A vanishing result for strictly $p$-convex domains

Abstract

In view of Andreotti and Grauert (Bull Soc Math France 90:193–259, 1962) vanishing theorem for $q$-complete domains in $\mathbb C ^{n}$, we reprove a vanishing result by Sha (Invent Math 83(3):437–447, 1986), and Wu (Indiana Univ Math J 36(3):525–548, 1987), for the de Rham cohomology of strictly $p$-convex domains in $\mathbb R ^n$ in the sense of Harvey and Lawson (The foundations of $p$-convexity and $p$-plurisubharmonicity in riemannian geometry. arXiv:1111.3895v1 [math.DG]). Our proof uses the ${L}^2$-techniques developed by Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Publishing Co, Amsterdam 1990), and Andreotti and Vesentini (Inst Hautes Études Sci Publ Math 25:81–130, 1965).

1 Introduction

A weaker condition than holomorphic convexity for domains in $\mathbb C ^n$ has been introduced by Andreotti and Grauert [1], defining $q$ -complete domains as domains in $\mathbb C ^n$ admitting a proper exhaustion function whose Levi form has $n-p+1$ positive eigenvalues.

In a recent series of foundational papers, [6, 7], and references therein, Harvey and Lawson raise the interest on generalizations of the concept of convexity for Riemannian manifolds, proving many important results for $p$ -convex manifolds: namely, starting with a Riemannian manifold $(X,\, g)$, they ask whether it admits an exhaustion function whose Hessian is positive definite or satisfies weaker positive conditions.

Interpolating between the classical notions of convex functions and pluri-sub-harmonic functions, in [7], they define the class of $p$ -pluri-sub-harmonic functions in terms of the positivity of the minors of their Hessian form, and they study $p$ -convex domains, which can be regarded as domains in $\mathbb R ^n$ endowed with a smooth $p$-pluri-sub-harmonic proper exhaustion function.

The notions of geometric pluri-sub-harmonicity and geometric convexity, introduced and studied by Harvey and Lawson [6], is closely related to holomorphic convexity and $q$-completeness in the sense of Andreotti and Grauert [1].

In the complex case, holomorphic convexity and, more in general, $q$-completeness provide vanishing theorems for the Dolbeault cohomology ([8], respectively [1, 2]).

We are concerned in studying vanishing results for strictly $p$-convex domains in $\mathbb R ^n$ in the sense of F. R. Harvey and H. B. Lawson. More precisely, we give a proof of the following result.

Theorem 3.1

Let $X$ be a strictly $p$-convex domain in $\mathbb R ^n$. Then, $H^k_{dR}(X;\mathbb R )=\{0\}$ for every $k\ge p$.

As pointed out to us by Harvey and Lawson, the above result was already known, as a consequence of [9, Theorem 1] by Sha, and [10, Theorem 1] by Wu, see also [7, Proposition 5.7]: more precisely, they prove, using Morse theory, that the existence of a smooth proper strictly $p$-pluri-sub-harmonic exhaustion function has consequences on the homotopy type of the domain.

In spite of this, our proof differs in the techniques, which are inspired by Andreotti and Vesentini [2]: in particular, the ${L}^2$-techniques used in our proof could be hopefully applied in a wider context, a fact which we would like to investigate further in future work.

The organization of the paper is as follows. In Sect. 1, we recall the main definitions introduced in [6, 7] and the results proven by Andreotti and Grauert in [1]. In Sect. 2, we prove some useful estimates, which will be used in Sect. 3 to prove Theorem 3.1.

2 The notion of $p$-convexity by Harvey and Lawson

Following Harvey and Lawson [6, 7], firstly, we recall point-wise definitions of $p$-positive symmetric endomorphisms; then, we will turn to manifolds, and finally, we will recall the notion of $p$-pluri-sub-harmonic (exhaustion) functions and (strictly) $p$-convex domains.

2.1 $p$-Positive (sections of) symmetric endomorphisms

Let $\left(V,\, \left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle \right)$ be an $n$-dimensional real inner product space. Let $G:V\rightarrow V^*$ denote the isomorphism defined as $G(v):= \left\langle {v} \,\left|\, {\cdot } \right. \right\rangle$.

Let $\mathrm{Sym}^{2}\left(V\right)$ denote the space of symmetric elements of $\left(V \otimes V\right)^*$; namely, $A\in \mathrm{Sym}^{2}\left(V\right)$ if and only if $A(v \otimes w) = A(w \otimes v)$, for any $v,w\in V$. By means of the inner product $\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle$, the space $\mathrm{Sym}^{2}\left(V\right)$ is isomorphic to the space of the $\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle$-symmetric endomorphisms of $V$: given $A \in \mathrm{Sym}^{2}\left(V\right)$, we denote by $G^{-1}A \in \mathrm{Hom}\left(V,\,V\right)$ the corresponding $\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle$-symmetric endomorphism.

The endomorphism $G^{-1}A\in \mathrm{Hom}\left(V,\,V\right)$ extends to $D_{G^{-1}A}^{[p]}\in \mathrm{Hom}\left(\wedge ^p V,\,\wedge ^p V\right)$; namely, on a simple vector $v_{i_1} \wedge \cdots \wedge v_{i_p}\in \wedge ^pV$, the endomorphism $D_{G^{-1}A}^{[p]}$ acts as

$$\begin{aligned} D_{G^{-1}A}^{[p]} \left( v_{i_1} \wedge \cdots \wedge v_{i_p} \right) \;:=\; \sum _{\ell =1}^{p} v_{i_1}\wedge \cdots \wedge v_{i_{\ell -1}}\wedge G^{-1}A \left(v_{i_\ell }\right) \wedge v_{i_{\ell +1}}\wedge \cdots \wedge v_{i_p} \;. \end{aligned}$$

Observe that $D_{G^{-1}A}^{[p]}\in \mathrm{Hom}\left(\wedge ^p V,\,\wedge ^p V\right)$ is a symmetric endomorphism with respect to the scalar product on $\wedge ^pV$ induced by $\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle$.

Finally, given a $\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle$-symmetric endomorphism $E\in \mathrm{Hom}\left(V,\,V\right)$, let $\mathrm{sgn}\left(E\right)$ denote the number of non-negative eigenvalues of $E$.

Notice that, given $A\in \mathrm{Sym}^{2}\left(V\right)$, and given two inner products on $V$ inducing, respectively, the isomorphisms $G_1$ and $G_2$, then there holds $\mathrm{sgn}\left(G_1^{-1}A\right) = \mathrm{sgn}\left(G_2^{-1}A\right)$; it is also important to notice that, for $p>1$, it might hold $\mathrm{sgn}\left(D_{G_1^{-1}A}^{[p]}\right) \ne \mathrm{sgn}\left(D_{G_2^{-1}A}^{[p]}\right)$, since the eigenvalues of $D_{G^{-1}A}^{[p]}$ are of the form

$$\begin{aligned} \lambda _{i_1} + \cdots + \lambda _{i_p} \qquad \text{ for} \quad i_1, \ldots , i_p \in \left\{ 1,\ldots ,n\right\} \text{ s.t.} i_1 < \cdots < i_p \;, \end{aligned}$$

where $\lambda _1, \, \ldots , \, \lambda _n$ are the eigenvalues of $G^{-1}A$.

Definition 1.1

[6, 7]

• Let $V$ be a $\mathbb R$-vector space endowed with an inner product $\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle$. Denote the space of $p$-positive forms of $k\text{ th}$ branch on $V$ as

  $$\begin{aligned} \mathcal P _{p}^{(k)}\left(V,\, \left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle \right):= \left\{ A \in \mathrm{Sym}^{2}\left(V\right) \;:\;\mathrm{sgn}\left( D_{G^{-1}A}^{[p]}\right) \ge \genfrac(){0.0pt}{}{n}{p} -k +1 \right\} \;. \end{aligned}$$

• Let $\left(X ,\, g \right)$ be a Riemannian manifold. Define the space of $p$ -positive sections of $k\text{ th}$ branch of the bundle $\mathrm{Sym}^{2}\left(TX\right)$ of symmetric endomorphisms of $TX$ as

  $$\begin{aligned} \mathcal P _p^{(k)}\left(X,\, g \right) \;:=\; \left\{ A \in \mathrm{Sym}^{2}\left(T X\right) \;:\;\forall x\in X, \; A_x\in \mathcal P _p^{(k)}\left(T_xX, g_x\right) \right\} . \end{aligned}$$

2.2 $p$-Pluri-sub-harmonic functions

In order to introduce an exhaustion of a given Riemannian manifold, we focus on special $p$-positive symmetric $2$-forms, those arising from the Hessian of smooth functions.

Thus, let $\left(X,\, g\right)$ be a Riemannian manifold, and let $u$ be a smooth real-valued function on $X$. Let $\nabla$ denote the Levi-Civita connection of the Riemannian metric $g$, and let

$$\begin{aligned} \mathrm{Hess}\,u \left(V, W\right) \;:=\; V\, W\, u - \left(\nabla _V W\right)\, u \;, \end{aligned}$$

where $V$ and $W$ are smooth sections of the tangent bundle $TX$. Thus, $\mathrm{Hess}\,u(x) \in \mathrm{Sym}^{2}\left(T_x X\right)$, for any $x \in X$.

