Solutions of the $\bar \partial $-equation on Stein and on K\"ahler manifold with compact support

TL;DR

This paper investigates solutions to the $ar ext{d}$-equation on Stein and K"ahler manifolds, providing $L^{r}$ and Sobolev estimates for solutions with compact support, and linking these to Poisson and Hodge Laplacian estimates.

Contribution

It establishes new $L^{r}$ and Sobolev estimates for $ar ext{d}$-equation solutions with compact support on Stein and K"ahler manifolds, extending previous results and improving Andreotti-Grauert type theorems.

Findings

Existence of solutions with compact support in Stein manifolds for $L^{r}$ $ar ext{d}$-closed forms.

Estimates on solutions to the Poisson equation with compact support in K"ahler manifolds.

Connection between $ar ext{d}$-equation solutions and Hodge Laplacian in K"ahler geometry.

Abstract

We study the $\bar \partial $-equation first in Stein manifold then in complete K\"ahler manifolds. The aim is to get $L^{r}$ and Sobolev estimates on solutions with compact support. In the Stein case we get that for any $(p,q)$-form $\omega $ in $L^{r}$ with compact support and $\bar \partial $-closed there is a $(p,q-1)$-form $u$ in $W^{1,r}$ with compact support and such that $\bar \partial u=\omega .$ In the case of K\"ahler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $\bar \partial $ equation is done by a classical theorem stating that the Hodge laplacian is twice the $\bar \partial $ (or Kohn) Laplacian in a K\"ahler manifold. This uses and improves, in special cases, our result on Andreotti-Grauert type theorem.

Equations

$$(\alpha ,\beta )dm:=\alpha \wedge \ast \bar{\beta };{\left|\alpha \right|}^{2}dm:=\alpha \wedge \overline{\ast \alpha ,}$$

$$⟨\alpha ,\beta ⟩={\int }_{\Omega }\alpha \wedge \overline{\ast \beta }.$$

$$\forall \alpha \in L_{n-p}^{r^{\prime }}(\Omega ,\eta ),\left|⟨u,\ast \alpha ⟩\right|\le C{\Vert \alpha \Vert }_{L^{r^{\prime }}(\Omega ,\eta )},$$

$$\forall \alpha \in L_{n-p}^{r^{\prime }}(\Omega ,\eta ),\exists \phi \in W_{n-p}^{2,r^{\prime }}(\Omega ),\Delta \phi =\alpha ::{\Vert \phi \Vert }_{W^{2,r^{\prime }}(\Omega )}\lesssim {\Vert \alpha \Vert }_{L^{r^{\prime }}(\Omega )}.$$

$$\forall \phi \in {\mathcal{D}}_{n-p}(\Omega ),\ll \phi ,\omega \gg =\ll \Delta \phi ,u_k\gg \underset{k\to \infty }{\to }\ll \Delta \phi ,u\gg \Rightarrow \ll \Delta \phi ,u\gg =\ll \phi ,\omega \gg$$

$${\Vert u\Vert }_{W^{2,r}(B^{\prime }x)}\le c_1{\Vert \Delta u\Vert }_{L^r(B_x)}+c_{2}R(x{)}^{-2}{\Vert u\Vert }_{L^r(B_x)}.$$

$${\Delta }_{\bar{\partial }}v=(\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial })v=\omega$$

Full text

In the Stein case we get that for any $(p,q)$-form $\omega$ in $L^{r}$ with compact support and $\bar{\partial}$-closed there is a $(p,q-1)$-form $u$ in $W^{1,r}$ with compact support and such that $\bar{\partial}u=\omega.$

In the case of Kähler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $\bar{\partial}$ equation is done by a classical theorem stating that the Hodge laplacian is twice the $\bar{\partial}$ (or Kohn) Laplacian in a Kähler manifold.

This uses and improves, in special cases, our result on Andreotti-Grauert type theorem.

1. 1 Introduction.

2. 2 On estimates for the $\displaystyle\bar{\partial}$ equation in complex manifolds.

3. 2.1 Weak $r$-regularity.

4. 2.2 On an improvement of regularity.

5. 3 Hodge laplacian on riemannian manifolds.

6. 4 Solution of the Poisson equation with compact support.

7. 4.1 Weighted $L^{r}$ spaces.

8. 5 Kähler manifold and Kohn laplacian.

9. 6 Appendix.

1. Introduction.

The study of $L^{r}$ solutions for the $\bar{\partial}$ equation is a long standing subject. By use of PDE methods, L. Hörmander [Hörmander, 1994] get his famous $L^{2}$ estimates, we shall still use here.

The next results were obtained by the use of solving kernels: Grauert-Lieb [Grauert and Lieb, 1970], Henkin [Henkin, 1970], Ovrelid [Ovrelid, 1971], Skoda [Skoda, 1976], Krantz [Krantz, 1976], in the case of strictly pseudo-convex domains with $\displaystyle{\mathcal{C}}^{\infty}$ smooth boundary in $\displaystyle{\mathbb{C}}^{n},$ with the exception of Kerzman [Kerzman, 1971] who worked in the case of $\displaystyle(0,1)$ forms in strictly pseudo-convex domains with $\displaystyle{\mathcal{C}}^{4}$ smooth boundary in Stein manifolds.

The case of smooth solutions with compact support goes to the work of Andreotti and

Grauert [Andreotti and Grauert, 1962]. Our aim here is to study the same problem but with $L^{r}$ estimates, as we did in [Amar, 2019].

In a first part, we recall some results on solutions of the $\bar{\partial}$ equation in complex manifolds. Then, modifying a result by C. Laurent-Thiébaut [Laurent-Thiébaut, 2015], we prove that:

Corollary 1.1**.**

Let $X$ be a complex manifold. Let $S$ be a $(p,q-1)$-current in $L_{p,q-1}^{r}(X)$ with compact support $W$ in $X.$ Suppose that $W\subset\Omega_{1}\Subset\Omega_{2},$ where $\Omega_{j},\ j=1,2$ are relatively compact pseudo-convex open sets with smooth ${\mathcal{C}}^{\infty}$ boundary in $X$ and such that there is a strictly pluri-subharmonic function $\rho_{1}$ in ${\mathcal{C}}^{3}(\bar{\Omega}_{1}).$

Moreover suppose that $\omega:=\bar{\partial}S$ is also in $L_{p,q}^{r}(X).$ Let $U$ be any neighborhood of $W$ contained in $\Omega_{1}.$ Then there is a $(p,q-1)$-current $u$ with compact support in $U$ such that $\bar{\partial}u=\omega$ and $u\in W_{p,q-1}^{1,r}(U).$

Then we get the following result which seems to end a question Guiseppe Tomassini ask me almost ten years ago (see  [Amar and Mongodi, 2014] and  [Amar, 2012]).

Let $M$ be a complex manifold and $\Lambda_{p,q}(\bar{M})$ the set of $(p,q)$-forms ${\mathcal{C}}^{\infty}$ in $\bar{M}.$

Recall that the Kohn laplacian $\Delta_{\bar{\partial}}$ is defined as:

$\forall\alpha\in L_{p,q}^{2}(M),\ \Delta_{\bar{\partial}}\alpha:=\bar{\partial}\bar{\partial}^{*}\alpha+\bar{\partial}^{*}\bar{\partial}\alpha.$

First let us define, as in p. 278 in [Kohn, 1973], the harmonic fields:

${\mathcal{H}}_{p,q}:=\{h\in\Lambda_{p,q}(\bar{M})::\bar{\partial}h=\bar{\partial}^{*}h=0\}.$

Then we have:

Theorem 1.2**.**

Let $X$ be a Stein manifold and $\omega$ be a $(p,q)$ form in $L^{r}(X),\ r>1$ with compact support in $X.$ Suppose that $\omega$ is such that:

$\bullet$* if $1\leq q<n,\ \bar{\partial}\omega=0$;*

$\bullet$* if $q=n,\ \forall V\subset X,\ \mathop{\rm Supp}\nolimits\omega\subset V,\ \omega\perp{\mathcal{H}}_{n-p,0}(V)$​​​​ .*

Then there is a $(p,q-1)$ form $u$ in $W^{1,r}(X)$ with compact support in $X$ such that $\bar{\partial}u=\omega$ as distributions and $\displaystyle{\left\|{u}\right\|}_{W^{1,r}(\Omega)}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}.$

In a second part we study this problem in a Kähler manifold. The method is completely different: we first study $L^{r}$ solutions with compact support for the Poisson equation in a riemannian manifold and we use the link done by the following classical theorem relying the Hodge laplacian and the $\bar{\partial}$ (or Kohn) laplacian. See for instance C. Voisin’s book [Voisin, 2002].

Theorem 1.3**.**

Let $(X,\kappa)$ be a kählerian manifold. Let $\Delta,\ \Delta_{\partial},\ \Delta_{\bar{\partial}}$ the laplacians associated to $d,\ \partial,\ \bar{\partial}$ respectively. Then we have the relations:

$\Delta=2\Delta_{\partial}=2\Delta_{\bar{\partial}}.$**

We get, with this time ${\mathcal{H}}_{q}(\Omega)=\{h\in\Lambda_{q}(\bar{\Omega})::\Delta h=0\}$:

Theorem 1.4**.**

Let $(X,\omega)$ be a complete kählerian manifold. Let $\Omega$ be a relatively compact domain in $X.$ Let $\displaystyle\omega\in L_{p,q}^{r}(\Omega),\ \bar{\partial}\omega=0$ in $\Omega$ and $\omega$ compactly supported in $\Omega.$ Suppose moreover that $\displaystyle\omega\perp{\mathcal{H}}_{2n-p-q}(\Omega).$

Then there is a $\displaystyle u\in W_{p,q-1}^{1,r}(\Omega)$ with compact support in $\Omega$ and such that $\bar{\partial}u=\omega.$

This result seems weaker than the previous one because we need that $\displaystyle\omega\perp{\mathcal{H}}_{2n-p-q}(\Omega),$ but, unless $X$ is weakly pseudo-convex, a compact set is not contained in a pseudo-convex one in general. Hence the method used for the proof of Theorem 1.2 cannot apply here.

This work is presented the following way.

For the first part:

$\bullet$ In Section 2 we recall results on estimates for the $\bar{\partial}$ equation.

$\bullet$ In Subsection 2.1 we recall the notion of $r$-regularity and its consequence in term of solution of the $\bar{\partial}$ equation with compact support.

$\bullet$ In Subsection 2.2 we show that, under some circumstances, the regularity of solutions of the $\bar{\partial}$ equation may increases.

This part is directly coming from a work of C. Laurent-Thiébaut [Laurent-Thiébaut, 2015].

