $V^{\prime}$ given by*

[TABLE]

and $D(A_{k})=\cal A_{k}^{-1}(H_{k})$ holds.

$2^{\circ}$* When (9) holds, then $A_{k}|_{X_{k}}\subset\tilde{\cal A}_{k}$, when $\tilde{\cal A}_{k}$ is the isomorphism $\tilde{V}_{k}\overset{\sim}{\longrightarrow}(\tilde{V}_{k})^{\prime}$, with $\tilde{V}_{k}$ denoting $V_{k}\cap X_{k}$ normed by $(\,d^{*}v\,|\,d^{*}v\,)_{\mu}^{1/2}$.*

Observe that when also $H_{k}\simeq H^{\prime}_{k}$ in this manner (i.e. $g\mapsto(\,\cdot\,|\,\overline{g}\,)_{\mu}$ instead of $(\,\cdot\,|\,g\,)$), then there are inclusions $V_{k}\overset{\iota}{\hookrightarrow}H_{k}\overset{\iota^{\prime}}{\hookrightarrow}V^{\prime}_{k}$ with dense ranges and $\iota^{\prime}$ equal to the transpose of $\iota$:

[TABLE]

Similarly $\tilde{V}_{k}\overset{\iota}{\hookrightarrow}X_{k}\overset{\iota^{\prime}}{\hookrightarrow}(\tilde{V}_{k})^{\prime}$. Thus it is meaningful to state the last part of $1^{\circ}$ or the corresponding fact that $D(A_{k}|_{X_{k}})=\tilde{\cal A}_{k}^{-1}(X_{k})$.

Proof.

$a_{k}(u,\overline{w})=(\,A_{k}u\,|\,\overline{w}\,)_{\mu}$ $\forall w\in V_{k}$, so (5.5) gives $1^{\circ}$. For $2^{\circ}$, identify $A_{k}|_{X_{k}}$ with the operator $\tilde{A}_{k}$ defined from $(X_{k},\tilde{V}_{k},(\,d^{*}\cdot\,|\,d^{*}\cdot\,)_{\mu})$; completeness of $\tilde{V}_{k}$ and denseness carry over from $(H_{k},V_{k},a_{k})$ by means of $P_{k}$ in Lemma 4.4. While $A_{k}|_{X_{k}}\subset\tilde{A_{k}}$ is clear, any $v\in D(\tilde{A}_{k})$ gives $(\,\tilde{A}_{k}v\,|\,w\,)_{\mu}=(\,d^{*}v\,|\,d^{*}w\,)_{\mu}$ for all $w$ in $V_{k}$ since $\tilde{A}_{k}v\perp(1-P_{k})w$ and $d^{*}(1-P_{k})w=0$. Since $v\in X_{k}$ we first get $(\,\tilde{A}_{k}v\,|\,w\,)_{\mu}=a_{k}(v,w)$, thereafter $\tilde{A}_{k}\subset A_{k}$. ∎

When applying this we shall need that $V^{\prime}_{k}$ or $\tilde{V}^{\prime}_{k}$ can receive the image $d(H_{k-1})$. To establish this it is necessary to make a precise identification of $H_{k}$, $V^{\prime}_{k}$ and $\tilde{V}^{\prime}_{k}$ with subspaces of $\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$, which roughly speaking consists of forms $\sum{}^{\prime}u_{J}\,dz^{J}$ with $u_{J}\in\cal D^{\prime}({{\mathbb{R}}}^{n})$.

In Appendix A below this is introduced concisely by means of a direct approach based on the finite dimension of $\wedge^{k}{{\mathbb{C}}}^{n}$ and the simplicity of the manifold ${{\mathbb{R}}}^{n}$. This should provide the reader with an alternative to the general and vast expositions of G. de Rham [dR55] and L. Schwartz [Sch59] and [Sch66, Ch. 9].

By the continuity of $\cal J$ in (A.1), the linear form $\Lambda_{f}(\varphi):=\sum{}^{\prime}\int f_{J}\varphi_{J}\,dx$, defined on $\varphi\in C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$, gives injections $L^{2}(\mu,\wedge^{k}{{\mathbb{C}}}^{n})\subset L^{2}_{\operatorname{loc}}(\wedge^{k}{{\mathbb{C}}}^{n})\overset{\Lambda}{\hookrightarrow}\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ for which

[TABLE]

so $\Lambda$ is the transpose of $M_{e^{-\Phi}}\colon C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})\to H_{k}$, multiplication by $e^{-\Phi}$.

From the last identity above $\Lambda\colon H_{k}\to\cal D^{\prime}$ is seen to be continuous in the topology induced by $V^{\prime}_{k}$, hence it extends to $V^{\prime}_{k}$ by the density of $H_{k}\subset V^{\prime}_{k}$:

Proposition 5.3**.**

The operator $\Lambda$ introduced above (5.6) extends by continuity to an embedding $\Lambda\colon V^{\prime}_{k}\hookrightarrow\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ and for this it holds that

[TABLE]

for every $v^{\prime}\in V^{\prime}_{k}$ and $\varphi\in C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ (when $\Lambda$ is suppressed).

Proof.

By taking closures, (5.7) clearly follows from the left- and rightmost parts of (5.6). Because $M_{e^{\Phi}}$ has dense range in $V^{\prime}_{k}$, the extended $\Lambda$ is an injection. ∎

The point of this proposition and (5.7) is of course to note the factor $e^{\Phi}$.