Definition 1.2

[6] Let $\left(X,\, g\right)$ be a Riemannian manifold.

• The space

  $$\begin{aligned} \mathrm{PSH}^{(k)}_p\left(X,\, g\right) \;:=\; \left\{ u\in \mathcal C ^\infty \left(X;\mathbb R \right) \;:\;\mathrm{Hess}\,u \in \mathcal P ^{(k)}_{p} \left( X,\, g\right) \right\} \;, \end{aligned}$$

  is called the space of $p$ -pluri-sub-harmonic functions of $k\text{ th}$ branch on $X$.

• The space

  $$\begin{aligned} \mathrm{int}\left({\mathrm{PSH}^{(k)}_p\left(X,\, g\right)}\right) \;:=\; \left\{ u\in \mathcal C ^\infty \left(X;\mathbb R \right) \;:\;\mathrm{Hess}\,u \in \mathrm{int}\left({\mathcal{P }^{(k)}_{p}}\right) \left( X,\, g\right) \right\} \;, \end{aligned}$$

  (where $\mathrm{int}\left({\mathcal{P }^{(k)}}\right)_{p} \left( X,\, g\right)$ denotes the interior of ${\mathcal{P }^{(k)}}_{p} \left( X,\, g\right)$) is called the space of strictly $p$ -pluri-sub-harmonic functions of $k\text{ th}$ branch on $X$.

2.3 (Strictly) $p$-convexity

We are now ready to recall the concept of $p$-convexity, which is central in [7]. Let $\left(X,\, g\right)$ be a Riemannian manifold. Let $K\subseteq X$ be a compact set. The $p$ -convex hull of $K$ is given by

$$\begin{aligned} {\widetilde{K}}^{\mathrm{PSH}^{(1)}_p \left( X ,\, g\right)} \;:=\; \left\{ x\in X \;:\;\forall \phi \in \mathrm{PSH}^{(1)}_p \left( X ,\, g\right) ,\; \phi (x) \le \max _{y\in K} \phi (y) \right\} \;. \end{aligned}$$

Definition 1.3

[6] Let $\left(X,\, g\right)$ be a Riemannian manifold. Then, $X$ is called $p$ -convex; if for any compact set $K\subseteq X$, then ${\widetilde{K}}^{\mathrm{PSH}^{(1)}_p \left( X ,\, g\right)}$ is relatively compact in $X$.

Define the $p$ -core of $X$, [6, Definition 4.1], as

$$\begin{aligned} \mathrm{Core}_p \left( X ,\, g \right) \;:=\; \left\{ x\in X \;:\;\text{ for} \text{ all} u\in \mathrm{PSH}^{(1)}_p\left(X,\, g\right),\right.\nonumber \\ \left.\mathrm{Hess}\,u(x)\not \in \mathrm{int}\left(\mathcal{P ^{(1)}}\right)_{p}\left(T_xX,\, g_x\right) \right\} \;. \end{aligned}$$

Definition 1.4

[6] Let $\left(X,\, g\right)$ be a Riemannian manifold. Then, $X$ is called strictly $p$ -convex if (i) $\mathrm{Core}_p \left( X ,\, g \right) = \varnothing$ and, (ii) for any compact set $K\subseteq X$, then ${\widetilde{K}}^{\mathrm{PSH}^{(1)}_p\left( X ,\, g\right)}$ is relatively compact in $X$.

2.4 (Strictly) $p$-convexity and (strictly) $p$-pluri-sub-harmonic exhaustion functions

The following correspondences come from [6].

Theorem 1.5

[6, Theorem 4.4, Theorem 4.8] Let $\left(X,\, g\right)$ be a Riemannian manifold. Then $X$ is $p$-convex (respectively, strictly $p$-convex) if and only if $X$ admits a smooth proper exhaustion function $u\in \mathrm{PSH}^{(1)}_p \left( X ,\, g\right)$ [ respectively, $u\in \mathrm{int}(\mathrm{PSH}^{(1)}_p(X ,\, g))$ ].

2.5 The $p$-convexity and the $q$-completeness

All along the definitions of the previous section, the special case that we had in mind is the following classical construction in Complex Analysis.

In [1], Andreotti and Grauert pointed out the following concept.

Definition 1.6

[1] Let $D \subseteq \mathbb C ^n$ be a domain, and let $\phi$ be a smooth real-valued function on $D$. The function $\phi$ is called $p$ -pluri-sub-harmonic (respectively, strictly $p$ -pluri-sub-harmonic) if and only if, for any $z\in D$, the Hermitian form defined, for $\xi :=:\left(\xi ^a\right)_{a\in \left\{ 1,\ldots ,n\right\} } \in \mathbb C ^n$, as

$$\begin{aligned} {L}(\phi )_z\,(\xi ) \;:=\; \sum _{a , b =1}^n \frac{\partial ^2 \phi }{\partial z^a \partial \bar{z}^{b}}(z)\, \xi ^a\, \overline{\xi ^{b}} \;, \end{aligned}$$

has $n-p+1$ non-negative (respectively, positive) eigenvalues.

Andreotti and Grauert [1], studied domains of $\mathbb C ^n$ admitting strictly $q$-pluri-sub-harmonic exhaustion functions (the so-called $q$ -complete domains), proving a vanishing theorem for the higher-degree Dolbeault cohomology groups of such domains; then Andreotti and Vesentini [2], reproved the same result extending the ${L}^2$-techniques by Hörmander [8].

Thus, in the same vein as A. Andreotti and H. Grauert, we would consider domains $X$ in $\mathbb R ^n$ endowed with an exhaustion function $u\in \mathcal C ^\infty (X;\mathbb R )$ whose Hessian is in $\mathrm{int}\left({\mathcal{P }_{p}^{(1)}}\right)(X,g)$, proving a vanishing result for the higher-degree de Rham cohomology groups for strictly $p$-convex domains in the sense of F. R. Harvey and H. B. Lawson.

3 Vanishing of the de Rham cohomology for strictly $p$-convex domains

Let $X$ be an oriented Riemannian manifold of dimension $n$, and denote by $g$ its Riemannian metric and by $\mathrm{vol}$ its volume. The Riemannian metric $g$ induces, for every $x\in X$, a point-wise scalar product $\left\langle \cdot \left| \cdot \cdot \right. \right\rangle _{g_x}:\wedge ^\bullet T^*_xX \times \wedge ^\bullet T^*_xX \rightarrow \mathbb R$.

Fix $\phi \in \mathcal C ^0(X;\mathbb R )$ a continuous function. For every $\varphi ,\, \psi \in \mathcal C ^\infty _{{c}}\left(X;\wedge ^{\bullet }T^*X\right)$, let

$$\begin{aligned} \left\langle {\varphi } \,\left|\, {\psi } \right. \right\rangle _{{L}^2_{\phi }} \;:=\; \int \limits _X \left\langle \varphi \left| \psi \right. \right\rangle _{g_x} \, \exp \left(-\phi \right)\, \mathrm{vol}\;\in \; \mathbb R \;, \end{aligned}$$

and, for $k\in \mathbb N$, define ${L}^{2}_{\phi }\left(X;\wedge ^{k}T^*X\right)$ as the completion of the space $\mathcal C ^\infty _{{c}}\left(X;\wedge ^{k}T^*X\right)$ of smooth $k$-forms with compact support, with respect to the metric induced by $\left\Vert{\cdot }\right\Vert_{{L}^2_{\phi }}:=\left\langle {\cdot } \,\left|\, {\cdot } \right. \right\rangle _{{L}^2_{\phi }}$. Therefore, the space ${L}^{2}_{\phi }\left(X;\wedge ^{k}T^*X\right)$ is a Hilbert space, endowed with the scalar product $\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle _{{L}^2_{\phi }}$, and $\mathcal C ^\infty _{{c}}\left(X;\wedge ^{k}T^*X\right)$ is dense in ${L}^{2}_{\phi }\left(X;\wedge ^{k}T^*X\right)$. For any $k\in \mathbb N$, let ${L}^{2}_{\text{ loc}}\left(X;\wedge ^{k}T^*X\right)$ denote the space of $k$-forms $f$ whose restriction $f\lfloor _{K}$ to every compact set $K\subseteq X$ belongs to ${L}^{2}_{}\left(K;\wedge ^{k}T^*X\right)$.