For the second part:

$\bullet$ In Section 3 we start with the Hodge laplacian on a riemannian manifold and we recall results we get in [Amar, 2018] concerning the Poisson equation.

$\bullet$ In Section 4 we study the solutions of the Poisson equation with compact support and we prove, using weighted estimates:

Theorem 1.5**.**

Let $X$ be a complete oriented riemannian manifold. Let $\Omega$ be a relatively compact domain in $X.$ Let $\omega\in L^{r}_{p}(\Omega)$ with compact support in $\Omega$ and such that $\omega$ is orthogonal to the harmonic $p$-forms $\displaystyle{\mathcal{H}}_{p}(\Omega).$ Then there is a $p$-form $u\in W^{2,r}_{p}(\Omega)$ with compact support in $\Omega$ such that $\Delta u=\omega$ as distributions and $\displaystyle{\left\|{u}\right\|}_{W^{2,r}_{p}(\Omega)}\leq C{\left\|{\omega}\right\|}_{L^{r}_{p}(\Omega)}.$

$\bullet$ In Section 5, using equality of the laplacians, we prove Theorem 1.4.

$\bullet$ Finally in the Appendix we prove certainly known results on the duality $L^{r}-L^{r^{\prime}}$ for $(p,q)$-forms in a complex manifold we needed.

2. On estimates for the $\displaystyle\bar{\partial}$

equation in complex manifolds.

Here we shall be interested in strictly $c$-convex (s.c.c. for short) domains $D$ in a complex manifold. Such a domain is defined by a function $\rho$ of class $\displaystyle{\mathcal{C}}^{3}$ in a neighbourhood $U$ of $\bar{D}$ and such that $\displaystyle i\partial\bar{\partial}\rho$ has at least $\displaystyle n-c+1$ strictly positive eigenvalues in $U.$

We have the following Theorem 1.1 from [Amar, 2017]:

Theorem 2.1**.**

Let $\Omega$ be a Stein manifold of dimension $n$ and a s.c.c. domain $D$ such that $D$ is relatively compact with smooth $\displaystyle{\mathcal{C}}^{3}$ boundary in $\Omega.$ Let $\omega$ be a $\displaystyle(p,q)$ form in $\displaystyle L^{r}_{p,q}(D),\ \bar{\partial}\omega=0$ with $\displaystyle 1<r<2n+2,\ c\leq q\leq n.$ Then there is a $\displaystyle(p,q-1)$ form $u$ in $\displaystyle L^{s}(D),$ with $\displaystyle\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2n+2},$ such that $\displaystyle\bar{\partial}u=\omega.$

If $\omega$ is in $\displaystyle L^{r}_{p,q}(D),\ \bar{\partial}\omega=0$ with $\displaystyle r\geq 2n+2,\ c\leq q\leq n,$ then there is a $\displaystyle(p,q-1)$ form $u$ in $\displaystyle\Lambda_{(p,q-1)}^{\epsilon}(\bar{D})$ such that $\displaystyle\bar{\partial}u=\omega$ with $\displaystyle\epsilon=\frac{1}{2}-\frac{n+1}{r}.$

The spaces $\displaystyle\Lambda_{(p,q-1)}^{\epsilon}(\bar{D})$ are the (isotropic) Lipschitz spaces of order $\epsilon$ and we set $\displaystyle\ \Lambda_{(p,q-1)}^{0}(\bar{D}):=L_{(p,q-1)}^{\infty}(D).$

This theorem has the obvious corollary:

Corollary 2.2**.**

Let $\Omega$ be a complex manifold of dimension $n$ and a domain $D$ relatively compact with smooth $\displaystyle{\mathcal{C}}^{3}$ boundary in $\Omega.$ Suppose moreover that $D:=\{\rho<0\},$ where $\rho$ is a strictly pluri-subharmonic function in ${\mathcal{C}}^{\infty}(\bar{D})$ with $\left|{\partial\rho}\right|>0$ on $\partial D.$ Let $\omega$ be a $\displaystyle(p,q)$ form in $\displaystyle L^{r}_{p,q}(D),\ \bar{\partial}\omega=0$ with $\displaystyle 1<r<2n+2,\ 1\leq q\leq n.$ Then there is a $\displaystyle(p,q-1)$ form $u$ in $\displaystyle L^{s}(D),$ with $\displaystyle\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2n+2},$ such that $\displaystyle\bar{\partial}u=\omega,$ with ${\left\|{u}\right\|}_{s}\leq C{\left\|{\omega}\right\|}_{r}.$

If $\omega$ is in $\displaystyle L^{r}_{p,q}(D),\ \bar{\partial}\omega=0$ with $\displaystyle r\geq 2n+2,\ 1\leq q\leq n,$ then there is a $\displaystyle(p,q-1)$ form $u$ in $\displaystyle\Lambda_{(p,q-1)}^{\epsilon}(\bar{D})$ such that $\displaystyle\bar{\partial}u=\omega$ with $\displaystyle\epsilon=\frac{1}{2}-\frac{n+1}{r}$ and ${\left\|{u}\right\|}_{\Lambda^{\epsilon}}\leq C{\left\|{\omega}\right\|}_{r}.$

Proof.

Take a convex increasing function $\chi$ on ${\mathbb{R}}^{-}$ such that $\displaystyle\chi(t)\rightarrow\infty$ when $t\rightarrow 0.$ The function $\varphi(z):=\chi\circ\rho(z)$ is still strictly pluri-subharmonic on $D$ and exhausting. So $D$ is a Stein manifold by Theorem 5.2.10 in [Hörmander, 1994]. A strictly pseudo-convex domain is a s.c.c. domain with $c=1,$ so we can apply Theorem 2.1. $\hfill\blacksquare$

Corollary 2.3**.**

Let $\Omega$ be a complex manifold of dimension $n$ and a domain $D$ relatively compact with smooth $\displaystyle{\mathcal{C}}^{3}$ boundary in $\Omega.$ Suppose moreover that $D:=\{\rho<0\},$ where $\rho$ is a strictly pluri-subharmonic function in ${\mathcal{C}}^{3}(\bar{D})$ with $\left|{\partial\rho}\right|>0$ on $\partial D.$ Let $\omega$ be a $\displaystyle(p,q)$ form in $\displaystyle L^{r}_{p,q}(D),\ \bar{\partial}\omega=0$ with $\displaystyle 1<r<\infty,\ 1\leq q\leq n.$ Then there is a $\displaystyle(p,q-1)$ form $u$ in $\displaystyle L^{r}(D),$ with ${\left\|{u}\right\|}_{r}\leq C{\left\|{\omega}\right\|}_{r},$ such that $\displaystyle\bar{\partial}u=\omega.$

Proof.

Because $D$ is relatively compact, if $\displaystyle u\in L^{s}(D)$ for $s\geq r$ then $\displaystyle u\in L^{r}(D).$ Then the Corollary 2.2 gives the result. $\hfill\blacksquare$

2.1. Weak $r$-regularity.

We shall need the definition, see [Amar, 2019]:

Definition 2.4**.**

Let $X$ be a complex manifold equipped with a Borel $\sigma$-finite measure $dm$ and $\Omega$ a domain in $X$; let $\displaystyle r\in[1,\ \infty],$ we shall say that $\Omega$ is $r$ regular if for any $p,q\in\{0,...,n\},\ q\geq 1,$ there is a constant $C=C_{p,q}(\Omega)$ such that for any $(p,q)$ form $\omega,\ \bar{\partial}$ closed in $\Omega$ and in $L^{r}(\Omega,dm)$ there is a $(p,q-1)$ form $u\in L^{r}(\Omega,dm)$ such that $\bar{\partial}u=\omega$ and $\ {\left\|{u}\right\|}_{L^{r}(\Omega)}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}.$

We shall say that $\Omega$ is weakly $r$ regular if for any compact set $K\Subset\Omega$ there are $3$ open sets $\Omega_{1},\Omega_{2},\Omega_{3}$ such that $K\Subset\Omega_{3}\subset\Omega_{2}\subset\Omega_{1}\subset\Omega_{0}:=\Omega$ and $3$ constants $C_{1},C_{2},C_{3}$ such that:

$\displaystyle\forall j=0,1,2,\ \forall p,q\in\{0,...,n\},\ q\geq 1,\ \forall\omega\in L_{p,q}^{r}(\Omega_{j},dm),\ \bar{\partial}\omega=0,$

$\displaystyle\exists u\in L_{p,q-1}^{r}(\Omega_{j+1},dm),\ \bar{\partial}u=\omega$

and $\ {\left\|{u}\right\|}_{L^{r}(\Omega_{j+1})}\leq C_{j+1}{\left\|{\omega}\right\|}_{L^{r}(\Omega_{j})}.$

I.e. we have a $3$ steps chain of resolution.

in [Amar, 2019] we prove the Theorem 3.5, p. 6, where $L^{r,c}(\Omega)$ means that the form is in $L^{r}(\Omega)$ with compact support in $\Omega$:

Theorem 2.5**.**

Let $\Omega$ be a weakly $r^{\prime}$ regular domain in a complex manifold and $\omega$ be a $(p,q)$ form in $L^{r,c}(\Omega),\ r>1.$ Suppose that $\omega$ is such that:

$\bullet$* if $1\leq q<n,\ \bar{\partial}\omega=0$;*

$\bullet$* if $q=n,\ \forall V\subset\Omega,\ \mathop{\rm Supp}\nolimits\omega\subset V,\ \omega\perp{\mathcal{H}}_{n-p}(V)$​​​​ .*

Then there is a $C>0$ and a $(p,q-1)$ form $u$ in $L^{r,c}(\Omega)$ such that $\bar{\partial}u=\omega$ as distributions and $\displaystyle{\left\|{u}\right\|}_{L^{r}(\Omega)}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}.$

In fact in [Amar, 2019] we made the general assumption that our complex manifold $X$ is Stein, just to be sure that any compact set is in a weakly regular domain, because we proved in [Amar, 2019] that a Stein manifold is weakly $r$-regular. The example 2.8 prove that this is not the case in general. But in Theorem 2.5, the proof works for $X$ being just a complex manifold.