From the boundedness of $d^{*}\colon V_{k}\to H_{k-1}$ follows the existence of a bounded transpose $d^{*\prime}\colon H_{k-1}\to V^{\prime}_{k}$, and by means of (5.7) and (4.11) this is seen to be a realisation of the distributional differential $d$: indeed for $f\in H_{k-1}$ and elements of the dense subset $C^{2}_{0}\subset V_{k}$ of the form $e^{\Phi}\varphi$ with $\varphi$ in $C^{\infty}_{0}$,

[TABLE]

where the last identity uses (A.5), (A.7); by (5.7) this means that $d^{*\prime}f=df$.

The space $\tilde{V}^{\prime}_{k}$ is normed by $\mathinner{\|}\cdot\,|V_{k}\|$ and moreover a closed subspace of $V^{\prime}_{k}$; indeed $\tilde{V}^{\prime}_{k}=P^{\prime}_{k}V^{\prime}_{k}$ because $P_{k}\in{{\mathbb{B}}}(V_{k})$ (the last fact follows from $d^{*}(1-P_{k})\equiv 0$).

For one thing this gives an embedding $\tilde{V}^{\prime}_{k}\hookrightarrow\cal D^{\prime}$ by the above construction for $V^{\prime}_{k}$, and for another that $d(H_{k-1})\subset\tilde{V}^{\prime}_{k}$. Indeed, $d^{*}_{k-1}$ restricts to a continuous map $\tilde{V}_{k}\to H_{k-1}$, so $d_{k-1}$ extends to a bounded operator $H_{k-1}\to\tilde{V}^{\prime}_{k}$; however, this is not a proper extension since $\tilde{V}^{\prime}_{k}\subset V^{\prime}_{k}$. Altogether we have:

Proposition 5.4**.**

The distributional differential is bounded $d\colon H_{k-1}\to V^{\prime}_{k}$; it is the transpose of $d^{*}\colon V_{k}\to H_{k-1}$, so for $f\in H_{k-1}$ and $\varphi\in C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$

[TABLE]

Moreover, the range $d(H_{k-1})$ is contained in the subspace $(V_{k}\cap X_{k})^{\prime}=P^{\prime}_{k}V^{\prime}_{k}$.

Remark 5.5*.*

Considering ${{\mathbb{R}}}^{n}$ as a manifold, it would be possible to use the $C^{2}$-density furnished by the measure $\mu=e^{-\Phi}\,dx$ (see e.g. [Hör85, Ch. 6] for the notions), but it is preferable to use the Lebesgue integral, for this gives an extension of the usual embeddings, such as $L^{2}(\mu)\subset L^{2}_{\operatorname{\operatorname{loc}}}({{\mathbb{R}}}^{n})\hookrightarrow\cal D^{\prime}({{\mathbb{R}}}^{n})$.

5.3. Continuation of Proofs

When $R(A_{0})$ is closed the implication (8)$\implies$(9) of Theorem 5.1 allows a renorming of $\tilde{V}:=V\cap X$ such that $A_{1}|_{X}\subset\tilde{\cal A}_{1}$; cf. $2^{\circ}$ of Corollary 5.2. From Proposition 5.4 it follows that $d^{*}\tilde{\cal A}_{1}^{-1}d$ is bounded

[TABLE]

and it coincides with $d^{*}A_{1}^{-1}d$ in $H^{1}(\mu)$ by the remark after (5.5) and therefore with $u\mapsto u-\int u\,d\mu$ by (4.37); extension by continuity gives (1.23) for all $u\in L^{2}(\mu)$. Closure of (4.34) similarly yields (1.2): indeed, for $g_{j}\in H^{1}(\mu)$ one can take $v=A_{1}|_{X_{1}}^{-1}dg_{1}$ and $f=d\overline{g_{2}}$ in formula (5.5) so (4.34) (and the obvious interpretation of gradients as differentials) gives

[TABLE]

The last equality extends to all $g_{j}$ in $L^{2}(\mu)$ in view of the density of $H^{1}(\mu)$ and the continuity of $d\colon L^{2}(\mu)\to\tilde{V}^{\prime}$ and $\tilde{\cal A}_{1}^{-1}d\colon L^{2}(\mu)\to\tilde{V}$; cf. Proposition 5.4.

For the Brascamp–Lieb inequality (4.38) with $f\in L^{2}(\mu)\cap H^{1}_{\operatorname{loc}}$, (4.39) still shows that $A_{1}>0$ in the uniformly strictly convex case; from (1)$\implies$(8) of Theorem 5.1 it follows that (1.2) is available for $g_{j}=f$. From (1)$\implies$(2) we see that $1^{\circ}$ of Corollary 5.2 applies. Since $\cal A_{1}$ is an isometry, $a_{1}(\cal A_{1}^{-1}\cdot,\cal A_{1}^{-1}\cdot)$ is an inner product on $V^{\prime}$ inducing the norm, so for $v\in V^{\prime}$,

[TABLE]

Because of the $H^{1}_{\operatorname{loc}}$-condition, $(\nabla f)^{T}(\Phi^{\prime\prime})^{-1}\overline{\nabla f}$ is a well defined function belonging to $L^{1}_{\operatorname{loc}}$; if it has finite integral, then $v=df$ is in $D(\Phi^{\prime\prime})\subset L^{2}(\mu,\wedge^{1}{{\mathbb{C}}}^{n})$ so that (4.39) may be invoked as in the argument for (4.42), which hence also holds in this case.