For every $\phi _1,\,\phi _2\in \mathcal C ^0(X;\mathbb R )$, the operator

$$\begin{aligned} \mathrm{d}:{L}^{2}_{\phi _1}\left(X;\wedge ^{\bullet }T^*X\right) \dashrightarrow {L}^{2}_{\phi _2}\left(X;\wedge ^{\bullet +1}T^*X\right) \end{aligned}$$

is densely defined and closed; denote by

$$\begin{aligned} \mathrm{d}^*_{\phi _2,\,\phi _1}:{L}^{2}_{\phi _2}\left(X;\wedge ^{\bullet +1}T^*X\right) \dashrightarrow {L}^{2}_{\phi _1}\left(X;\wedge ^{\bullet }T^*X\right) \end{aligned}$$

its adjoint, which is a densely defined closed operator.

Recall that on a domain $X$ in $\mathbb R ^n$, fixed $k\in \mathbb N$, $s\in \mathbb N$, and $\phi \in \mathcal C ^\infty \left(X;\mathbb R \right)$, the Sobolev space ${W}^{s, 2}_{\phi }\left(X;\wedge ^{k}T^*X\right)$ is the space of $k$-forms $f:=:\widetilde{\sum _{|I|=k}}f_I\,\mathrm{d}x^I$ such that $\frac{\partial ^{\ell _1+\cdots +\ell _n} f_I}{\partial ^{\ell _1} x^1\cdots \partial ^{\ell _n}x^n}\in {L}^{2}_{\phi }\left(X;\wedge ^{k}T^*X\right)$ for every multi-index $\left(\ell _1,\ldots ,\ell _n\right)\in \mathbb N ^n$ such that $\ell _1+\cdots +\ell _n\le s$ and for every strictly increasing multi-index $I$ such that $|I|=k$. The space ${W}^{s, 2}_\mathrm{loc }\left(X;\wedge ^{k}T^*X\right)$ is defined as the space of $k$-forms $f$ whose restriction $f\lfloor _K$ to every compact set $K\subseteq X$ belongs to ${W}^{s, 2}_{}\left(K;\wedge ^{k}T^*X\right)$.

As a matter of notation, the symbol $\widetilde{\sum _{|I|=k}}$ denotes the sum over the strictly increasing multi-indices $I:=:\left(i_1,\ldots ,i_k\right)\in \mathbb N ^k$ (that is, the multi-indices such that $0<i_1<\cdots <i_k$) of length $k$. Given $I_1$ and $I_2$ two multi-indices of length $k$, let $\mathrm{sign}\left(\begin{array}{c} I_1 \\ I_2\end{array}\right)\,$ be the sign of the permutation $\left(\begin{array}{c}I_1\\ I_2\end{array}\right)$ if $I_1$ is a permutation of $I_2$ and zero otherwise.

3.1 Some preliminary computations

Let $X$ be a domain in $\mathbb R ^n$, that is, an open connected subset of $\mathbb R ^n$ endowed with the metric and the volume induced, respectively, by the Euclidean metric and the standard volume of $\mathbb R ^n$.

For $\phi _1,\, \phi _2\in \mathcal C ^\infty \left(X;\mathbb R \right)$, consider $\mathrm{d}:{L}^{2}_{\phi _1}\left(X;\wedge ^{k-1}T^*X\right) \dashrightarrow {L}^{2}_{\phi _2}\left(X;\wedge ^{k}T^*X\right)$. The following lemma gives an explicit expression of the adjoint $\mathrm{d}^*_{\phi _2,\,\phi _1}:{L}^{2}_{\phi _2}\left(X;\wedge ^{k}T^*X\right) \dashrightarrow {L}^{2}_{\phi _1}\left(X;\wedge ^{k-1}T^*X\right)$ (compare, e.g., with [3, §8.2.1], [5, Lemma O.2] in the complex case).

Lemma 2.1

Let $X$ be a domain in $\mathbb R ^n$. Let $\phi _1,\,\phi _2\in \mathcal C ^\infty \left(X;\mathbb R \right)$ and consider

Let

$$\begin{aligned} v \;:=:\; \widetilde{\sum _{|I|=k}} v_I\, \mathrm{d}x^I \in {L}^{2}_{\phi _2}\left(X;\wedge ^{k}T^*X\right) \end{aligned}$$

and suppose that $v\in \mathrm{dom}\,\mathrm{d}^*_{\phi _2,\,\phi _1}$. Then

$$\begin{aligned} \mathrm{d}^*_{\phi _2,\,\phi _1} v&= \exp \left(\phi _1\right)\, \mathrm{d}^*_{0,\,0}\left(\exp \left(-\phi _2\right)\, v\right) \\&= \widetilde{\sum _{|J|=k-1}} \left(-\exp \left(\phi _1\right)\, \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\,\, \frac{\partial \left(v_I\exp \left(-\phi _2\right)\,\right)}{\partial x^{\ell }}\, \right) \, \mathrm{d}x^J \;. \end{aligned}$$

Proof

By definition of $\mathrm{d}^*_{\phi _2,\,\phi _1}$, for every $u\in \mathrm{dom}\,\mathrm{d}$, one has $\left\langle {\mathrm{d}u} \,\left|\, {v} \right. \right\rangle _{{L}^2_{\phi _2}} = \left\langle {u} \,\left|\, {\mathrm{d}^*_{\phi _2,\,\phi _1} v} \right. \right\rangle _{{L}^2_{\phi _1}}$. Hence, consider

$$\begin{aligned} u \;:=:\; \widetilde{\sum _{|J|=k-1}} u_J\, \mathrm{d}x^J \;\in \; \mathcal C ^\infty _{{c}}\left(X;\wedge ^{k-1}T^*X\right) \;, \end{aligned}$$

and compute

$$\begin{aligned} \mathrm{d}u \;=\; \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I|=k \end{matrix}}}\sum _{\ell =1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \frac{\partial u_J}{\partial x^{\ell }}\, \mathrm{d}x^I \;. \end{aligned}$$

The statement follows by computing

$$\begin{aligned} \left\langle {\mathrm{d}u} \,\left|\, {v} \right. \right\rangle _{{L}^2_{\phi _2}}&= \int \limits _X \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I|=k \end{matrix}}} \sum _{\ell =1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \frac{\partial u_J}{\partial x^{\ell }}\, v_I \, \exp \left(-\phi _2\right)\,\, \mathrm{vol}\\&= -\int \limits _X \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I|=k \end{matrix}}} \sum _{\ell =1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \frac{\partial \left(v_I \, \exp \left(-\phi _2\right)\,\right)}{\partial x^{\ell }}\, u_J\, \mathrm{vol}\end{aligned}$$

and

$$\begin{aligned} \left\langle {u} \,\left|\, {\mathrm{d}^*_{\phi _2,\,\phi _1} v} \right. \right\rangle _{{L}^2_{\phi _1}} \;=\; \int \limits _X \widetilde{\sum _{|J|=k-1}} \left(\mathrm{d}^*_{\phi _2,\,\phi _1} v\right)_J\, u_J\, \exp \left(-\phi _1\right)\,\, \mathrm{vol}\;, \end{aligned}$$

where $\mathrm{d}^*_{\phi _2,\,\phi _1} v=:\widetilde{\sum _{|J|=k-1}} \left(\mathrm{d}^*_{\phi _2,\,\phi _1} v\right)_J\,\mathrm{d}x^J$. $\square$

For any fixed $\phi \in \mathcal C ^\infty \left(X;\mathbb R \right)$ and for any $j\in \left\{ 1,\ldots ,n\right\}$, define the operator

$$\begin{aligned} \delta ^\phi _j :\mathcal C ^\infty \left(X;\mathbb R \right) \rightarrow \mathcal C ^\infty \left(X;\mathbb R \right) \;, \end{aligned}$$

where

$$\begin{aligned} \delta ^\phi _j(f) \;:=\; -\exp \left(\phi \right)\,\,\frac{\partial \left(f\,\exp \left(-\phi \right)\,\right)}{\partial x^{j}}\, \;=\; \frac{\partial \phi }{\partial x^{j}}\,\cdot f-\frac{\partial f}{\partial x^{j}}\, \;. \end{aligned}$$

The following lemma states that $\delta ^\phi _j$ is the adjoint of $\frac{\partial }{\partial x^{j}}\,$ in ${L}^{2}_{\phi }\left(X;\wedge ^{0}T^*X\right)$ and computes the commutator between $\delta ^\phi _j$ and $\frac{\partial }{\partial x^{k}}\,$ (compare with, e.g., [8, pages 83–84]).

Lemma 2.2

Let $X$ be a domain in $\mathbb R ^n$. Let $\phi \in \mathcal C ^\infty \left(X;\mathbb R \right)$ and $j\in \left\{ 1,\ldots ,n\right\}$, and consider the operator $\delta ^\phi _j:\mathcal C ^\infty \left(X;\mathbb R \right) \rightarrow \mathcal C ^\infty \left(X;\mathbb R \right)$. Then:

• for every $w_1,w_2\in \mathcal C ^\infty _{{c}}\left(X;\mathbb R \right)$,

  $$\begin{aligned} \int \limits _X w_1\cdot \frac{\partial w_2}{\partial x^{k}}\,\exp \left(-\phi \right)\,\mathrm{vol}\;=\; \int \limits _X\delta ^\phi _k(w_1)\cdot w_2\,\exp \left(-\phi \right)\,\mathrm{vol}\;; \end{aligned}$$

• for any $k\in \left\{ 1,\ldots ,n\right\}$, the following commutation formula holds in $\mathrm{End}\left(\mathcal C ^\infty _{{c}}\left(X;\mathbb R \right)\right)$:

  $$\begin{aligned} \left[\delta ^\phi _j,\, \frac{\partial }{\partial x^{k}}\,\right] \;=\; -\frac{\partial ^2\phi }{\partial x^j\partial x^k}\cdot \;. \end{aligned}$$

Finally, we prove the following estimate, which will be used in the proof of Theorem 3.1 (we refer to [8, Sect. 4.2], or, e.g., [5, Lemma O.3] and [3, Sect. 8.3.1] for its complex counterpart).