As a corollary we get:

Corollary 2.6**.**

Let $\Omega$ be a complex manifold of dimension $n$ and a domain $D$ relatively compact with smooth $\displaystyle{\mathcal{C}}^{3}$ boundary in $\Omega.$ Suppose moreover that $D:=\{\rho<0\},$ where $\rho$ is a strictly pluri-subharmonic function in ${\mathcal{C}}^{3}(\bar{D})$ with $\left|{\partial\rho}\right|>0$ on $\partial D.$ Then $D$ is $r^{\prime}$-regular. Moreover suppose that $\omega$ is a $(p,q)$ form in $L^{r,c}(D),\ r>1$ such that:

$\bullet$* if $1\leq q<n,\ \bar{\partial}\omega=0$;*

$\bullet$* if $q=n,\ \forall V\subset D,\ \mathop{\rm Supp}\nolimits\omega\subset V,\ \omega\perp{\mathcal{H}}_{n-p}(V)$​​​​ .*

Then there is a $C>0$ and a $(p,q-1)$ form $u$ in $L^{r,c}(D)$ such that $\bar{\partial}u=\omega$ as distributions and $\displaystyle{\left\|{u}\right\|}_{L^{r}(D)}\leq C{\left\|{\omega}\right\|}_{L^{r}(D)}.$

Proof.

The Corollary 2.3 gives that $D$ is $r^{\prime}$-regular for any $1<r^{\prime}<\infty.$ Hence we can apply Theorem 2.5 to the domain $D.$ $\hfill\blacksquare$

2.2. On an improvement of regularity.

This section is coming from C. Laurent-Thiébaut [Laurent-Thiébaut, 2015], Proposition 1.4 p. 257.

To use Theorem 4 in [Beals et al., 1987] we need a compact complex manifold with a smooth ${\mathcal{C}}^{\infty}$ boundary having property $Z(q)$ and a $(p,q)$-form in the range of the Kohn laplacian, which means that the form must be orthogonal to the harmonic fields.

In [Laurent-Thiébaut, 2015] the author used twice this Theorem without any references to these two conditions. Because I was unable to understand why they are fulfilled, I prove here a weaker result fitting well enough with my purpose.

The nice idea to work with exact forms the regularity of which being increased is due to C. Laurent-Thiébaut.

Let us define, as in p. 278 in [Kohn, 1973], the harmonic fields:

${\mathcal{H}}_{p,q}:=\{h\in\Lambda_{p,q}(\bar{M})::\bar{\partial}h=\bar{\partial}^{*}h=0\}.$

Then we have in [Kohn, 1973], that:

$\bullet$ ${\mathcal{H}}_{p,q}=\mathrm{k}\mathrm{e}\mathrm{r}\Delta_{\bar{\partial}};$

$\bullet$ ${\mathcal{H}}_{p,q}$ is a finite dimensional subspace of $\Lambda_{p,q}(\bar{M}).$

We shall denote $H$ the orthogonal projection of $L_{p,q}^{2}(M)$ onto ${\mathcal{H}}_{p,q}.$

And we have a Hodge decomposition, eq. 2.26, p. 278 in [Kohn, 1973]:

$\forall\alpha\in L_{p,q}^{2}(M),\ \alpha=\bar{\partial}\bar{\partial}^{*}\alpha+\bar{\partial}^{*}\bar{\partial}\alpha+H\alpha.$

Moreover if $\bar{\partial}\alpha=0$ and $\alpha\perp{\mathcal{H}}_{n-p,n-q}$ then, eq. 2.27, p. 278 in [Kohn, 1973]:

$\alpha=\bar{\partial}\bar{\partial}^{*}N\alpha$ and $\varphi=\bar{\partial}^{*}N\alpha$ is the unique solution of the equation $\alpha=\bar{\partial}\varphi$ orthogonal to ${\mathcal{H}}_{n-p,n+1-q}.$

Recall also that a pseudo-convex set with smooth ${\mathcal{C}}^{\infty}$ boundary has the $Z(q)$ property for any $q\geq 1.$ This is why we shall use mainly this notion.

Theorem 2.7**.**

Let $X$ be a complex manifold and $q\geq 1.$ Let $S$ be a $(p,q-1)$-current in $L_{p,q-1}^{r}(X).$ with compact support $W$ in $X.$ Suppose that $W\subset\Omega_{1}\Subset\Omega_{2},$ where $\Omega_{j},\ j=1,2$ are relatively compact pseudo-convex open sets with smooth ${\mathcal{C}}^{\infty}$ boundary in $X$ and such that $S\perp{\mathcal{H}}_{n-p,n+1-q}(\Omega_{1}).$

Moreover suppose that $\omega:=\bar{\partial}S$ is also in $L_{p,q}^{r}(X).$ Let $U$ be any neighborhood of $W$ contained in $\Omega_{1}.$ Then there is a $(p,q-1)$-current $u$ with compact support in $U$ such that $\bar{\partial}u=\omega$ and $u\in W_{p,q-1}^{1,r}(U).$

Proof.

As done in [Laurent-Thiébaut, 2015] we shall use Theorem 4 in [Beals et al., 1987].

We have to see that $\omega$ is orthogonal to ${\mathcal{H}}_{n-p,n-q}(\Omega_{2}),$ and this is a necessary condition (see [Kohn, 1973]). Because $h\in{\mathcal{H}}_{n-p,n-q}(\Omega_{2})$ implies that $h\in\Lambda_{n-p,n-q}(\bar{\Omega}_{2}),$ the scalar product ${\left\langle{\omega,h}\right\rangle}$ is well defined and we have:

$\forall h\in{\mathcal{H}}_{n-p,n-q}(\Omega_{2}),\ {\left\langle{\omega,h}\right\rangle}_{\Omega_{2}}={\left\langle{\bar{\partial}S,h}\right\rangle}_{\Omega_{2}}={\left\langle{S,\bar{\partial}^{*}h}\right\rangle}_{\Omega_{2}}=0.$

Hence $\omega$ is in the range of $\Delta_{\bar{\partial}},$ so noting $N$ as usual the inverse of $\Delta_{\bar{\partial}},$ we get that $N$ is well defined on $\omega$ and Theorem 4 in [Beals et al., 1987] gives that there is a $(p,q-1)$-current $g_{0}\in W_{p,q-1}^{1/2,r}(\Omega_{2})$ such that $\bar{\partial}g_{0}=\omega.$ Moreover, on any compact set $K\Subset\Omega_{2},$ we have $g_{0}\in W_{p,q-1}^{1,r}(K)$ because on $K$ any vectors field can be extended to $\Omega_{2}$ as an admissible vectors field.

In particular we can choose $K:=\bar{\Omega}_{1},$ so we have that $g_{0}\in W_{p,q-1}^{1,r}(\bar{\Omega}_{1}).$

Now we have $\bar{\partial}(S-g_{0})=\omega-\bar{\partial}g_{0}=0$ in $\Omega_{2}.$

If $q=1,$ then $S-g_{0}$ is holomorphic in $\Omega_{2},$ hence ${\mathcal{C}}^{\infty}$ in $\bar{\Omega}_{1},$ so we have directly that $S\in W_{p,q-1}^{1,r}(\bar{\Omega}_{1}).$

Suppose now that $q\geq 2.$ Because $\Omega_{1}\backslash U$ is not in general pseudo-convex even if $\Omega_{1}$ is, we cannot end the proof as in [Laurent-Thiébaut, 2015].

So again we want to apply Theorem 4 from  [Beals et al., 1987] to $\omega^{\prime}:=S-g_{0}$ in $\Omega_{1}.$ We have to verify that $\omega^{\prime}$ is orthogonal to ${\mathcal{H}}_{n-p,n+1-q}(\Omega_{1}).$

But recall that in $\Omega_{2},\ g_{0}:=\bar{\partial}^{*}N\omega$ hence, because $\forall h\in{\mathcal{H}}_{n-p,n+1-q}(\Omega_{1})\Rightarrow h\in\mathrm{d}\mathrm{o}\mathrm{m}(\bar{\partial})\cap\mathrm{d}\mathrm{o}\mathrm{m}(\bar{\partial}^{*})$ and $\bar{\partial}h=0$:

${\left\langle{g_{0},h}\right\rangle}_{\Omega_{1}}={\left\langle{\bar{\partial}^{*}N\omega,h}\right\rangle}_{\Omega_{1}}={\left\langle{N\omega,\bar{\partial}h}\right\rangle}_{\Omega_{1}}=0.$

By assumption $S\perp{\mathcal{H}}_{n-p,n+1-q}(\Omega_{1})$ hence $\omega^{\prime}\perp{\mathcal{H}}_{n-p,n-q}(\Omega_{1}).$

So again there is a $(p,q-2)$-current $g_{1}\in W_{p,q-2}^{1/2,r}(\Omega_{1})$ such that $\bar{\partial}g_{1}=S-g_{0}.$ And again, on any compact set $K\Subset\Omega_{1},$ we have $g_{1}\in W_{p,q-2}^{1,r}(K).$

Let $\chi\in{\mathcal{C}}^{\infty}(X)$ such that $\chi=0$ near the support $W$ of $S,$ and $\chi=1$ in a neighborhood of $X\backslash U.$ Then the form $u:=g_{0}+\bar{\partial}(\chi g_{1})$ verifies:

$\bar{\partial}u=\bar{\partial}g_{0}=\omega$ because $\bar{\partial}^{2}=0.$

Now

$\bar{\partial}(\chi g_{1})=\bar{\partial}\chi\wedge g_{1}+\chi\bar{\partial}g_{1}=\bar{\partial}\chi\wedge g_{1}+\chi\bar{\partial}g_{1}=\bar{\partial}\chi\wedge g_{1}+\chi(S-g_{0}).$

So we get, because $\bar{\partial}\chi=0$ and $S=0$ outside $U$

$\bar{\partial}(\chi g_{1})=-g_{0}$ outside $U$

hence $u=0$ outside $U.$ Hence $u$ has its support in $U.$

Now in $U$ we have $\chi=0$ near $W$ so

$\bar{\partial}(\chi g_{1})=\bar{\partial}\chi\wedge g_{1}-\chi g_{0}$

and, because $\chi\in{\mathcal{C}}^{\infty}(X)$ and $g_{0},g_{1}\in W_{p,q-1}^{1,r}(K)$ for any compact $K$ in $\Omega_{1},$ we get

$u=\bar{\partial}\chi\wedge g_{1}-\chi g_{0}(1-\chi)g_{0}\in W_{p,q-1}^{1,r}(K)$ for any compact $K$ in $\Omega_{1}.$ In particular, because $u$ has its support in $U,$ we get that $u\in W_{p,q-1}^{1,r}(X).$

The origin of this method of control of the support is in section 3.5, p. 9 of [Amar, 2012].

The proof is complete. $\hfill\blacksquare$

Let us see the following example.

Example 2.8**.**

There is a bounded open set with smooth boundary $\Omega$ in ${\mathbb{C}}^{n}$ and a compact set $K\subset\Omega$ such that there is no pseudo-convex set $D$ contained in $\Omega$ and containing $K.$

Proof.