When $\Phi$ is merely strictly convex, the reduction to the uniform case carries over verbatim.

6. Criteria for Closed Range

Because $Z(A_{0})$ has finite dimension, the closed-range requirement in (4.33) is satisfied when $0\notin\sigma_{\operatorname{ess}}(A_{0})$, which holds when $\Phi(x)$ is well behaved near $\infty$:

Proposition 6.1**.**

If $\Phi$ in addition to (1.5) satisfies

• (I)

$\frac{1}{2}|\Phi^{\prime}|^{2}-\!\operatorname{\Delta}\Phi\geq c>0$* in a neighbourhood of $\infty$,*

then $0\notin\sigma_{\operatorname{ess}}(A_{0})$, so (4.33) and the projection identity (4.34) hold.

Proof.

$A_{0}$ is equivalent to $B_{0}=-\!\operatorname{\Delta}+\tfrac{1}{4}|\Phi^{\prime}|^{2}-\tfrac{1}{2}\operatorname{\Delta}\Phi$, cf. Section 4.2. The latter is essentially self-adjoint by Kato’s result [Kat73], so the Persson–Agmon formula [Per60], [Agm78, Thm. 3.2] may without ambiguity be used for the estimate:

[TABLE]

when (I) holds in ${{\mathbb{R}}}^{n}\setminus K_{0}$. This yields (4.33). ∎

In addition, (I) combined with a growth condition implies the stronger fact that $\sigma_{\operatorname{ess}}(A_{0})=\emptyset$, as shown below. Note that whenever $0<\eta<1$ and $\theta\in\,]\eta,1[$,

[TABLE]

so if $|\Phi^{\prime}(x)|\to\infty$ for $|x|\to\infty$ and the first term on the right hand side is known to have finite infimum, consequently the left hand side tends to $\infty$ for $x\to\infty$. Taking $\eta=1/2$, this shows that (I) together with $\lim_{|x|\to\infty}|\Phi^{\prime}(x)|=\infty$ implies condition (II) below. Similarly one finds that (II) is more general than the results one would obtain from the techniques of J.-M. Kneib and F. Mignot in [KM94, Lem. 5] (where the proof contains a minor flaw).

Proposition 6.2**.**

If $\Phi$ satisfies the following condition

• (II)

$\exists\theta\in\,]0,1[\colon\lim_{|x|\to\infty}\theta|\Phi^{\prime}(x)|^{2}-\!\operatorname{\Delta}\Phi(x)=\infty$,

then $H^{1}(\mu)\hookrightarrow L^{2}(\mu)$ is compact, and consequently $\sigma_{\operatorname{ess}}(A_{0})=\emptyset$.

That (II) is sufficient may be proved along the lines of P. Bolley, Dauge and Helffer [BDH89] (even directly, that is without the unitary transformation in Section 4.2); because of this reference’s inaccessibility we shall supply the details.

Proof.

Introducing the vector fields $X_{j}=\partial_{j}$ and their formal adjoints $X^{*}_{j}=-\partial_{j}+\Phi^{\prime}_{j}$, one has when $u\in C^{\infty}_{0}({{\mathbb{R}}}^{n})$ for their sum and commutator

[TABLE]

Now it is straightforward to see that

[TABLE]

so that a linear combination of these formulae gives for any $\varepsilon>0$

[TABLE]

Because $C^{\infty}_{0}({{\mathbb{R}}}^{n})$ is dense in $H^{1}(\mu)$, this inequality is valid for all $u\in H^{1}(\mu)$. Indeed, letting $\mu^{\prime}=(|\Phi^{\prime}|^{2}-(1+\varepsilon)\operatorname{\Delta}\Phi)\mu$, we infer from (6.6) that a fundamental sequence in $H^{1}(\mu)$ also converges in $L^{2}({{\mathbb{R}}}^{n},\mu^{\prime})$, and necessarily to the same limit since both spaces are embedded into $\cal D^{\prime}({{\mathbb{R}}}^{n})$.

If $u_{k}\to u$ weakly in $H^{1}(\mu)$, assumption (II) implies that $\Psi:=|\Phi^{\prime}|^{2}-(1+\varepsilon)\operatorname{\Delta}\Phi$ is positive in a neighbourhood of $\infty$ when $\theta=(1+\varepsilon)^{-1}$, so by (6.6),

[TABLE]

Hence (II) and the compactness of $H^{1}(B(0,R))\hookrightarrow L^{2}(B(0,R))$ show that a subsequence of $u_{k}$ tends to [math] in $L^{2}(\mu)$.

If $\lambda\in\sigma_{\operatorname{ess}}$ there is $u_{k}\in D(A_{0})$ such that $\|u_{k}\|_{\mu}=1$ while $u_{k}\to 0$ weakly and $\|(A_{0}-\lambda)u_{k}\|_{\mu}\to 0$. Since

[TABLE]

the sequence $(u_{k})$ is bounded in $H^{1}(\mu)$, but without convergent subsequences in $L^{2}(\mu)$, so the embedding is non-compact. Thus $\sigma_{\operatorname{ess}}(A_{0})=\emptyset$ is shown. ∎

7. A Pseudo-differential View Point

As shown in the following, a few extra assumptions on $\Phi(x)$ lead to domain characterisations, essential self-adjointness of the $A_{j}$ and positivity of $A_{1}$ (in addition to closed ranges).