Proposition 2.3

Let $X$ be a domain in $\mathbb R ^n$ and $\phi ,\,\psi \in \mathcal C ^\infty \left(X;\mathbb R \right)$. Consider

Then, for any $\eta :=: \widetilde{\sum _{|I|=k}}\eta _I \, \mathrm{d}x^I \in \mathcal C ^\infty _{{c}}\left(X;\wedge ^{k}T^*X\right)$, one has

$$\begin{aligned}&{\int \limits _X \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\,\mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial ^2\phi }{\partial x^{\ell _1}\,\partial x^{\ell _2}}\, \eta _{I_1}\, \eta _{I_2} \, \exp \left(-\phi \right)\, \mathrm{vol}} \\&\le \; C \cdot \left( \left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi } \eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2 + \left\Vert{\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}^2 + \int \limits _X \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\left|\frac{\partial \psi }{\partial x^\ell }\right|^2\,\left|\eta _I\right|^2\, \exp \left(-\phi \right)\, \mathrm{vol}\right) \;, \end{aligned}$$

where $C:=:C(k,n)\in \mathbb N$ is a constant depending just on $k$ and $n$.

Proof

It is straightforward to compute

$$\begin{aligned} \mathrm{d}\eta \;=\; \widetilde{\sum _{\begin{matrix} |I|=k\\ |H|=k+1 \end{matrix}}}\sum _{\ell =1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell I \\ H\end{array}\right)\,\frac{\partial \eta _I}{\partial x^{\ell }}\,\mathrm{d}x^H \end{aligned}$$

and, using Lemma ,

$$\begin{aligned} \mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi } \eta&= -\exp \left(-\psi \right)\, \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I|=k \end{matrix}}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \left(\frac{\partial \eta _I}{\partial x^{\ell }}\,-\frac{\partial \left(\phi -\psi \right)}{\partial x^{\ell }}\,\eta _I\right) \, \mathrm{d}x^J \\&= \exp \left(-\psi \right)\, \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I|=k \end{matrix}}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \left(\delta ^\phi _\ell \left(\eta _I\right)-\frac{\partial \psi }{\partial x^{\ell }}\,\,\eta _I\right) \, \mathrm{d}x^J \;. \end{aligned}$$

For every $J$ such that $|J|=k-1$, the previous equality gives

$$\begin{aligned} \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \delta ^\phi _\ell \left(\eta _I\right)= \exp \left(\psi \right)\, \left(\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi } \eta \right)_J + \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \frac{\partial \psi }{\partial x^{\ell }}\,\,\eta _I \;, \end{aligned}$$

where $\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta =: \widetilde{\sum _{|J|=k-1}}\left(\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta \right)_J\mathrm{d}x^J$.

By the arithmetic mean–geometric mean inequality, one gets

$$\begin{aligned}&{\int \limits _X \widetilde{\sum _{|J|=k-1}} \left| \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \delta ^\phi _\ell \left(\eta _I\right) \right|^2 \, \exp \left(-\phi \right)\, \mathrm{vol}} \nonumber \\&\quad \le \; 2\, \int \limits _X \widetilde{\sum _{|J|=k-1}}\left(\left|\left(\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta \right)_J\right|^2\, \exp \left(2\psi \right)\,\right.\nonumber \\&\qquad \left.+ \left|\widetilde{\sum _{|I|=k}}\sum _{\ell =1}^n\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \frac{\partial \psi }{\partial x^{\ell }}\,\eta _I\right|^2 \right)\, \exp \left(-\phi \right)\, \mathrm{vol}\nonumber \\&\quad \le \; C \, \left(\left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2+\int \limits _X\widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\left|\frac{\partial \psi }{\partial x^{\ell }}\,\right|^2\cdot \left|\eta _I\right|^2\, \exp \left(-\phi \right)\, \mathrm{vol}\right) , \end{aligned}$$

(1)

where $C:=:C(k,n)\in \mathbb N$ depends on $k$ and $n$ only.

Now, using Lemma 2.2, one computes

$$\begin{aligned}&{\int \limits _X \widetilde{\sum _{|J|=k-1}} \left| \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \delta ^\phi _\ell \left(\eta _I\right) \right|^2 \, \exp \left(-\phi \right)\, \mathrm{vol}} \nonumber \\&=\; \widetilde{\sum _{|J|=k-1}} \widetilde{\sum _{\begin{matrix} |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\, \mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \int \limits _X \delta ^\phi _{\ell _1}\left(\eta _{I_1}\right) \cdot \delta ^\phi _{\ell _2}\left(\eta _{I_2}\right) \exp \left(-\phi \right)\, \mathrm{vol}\nonumber \\&=\; \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\, \mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \nonumber \\&\quad \times \int \limits _X\left(\frac{\partial \eta _{I_1}}{\partial x^{\ell _2}}\,\,\frac{\partial \eta _{I_2}}{\partial x^{\ell _1}}\,+\frac{\partial ^2\phi }{\partial x^{\ell _1}\,\partial x^{\ell _2}}\,\eta _{I_1}\,\eta _{I_2}\right) \, \exp \left(-\phi \right)\, \mathrm{vol}. \end{aligned}$$

(2)

Now, note that

$$\begin{aligned} \left|\mathrm{d}\eta \right|^2&= \widetilde{\sum _{|H|=k+1}} \left|\widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell I \\ H\end{array}\right)\,\frac{\partial \eta _I}{\partial x^{\ell }}\,\right|^2 \nonumber \\&= \widetilde{\sum _{|H|=k+1}} \left(\widetilde{\sum _{\begin{matrix} |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell _1 I_1 \\ H\end{array}\right)\,\mathrm{sign}\left(\begin{array}{c} \ell _2 I_2 \\ H\end{array}\right)\,\frac{\partial \eta _{I_1}}{\partial x^{\ell _1}}\,\frac{\partial \eta _{I_2}}{\partial x^{\ell _2}}\,\right) \nonumber \\&= \widetilde{\sum _{\begin{matrix} |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell _1 I_1 \\ \ell _2 I_2\end{array}\right)\,\frac{\partial \eta _{I_1}}{\partial x^{\ell _1}}\,\frac{\partial \eta _{I_2}}{\partial x^{\ell _2}}\, \nonumber \\&= \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n} \left|\frac{\partial \eta _I}{\partial x^{\ell }}\,\right|^2 - \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_2|=k \end{matrix}}} \sum _{\ell _1,\,\ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\, \mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial \eta _{I_1}}{\partial x^{\ell _2}}\, \frac{\partial \eta _{I_2}}{\partial x^{\ell _1}}\,. \end{aligned}$$

(3)

Hence, in view of (3), (2), (1), we get

$$\begin{aligned}&{\int \limits _X \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\,\mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial ^2\phi }{\partial x^{\ell _1}\,\partial x^{\ell _2}}\, \eta _{I_1}\, \eta _{I_2} \, \exp \left(-\phi \right)\, \mathrm{vol}} \\&\quad \le \; \int \limits _X \left(\widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\,\mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial ^2\phi }{\partial x^{\ell _1}\,\partial x^{\ell _2}}\, \eta _{I_1}\, \eta _{I_2}\right.\nonumber \\&\quad \left.+\widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\left|\frac{\partial \eta _I}{\partial x^\ell }\right|^2 \right) \, \exp \left(-\phi \right)\, \mathrm{vol}\\&\quad =\; \int \limits _X \left(\widetilde{\sum _{|J|=k-1}} \left| \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \delta ^\phi _\ell \left(\eta _I\right) \right|^2 + \widetilde{\sum _{|H|=k+1}}\left|\left(\mathrm{d}\eta \right)_H\right|^2\right) \, \exp \left(-\phi \right)\, \mathrm{vol}\\&\quad \le \; C \cdot \left( \left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2 + \left\Vert{\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}^2 + \int \limits _X \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\left|\frac{\partial \psi }{\partial x^\ell }\right|^2\,\left|\eta _I\right|^2\, \exp \left(-\phi \right)\, \mathrm{vol}\right) \;, \end{aligned}$$

concluding the proof. $\square$

Remark 2.4

The argument in the proof of Proposition 2.3 actually proves the following stronger estimate, which will be used in the regularization process in Theorem 3.1.