Take a bounded open cooking pot as $\Omega$ in ${\mathbb{C}}^{2}$ and a compact one $K$ in $\Omega$ ( one can smoothed the boundaries) see Figure 1.

Rotate the picture around the vertical axis in ${\mathbb{R}}^{4}={\mathbb{C}}^{2}$ to get $\Omega.$

Suppose that there is a pseudo-convex set $D$ in $\Omega$ and containing $K.$ Take any holomorphic function $h$ in $D.$ Then $h$ is holomorphic in a neighborhood of the boundary of $K.$ By the Cartan-Thullen Theorem $h$ extends in the red part, hence outside $D,$ so $D$ is not a domain of holomorphy hence is not pseudo-convex. $\hfill\blacksquare$

Remark 2.9**.**

Because of Example 2.8, and the fact that an open set in ${\mathbb{C}}^{n}$ is a complex manifold, it seems difficult to get rid of the assumption that the support of $S$ must be in a pseudo-convex domain of $X.$

Also the condition that the forms we want to solve this way be in the range of $\Delta_{\bar{\partial}}$ is necessary.

Remark 2.10**.**

Because the Theorem of Beals and all. is valid for domains having $Z(q)$ boundary, it is enough to suppose that $\Omega_{2}$ and $\Omega_{1}$ be of type $Z(q)$ and $Z(q-1).$ The condition to belong to the range of $\Delta_{K}$ being the same as for the pseudo-convex case, by Theorem 3.2.2 p. 57 and the results at the beginning of p. 51 in [Folland and Kohn, 1972].

The next corollary says that we can suppress the assumption $S\perp{\mathcal{H}}_{n-p,n+1-q}(\Omega_{1})$ provided that, on $\bar{\Omega}_{1},$ is defined a smooth strictly pluri-subharmonic function.

Corollary 2.11**.**

Let $X$ be a complex manifold. Let $S$ be a $(p,q-1)$-current in $L_{p,q-1}^{r}(X).$ with compact support $W$ in $X.$ Suppose that $W\subset\Omega_{1}\Subset\Omega_{2},$ where $\Omega_{j},\ j=1,2$ are pseudo-convex open sets with smooth ${\mathcal{C}}^{\infty}$ boundary in $X$ and such that there is a strictly pluri-subharmonic function $\rho_{1}$ in ${\mathcal{C}}^{3}(\bar{\Omega}_{1}).$

Moreover suppose that $\omega:=\bar{\partial}S$ is also in $L_{p,q}^{r}(X).$ Let $U$ be any neighborhood of $W$ contained in $\Omega_{1}.$ Then there is a $(p,q-1)$-current $u$ with compact support in $U$ such that $\bar{\partial}u=\omega$ and $u\in W_{p,q-1}^{1,r}(U).$

Proof.

In fact we shall prove that, for $q\geq 1,\ {\mathcal{H}}_{p,q}(\Omega_{1})=\{0\}.$ So let $h\in{\mathcal{H}}_{p,q}(\Omega_{1}).$ then $h\in\Lambda_{p,q}(\bar{\Omega}_{1}),\ \bar{\partial}h=\bar{\partial}^{*}h=0.$

Because $h\in\Lambda_{p,q}(\bar{\Omega}_{1})$ and $\bar{\Omega}_{1}$ is compact, we have $h\in L_{p,q}^{2}(\Omega_{1}).$

If $\Omega_{1}$ is strongly pseudo-convex i.e. $\Omega_{1}:=\{z\in X,\ \rho_{1}(z)<0\}$ with $\rho_{1}\in{\mathcal{C}}^{2}(\bar{\Omega})$ and the smallest eigenvalue of the form $\partial\bar{\partial}\rho_{1}$ is bounded below by $c_{\rho}>0$ by the continuity of $\partial\bar{\partial}\rho_{1}$ in $\bar{\Omega},$ we can apply Corollary 2.3 with $r=2.$

With just the existence of $\rho_{1}$ as in the statement of the theorem, we can apply a well known $L^{2}$ Theorem of [Hörmander, 1994]:

$\exists k\in L_{p,q-1}^{2}(\Omega_{1})::\bar{\partial}k=h.$

So we have, because $h\in\mathrm{d}\mathrm{o}\mathrm{m}(\bar{\partial}^{*})$ and $\bar{\partial}^{*}h=0,$

${\left\|{h}\right\|}^{2}={\left\langle{h,h}\right\rangle}={\left\langle{h,\bar{\partial}k}\right\rangle}={\left\langle{\bar{\partial}^{*}h,k}\right\rangle}=0.$

The proof is complete. $\hfill\blacksquare$

Remark 2.12**.**

Because the Theorem 3.4.10 p. 145 in [Hörmander, 1965] is valid for domains having $Z(q)$ boundary (called $a_{q}$ in [Hörmander, 1965]), we have the same kind of corollary for these domains, provided that the defining function $\varphi_{j}$ for $\Omega_{j}$ is defined in a neighborhood of $\bar{\Omega}_{j},$ verifies the condition $Z(q)$ outside of $\Omega_{j,c}:=\{z\in\Omega_{j},\ \varphi(z)<c\}$ for some $c>c_{0}$ and is exhausting in $\Omega_{j},\ j=1,2.$

So adding the results of Corollary 2.6 and of Theorem 2.7, we get:

Theorem 2.13**.**

Let $\Omega$ be a strictly pseudo-convex domain in a complex manifold and $\omega$ be a $(p,q)$ form in $L^{r}(\Omega),\ r>1$ with compact support in $\Omega.$ Suppose that $\omega$ is such that:

$\bullet$* if $1\leq q<n,\ \bar{\partial}\omega=0$;*

$\bullet$* if $q=n,\ \forall V\subset\Omega,\ \mathop{\rm Supp}\nolimits\omega\subset V,\ \omega\perp{\mathcal{H}}_{n-p}(V)$​​​​ .*

Then there is a $(p,q-1)$ form $u$ in $W^{1,r}(\Omega)$ with compact support in $\Omega$ such that $\bar{\partial}u=\omega$ as distributions and $\displaystyle{\left\|{u}\right\|}_{W^{1,r}(\Omega)}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}.$

And the natural corollary:

Corollary 2.14**.**

Let $X$ be a Stein manifold and $\omega$ be a $(p,q)$ form in $L^{r}(X),\ r>1$ with compact support in $X.$ Suppose that $\omega$ is such that:

$\bullet$* if $1\leq q<n,\ \bar{\partial}\omega=0$;*

$\bullet$* if $q=n,\ \forall V\subset X,\ \mathop{\rm Supp}\nolimits\omega\subset V,\ \omega\perp{\mathcal{H}}_{n-p}(V)$​​​​ .*

Then there is a $(p,q-1)$ form $u$ in $W^{1,r}(X)$ with compact support in $X$ such that $\bar{\partial}u=\omega$ as distributions and $\displaystyle{\left\|{u}\right\|}_{W^{1,r}(\Omega)}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}.$

Proof.

Because $X$ is a Stein manifold, for any compact set $K$ in $X$ there is a relatively compact strictly pseudo-convex set $\Omega$ containing $K.$ So we can apply Theorem 2.13. $\hfill\blacksquare$

3. Hodge laplacian on riemannian manifolds.

A riemannian manifold $(M,g)$ is a real, smooth manifold $M$ equipped with an inner product $g_{x}$ on the tangent space $T_{x}M$ at each point $x$ that varies smoothly from point to point in the sense that if $X$ and $Y$ are differentiable vector fields on $M,$ then $x\rightarrow g_{x}(X(x),Y(x))$ is a smooth function. The family $g_{x}$ of inner products is called a riemannian metric.

Let $X$ be a complete oriented riemannian manifold and $\Omega$ a relatively compact domain in $X.$

We shall denote by $\Lambda^{p}(\Omega)$ the set of ${\mathcal{C}}^{\infty}$ smooth $p$-forms in $\Omega$ and by $L^{r}_{p}(\Omega)$ its closure in the Lebesgue space $L^{r}(\Omega)$ with respect to the riemannian volume measure $dm$ on $X.$

We shall take the following notation from the book by C. Voisin [Voisin, 2002].

To a $p$-form $\alpha$ on $\Omega$ we associate its Hodge $*\ (n-p)$-form $*\alpha.$ This gives us a pointwise scalar product and a pointwise modulus:

[TABLE]

because $\displaystyle\alpha\wedge*\bar{\beta}$ is a $n$-form hence is a function time the volume form $dm.$

With the volume measure, we have a scalar product ${\left\langle{\alpha,\beta}\right\rangle}$ on $p$-forms such that$\displaystyle\int_{\Omega}{\left|{\alpha}\right|^{2}dm}<\infty.$ The link between these notions is given by [Voisin, 2002, Lemme 5.8, p. 119]:

[TABLE]

We shall define now $L_{p}^{r}(\Omega)$ to be the set of $p-$forms $\alpha$ defined on $\Omega$ such that

$\ {\left\|{\alpha}\right\|}_{L_{p}^{r}(\Omega)}^{r}:=\int_{\Omega}{\left|{\alpha(z)}\right|^{r}dm(z)}<\infty,$

where $\left|{\alpha}\right|$ is defined by (3.1).

As usual let $\displaystyle{\mathcal{D}}_{p}(\Omega)$ be the set of ${\mathcal{C}}^{\infty}\ p$-forms with compact support in $\Omega.$

On the manifold $M$ we have the exterior derivative $d$ on $p$-forms. To it we associate its formal adjoint $d^{*}$ defined by:

$\forall u\in L^{r}_{p}(\Omega),\ \forall\varphi\in{\mathcal{D}}_{p-1}(\Omega),\ {\left\langle{d^{*}u,\varphi}\right\rangle}:={\left\langle{u,d\varphi}\right\rangle}.$

Now we define the Hodge laplacian to be

$\Delta:=dd^{*}+d^{*}d.$

This operator sends $p$-form to $p$-form and is essentially self-adjoint. In case $p=0,$ i.e. on functions, $\Delta$ is the usual Laplace-Beltrami operator on $M.$

We proved the following theorem [Amar, 2018, Theorem 1.1], written here in the special case of the Hodge laplacian:

Theorem 3.1**.**

Let $(M,g)$ be a ${\mathcal{C}}^{\infty}$ smooth compact riemannian manifold without boundary. Let $\Delta:\Lambda_{p}\rightarrow\Lambda_{p}$ be the Hodge laplacian acting on the $p$-forms over $M.$ Let $\omega\in L^{r}_{p}(M)\cap\mathrm{(}\mathrm{k}\mathrm{e}\mathrm{r}\Delta\mathrm{)}^{\perp}$ with $r\in(1,\infty).$ Then there is a bounded linear operator $S:L^{r}_{p}(M)\cap\mathrm{(}\mathrm{k}\mathrm{e}\mathrm{r}\Delta\mathrm{)}^{\perp}\rightarrow W^{2,r}_{p}(M)$ such that $\displaystyle\Delta S(\omega)=\omega$ on $M.$ So, with $u:=S\omega$ we get $\displaystyle\Delta u=\omega$ and $\displaystyle u\in W^{2,r}_{p}(M).$ Moreover we have $\displaystyle{\left\|{u}\right\|}_{W_{p}^{2,r}(M)}\leq c{\left\|{\omega}\right\|}_{L_{p}^{r}(M)}.$

We also proved the following theorem [Amar, 2018, Theorem 4.3, p. 14], as a consequence of the Local Increasing Regularity Method. We just need to know here that the WMP is a weaker property than the Unique Continuation Property.