Actually $C^{\infty}$-smoothness with a little control of the higher order derivatives is enough to invoke the calculus in [Hör85, 18.4–6], and in this framework $A_{0}$ and $A_{1}$ are easily seen to be Fredholm operators if $|\Phi^{\prime}|$ tends to $\infty$ at infinity. Therefore it is assumed in this section that

• (III)

$\Phi\in C^{\infty}({{\mathbb{R}}}^{n},{{\mathbb{R}}})$,

• (IV)

$|\Phi^{\prime}(x)|\to\infty$ for $|x|\to\infty$,

• (V)

for $|\alpha|\geq 1$ there are constants $C_{\alpha}$ such that

[TABLE]

• (VI)

$D^{\beta}\Phi$ is bounded on ${{\mathbb{R}}}^{n}$ when $\beta$ has a fixed length, say $M\in{\mathbb{N}}$.

This implies that $\Phi(x)$ is slowly increasing, $\Phi\in\cal O_{M}({{\mathbb{R}}}^{n})$, so $\Phi$ of, say exponential growth is ruled out; thus the stronger conclusions of this section have their price.

7.1. An auxiliary Schrödinger operator

To exploit (III)–(VI) above, we shall henceforth work in the unweighted space $L^{2}({{\mathbb{R}}}^{n})$ and with the Witten-Laplacians ensuing after the unitary transformation in Section 4.2. That is, we shall consider

[TABLE]

which act in the distribution sense, and provide them with their maximal domains in $L^{2}({{\mathbb{R}}}^{n})$ and $L^{2}({{\mathbb{R}}}^{n},\wedge^{1}{{\mathbb{C}}}^{n})$, respectively.

For convenience one can here study the auxiliary operator

[TABLE]

with the domain

[TABLE]

To analyse this, let the pseudo-differential operators $p(x,D)$ and $q(x,D)$ have symbols

[TABLE]

where $\chi\in C^{\infty}_{0}({{\mathbb{R}}}^{2n})$ is positive and $\equiv 1$ on a compact set $K$ such that

[TABLE]

This makes $q(x,\xi)$ well defined in $C^{\infty}({{\mathbb{R}}}^{2n})$.

The calculus in [Hör85, Ch. 18.4–6] applies to this case with

[TABLE]

when the weight and metric equal, respectively,

[TABLE]

Hörmander’s notation and terminology is used here and below. When applying this theory, condition (VI) is posed in order to show that $g$ is $\sigma$-temperate.

From the calculus we next infer that $q(x,D)$ acts as a parametrix of $p(x,D)$, i.e.

[TABLE]

where $K_{j}\in\operatorname{OPS}(m^{-1},g)$. By [Hör85, Thm. 18.6.6] the $K_{j}$ are compact in $L^{2}({{\mathbb{R}}}^{n})$ because $m^{-1}$ according to (IV) tends to [math] at infinity, for with the choice of $g$ made above we have $g\leq g^{\sigma}$.

To see the latter fact, note that by definition

[TABLE]

so the isometry of the Hilbert space $({{\mathbb{R}}}^{2n},g_{x,\xi}(\cdot,\cdot))$ onto its dual gives

[TABLE]

This shows for one thing the claim that $g\leq g^{\sigma}$, because $m\geq 1$, and for another that

[TABLE]

Using [Hör85, 18.5.10] we can pass to the Weyl calculus and conclude that

[TABLE]

with the remainder information that, since $h\cdot m^{2}=m$,

[TABLE]

Moreover, by [Hör85, 18.4.3] and [Hör85, 18.5.4],

[TABLE]

This gives finally, by the compact support of $\chi$,

[TABLE]

and hence the relations

[TABLE]

with $K_{1}(x,\xi)$ in $S(m^{-1},g)$ as claimed. Similarly, because $R_{1}(b,a)\in S(m^{-1},g)$, one finds that $q(x,\xi)p(x,\xi)-1\in\operatorname{OPS}(m^{-1},g)$.

Note that in a similar fashion one has:

Proposition 7.1**.**

Let $g$ be a $\sigma$-temperate metric fulfilling $g\leq g^{\sigma}$ and $g_{x,\xi}(t,\tau)\equiv g_{x,\xi}(t,-\tau)$, and let $m_{1}$ and $m_{2}$ be $\sigma$, $g$-temperate weights. Then

[TABLE]

Proof.

One may use [Hör85, Th. 18.5.10] and the remark thereafter to express $p(x,D)q(x,D)$ as $(a\#b)^{W}(x,D)$ and then apply 18.5.4 and 18.5.10. ∎

Obviously the maximal domain of $p(x,D)$ as an operator in $L^{2}({{\mathbb{R}}}^{n})$ is the set $D(p)$ consisting of those $u\in L^{2}$ for which also $p(x,D)u\in L^{2}$. However this equals $D(P)$ in (7.5): for if $u\in D(p)$ then (7.10) yields, with $f=p(x,D)u$ in $L^{2}$,