Let $X$ be a domain in $\mathbb R ^n$ and $\phi ,\,\psi \in \mathcal C ^\infty \left(X;\mathbb R \right)$. Consider

Then, for any $\eta :=: \widetilde{\sum _{|I|=k}}\eta _I \, \mathrm{d}x^I \in \mathcal C ^\infty _{{c}}\left(X;\wedge ^{k}T^*X\right)$, one has

$$\begin{aligned}&\int \limits _X \left(\widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\,\mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial ^2\phi }{\partial x^{\ell _1}\,\partial x^{\ell _2}}\, \eta _{I_1}\, \eta _{I_2}\right.\nonumber \\&\left.+\widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\left|\frac{\partial \eta _I}{\partial x^\ell }\right|^2\right) \, \exp \left(-\phi \right)\, \mathrm{vol}\\&\quad \le \; C \cdot \left( \left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2 + \left\Vert{\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}^2 + \int \limits _X \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\left|\frac{\partial \psi }{\partial x^\ell }\right|^2\,\left|\eta _I\right|^2\, \exp \left(-\phi \right)\, \mathrm{vol}\right) \;, \end{aligned}$$

where $C:=:C(k,n)\in \mathbb N$ is a constant depending just on $k$ and $n$.

4 Proof of the main theorem

We are ready to prove the following vanishing theorem for the higher-degree de Rham cohomology groups of a strictly $p$-convex domain in $\mathbb R ^n$ (for a different proof, involving Morse theory, compare [9, Theorem 1] by Sha, and [10, Theorem 1] by Wu, see also [7, Proposition 5.7]).

Theorem 3.1

Let $X$ be a strictly $p$-convex domain in $\mathbb R ^n$. Then $H^k_{dR}(X;\mathbb R )=\{0\}$ for every $k\ge p$.

Proof

We are going to prove that every $\mathrm{d}$-closed $k$-form $\eta \in \mathcal C ^\infty \left(X;\wedge ^{k}T^*X\right)$ is $\mathrm{d}$-exact, namely, there exists $\alpha \in \mathcal C ^\infty \left(X;\wedge ^{k-1}T^*X\right)$ such that $\eta =\mathrm{d}\alpha$; the statement of the theorem is a direct consequence of this result. Let us split the proof in the following steps.

Step 1—Definitions of the weight functions and other notations. Being $X$ a strictly $p$-convex domain in $\mathbb R ^n$, by Harvey and Lawson’s [6, Theorem 4.8] (see also [7, Theorem 5.4]), there exists a smooth proper strictly $p$-pluri-sub-harmonic exhaustion function

$$\begin{aligned} \rho \;\in \; \mathrm{int}\left({\mathrm{PSH}^{(1)}_p\left( X,\, g\right)}\right) \cap \mathcal C ^\infty \left(X;\mathbb R \right) \;, \end{aligned}$$

where $g$ is the metric on $X$ induced by the Euclidean metric on $\mathbb R ^n$.

For every $m\in \mathbb N$, consider the compact set

$$\begin{aligned} K^{(m)} \;:=\; \left\{ x\in X \;:\;\rho (x)\le m \right\} \;, \end{aligned}$$

and define

$$\begin{aligned} L^{(m)} \;:=\; \min _{K^{(m)}} \lambda _{1}^{[k]} \;>\; 0 \;, \end{aligned}$$

where, for every $x\in X$, the real numbers $\lambda _{1}^{[k]}(x)\le \cdots \le \lambda _{\genfrac(){0.0pt}{}{n}{k}}^{[k]}(x)$ are the ordered eigenvalues of $D^{[k]}_{g^{-1}\mathrm{Hess}\,\rho (x)}\in \mathrm{Hom}\left(\wedge ^kT_xX,\,\wedge ^kT_xX\right)$, and $\lambda _1(x)\le \cdots \le \lambda _n(x)$ are the ordered eigenvalues of $g^{-1}\mathrm{Hess}\,\rho (x)\in \mathrm{Hom}\left(T_xX,\,T_xX\right)$; indeed, note that, for every $x\in X$,

$$\begin{aligned} \lambda _{1}^{[k]}(x) \;=\; \lambda _1(x)+\cdots +\lambda _{k}(x) \;\ge \; \lambda _1(x)+\cdots +\lambda _p(x) \;>\; 0 \;, \end{aligned}$$

being $\rho$ strictly $p$-pluri-sub-harmonic and that the function $X\ni x\mapsto \lambda _{1}^{[k]}(x)\in \mathbb R$ is continuous.

Fix $\left\{ \rho _\nu \right\} _{\nu \in \mathbb N }\subset \mathcal C ^\infty _{{c}}\left(X;\mathbb R \right)$ such that (i) $0\le \rho _\nu \le 1$ for every $\nu \in \mathbb N$, and (ii) for every compact set $K\subseteq X$, there exists $\nu _0:=:\nu _0(K)\in \mathbb N$ such that $\rho _\nu \lfloor _{K}=1$ for every $\nu \ge \nu _0$.

Then, we can choose $\psi \in \mathcal C ^\infty \left(X;\mathbb R \right)$ such that for every $\nu \in \mathbb N$,

$$\begin{aligned} \left|\mathrm{d}\rho _\nu \right|^2 \;\le \; \exp \left(\psi \right)\, \;. \end{aligned}$$

For every $m\in \mathbb N$, set

$$\begin{aligned} \gamma ^{(m)} \;:=\; \max _{K^{(m)}} \left(C\cdot \left|\mathrm{d}\psi \right|^2+\exp \left(\psi \right)\,\right) \;, \end{aligned}$$

where $C:=:C(n,k)$ is the constant in Proposition 2.3.

Fix $\chi \in \mathcal C ^\infty \left(\mathbb R ;\mathbb R \right)$ such that $(i)\chi ^{\prime }>0, (ii)\chi ^{\prime \prime }>0$, and $(iii) \chi ^{\prime }\lfloor _{\left(-\infty ,\,m\right]}>\frac{\gamma ^{(m)}}{L^{(m)}}$, for every $m\in \mathbb N$. Define

$$\begin{aligned} \phi \;:=\; \chi \circ \rho \;: \end{aligned}$$

then, $\phi \in \mathrm{int}\left({\mathrm{PSH}^{(1)}_p\left( X,\, g\right)}\right) \cap \mathcal C ^\infty \left(X;\mathbb R \right)$; furthermore

$$\begin{aligned} \frac{\partial ^2\phi }{\partial x^{\ell _1}\partial x^{\ell _2}} \;=\; \chi ^{\prime \prime }\circ \rho \cdot \frac{\partial \rho }{\partial x^{\ell _1}} \cdot \frac{\partial \rho }{\partial x^{\ell _2}}+\chi ^{\prime }\circ \rho \cdot \frac{\partial ^2\rho }{\partial x^{\ell _1}\partial x^{\ell _2}} \;. \end{aligned}$$

Choose $\mu \in \mathcal C ^\infty \left(X;\mathbb R \right)$ such that, for every $m\in \mathbb N$,

$$\begin{aligned} \chi ^{\prime }\circ \rho \lfloor _{K^{(m)}}\cdot L^{(m)} \;\ge \; \mu \lfloor _{K^{(m)}} \;\ge \; \gamma ^{(m)} \;. \end{aligned}$$

Step 2—For every $\eta \in \mathcal C ^\infty _{{c}}\left(X;\wedge ^kT^*X\right)$, it holds $\left\Vert{\eta }\right\Vert_{{L}^2_{\phi -\psi }}^2\le C \cdot \Bigg (\!\left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2\!+ \left\Vert{\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}^2 \Bigg )$. Since

$$\begin{aligned}&D^{\left[k\right]}_{g^{-1}\mathrm{Hess}\,\rho } \;=\; \left(\widetilde{\sum _{|J|=k-1}}\sum _{\ell _1,\, \ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\, \mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial ^2\rho }{\partial x^{\ell _1}\partial x^{\ell _2}}\right)_{I_1, I_2}\\&\quad \in \mathrm{Hom}\left(\wedge ^{k}TX,\,\wedge ^{k}TX\right) \;, \end{aligned}$$

one estimates

$$\begin{aligned}&{\widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_1|=k \end{matrix}}}\sum _{\ell _1,\, \ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\, \mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial ^2\phi }{\partial x^{\ell _1} \partial x^{\ell _2}}\,\eta _{I_1}\,\eta _{I_2}} \\&\quad = \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_1|=k \end{matrix}}}\sum _{\ell _1,\, \ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\, \mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \chi ^{\prime \prime }\circ \rho \cdot \frac{\partial \rho }{\partial x^{\ell _1}}\,\frac{\partial \rho }{\partial x^{\ell _2}}\,\eta _{I_1}\,\eta _{I_2} \\&\qquad +\; \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_1|=k \end{matrix}}}\sum _{\ell _1,\, \ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\, \mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \chi ^{\prime }\circ \rho \cdot \frac{\partial ^2\rho }{\partial x^{\ell _1}\partial x^{\ell _2}}\, \eta _{I_1}\, \eta _{I_2} \\&\quad = \widetilde{\sum _{|J|=k-1}}\chi ^{\prime \prime }\circ \rho \cdot \left|\widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\,\frac{\partial \rho }{\partial x^{\ell }}\,\eta _I \right|^2 \\&\qquad +\; \chi ^{\prime }\circ \rho \cdot \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_1|=k \end{matrix}}}\sum _{\ell _1,\, \ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\, \mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial ^2\rho }{\partial x^{\ell _1}\partial x^{\ell _2}}\, \eta _{I_1}\, \eta _{I_2} \\&\quad \ge \chi ^{\prime }\circ \rho \cdot \lambda _{1}^{[k]}(x) \cdot \widetilde{\sum _{|I|=k}}\left|\eta _I\right|^2 \\&\quad \ge \mu \cdot \widetilde{\sum _{|I|=k}}\left|\eta _I\right|^2 \;. \end{aligned}$$