Theorem 3.2**.**

Let $N$ be a smooth compact riemannian manifold with smooth boundary $\partial N.$ Let $D:G\rightarrow G$ be an elliptic linear differential operator of order $m$ with ${\mathcal{C}}^{1}$ coefficients acting on sections of a vector bundle $G:=(H,\pi,M)$ on $N.$ Let $\displaystyle\omega\in L^{r}_{G}(N)$ be such a section. There is a $G$-section $u\in W^{m,r}_{G}(N),$ such that $Du=\omega$ and ${\left\|{u}\right\|}_{W^{m,r}_{G}(N)}\leq c{\left\|{\omega}\right\|}_{L^{r}_{G}(N)},$ provided that the operator $D$ has the WMP for the $D$-harmonic $G$-forms.

Taking here $G:=\Lambda_{p}(N)$ the set of $p$-forms on $N,$ and $D:=\Delta$ the Hodge laplacian, we have that $\Delta$ verifies the Unique Continuation Property by a difficult result by N. Aronszajn, A. Krzywicki and J. Szarski [Aronszajn et al., 1962] hence it has the WMP too.

So we get in this special case:

Theorem 3.3**.**

Let $N$ be a smooth compact riemannian manifold with smooth boundary $\partial N.$ Let $\Delta$ be the Hodge Laplacian acting on $p$-forms on $N.$ Let $\displaystyle\omega\in L^{r}_{p}(N).$ There is a $p$-form $u\in W^{2,r}_{p}(N),$ such that $\Delta u=\omega$ and ${\left\|{u}\right\|}_{W^{2,r}_{p}(N)}\leq c{\left\|{\omega}\right\|}_{L^{r}_{p}(N)}.$

This Theorem has the easy corollary:

Corollary 3.4**.**

Let $X$ be a complete smooth riemannian manifold without boundary. Let $\Omega$ a relatively compact domain in $X.$ Let $\omega\in L^{r}_{p}(\Omega).$ There is a $p$-form $u\in W^{2,r}_{p}(\Omega),$ such that $\Delta u=\omega$ and ${\left\|{u}\right\|}_{W^{2,r}_{p}(\Omega)}\leq c{\left\|{\omega}\right\|}_{L^{r}_{p}(\Omega)}.$

Proof.

Put $\bar{\Omega}$ in a compact sub manifold $N$ of $X$ with a smooth boundary. Extend $\omega$ by [math] outside $\Omega,$ then this extension $\displaystyle\tilde{\omega}$ is still in $\displaystyle L^{r}_{p}(N).$ We can apply Theorem 3.3 to get a $p$-form $\tilde{u}\in W^{2,r}_{p}(N)$ such that $\Delta\tilde{u}=\tilde{\omega}.$ Now we let $u$ to be the restriction of $\tilde{u}$ to $\Omega.$ This ends the proof of the corollary. $\hfill\blacksquare$

Remark 3.5**.**

In the case where $\Omega$ is a bounded domain in ${\mathbb{R}}^{n},$ to get this solution we just have to use the Newton kernel on $\omega$ and apply  [Gilbarg and Trudinger, 1998, Theorem 9.9, p. 230]. In the riemannian case we have to add a difficult result by N. Aronszajn, A. Krzywicki and J. Szarski [Aronszajn et al., 1962] to get the UCP.

4. Solution of the Poisson equation with compact support.

Firts we shall study a duality between currents inspired by the Serre duality [Serre, 1955].

Because using Theorem 3.1, the following results are easy, we shall assume from now on that $X$ is non compact.

So let $X$ be an oriented non compact riemannian manifold of dimension $n.$ It has a volume form $dm$ and we denote also by $dm$ the associated volume measure on $X.$ We shall denote by $r^{\prime}$ the conjugate exponent of $r\in(1,\infty),\ \frac{1}{r}+\frac{1}{r^{\prime}}=1.$

4.1. Weighted $L^{r}$ spaces.

Let $\Omega$ be a domain in $X.$

Lemma 4.1**.**

Let $\eta>0$ be a weight. If $u$ is a $p$-current defined on $(n-p)$-forms $\alpha$ in $L^{r^{\prime}}(\Omega,\eta)$ and such that

[TABLE]

then $\ {\left\|{u}\right\|}_{L_{p}^{r}(\Omega,\eta^{1-r})}\leq C.$

Proof.

Set $\tilde{\alpha}:=\eta^{1/r^{\prime}}\alpha;\ \tilde{u}:=\frac{1}{\eta^{1/r^{\prime}}}u$ then we have

$\displaystyle{\left\langle{u,*\alpha}\right\rangle}=\int_{\Omega}{u\wedge{\overline{\alpha}}}=\int_{\Omega}{\tilde{u}\wedge{\overline{\tilde{\alpha}}}}={\left\langle{\tilde{u},*\tilde{\alpha}}\right\rangle}$

and ${\left\|{\tilde{\alpha}}\right\|}_{L^{r^{\prime}}(\Omega)}={\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega,\eta)}.$

We notice that ${\left\|{\tilde{\alpha}}\right\|}_{L^{r^{\prime}}(\Omega)}={\left\|{*\tilde{\alpha}}\right\|}_{L^{r^{\prime}}(\Omega)}$ because we have $(*\tilde{\alpha},*\tilde{\alpha})dm=*\tilde{\alpha}\wedge{\overline{**\tilde{\alpha}}}$ but $**\tilde{\alpha}=(-1)^{p(n-p)}\tilde{\alpha},$ by  [Voisin, 2002, Lemma 5.5], hence, because $\displaystyle(*\tilde{\alpha},*\tilde{\alpha})$ is positive, $(*\tilde{\alpha},*\tilde{\alpha})=\left|{\tilde{\alpha}}\right|^{2}.$

By use of the duality $L^{r}_{p}(\Omega)-L^{r^{\prime}}_{n-p}(\Omega),$ done in Lemma 6.3, we get

$\displaystyle{\left\|{\tilde{u}}\right\|}_{L_{p}^{r}(\Omega)}=\sup_{\alpha\in L_{n-p}^{r^{\prime}}(\Omega),\ \alpha\neq 0}\frac{\left|{{\left\langle{\tilde{u},*\tilde{\alpha}}\right\rangle}}\right|}{{\left\|{\tilde{\alpha}}\right\|}_{L^{r^{\prime}}(\Omega)}}.$

But

$\displaystyle{\left\|{\tilde{u}}\right\|}_{L_{p}^{r}(\Omega)}^{r}:=\int_{\Omega}{\left|{u}\right|^{r}\eta^{-\frac{r}{r^{\prime}}}dm}=\int_{\Omega}{\left|{u}\right|^{r}\eta^{1-r}dm}={\left\|{u}\right\|}_{L^{r}(\Omega,\eta^{1-r})}^{r}.$

So we get

$\displaystyle{\left\|{u}\right\|}_{L_{p}^{r}(\Omega,\eta^{1-r})}=\sup_{*\alpha\in L_{p}^{r^{\prime}}(\Omega,\eta),\ \alpha\neq 0}\frac{\left|{{\left\langle{u,*\alpha}\right\rangle}}\right|}{{\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega,\eta)}}.$

The proof is complete. $\hfill\blacksquare$

Let ${\mathcal{H}}_{p}(\Omega)$ be the set of all $p$ harmonic forms, i.e. $h\in{\mathcal{H}}_{p}(\Omega)\iff\Delta h=0$ in $\Omega.$

In order to simplify notation, we note the pairing for $\alpha$ a $p$-form and $\beta$ a $(n-p)$-form by:

$\displaystyle\ll\alpha,\beta\gg:=\int_{\Omega}{\alpha\wedge\beta}.$

With this notation we also have ${\left\langle{\alpha,\beta}\right\rangle}=\ll\alpha,{\overline{*\beta}}\gg.$

Lemma 4.2**.**

We have $\Delta({\overline{*u}})={\overline{*\Delta u}}.$ And $\displaystyle\ll\Delta\alpha,\beta\gg=\ll\alpha,\Delta\beta\gg$ provided that $\alpha$ or $\beta$ has compact support. Moreover we have

$\omega\in L^{r}_{p}(\Omega),\ \omega\perp{\mathcal{H}}_{p}^{r^{\prime}}(\Omega)\iff\omega\perp{\mathcal{H}}_{n-p}^{r^{\prime}}(\Omega).$

with the suitable notion of orthogonality:

$\displaystyle\omega\in L^{r}_{p}(\Omega),\ \omega\perp{\mathcal{H}}_{p}^{r^{\prime}}(\Omega)\iff\forall h\in{\mathcal{H}}_{p}^{r^{\prime}}(\Omega),\ {\left\langle{\omega,h}\right\rangle}=0$

and

$\displaystyle\omega\in L^{r}_{p}(\Omega),\ \omega\perp{\mathcal{H}}_{n-p}^{r^{\prime}}(\Omega)\iff\forall h\in{\mathcal{H}}_{n-p}^{r^{\prime}}(\Omega),\ \ll\omega,h\gg=0.$**

Proof.

We have $\Delta\varphi=dd^{*}\varphi+d^{*}d\varphi.$ The definition of $d^{*}$ in [Voisin, 2002, Section 5.1.2, p. 118] gives:

$d^{*}=(-1)^{p}*^{-1}d*$ on $\Lambda^{p}.$

We also have by [Voisin, 2002, Lemme 5.5, p. 117]:

$*^{2}=(-1)^{p(n-p)}$ on $\Lambda^{p}.$

These facts give:

$\displaystyle d(*\varphi)=**^{-1}d(*\varphi)=(-1)^{p}*d^{*}\varphi.$

And, replacing the first $d^{*},$

$d^{*}d(*\varphi)=(-1)^{p}d^{*}*d^{*}\varphi=(-1)^{p}(-1)^{p}*^{-1}d**d^{*}\varphi=$

$\displaystyle=(-1)^{2p}(-1)^{2p(n-p)}*dd^{*}\varphi=*dd^{*}\varphi,$

because $\displaystyle*^{2}=(-1)^{p(n-p)}\Rightarrow*^{-1}=(-1)^{p(n-p)}*.$ Hence $\displaystyle d^{*}d(*\varphi)=*dd^{*}\varphi.$

The same way we get $dd^{*}(*\varphi)=*d^{*}d\varphi.$ Because the laplacian is real the bar gets out.