[TABLE]

and by application of $1+K_{2}(x,D)$

[TABLE]

by the proposition both $K_{2}(x,D)^{2}$ and $(1+K_{2}(x,D))q(x,D)$ are in $\operatorname{OPS}(m^{-2},g)$, so that $D^{\alpha}u$ and $V_{0}u$ are in $L^{2}$ when $|\alpha|\leq 2$ by combined application of Proposition 7.1 and [Hör85, 18.6.3]. Since also $(\Phi^{\prime}_{j_{1}}(x))^{k_{1}}(\Phi^{\prime}_{j_{2}}(x))^{k_{2}}\xi^{\alpha}$ is in $S(m^{2},g)$ when $k_{1}+k_{2}+|\alpha|\leq 2$, it is seen that $(\Phi^{\prime})^{\beta}D^{\alpha}u\in L^{2}$, so that we for later use have the more precise result:

Lemma 7.2**.**

Both $D(P)$ and $D(p(x,D)_{\operatorname{max}})$ coincide with the space of $u$ for which

[TABLE]

for all $\alpha$ and $\beta\in{\mathbb{N}}_{0}^{n}$ for which $|\alpha|+|\beta|\leq 2$; and $(\sum_{|\alpha+\beta|\leq 2}|(\Phi^{\prime})^{\beta}D^{\alpha}u|^{2})^{1/2}$ is equivalent to the graph norm of $P$.

Thereby $P=p(x,D)_{\operatorname{max}}$, so by duality and symmetry of $P$,

[TABLE]

so that $D(P)$ is both the minimal and maximal domain of $p(x,D)$. Consequently $P$ is essentially self-adjoint on $C^{\infty}_{0}({{\mathbb{R}}}^{n})$.

Furthermore $D(P)$ is a Hilbert space in $P$’s graph norm with $P$ and $q(x,D)$ belonging to ${{\mathbb{B}}}(D(P),L^{2})$ and ${{\mathbb{B}}}(L^{2},D(P))$, respectively. For this reason $\dim\operatorname{coker}P\leq\dim\operatorname{coker}(1-K_{1})<\infty$, where the last inequality is obtained from the compactness of $K_{1}$, and hence $R(P)$ is necessarily closed; cf. [Hör85, 19.1.1].

Returning to $\operatorname{\Delta}^{(0)}_{\Phi}$, the perturbation $-\!\operatorname{\Delta}\Phi(x)$ of $P$ is easily handled; first we show that the domain of $\operatorname{\Delta}^{(0)}_{\Phi}$ equals $D(P)$. Clearly it contains $D(P)$ in view of (V) (when this is used to get the elementary estimate $|\operatorname{\Delta}\Phi|\leq c(1+|\Phi^{\prime}|^{2})^{1/2}\leq c(1+|\Phi^{\prime}|^{2})$, which entails $|u\operatorname{\Delta}\Phi|\leq c(1+|\Phi^{\prime}|^{2})|u|\in L^{2}$), and if $f:=\operatorname{\Delta}^{(0)}_{\Phi}u$ is in $L^{2}$ for some $u\in L^{2}$, one has in $\cal S^{\prime}({{\mathbb{R}}}^{n})$

[TABLE]

where

[TABLE]

By application of $1+K^{\prime}_{2}(x,D)$

[TABLE]

so, like for $P$ above, we find that $u$ is in $H^{2}$ with $|\Phi^{\prime}|^{2}u$ in $L^{2}$.

For the range we get that

[TABLE]

where $K^{\prime}_{1}(x,D)\in\operatorname{OPS}(m^{-1},g)$ since both $K_{1}(x,D)$ and $(\operatorname{\Delta}\Phi)q(x,D)$ are so; the latter fact is by Proposition 7.1 because $\operatorname{\Delta}\Phi$ is in $S(m,g)$ in view of condition (V). Using (IV) as above $K^{\prime}_{1}(x,D)$ is compact in $L^{2}$, and since $q(x,D)$ maps all of $L^{2}({{\mathbb{R}}}^{n})$ into $D(\operatorname{\Delta}^{(0)}_{\Phi})$ by the just shown characterisation of this set, $\operatorname{\Delta}^{(0)}_{\Phi}$ has closed range.

Using that $D(\operatorname{\Delta}^{(0)}_{\Phi})=D(P)$, it is straightforward to see that $\operatorname{\Delta}^{(0)}_{\Phi}$ is self-adjoint: for if $u\in D(\operatorname{\Delta}^{(0)}_{\Phi})$, then $d_{\Phi}u$ is in $L^{2}$ by (V) and the just shown characterisation of $D(-\!\operatorname{\Delta}^{(0)}_{\Phi})$, and in addition we may by Lemma 4.1 approximate any $v$ in the domain of $d_{\Phi}$ by functions $\varphi_{k}\in C^{\infty}_{0}({{\mathbb{R}}}^{n})$ and get

[TABLE]

so that $\operatorname{\Delta}^{(0)}_{\Phi}$ coincides with the self-adjoint Lax–Milgram operator $B_{0}$ defined in Section 4.2. Because of this it is moreover essentially self-adjoint on $C^{\infty}_{0}({{\mathbb{R}}}^{n})$.

Altogether we have the following:

Theorem 7.3**.**

Let $\Phi(x)$ have the properties in (1.5) and (III)–(VI) above. Then $A_{0}$ has closed range in $L^{2}({{\mathbb{R}}}^{n},\mu,{{\mathbb{C}}})$, so that the conclusions of Theorem 1.4 are valid, and $A_{0}$ is essentially self-adjoint on $C^{\infty}_{0}({{\mathbb{R}}}^{n})$.