Hence, using Proposition 2.3, we get that, for every $\eta \in \mathcal C ^\infty _{{c}}\left(X;\wedge ^{k}T^*X\right)$,

$$\begin{aligned} \left\Vert{\eta }\right\Vert_{{L}^2_{\phi -\psi }}^2&= \int \limits _X \widetilde{\sum _{|I|=k}}\left|\eta _I\right|^2\, \exp \left(-\left(\phi -\psi \right)\right)\,\mathrm{vol}\\&\le \int \limits _X \widetilde{\sum _{|I|=k}} \left(\mu -C\cdot \sum _{\ell =1}^{n}\left|\frac{\partial \psi }{\partial x^\ell }\right|^2\right)\cdot \left|\eta _I\right|^2 \, \exp \left(-\phi \right)\, \mathrm{vol}\\&\le \int \limits _X \left(\widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I_1|=k\\ |I_2|=k \end{matrix}}}\sum _{\ell _1,\,\ell _2=1}^{n} \mathrm{sign}\left(\begin{array}{c} \ell _1 J \\ I_1\end{array}\right)\,\mathrm{sign}\left(\begin{array}{c} \ell _2 J \\ I_2\end{array}\right)\, \frac{\partial ^2\phi }{\partial x^{\ell _1}\,\partial x^{\ell _2}}\, \eta _{I_1}\, \eta _{I_2} \right. \\&\left. - C\cdot \widetilde{\sum _{|I|=k}}\sum _{\ell =1}^{n}\left|\frac{\partial \psi }{\partial x^\ell }\right|^2\,\left|\eta _I\right|^2\right)\, \exp \left(-\phi \right)\, \mathrm{vol}\\&\le C \cdot \left( \left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2 + \left\Vert{\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}^2 \right) \;, \end{aligned}$$

where $C:=:C(k,n)\in \mathbb N$ is the constant in Proposition 2.3, depending just on $k$ and $n$.

Step 3—$\mathcal C ^\infty _\mathrm{{c}}\left(X;\,\wedge ^kT^*X\right)$ is dense in $\left(\mathrm{dom}\,\mathrm{d}\cap \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi },\, \left\Vert{\cdot }\right\Vert_{{L}^2_{\phi -\psi }}\right.$ $\left.+\left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi } \cdot }\right\Vert_{{L}^2_{\phi -2\psi }}+\left\Vert{\mathrm{d}\cdot }\right\Vert_{{L}^2_{\phi }}\right)$. Consider

Fix $\eta \in \mathrm{dom}\,\mathrm{d}\cap \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\subseteq {L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)$. Firstly, we prove that $\left\{ \rho _\nu \,\eta \right\} _{\nu \in \mathbb N }\subset \mathrm{dom}\,\mathrm{d}\cap \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\subseteq {L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)$ (where $\left\{ \rho _\nu \right\} _{\nu \in \mathbb N }\subset \mathcal C ^\infty _{{c}}\left(X;\mathbb R \right)$ has been defined in Step 1) is a sequence of functions having compact support and converging to $\eta$ in the graph norm $\left\Vert{\cdot }\right\Vert_{{L}^2_{\phi -\psi }}+\left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\cdot }\right\Vert_{{L}^2_{\phi -2\psi }}+\left\Vert{\mathrm{d}\cdot }\right\Vert_{{L}^2_{\phi }}$. Indeed,

$$\begin{aligned} \left| \mathrm{d}\left(\rho _\nu \,\eta \right)-\rho _\nu \,\mathrm{d}\eta \right|^2 \,\exp \left(-\phi \right)\,&= \left|\eta \right|^2\cdot \left|\mathrm{d}\rho _\nu \right|^2 \,\exp \left(-\phi \right)\, \\&\le \left|\eta \right|^2\,\exp \left(-\left(\phi -\psi \right)\right)\, \;\in \; {L}^{2}\left(X;\wedge ^{k}T^*X\right) \;, \end{aligned}$$

hence, by Lebesgue’s dominated convergence theorem, $\left\Vert{\mathrm{d}\left(\rho _\nu \,\eta \right)-\rho _\nu \,\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}\rightarrow 0$ as $\nu \rightarrow +\infty$. Furthermore, for every $\nu \in \mathbb N$, note that $\rho _\nu \,\eta \in \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }$, since the map

$$\begin{aligned} {L}^{2}_{\phi -2\psi }\left(X;\wedge ^{k-1}T^*X\right) \;\supseteq \; \mathrm{dom}\,\mathrm{d}\;\ni \; u \mapsto \left\langle {\rho _\nu \,\eta } \,\left|\, {\mathrm{d}u} \right. \right\rangle _{{L}^2_{\phi -\psi }} \;\in \; \mathbb R \end{aligned}$$

is continuous, being

$$\begin{aligned} \left\langle {\rho _\nu \,\eta } \,\left|\, {\mathrm{d}u} \right. \right\rangle _{{L}^2_{\phi -\psi }}&= \left\langle {\eta } \,\left|\, {\mathrm{d}\left(\rho _\nu \, u\right)} \right. \right\rangle _{{L}^2_{\phi -\psi }} - \left\langle {\eta } \,\left|\, {\mathrm{d}\rho _\nu \wedge u} \right. \right\rangle _{{L}^2_{\phi -\psi }} \\&= \left\langle {\rho _\nu \,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta } \,\left|\, {u} \right. \right\rangle _{{L}^2_{\phi -2\psi }} - \left\langle {\eta } \,\left|\, {\mathrm{d}\rho _\nu \wedge u} \right. \right\rangle _{{L}^2_{\phi -\psi }} \;, \end{aligned}$$

hence, by the Riesz representation theorem, there exists $\tilde{\eta }=:\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\left(\rho _\nu \,\eta \right)\in {L}^{2}_{\phi -2\psi }\left(X;\wedge ^{k-1}T^*X\right)$ such that, for every $u\in \mathrm{dom}\,\mathrm{d}\subseteq {L}^{2}_{\phi -2\psi }\left(X;\wedge ^{k-1}T^*X\right)$, it holds $\left\langle {\rho _\nu \,\eta } \,\left|\, {\mathrm{d}u} \right. \right\rangle _{{L}^2_{\phi -\psi }}=\left\langle {\tilde{\eta }} \,\left|\, {u} \right. \right\rangle _{{L}^2_{\phi -2\psi }}$. Lastly, note that, for every $u\in \mathrm{dom}\,\mathrm{d}\subseteq {L}^{2}_{\phi -2\psi }\left(X;\wedge ^{k-1}T^*X\right)$,

$$\begin{aligned}&\left|\left\langle {\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\left(\rho _\nu \,\eta \right)-\rho _\nu \,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\,\eta } \,\left|\, {u} \right. \right\rangle _{{L}^2_{\phi -2\psi }}\right|\\&\quad = \left|\left\langle {\rho _\nu \,\eta } \,\left|\, {\mathrm{d}u} \right. \right\rangle _{{L}^2_{\phi -\psi }}-\left\langle {\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta } \,\left|\, {\rho _\nu \,u} \right. \right\rangle _{{L}^2_{\phi -2\psi }}\right| \\&\quad = \left|\left\langle {\eta } \,\left|\, {\mathrm{d}\rho _\nu \wedge u} \right. \right\rangle _{{L}^2_{\phi -\psi }}\right| \\&\quad \le \left\Vert{\eta }\right\Vert_{{L}^2_{\phi -\psi }}\cdot \left\Vert{\mathrm{d}\rho _\nu \wedge u}\right\Vert_{{L}^2_{\phi -\psi }} \,, \end{aligned}$$

hence, by Lebesgue’s dominated convergence theorem, $\Big \Vert \mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\left(\rho _\nu \, \eta \right)-\rho _\nu \,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta \Big \Vert _{{L}^2_{\phi -2\psi }} \rightarrow 0$ as $\nu \rightarrow +\infty$. This shows that $\rho _\nu \,\eta \rightarrow \eta$ as $\nu \rightarrow +\infty$ with respect to the graph norm.