Now suppose that $\alpha$ has compact support we have:

$\ll\Delta\alpha,\beta\gg={\left\langle{\Delta\alpha,{\overline{*\beta}}}\right\rangle}={\left\langle{\alpha,\Delta({\overline{*\beta}})}\right\rangle}={\left\langle{\alpha,{\overline{*\Delta\beta}}}\right\rangle}=\ll\alpha,\Delta\beta\gg,$

the second equality because $\Delta$ is essentially self-adjoint and the third one by the first part of this lemma.

For the "moreover", we have $h\in{\mathcal{H}}_{p}^{r^{\prime}}(\Omega)\iff{\overline{*h}}\in{\mathcal{H}}_{n-p}^{r^{\prime}}(\Omega)$ because the first part of the lemma gives:

$\displaystyle\Delta({\overline{*h}})={\overline{*\Delta h}}=0.$

Now take $\displaystyle\omega\in L^{r}_{p}(\Omega)$ and $h\in{\mathcal{H}}_{p}^{r^{\prime}}(\Omega)$ such that ${\left\langle{\omega,h}\right\rangle}=0$ then

$\displaystyle 0={\left\langle{\omega,h}\right\rangle}=\ll\omega,{\overline{*h}}\gg$

and the same for the converse, starting with $\displaystyle h\in{\mathcal{H}}_{n-p}^{r^{\prime}}(\Omega)$ and $\displaystyle\ll\omega,h\gg=0$ we get $\displaystyle{\left\langle{\omega,{\overline{*h}}}\right\rangle}=0.$

The proof is complete. $\hfill\blacksquare$

Suppose that $\Omega$ is relatively compact in $X.$ Let $\omega\in L_{p}^{r}(\Omega)$ with compact support in $\Omega,\ \omega\in L_{p}^{r,c}(\Omega).$

Set the weight $\eta=\eta_{\epsilon}:={{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{\Omega_{1}}(z)+\epsilon{{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{\Omega\backslash\Omega_{1}}(z)$ for a fixed $\epsilon>0,$ with $\mathop{\rm Supp}\nolimits\omega\subset\Omega_{1}\Subset\Omega.$

Let $\alpha\in L^{r^{\prime}}_{p}(\Omega,\eta),$ with $r^{\prime}$ conjugate to $r.$ Because $\epsilon>0$ we have $\alpha\in L^{r^{\prime}}(\Omega,\eta)\Rightarrow\alpha\in L^{r^{\prime}}(\Omega).$

By Corollary 3.4 we get:

[TABLE]

Lemma 4.3**.**

Let $\omega\in L_{p}^{r}(\Omega_{1})\cap{\mathcal{H}}_{n-p}(\Omega_{1})^{\perp}$ with compact support in $\Omega_{1}$ and define:

$\displaystyle\forall\alpha\in L^{r^{\prime}}_{n-p}(\Omega,\eta),\ {\mathcal{L}}(\alpha):=\ll\varphi,\omega\gg,$

where $\varphi$ is a solution in $\Omega$ of (4.3).

Then ${\mathcal{L}}$ is well defined and linear on $\displaystyle L^{r^{\prime}}_{n-p}(\Omega,\eta).$

Proof.

In order for ${\mathcal{L}}(\alpha)$ to be well defined, we need that if $\varphi^{\prime}$ is another solution of $\Delta\varphi^{\prime}=\alpha,$ then $\ll\varphi-\varphi^{\prime},\omega\gg=0;$ hence we need that $\omega$ must be "orthogonal" to $(n-p)$-forms $\varphi$ such that $\Delta\varphi=0$ in $\Omega,$ which is contained in our assumption.

Hence we have that ${\mathcal{L}}(\alpha)$ is well defined.

The linearity of ${\mathcal{L}}$ is clear because if $\alpha=\alpha_{1}+\alpha_{2}$ take $\varphi_{j}::\Delta\varphi_{j}=\alpha_{j}$ then $\varphi:=\varphi_{1}+\varphi_{2}$ implies $\displaystyle\Delta\varphi=\alpha_{1}+\alpha_{2}$ and

${\mathcal{L}}(\alpha):=\ll\varphi,\omega\gg=\ll\varphi_{1},\omega\gg+\ll\varphi_{2},\omega\gg={\mathcal{L}}(\alpha_{1})+{\mathcal{L}}(\alpha_{2}).$

The same for $\lambda\alpha.$ The proof is complete. $\hfill\blacksquare$

By the Hölder inequalities done in Lemma 6.1 we get, because $\omega$ has its support in $\Omega_{1},$

$\left|{\ll\varphi,\omega\gg}\right|=\left|{{\left\langle{\varphi,{\overline{*\omega}}}\right\rangle}}\right|\leq{\left\|{\omega}\right\|}_{L^{r}(\Omega_{1})}{\left\|{\varphi}\right\|}_{L^{r^{\prime}}(\Omega_{1})}.$

Let $\alpha,\varphi$ be as in (4.3), then

${\left\|{\varphi}\right\|}_{L^{r^{\prime}}(\Omega)}\leq{\left\|{\varphi}\right\|}_{W^{2,r^{\prime}}(\Omega)}\leq C{\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega)}.$

But $\displaystyle{\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega)}$ can be very big compared to $\displaystyle{\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega_{1})}.$ So let $\psi$ be such that $\Delta\psi=\alpha$ in $\Omega_{1}$ and with ${\left\|{\psi}\right\|}_{W^{2,r^{\prime}}(\Omega_{1})}\leq C{\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega_{1})}.$ This is possible by Corollary 3.4, $\bar{\Omega}_{1}$ being compact.

Then, because $\Delta(\varphi-\psi)=0$ in $\Omega_{1}$ and $\omega\perp{\mathcal{H}}_{n-p}(\Omega_{1}),$ we get

$\displaystyle{\mathcal{L}}(\alpha):=\ll\varphi,\omega\gg=\ll\psi,\omega\gg.$

Hence

$\ \left|{{\mathcal{L}}(\alpha)}\right|\leq{\left\|{\omega}\right\|}_{L^{r}(\Omega_{1})}{\left\|{\psi}\right\|}_{L^{r^{\prime}}(\Omega_{1})}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega_{1})}{\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega_{1})}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega_{1})}{\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega,\eta)},$

because $\eta_{\epsilon}=1$ on $\Omega_{1}\supset\mathop{\rm Supp}\nolimits\omega,$ hence ${\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega_{1})}\leq{\left\|{\alpha}\right\|}_{L^{r^{\prime}}(\Omega,\eta)}.$

So we have that the norm of ${\mathcal{L}}$ is bounded on $\displaystyle L_{n-p}^{r^{\prime}}(\Omega,\eta).$ The bound of ${\mathcal{L}}$ is $C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}$ which is independent of $\eta$ hence of $\epsilon.$

This means, by the definition of currents, that there is a $p$ current $u$ which represents the form ${\mathcal{L}}$: $\displaystyle{\mathcal{L}}(\alpha)=\ll\alpha,u\gg.$ So if $\alpha:=\Delta\varphi$ with $\varphi\in{\mathcal{C}}^{\infty}$ with compact support in $\Omega,$ we get

$\displaystyle\ll\varphi,\omega\gg={\mathcal{L}}(\alpha)=\ll\alpha,u\gg=\ll\Delta\varphi,u\gg.$

Now we use Lemma 4.2 to get $\displaystyle\ll\varphi,\omega\gg=\ll\varphi,\Delta u\gg$ and we have $\Delta u=\omega$ in the sense of distributions.

Moreover we have

$\displaystyle\sup_{\alpha\in L^{r^{\prime}}(\Omega,\eta),\ {\left\|{\alpha}\right\|}=1}\ \left|{\ll\alpha,u\gg}\right|\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}$

and by Lemma 4.1 with the weight $\eta,$ this implies

$\ {\left\|{u}\right\|}_{L^{r}(\Omega,\eta^{1-r})}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}.$

So we proved

Proposition 4.4**.**

Let $\Omega_{1}\Subset\Omega$ and $\omega\in L^{r}(\Omega_{1})$ with compact support in $\Omega_{1}$ and such that $\omega\perp{\mathcal{H}}_{n-p}(\Omega_{1}).$ Let also $\eta=\eta_{\epsilon}:={{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{\Omega_{1}}(z)+\epsilon{{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{\Omega\backslash\Omega_{1}}(z).$ Then there is a $p$ form $u\in L^{r}(\Omega,\eta^{1-r})$ such that $\displaystyle\Delta u=\omega$ and $\displaystyle{\left\|{u}\right\|}_{L^{r}(\Omega,\eta^{1-r})}\leq C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}.$

Now we are in position to prove:

Theorem 4.5**.**

Let $X$ be a complete oriented riemannian manifold. Let $\Omega$ be a relatively compact domain in $X$ and $\Omega_{1}\Subset\Omega.$ Let $\omega\in L^{r}_{p}(\Omega_{1})$ with compact support in $\Omega_{1}$ and such that $\omega\perp{\mathcal{H}}_{n-p}(\Omega_{1}).$ Then there is a $p$-form $u\in L^{r}_{p}(\Omega)$ with compact support in $\Omega_{1}$ such that $\Delta u=\omega$ as distributions and $\displaystyle{\left\|{u}\right\|}_{L^{r}_{p}(\Omega)}\leq C{\left\|{\omega}\right\|}_{L^{r}_{p}(\Omega_{1})}.$

Proof.