Moreover, $D(A_{0})$ consists of the functions $u$ for which $(\Phi^{\prime})^{\beta}D^{\alpha}u$ belongs to $L^{2}({{\mathbb{R}}}^{n},\mu)$ for all $\alpha$ and $\beta\in{\mathbb{N}}_{0}^{n}$ such that $|\alpha+\beta|\leq 2$.

Observe that the closed range, and even $\sigma_{\operatorname{ess}}(A_{0})=\emptyset$, is an immediate consequence of Proposition 6.2 since (IV)–(V) imply condition (II) there.

However, the density of $C^{\infty}_{0}({{\mathbb{R}}}^{n})$ in the graph norm has only been obtained because the pseudo-differential techniques made an analysis of the maximal domains possible.

7.2. Applications to $\boldsymbol{A_{1}}$

Using the same line of thought as for $A_{0}$ one finds:

Theorem 7.4**.**

Let $\Phi(x)$ satisfy (1.5) together with (III)–(VI) above. Then the domain of $A_{1}$ is given by

[TABLE]

and $A_{1}$ is essentially self-adjoint from $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{1}{{\mathbb{C}}}^{n})$ and has closed range.

Indeed, that $D(\operatorname{\Delta}^{(1)}_{\Phi})$ contains $D(\operatorname{\Delta}^{(0)}_{\Phi})\times\dots\times D(\operatorname{\Delta}^{(0)}_{\Phi})$ (cf. the set in (7.31)) is clear by (7.3) and (V); if conversely $w:=\operatorname{\Delta}^{(1)}_{\Phi}v$ is in $L^{2}({{\mathbb{R}}}^{n},\wedge^{1}{{\mathbb{C}}}^{n})$ for some $v$ there, the procedure in (7.26)–(7.28) gives, when $q(x,D)$ is tensored with $I$,

[TABLE]

where $\tilde{K}_{2}(x,D)$ equals $K_{2}^{\prime}(x,D)\otimes I-q(x,D)\otimes I\cdot\Phi^{\prime\prime}$ and has all of its entries in $\operatorname{OPS}(m^{-1},g)$. Therefore the inclusion from the left to the right in (7.31) follows.

When applying $q(x,\xi)\otimes I$ as a right-parametrix we find

[TABLE]

where each entry of $\tilde{K}_{1}(x,D)$ is in $\operatorname{OPS}(m^{-1},g)$, and hence compact in $L^{2}({{\mathbb{R}}}^{n})$. Now $q\otimes I$ sends $L^{2}({{\mathbb{R}}}^{n},\wedge^{1}{{\mathbb{C}}}^{n})$ into $D(\operatorname{\Delta}^{(1)}_{\Phi})$, so this shows that $\operatorname{\Delta}^{(1)}_{\Phi}$ has closed range; cf. the argument for $P$ above.

To show the self-adjointness one can identify $\operatorname{\Delta}^{(1)}_{\Phi}$ with $UA_{1}U^{*}$, where $U=U\otimes I$; cf. Section 4.2. Now $d_{\Phi}$ and $d_{\Phi}^{*}$ map any $v\in D(\operatorname{\Delta}^{(1)}_{\Phi})$ into $L^{2}$ by the characterisation of $D(\operatorname{\Delta}^{(1)}_{\Phi})$, so as in (7.30) one finds that $\operatorname{\Delta}^{(1)}_{\Phi}$ is contained in the self-adjoint Lax–Milgram operator defined from $(L^{2}({{\mathbb{R}}}^{n},\wedge^{1}{{\mathbb{C}}}^{n}),V_{\Phi},b_{1})$. Hence $\operatorname{\Delta}^{(1)}_{\Phi}$ equals this as well as the minimal realisation of the expression in (7.3). Consequently $\operatorname{\Delta}^{(1)}_{\Phi}$ and $A_{1}$ are essentially self-adjoint.

Note that $D(A_{1})\hookrightarrow H^{1}(\mu)^{n}$ by Lemma 7.2, so that $\sigma_{\operatorname{ess}}(A_{1})$ is empty.

Injectivity of $A_{1}$ may be obtained in the set-up above as soon as (IV) is strengthened to a specific growth rate at infinity; that is when (IV) is replaced by:

• (IVω)

There exist $\omega>0$ and $C>0$ such that

[TABLE]

Since $C|\Phi^{\prime}|\geq|x|^{\omega}$ holds a fortiori, $A_{1}$ is then moreover strictly positive because of the closed range obtained in Theorem 7.4:

Theorem 7.5**.**

Let $\Phi$ satisfy (1.5), (III)–(VI) and (IVω). Then $A_{1}>0$ on $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{1}{{\mathbb{C}}}^{n})$.

Proof.

As remarked it suffices to show injectivity of $\operatorname{\Delta}^{(1)}_{\Phi}$, and for this it is enough that

[TABLE]

Indeed, given $v\in Z(\operatorname{\Delta}^{(1)}_{\Phi})\cap\cal S({{\mathbb{R}}}^{n},{{\mathbb{C}}}^{n})$, condition (IVω) will imply that

[TABLE]

defines an element $f(x)$ of $L^{2}({{\mathbb{R}}}^{n})$ for which $d_{\Phi}f=v$; since $v\in Z(\operatorname{\Delta}^{(1)}_{\Phi})\subset Z(d_{\Phi}^{*})=R(d_{\Phi})^{\perp}$ this will give $v=0$ as desired.