Hence, we may suppose that $\eta \in \mathrm{dom}\,\mathrm{d}\cap \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\subseteq {L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)$ has compact support. Let $\left\{ \Phi _\varepsilon \right\} _{\varepsilon \in \mathbb R }\subseteq \mathcal C ^\infty \left(\mathbb R ^n;\mathbb R \right)$ be a family of positive mollifiers, that is, $\Phi _\varepsilon :=\varepsilon ^{-n}\,\Phi \left(\frac{\cdot }{\varepsilon }\right)$, where $(i) \Phi \in \mathcal C ^\infty _\mathrm{c }\left(\mathbb R ^n;\mathbb R \right), (ii) \int _\mathbb{R ^n}\Phi \mathrm{vol}_\mathbb{R ^n} =1, (iii) \lim _{\varepsilon \rightarrow 0}\Phi _\varepsilon =\delta$, where $\delta$ is the Dirac delta function, and $(iv) \Phi \ge 0$.

Consider the convolution $\left\{ \eta * \Phi _\varepsilon \right\} _{\varepsilon \in \mathbb R }\subset \mathcal C ^\infty _{{c}}\left(X;\wedge ^{k}T^*X\right)$; we prove that $\eta * \Phi _\varepsilon \rightarrow \eta$ as $\varepsilon \rightarrow 0$ with respect to the graph norm. Clearly, $\left\Vert{\eta -\eta *\Phi _\varepsilon }\right\Vert_{{L}^2_{\phi -\psi }}\rightarrow 0$ as $\varepsilon \rightarrow 0$. Since $\mathrm{d}\left(\eta *\Phi _\varepsilon \right)=\mathrm{d}\eta *\Phi _\varepsilon$, one has that $\left\Vert{\mathrm{d}\left(\eta *\Phi _\varepsilon \right)-\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}\rightarrow 0$ as $\varepsilon \rightarrow 0$. Lastly, write

$$\begin{aligned} \mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi } \;=\; \exp \left(-\psi \right)\, \left(\mathrm{d}^*_{0,\,0}+A_{\phi -\psi ,\,\phi -2\psi }\right) \;, \end{aligned}$$

where $\mathrm{d}^*_{0,\,0}$ is a differential operator with constant coefficients, and $A_{\phi -\psi ,\,\phi -2\psi }$ is a differential operator of order zero defined, for every $v\in {L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)$, as

$$\begin{aligned} A_{\phi -\psi ,\,\phi -2\psi }\left(v\right) \;:=\; \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I|=k \end{matrix}}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\, \frac{\partial \left(\phi -\psi \right)}{\partial x^{\ell }}\,\cdot \eta \mathrm{d}x^J \;; \end{aligned}$$

hence

$$\begin{aligned}&\left(\mathrm{d}^*_{0,\,0} +A_{\phi -\psi ,\,\phi -2\psi }\right) \left(\eta *\Phi _\varepsilon \right)\\&\quad = \left(\left(\mathrm{d}^*_{0,\,0}+A_{\phi -\psi ,\,\phi -2\psi }\right) \left(\eta \right)\right) *\Phi _\varepsilon - \left(A_{\phi -\psi ,\,\phi -2\psi }\eta \right)*\Phi _\varepsilon + A_{\phi -\psi ,\,\phi -2\psi }\left(\eta *\Phi _\varepsilon \right) \\&\quad \rightarrow \left(\mathrm{d}^*_{0,\,0}+A_{\phi -\psi ,\,\phi -2\psi }\right)\left(\eta \right) \end{aligned}$$

as $\varepsilon \rightarrow 0$ in ${L}^{2}_{\phi -2\psi }\left(X;\wedge ^{k-1}T^*X\right)$; having $\eta$ compact support, it follows that $\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }$ $\left(\eta *\Phi _\varepsilon \right)\rightarrow \mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\left(\eta \right)$ as $\varepsilon \rightarrow 0$ in ${L}^{2}_{\phi -2\psi }\left(X;\wedge ^{k-1}T^*X\right)$.

Step 4—If $\left\Vert{\eta }\right\Vert_{{L}^2_{\phi -\psi }}^2\le C\cdot \left(\left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2+\left\Vert{\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}^2\right)$ holds for every $\mathcal C ^\infty _\mathrm{{c}}\left(X;\,\wedge ^kT^*X\right)$, then it holds for every $\eta \in \mathrm{dom}\,\mathrm{d}\cap \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }$. Let $\eta \in \mathrm{dom}\,\mathrm{d}\cap \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }$. By Step 3, take $\left\{ \eta _j\right\} _{j\in \mathbb N }\subset \mathcal C ^\infty _{{c}}\left(X;\wedge ^{k}T^*X\right)$ such that $\eta _j\rightarrow \eta$ as $j\rightarrow +\infty$ in the graph norm. Since, for every $j\in \mathbb N$, one has $\left\Vert{\eta _j}\right\Vert_{{L}^2_{\phi -\psi }}^2\le C\cdot \left(\left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta _j}\right\Vert_{{L}^2_{\phi -2\psi }}^2+\left\Vert{\mathrm{d}\eta _j}\right\Vert_{{L}^2_{\phi }}^2\right)$, and since $\left\Vert{\eta _j-\eta }\right\Vert_{{L}^2_{\phi -\psi }}\rightarrow 0$, $\left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta _j-\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}\rightarrow 0$ and $\left\Vert{\mathrm{d}\eta _j-\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}\rightarrow 0$ as $j\rightarrow +\infty$, we get that also $\left\Vert{\eta }\right\Vert_{{L}^2_{\phi -\psi }}^2\le C\cdot \left(\left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2+\left\Vert{\mathrm{d}\eta }\right\Vert_{{L}^2_{\phi }}^2\right)$.

Step 5—Existence of a solution in $\mathrm {L}^2_\mathrm{{loc}}\left(X;\wedge ^{k}T^*X\right)$. We prove here that the operator

$$\begin{aligned} \mathrm{d}:{L}^{2}_{\phi -2\psi }\left(X;\wedge ^{k-1}T^*X\right) \dashrightarrow \ker \left(\mathrm{d}:{L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)\dashrightarrow {L}^{2}_{\phi }\left(X;\wedge ^{k+1}T^*X\right)\right) \end{aligned}$$

is surjective, hence, for every $\eta \in \ker \left(\mathrm{d}:{L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)\dashrightarrow {L}^{2}_{\phi }\left(X;\wedge ^{k+1}T^*X\right)\right)$, the equation $\mathrm{d}\alpha =\eta$ has a solution $\alpha$ in ${L}^{2}_{\phi -\psi }\left(X;\wedge ^{k-1}T^*X\right)\subseteq {L}^{2}_{\text{ loc}}\left(X;\wedge ^{k-1}T^*X\right)$.

We recall (see, e.g., [8, Lemma 4.1.1]) that given two Hilbert spaces $\left(H_1,\,\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle _{{L}^2_{H_1}}\right)$ and $\left(H_2,\,\left\langle {\cdot } \,\left|\, {\cdot \cdot } \right. \right\rangle _{{L}^2_{H_2}}\right)$, and a densely defined closed operator $T:H_1 \dashrightarrow H_2$, whose adjoint is $T^*:H_2\dashrightarrow H_1$, if $F\subseteq H_2$ is a closed subspace such that $\mathrm{im}\,T\subseteq F$, then the following conditions are equivalent:

• (i) $\mathrm{im}\,T=F$;

• (ii) there exists $C>0$ such that, for every $y\in \mathrm{dom}\,T^* \cap F$,

  $$\begin{aligned} \left\Vert{y}\right\Vert_{{L}^2_{H_2}} \;\le \; C\cdot \left\Vert{T^*y}\right\Vert_{{L}^2_{H_1}} \;. \end{aligned}$$

Hence, consider

$$\begin{aligned} \mathrm{d}:{L}^{2}_{\phi -2\psi }\left(X;\wedge ^{k-1}T^*X\right) \dashrightarrow {L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right) \end{aligned}$$

and

$$\begin{aligned} {L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right) \;\supseteq \; F&:= \ker \left(\mathrm{d}:{L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)\dashrightarrow {L}^{2}_{\phi }\left(X;\wedge ^{k+1}T^*X\right)\right) \\&\supseteq \mathrm{im}\,\left(\mathrm{d}:{L}^{2}_{\psi -2\psi }\left(X;\wedge ^{k-1}T^*X\right)\dashrightarrow {L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)\right) \;. \end{aligned}$$

By Step 4, for every $\eta \in \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\cap F \subseteq \mathrm{dom}\,\mathrm{d}\cap \mathrm{dom}\,\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }$, it holds that

$$\begin{aligned} \left\Vert{\eta }\right\Vert_{{L}^2_{\phi -\psi }}^2 \;\le \; C \, \left\Vert{\mathrm{d}^*_{\phi -\psi ,\,\phi -2\psi }\eta }\right\Vert_{{L}^2_{\phi -2\psi }}^2 \;, \end{aligned}$$

from which it follows that

$$\begin{aligned} F \;=\; \mathrm{im}\,\left(\mathrm{d}:{L}^{2}_{\psi -2\psi }\left(X;\wedge ^{k-1}T^*X\right)\dashrightarrow {L}^{2}_{\phi -\psi }\left(X;\wedge ^{k}T^*X\right)\right) \;. \end{aligned}$$