For $\epsilon>0$ with $\eta_{\epsilon}(z):={{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{\Omega_{1}}(z)+\epsilon{{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{\Omega\backslash\Omega_{1}}(z),$ let $u_{\epsilon}\in L^{r}(\Omega,\eta_{\epsilon}^{1-r})$ be the solution given by Proposition 4.4, then

$\displaystyle\ {\left\|{u_{\epsilon}}\right\|}_{L^{r}(\Omega,\eta_{\epsilon}^{1-r})}^{r}\leq\int_{\Omega}{\left|{u_{\epsilon}}\right|^{r}\eta^{1-r}dm}\leq C^{r}{\left\|{\omega}\right\|}_{L^{r}(\Omega)}^{r}.$

Replacing $\eta$ by its value we get

$\displaystyle\ \int_{\Omega_{1}}{\left|{u_{\epsilon}}\right|^{r}dm}+\ \int_{\Omega\backslash\Omega_{1}}{\left|{u_{\epsilon}}\right|^{r}\epsilon^{1-r}dm}\leq C^{r}{\left\|{\omega}\right\|}_{L^{r}(\Omega)}^{r}$ $\displaystyle\Rightarrow\int_{\Omega\backslash\Omega_{1}}{\left|{u_{\epsilon}}\right|^{r}\epsilon^{1-r}dm}\leq C^{r}{\left\|{\omega}\right\|}_{L^{r}(\Omega)}^{r}$

hence

$\displaystyle\ \int_{\Omega\backslash\Omega_{1}}{\left|{u_{\epsilon}}\right|^{r}dm}\leq C^{r}\epsilon^{r-1}{\left\|{\omega}\right\|}_{L^{r}(\Omega)}^{r}.$

Because $C$ and the norm of $\omega$ are independent of $\epsilon,$ we have that $\ {\left\|{u_{\epsilon}}\right\|}_{L^{r}(\Omega)}$ is uniformly bounded and $r>1$ implies that $\displaystyle L_{p}^{r}(\Omega)$ is a dual by Lemma 6.3, hence there is a sub-sequence $\{u_{\epsilon_{k}}\}_{k\in{\mathbb{N}}}$ of $\{u_{\epsilon}\}$ which converges weakly to a $p$-form $u$ in $L_{p}^{r}(\Omega),$ when $\epsilon_{k}\rightarrow 0,$ still with $\ {\left\|{u}\right\|}_{L_{p}^{r}(\Omega)}\leq C{\left\|{\omega}\right\|}_{L_{p}^{r}(\Omega)}.$ Let us note $\displaystyle u_{k}:=u_{\epsilon_{k}}.$

To see that this form $u$ is $0\ a.e.$ on $\Omega\backslash\Omega_{1}$ let us write the weak convergence:

$\displaystyle\forall\alpha\in L_{p}^{r^{\prime}}(\Omega),\ {\left\langle{u_{k},\alpha}\right\rangle}=\int_{\Omega}{u_{k}\wedge{\overline{*\alpha}}}\mathrel{\mathop{\kern 0.0pt\rightarrow}\limits_{k\rightarrow\infty}}{\left\langle{u,\alpha}\right\rangle}=\int_{\Omega}{u\wedge{\overline{*\alpha}}}.$

As usual take $\displaystyle\alpha:=\frac{u}{\left|{u}\right|}{{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{E}$ where $E:=\{\left|{u}\right|>0\}\cap(\Omega\backslash\Omega_{1})$ then we get

$\displaystyle\int_{\Omega}{u\wedge{\overline{*\alpha}}}=\int_{E}{\left|{u}\right|dm}=\lim_{k\rightarrow\infty}\int_{\Omega}{u_{k}\wedge{\overline{*\alpha}}}=\lim_{k\rightarrow\infty}\int_{E}{\frac{u_{k}\wedge{\overline{*u}}}{\left|{u}\right|}}.$

Now we have by Hölder inequalities:

$\displaystyle\left|{\int_{E}{\frac{u_{k}\wedge{\overline{*u}}}{\left|{u}\right|}}}\right|\leq{\left\|{u_{k}}\right\|}_{L^{r}(E)}{\left\|{{{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{E}}\right\|}_{L^{r^{\prime}}(E)}.$

But

$\displaystyle\ {\left\|{u_{k}}\right\|}_{L^{r}(E)}^{r}\leq\int_{\Omega\backslash\Omega_{1}}{\left|{u_{k}}\right|^{r}dm}\leq(\epsilon_{k})^{r-1}C{\left\|{\omega}\right\|}_{L^{r}(\Omega)}\mathrel{\mathop{\kern 0.0pt\rightarrow}\limits_{k\rightarrow\infty}}0$

and ${\left\|{{{\rm\kern 1.99997pt\vrule width=0.20004pt,height=6.32915pt,depth=0.0pt\kern-3.69995pt1}}_{E}}\right\|}_{L^{r^{\prime}}(E)}=(m(E))^{1/r^{\prime}}.$

Hence

$\displaystyle\ \left|{\ \int_{E}{\left|{u}\right|dm}}\right|=\lim_{k\rightarrow\infty}\int_{E}{\frac{u_{k}\wedge{\overline{*u}}}{\left|{u}\right|}}\leq\ \lim_{k\rightarrow\infty}C^{r}(m(E))^{1/r^{\prime}}(\epsilon_{k})^{r-1}{\left\|{\omega}\right\|}_{L^{r}(\Omega)}^{r}=0,$

so $\int_{E}{\left|{u}\right|dm}=0$ which implies $m(E)=0$ because on $E,\ \left|{u}\right|>0.$

Hence we get that the form $u$ is $0\ a.e.$ on $\Omega\backslash\Omega_{1}.$

So we proved

[TABLE]

hence again by use of Lemma 4.2 we get $\Delta u=\omega$ in the sense of distributions. $\hfill\blacksquare$

Lemma 4.6**.**

Let $X$ be a complete riemannian manifold. Let $\Omega$ be a relatively compact domain in $X$ and $\Omega_{1}\Subset\Omega.$ Let $u\in L^{r}_{p}(\Omega)$ such that $\Delta u\in L^{r}_{p}(\Omega).$ Then we have the interior elliptic regularity:

$\displaystyle{\left\|{u}\right\|}_{W^{2,r}_{p}(\Omega_{1})}\leq C({\left\|{\Delta u}\right\|}_{L^{r}_{p}(\Omega)}+{\left\|{u}\right\|}_{L^{r}_{p}(\Omega)}).$**

Proof.

The interior elliptic inequalities [Amar, 2018], Theorem 3.4, valid in the complete riemannian manifold $M$ give that for any $x\in M,$ there is a ball $B_{x}:=B(x,R)$ and a smaller ball $B^{\prime}_{x}$ relatively compact in $B_{x},$ such that:

[TABLE]

Moreover the constants $c_{j},\ j=1,2,$ are independent of the radius $R(x)$ of the ball $B_{x}.$

Because $\bar{\Omega}_{1}$ is compact in $\Omega,$ there is a $\delta>0$ such that:

$\bigcup_{x\in\Omega_{1}}{B(x,\delta)}\subset\Omega.$

For all $x\in\bar{\Omega}_{1},$ choose $R^{\prime}(x)=\min(\delta,R(x))$ for the $R(x)$ given in (4.5).

We cover the compact set $\bar{\Omega}_{1}$ by a finite set of balls $B^{\prime}_{x_{j}}$ associated to $B(x_{j},R^{\prime}(x)).$ So we get, by (4.5),

${\left\|{u}\right\|}_{W^{2,r}(\Omega_{1})}\leq\sum_{j=1}^{N}{}{\left\|{u}\right\|}_{W^{2,r}(B^{\prime}_{x_{j}})}\leq c_{1}\sum_{j=1}^{N}{}{\left\|{\Delta u}\right\|}_{L^{r}(B_{x_{j}})}+c_{2}\sum_{j=1}^{N}{}R^{\prime}(x_{j})^{-2}{\left\|{u}\right\|}_{L^{r}(B_{x_{j}})}.$

Set $c:=\max_{j=1,..,N}R^{\prime}(x_{j})^{-2},$ we get:

${\left\|{u}\right\|}_{W^{2,r}(\Omega_{1})}\leq c_{1}\sum_{j=1}^{N}{}{\left\|{\Delta u}\right\|}_{L^{r}(B_{x_{j}})}+cc_{2}\sum_{j=1}^{N}{}{\left\|{u}\right\|}_{L^{r}(B_{x_{j}})}.$

Now we have

$\sum_{j=1}^{N}{}{\left\|{f}\right\|}_{L^{r}(B_{x_{j}})}=\sum_{j=1}^{N}{}\int_{B_{x_{j}}}{\left|{f}\right|^{r}}dm\leq\sum_{j=1}^{N}{}\int_{\Omega}{\left|{f}\right|^{r}}dm\leq N\int_{\Omega}{\left|{f}\right|^{r}}dm,$

because, by the choice of $R^{\prime}(x)$ we have that $B(x_{j},R^{\prime})\subset\Omega.$

Applying this with $f=u$ and $f=\Delta u$ we get

${\left\|{u}\right\|}_{W^{2,r}(\Omega_{1})}\leq c_{1}N{\left\|{\Delta u}\right\|}_{L^{r}(\Omega)}+cc_{2}N{\left\|{u}\right\|}_{L^{r}(\Omega)}.$

The proof is complete. $\hfill\blacksquare$

This lemma allows the better estimates:

Corollary 4.7**.**

Let $X$ be a complete oriented riemannian manifold. Let $\Omega$ be a relatively compact domain in $X$ and $\Omega_{1}\Subset\Omega.$ Let $\omega\in L^{r}_{p}(\Omega_{1})$ with compact support in $\Omega_{1}$ and such that $\omega\perp{\mathcal{H}}_{n-p}(\Omega_{1}).$ Then there is a $p$-form $u\in W^{2,r}_{p}(\Omega)$ with compact support in $\Omega_{1}$ such that $\Delta u=\omega$ as distributions and $\displaystyle{\left\|{u}\right\|}_{W^{2,r}_{p}(\Omega_{1})}\leq C{\left\|{\omega}\right\|}_{L^{r}_{p}(\Omega_{1})}.$

Proof.

We can apply Theorem 4.5 so we have a $p$-form $u\in L^{r}_{p}(\Omega)$ with compact support in $\Omega_{1}$ such that $\Delta u=\omega$ as distributions and ${\left\|{u}\right\|}_{L^{r}_{p}(\Omega)}\lesssim{\left\|{\omega}\right\|}_{L^{r}(\Omega)}.$

Now we apply the interior elliptic regularity with $\Delta u=\omega$:

$\displaystyle{\left\|{u}\right\|}_{W^{2,r}_{p}(\Omega_{1})}\leq C({\left\|{\omega}\right\|}_{L^{r}_{p}(\Omega)}+{\left\|{u}\right\|}_{L^{r}_{p}(\Omega)}).$

But ${\left\|{u}\right\|}_{L^{r}_{p}(\Omega)}\lesssim{\left\|{\omega}\right\|}_{L^{r}(\Omega)}$ so we get

$\displaystyle{\left\|{u}\right\|}_{W^{2,r}_{p}(\Omega_{1})}\leq C{\left\|{\omega}\right\|}_{L^{r}_{p}(\Omega)}.$

Because $\omega$ has compact support in $\Omega_{1},$ we get $\displaystyle{\left\|{u}\right\|}_{W^{2,r}_{p}(\Omega_{1})}\leq C{\left\|{\omega}\right\|}_{L^{r}_{p}(\Omega_{1})}.$

The proof is complete. $\hfill\blacksquare$

Remark 4.8**.**

The condition of orthogonality to ${\mathcal{H}}_{p}(\Omega_{1})$ is necessary: suppose there is a $p$-current $u\in W^{2,r}_{p}(\Omega)$ such that $\Delta u=\omega$ and $u$ with compact support in $\Omega,$ then if $h\in{\mathcal{H}}_{n-p}(\Omega),$ we have

$h\in{\mathcal{H}}_{n-p}(\Omega),\ \ll\omega,h\gg=\ll\Delta u,h\gg=\ll u,\Delta h\gg=0,$

because $u$ is compactly supported.