It is straightforward to see that $f\in C^{0}({{\mathbb{R}}}^{n})$ for such $v$, and (7.36) gives, cf. (4.26),

[TABLE]

hence that $dU^{*}f=U^{*}v$; therefore $f=e^{-\Phi/2}U^{*}f$ is in $C^{\infty}({{\mathbb{R}}}^{n})$ by (7.35) and (III). This also yields $v=UdU^{*}f=d_{\Phi}UU^{*}f=d_{\Phi}f$ as claimed.

That $f$ is in $L^{2}({{\mathbb{R}}}^{n})$ follows if $|x|^{k}f(x)$ is bounded for large $k$. Using (IVω) we get, for $|x|>C$ and $t|x|>C$,

[TABLE]

so the fact that $(1-t)^{1+\omega}\leq 1-t^{1+\omega}$ on $[0,1]$ (since these functions are convex and concave, respectively, on this interval) leads to the conclusion that

[TABLE]

Next one may for each $k\in{\mathbb{N}}$ deduce the existence of a constant $C_{1}$ such that when $|x|\geq C$, then (with $\langle x\rangle:=(1+|x|^{2})^{\tfrac{1}{2}}$ in the rest of this proof)

[TABLE]

when $\operatorname{ch}A$ denotes the convex hull of $A$. Indeed, if $|y|\geq C$ the inequality in (7.39) yields that the left hand side, when $r:=\langle x-y\rangle\in{{\mathbb{R}}}_{+}$, is estimated by

[TABLE]

when $|y|<C$ one can let $z=\tfrac{C}{|x|}x$ and reduce to the case $|y|\geq C$, using the inequalities

[TABLE]

For $f(x)$ this now gives, since $|x|<C$ is easy,

[TABLE]

It remains, therefore, to show (7.35). If $v\in Z(\operatorname{\Delta}^{(1)}_{\Phi})$, then $v=\tilde{K}_{2}(x,D)v$, hence

[TABLE]

cf. (7.32) ff. Since $\tilde{K}_{2}(x,D)^{N}$ by Proposition 7.1 has entries in $\operatorname{OPS}(m^{-N},g)$, it follows that $|\Phi^{\prime}(x)|^{N}D^{\beta}v_{j}(x)$ is in $L^{2}({{\mathbb{R}}}^{n})$ for all multiindices $\beta$ and all $j$ and $N$. Moreover,

[TABLE]

when $N^{\prime}\geq N/\omega$; cf. (IVω). It is thus shown that $v$ has coefficients in $\cal S({{\mathbb{R}}}^{n})$, and altogether this shows the theorem. ∎

7.3. Example

Consider the potential, as done by Helffer [Hel98, Hel97a] and many others,

[TABLE]

whereby $x_{n+1}=x_{1}$ as a convention. Here $h>0$ and $\cal I>0$ while

[TABLE]

Therefore $\Phi(x)$ is not convex, so the Brascamp–Lieb inequality does not apply to this case. The condition $\int e^{-\Phi}\,dx=1$ may be fulfilled by adding an $h$-dependent constant. Moreover, (III) and (V) clearly hold. Concerning (IVω), Euler’s formula gives

[TABLE]

when $|x|\geq C$ for some sufficiently large $C=C(h,n,\lambda,\nu)$; here it is used that $|x_{j}|^{2}\geq|x|^{2}/n$ for some $j\in\{1,\dots,n\}$. Consequently (IVω) holds for all $\omega\in\,]0,3]$ for the above $\Phi(x)$. (By comparison, the assumptions in [Sjö96] are unfulfilled since the $\Phi^{\prime\prime}_{jk}$ are unbounded on ${{\mathbb{R}}}^{n}$.)

Because of this, the corresponding operators $A_{0}$ and $A_{1}$ have the properties given in Theorems 1.4, 7.3, 7.4 and 7.5, in particular (1.2) holds because $A_{1}>0$.

The lower bound $m(A_{1})$ can moreover, for certain $h$ and $\cal I$, be estimated in various ways, see for example [Hel98, Hel97a].

8. Final Remark

The essential self-adjointness of $A_{0}$ and $A_{1}$ (or $A_{k}$) holds in a greater generality than that established in Section 7. For scalar Schrödinger operators this is well known from works of T. Kato [Kat73] and S. Agmon [Agm78], but especially C. Simader’s note [Sim78] appears useful for an extension to ‘systems’ like $\operatorname{\Delta}^{(k)}_{\Phi}$. In fact, Simader’s argument for $-\!\operatorname{\Delta}+V$ specialised to the case $V\in C^{0}({{\mathbb{R}}}^{n},{{\mathbb{R}}})$ appears in a recent lecture note [Hel99, Thm. 9.4.1], and in this form it is straightforward to carry over to $-\!\operatorname{\Delta}\otimes I+V$ with $V\in C^{0}({{\mathbb{R}}}^{n},{{\mathbb{R}}}^{n^{2}})$, when this operator is positive on $C_{0}^{\infty}({{\mathbb{R}}}^{n},{{\mathbb{R}}}^{n})$, hence to $\operatorname{\Delta}^{(1)}_{\Phi}$ and $A_{1}$.

However, the domain characterisations and the corollary on the compact resolvent (in particular of $A_{1}$) should in any case motivate the given applications of the Weyl calculus.