Step 6—Sobolev regularity of the solutions with compact support. We prove that, for every $\alpha \in {L}^{2}\left(X;\wedge ^{k-1}T^*X\right)$ with compact support, if $\mathrm{d}\alpha \in {L}^{2}\left(X;\wedge ^{k}T^*X\right)$ and $\mathrm{d}^*_{0,\,0}\alpha \in {L}^{2}\left(X;\wedge ^{k-2}T^*X\right)$, then $\alpha \in {W}^{1, 2}\left(X;\wedge ^{k-1}T^*X\right)$. Indeed, take $\left\{ \Phi _\varepsilon \right\} _{\varepsilon \in \mathbb R }$ a family of positive mollifiers and, for every $\varepsilon \in \mathbb R$, consider $\alpha *\Phi _\varepsilon \in \mathcal C ^\infty _{{c}}\left(X;\wedge ^{k-1}T^*X\right)$; by Remark 2.4 with $\phi :=0$ and $\psi :=0$, we get that, for any multi-index $I$ such that $|I|=k-1$ and for any $\ell \in \left\{ 1,\ldots ,n\right\}$,

$$\begin{aligned} \int \limits _X\left|\frac{\partial \left(\alpha _I*\Phi _\varepsilon \right)}{\partial x^\ell }\right|^2\,\mathrm{vol}\;\le \; C\cdot \left(\left\Vert{\mathrm{d}^*_{0,\,0} \left(\alpha *\Phi _\varepsilon \right)}\right\Vert_{{L}^2}^2+\left\Vert{\mathrm{d}\left(\alpha *\Phi _\varepsilon \right)}\right\Vert_{{L}^2}^2\right) \;, \end{aligned}$$

where $C\!:=:\!C(k,n)$ is a constant depending just on $k$ and $n$; since, for every multi-index $I$ such that $|I|=k-1$, and for every $\ell \in \left\{ 1,\ldots ,n\right\}$, it holds that $\lim _{\varepsilon \rightarrow 0}\int _X\left|\frac{\partial \left(\alpha _I*\Phi _\varepsilon \right)}{\partial x^\ell } - \frac{\partial \alpha _I}{\partial x^\ell }\right|^2\,\mathrm{vol}=\lim _{\varepsilon \rightarrow 0}\left\Vert{\mathrm{d}^*_{0,\,0}\left(\alpha *\Phi _\varepsilon \right) - \mathrm{d}^*_{0,\,0} \alpha }\right\Vert_{{L}^2}=\lim _{\varepsilon \rightarrow 0}\left\Vert{\mathrm{d}{\left(\alpha *\Phi _\varepsilon \right) - \mathrm{d}\alpha } }\right\Vert_{{L}^2}=0$, we get that

$$\begin{aligned} \int \limits _X\left|\frac{\partial \alpha _I}{\partial x^\ell }\right|^2\,\mathrm{vol}\;\le \; C\cdot \left(\left\Vert{\mathrm{d}^*_{0,\,0} \alpha }\right\Vert_{{L}^2}^2+\left\Vert{\mathrm{d}\alpha }\right\Vert_{{L}^2}^2\right) \;, \end{aligned}$$

proving the claim.

Step 7—Regularization of the solution. By Step 5, if $\eta \in \mathcal C ^\infty \left(X;\wedge ^{k}T^*X\right)$ is such that $\mathrm{d}\eta =0$, then the equation $\mathrm{d}\alpha =\eta$ has a solution $\alpha \in {L}^{2}_{\text{ loc}}\left(X;\wedge ^{k-1}T^*X\right)$; we prove that actually $\alpha \in \mathcal C ^\infty \left(X;\wedge ^{k-1}T^*X\right)$.

Note that we may suppose that the solution $\alpha \in {L}^{2}_{\text{ loc}}\left(X;\wedge ^{k-1}T^*X\right)$ satisfies

$$\begin{aligned} \alpha \;\in \; \left(\ker \mathrm{d}\right)^{\perp _{{L}^{2}_{\text{ loc}}\left(X;\wedge ^{k-1}T^*X\right)}} \;=\; \overline{\mathrm{im}\,\mathrm{d}^*_{0,\,0}} \;=\; \mathrm{im}\,\mathrm{d}^*_{0,\,0} \;\subseteq \; \ker \mathrm{d}^*_{0,\,0} \;; \end{aligned}$$

hence, $\alpha$ satisfies the system of differential equation

$$\begin{aligned} \left\{ \begin{array}{rcl} \mathrm{d}\alpha&= \eta \\ \mathrm{d}^*_{0,\,0} \alpha&= 0 \end{array} \right. \;. \end{aligned}$$

We prove, by induction on $s\in \mathbb N$, that $\alpha \in {W}^{s, 2}_\mathrm{loc }\left(X;\wedge ^{k-1}T^*X\right)$ for every $s\in \mathbb N$. Indeed, we have by Step 5 that $\alpha \in {W}^{0, 2}_\mathrm{loc }\left(X;\wedge ^{k-1}T^*X\right)={L}^{2}_{\text{ loc}}\left(X;\wedge ^{k-1}T^*X\right)$. Suppose now that $\alpha \in {W}^{s, 2}_\mathrm{loc }\left(X;\wedge ^{k-1}T^*X\right)$ and prove that $\alpha \in {W}^{s+1, 2}_\mathrm{loc }\left(X;\wedge ^{k-1}T^*X\right)$. Clearly, $\eta \in \mathcal C ^\infty \left(X;\wedge ^{k}T^*X\right)\subseteq {W}^{\sigma , 2}_\mathrm{loc }\left(X;\wedge ^{k}T^*X\right)$ for every $\sigma \in \mathbb N$. Take $K$ a compact subset of $X$, and choose $\widehat{\chi }\in \mathcal C ^\infty _{{c}}\left(X;\mathbb R \right)$ such that $\mathrm{supp}\widehat{\chi }\supset K$. For any multi-index $L:=:\left(\ell _1,\ldots ,\ell _n\right)\in \mathbb N ^n$ such that $\ell _1+\cdots +\ell _n=s$, being

$$\begin{aligned} \mathrm{d}\left(\widehat{\chi }\cdot \frac{\partial ^{s}\alpha }{\partial ^{\ell _1} x^1\cdots \partial ^{\ell _n}x^n}\right) = \mathrm{d}\widehat{\chi }\wedge \frac{\partial ^{s}\alpha }{\partial ^{\ell _1} x^1\cdots \partial ^{\ell _n}x^n}\!+\!\widehat{\chi }\cdot \frac{\partial ^{s}\eta }{\partial ^{\ell _1} x^1\cdots \partial ^{\ell _n}x^n} \in {L}^{2}\left(K;\wedge ^{k}T^*X\right) \end{aligned}$$

and

$$\begin{aligned} \mathrm{d}^*_{0,\,0}\left(\widehat{\chi }\cdot \frac{\partial ^{s}\alpha }{\partial ^{\ell _1} x^1\ldots \partial ^{\ell _n}x^n}\right)&= - \widetilde{\sum _{\begin{matrix} |J|=k-1\\ |I|=k \end{matrix}}}\sum _{\ell =1}^{n}\mathrm{sign}\left(\begin{array}{c} \ell J \\ I\end{array}\right)\,\frac{\partial \widehat{\chi }}{\partial x^\ell }\cdot \frac{\partial ^{s}\alpha _I}{\partial ^{\ell _1} x^1\ldots \partial ^{\ell _n}x^n} \mathrm{d}x^J\\&\in {L}^{2}\left(K;\wedge ^{k-2}T^*X\right) \end{aligned}$$

we get that $\widehat{\chi }\cdot \frac{\partial ^{s}\alpha }{\partial ^{\ell _1} x^1\ldots \partial ^{\ell _n}x^n}\in {W}^{1, 2}\left(K;\wedge ^{k-1}T^*X\right)$, that is, $\alpha \in {W}^{s+1, 2}\left(K;\wedge ^{k-1}T^*X\right)$. Hence, $\alpha \in {W}^{s+1, 2}_\mathrm{loc }\left(X;\wedge ^{k-1}T^*X\right)$. Since ${W}^{\sigma , 2}_\mathrm{loc }\left(X;\wedge ^{k-1}T^*X\right) \hookrightarrow \mathcal C ^m\left(X;\wedge ^{k-1}T^*X\right)$ for every $0\le m< \sigma -\frac{n}{2}$, see [4, Corollary 7.11], we get that $\alpha \in \mathcal C ^\infty \left(X;\wedge ^{k-1}T^*X\right)$, concluding the proof of the theorem. $\square$