Appendix A Forms with distributions as coefficients

The general framework for distribution-valued differential forms, so-called currents, is given by G. de Rham [dR55] and L. Schwartz [Sch59]. However, the definition of $E$-valued distributions as continuous linear maps $\cal D(\Omega)\to E$ given in [Sch59, Ch. 1 §2] leads to severe difficulties (cf. the introduction of [Sch59]) in the proof that $\cal D^{\prime}(\Omega,E)$ is the dual of $C^{\infty}_{0}(\Omega,E^{\prime})$; for the finite-dimensional example $E=\wedge^{k}{{\mathbb{C}}}^{n}$, a much more direct approach is given in Schwartz’ book [Sch66, Ch. 9] where differential forms on manifolds are treated.

In the present article where $\Omega={{\mathbb{R}}}^{n}$ is a flat, oriented manifold, further simplifications are given below for the reader’s sake. The definition of $\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ as the dual of $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ is a little unconventional (testfunctions valued in $\wedge^{n-k}{{\mathbb{C}}}^{n}$ is common), but this choice is consistent with the made identification of $L^{2}({{\mathbb{R}}}^{n},\mu,\wedge^{k}{{\mathbb{C}}}^{n})$ and its dual.

For precision, $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ denotes the compactly supported, infinitely differentiable maps ${{\mathbb{R}}}^{n}\to\wedge^{k}{{\mathbb{C}}}^{n}$ (i.e. into the space of anti-symmetric $k$-linear forms on ${{\mathbb{C}}}$). The canonical coordinates $z_{1}$,…,$z_{n}$ in ${{\mathbb{C}}}^{n}$ lead to a basis for $\wedge^{k}{{\mathbb{C}}}^{n}$ consisting of $dz^{J}:=dz_{j_{1}}\wedge\dots\wedge dz_{j_{k}}$, where $J=(j_{1},\dots,j_{k})$ is an increasing $k$-tuple. Therefore any $\varphi\in C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ equals $\sum^{\prime}\varphi_{J}\,dz^{J}$ with unique $\varphi_{J}\in C^{\infty}_{0}({{\mathbb{R}}}^{n})$. Thus there is a bijection

[TABLE]

and there is a unique topology on the domain which makes $\cal J$ a homeomorphism, when the codomain has the product topology. (For brevity, indexation on $\prod$ is suppressed below.)

The dual of $\cal J$’s codomain is isomorphic to $\prod\cal D^{\prime}({{\mathbb{R}}}^{n})$, for any continuous linear functional $F$ acts on $\varphi=(\varphi_{j})$ as $F(\varphi)=\sum F\circ I_{j}(\varphi_{j})$ where $I_{j}$ sends $\varphi_{j}$ into $(0,\dots,\varphi_{j},\dots,0)$; since $F\circ I_{j}$ is continuous it is in $\cal D^{\prime}({{\mathbb{R}}}^{n})$.

Now $\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ may be defined as the dual of $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$; equipping dual spaces with their $\operatorname{w}^{*}$-topologies, there is by transposition a linear homeomorphism

[TABLE]

so for $(u_{J})$ in $\prod\cal D^{\prime}({{\mathbb{R}}}^{n})$ and $\varphi=\sum{}^{\prime}\varphi_{J}\,dz^{J}$ in $C^{\infty}_{0}({{\mathbb{R}}}^{n}\wedge^{k}{{\mathbb{C}}}^{n})$,

[TABLE]

Indeed, $\cal J^{\prime}$ is surjective because any $u\in\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ gives rise to the continuous linear functional $u\circ\cal J^{-1}$, which is in $\prod\cal D^{\prime}({{\mathbb{R}}}^{n})$, in view of (A.1) ff, so that for some $(u_{J})$ it holds for all $(\varphi_{J})$ in $\prod C^{\infty}_{0}({{\mathbb{R}}}^{n})$ that

[TABLE]

Therefore $u=\cal J^{\prime}(u_{J})$; the rest of (A.2) is straightforward.

It is natural to write $u=\sum{}^{\prime}u_{J}\,dz^{J}$ instead of $u=\cal J^{\prime}(u_{J})$, and thereby (A.3) attains the following more intuitive form,

[TABLE]

The usual denseness of $C^{\infty}_{0}$ in $\cal D^{\prime}$ carries over to the $k$-form-valued spaces by (A.1) and (A.2), and therefore multiplication, $M_{\psi}$, by $\psi\in C^{\infty}({{\mathbb{R}}}^{n})$ and the operators $\partial_{j}$ extend in a unique way to $\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ as usual. More precisely, $M_{\psi}$ and $\partial_{j}$ are both continuous on $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ because their definitions show that they act on each $\varphi_{J}$ (i.e. they commute with $\cal J$), so the transposed operators $(M_{\psi})^{\prime}$ and $\partial_{j}^{\prime}$ act as $M_{\psi}$ and $-\partial_{j}$ on each $u_{J}$ in (A.5), respectively, and they are therefore denoted by the latter symbols throughout.

In this way one finds that $\cal D^{\prime}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$ is a $C^{\infty}({{\mathbb{R}}}^{n})$-module and that any differential operator $P(\partial)$ with coefficients in $C^{\infty}({{\mathbb{R}}}^{n})$ is well defined by its action on each $u_{J}$ (and independent of the canonical choice of $dz^{J}$); transposition moreover follows the usual rule. In particular this hold for the exterior derivative $d_{k}$, and for this the identies

[TABLE]

are obtained by transposition and by closure from $C^{\infty}_{0}({{\mathbb{R}}}^{n},\wedge^{k}{{\mathbb{C}}}^{n})$, respectively.