5. Kähler manifold and Kohn laplacian.

A Kähler manifold is a complex manifold $X$ with a Hermitian metric $h$ whose associated 2-form $\kappa$ is closed. In more detail, $h$ gives a positive definite Hermitian form on the tangent space $T_{x}$ at each point $x$ of $X,$ and the 2-form $\kappa$ is defined by

$\kappa(u,v):=\Re h(iu,v)$

for tangent vectors $u$ and $v$ (where $i$ is the complex number ${\sqrt{-1}}$ ). For a Kähler manifold $X,$ the Kähler form $\kappa$ is a real closed (1,1)-form. A Kähler manifold can also be viewed as a Riemannian manifold, with the Riemannian metric $g$ defined by

$\displaystyle g(u,v):=\Re h(u,v).$

On $X$ the $(p,q)$-forms are defined and so is the $\bar{\partial}$ operator. The Hodge $*$ operator is also defined, see C. Voisin [Voisin, 2002, Section 5.1.4, p. 121].

Recall the $\bar{\partial}$ (or Kohn) laplacian, acting from $(p,q)$-forms to $(p,q)$-forms is:

$\Delta_{\bar{\partial}}f:=(\bar{\partial}\bar{\partial}^{*}+\bar{\partial}^{*}\bar{\partial})f,$

where $\bar{\partial}^{*}$ is the formal adjoint to $\bar{\partial},$ i.e.

$\forall\varphi\in{\mathcal{D}}_{p,q-1},\ \forall u\in L^{r}_{p,q},\ {\left\langle{\bar{\partial}^{*}u,\ \varphi}\right\rangle}:={\left\langle{u,\bar{\partial}\varphi}\right\rangle}.$

The space ${\mathcal{H}}^{r}_{q}(\Omega):=\{h\in L_{q}^{r}(\Omega)::\Delta h=0\}$ is the space of harmonic $q$-forms in the set $\Omega.$

Because $X$ is a complex manifold, it is canonically oriented and we also note $dm$ the volume $(n,n)$ form on $X.$

Now our aim is to prove Theorem  1.4:

Theorem 5.1**.**

Let $(X,\omega)$ be a complete Kähler manifold. Let $\Omega$ be a relatively compact domain in $X.$ Let $\displaystyle\omega\in L_{p,q}^{r}(\Omega),\ \bar{\partial}\omega=0$ in $\Omega$ and $\omega$ compactly supported in $\Omega.$ Suppose moreover that $\displaystyle\omega\perp{\mathcal{H}}^{r^{\prime}}_{2n-p-q}(\Omega).$

Then there is a $\displaystyle u\in W_{p,q-1}^{1,r}(\Omega)$ with compact support in $\Omega$ and such that $\bar{\partial}u=\omega.$

Proof.

Let us see $X$ as a riemannian manifold, then we can apply Corollary 4.7 to get the existence of a $\displaystyle\tilde{v}\in W_{p+q}^{2,r}(\Omega)$ such that $\Delta\tilde{v}=\omega$ and $\tilde{v}$ compactly supported in $\Omega.$

By use of Theorem 1.3 we get that $\Delta_{\bar{\partial}}\tilde{v}=\frac{1}{2}\omega.$ So, setting $v:=\frac{1}{2}\tilde{v}$ we get:

$\displaystyle v\in W_{p+q}^{2,r,c}(\Omega)::\Delta_{\bar{\partial}}v=\omega.$

Now we have

[TABLE]

this implies, taking $\bar{\partial}$ on both sides,

$\bar{\partial}\bar{\partial}^{*}\bar{\partial}v=\bar{\partial}\omega=0,$

because $\bar{\partial}^{2}=0.$ Then

$0={\left\langle{\bar{\partial}\bar{\partial}^{*}\bar{\partial}v,\bar{\partial}v}\right\rangle}={\left\langle{\bar{\partial}^{*}\bar{\partial}v,\bar{\partial}^{*}\bar{\partial}v}\right\rangle}={\left\|{\bar{\partial}^{*}\bar{\partial}v}\right\|}_{L^{2}(\Omega)}^{2}$

because $v$ being compactly supported in $\Omega,$ so is $\bar{\partial}v,$ and we can shift the $\bar{\partial}$ operator on the right hand side.

From (5.6) we get $\displaystyle\ \bar{\partial}\bar{\partial}^{*}v=\omega,$ because $\displaystyle\bar{\partial}^{*}\bar{\partial}v=0.$ Now we set $u:=\bar{\partial}^{*}v$ and we get a $u$ with support in $\Omega,$ such that:

$u\in W_{p,q-1}^{1,r}(\Omega),\ \bar{\partial}u=\omega,$

because $\bar{\partial}^{*}$ is a first order differential operator and $\displaystyle v\in W_{p+q}^{2,r}(\Omega)$ with support in $\Omega.$

The proof is complete. $\hfill\blacksquare$

Remark 5.2**.**

1) In the case of bounded open sets in ${\mathbb{C}}^{n}$ and for the $L^{2}$ theory, this idea to use the usual laplacian to get estimates for the $\bar{\partial}$ equation was already done in the nice book by E. Straube [Straube, 2010, Section 2.9].

2) This method improves the regularity of the solution: from $\displaystyle L^{r,c}_{p,q-1}(\Omega)$ to $\displaystyle W_{p,q-1}^{1,r,c}(\Omega).$ The price is that $\displaystyle\omega\perp{\mathcal{H}}_{2n-p-q}(\Omega)$ but there is no pseudo-convexity condition on $\Omega.$

6. Appendix.

For the reader’s convenience we shall prove certainly known results on the duality $L^{r}-L^{r^{\prime}}$ for $(p,q)$-forms in a complex manifold.

Recall we have a pointwise scalar product and a pointwise modulus:

$\displaystyle(\alpha,\beta)dm:=\alpha\wedge{\overline{*\beta}};\ \ \left|{\alpha}\right|^{2}dm:=\alpha\wedge{\overline{*\alpha.}}$

By the Cauchy-Schwarz inequality for scalar product we get:

$\displaystyle\forall x\in X,\ \left|{(\alpha,\beta)(x)}\right|\leq\left|{\alpha(x)}\right|\left|{\beta(x)}\right|.$

This gives Hölder inequalities for $(p,q)$-forms:

Lemma 6.1**.**

(Hölder inequalities) Let $\alpha\in L^{r}_{p,q}(\Omega)$ and $\displaystyle\beta\in L^{r^{\prime}}_{p,q}(\Omega).$ We have

$\displaystyle\left|{{\left\langle{\alpha,\beta}\right\rangle}}\right|\leq{\left\|{\alpha}\right\|}_{L^{r}(\Omega)}{\left\|{\beta}\right\|}_{L^{r^{\prime}}(\Omega)}.$**

Proof.

We start with ${\left\langle{\alpha,\beta}\right\rangle}=\int_{\Omega}{(\alpha,\beta)(x)dm(x)}$ hence

$\displaystyle\left|{{\left\langle{\alpha,\beta}\right\rangle}}\right|\leq\int_{\Omega}{\left|{(\alpha,\beta)(x)}\right|dm}\leq\int_{\Omega}{\left|{\alpha(x)}\right|\left|{\beta(x)}\right|dm(x)}.$

By the usual Hölder inequalities for functions we get

$\displaystyle\int_{\Omega}{\left|{\alpha(x)}\right|\left|{\beta(x)}\right|dm(x)}\leq{\left({\int_{\Omega}{\left|{\alpha(x)}\right|^{r}dm}}\right)}^{1/r}{\left({\int_{\Omega}{\left|{\beta(x)}\right|^{r^{\prime}}dm}}\right)}^{1/r^{\prime}}$

which ends the proof of the lemma. $\hfill\blacksquare$

Lemma 6.2**.**

Let $\alpha\in L_{p,q}^{r}(\Omega)$ then

$\displaystyle{\left\|{\alpha}\right\|}_{L_{p,q}^{r}(\Omega)}=\sup_{\beta\in L_{p,q}^{r^{\prime}}(\Omega),\ \beta\neq 0}\frac{\left|{{\left\langle{\alpha,\beta}\right\rangle}}\right|}{{\left\|{\beta}\right\|}_{L^{r^{\prime}}(\Omega)}}.$**

Proof.

We choose $\beta:=\alpha\left|{\alpha}\right|^{r-2},$ then:

$\displaystyle\left|{\beta}\right|^{r^{\prime}}=\left|{\alpha}\right|^{r^{\prime}(r-1)}=\left|{\alpha}\right|^{r}\Rightarrow{\left\|{\beta}\right\|}_{L^{r^{\prime}}(\Omega)}^{r^{\prime}}={\left\|{\alpha}\right\|}_{L^{r}(\Omega)}^{r}.$

Hence

$\displaystyle{\left\langle{\alpha,\beta}\right\rangle}={\left\langle{\alpha,\alpha\left|{\alpha}\right|^{r-2}}\right\rangle}=\int_{\Omega}{(\alpha,\alpha)\left|{\alpha}\right|^{r-2}dm}={\left\|{\alpha}\right\|}_{L^{r}(\Omega)}^{r}.$

On the other hand we have

$\displaystyle{\left\|{\beta}\right\|}_{L^{r^{\prime}}(\Omega)}={\left\|{\alpha}\right\|}_{L^{r}(\Omega)}^{r/r^{\prime}}={\left\|{\alpha}\right\|}_{L^{r}(\Omega)}^{r-1},$

so

$\displaystyle{\left\|{\alpha}\right\|}_{L^{r}(\Omega)}{\times}{\left\|{\beta}\right\|}_{L^{r^{\prime}}(\Omega)}={\left\|{\alpha}\right\|}_{L^{r}(\Omega)}^{r}={\left\langle{\alpha,\beta}\right\rangle}.$

Hence

$\displaystyle{\left\|{\alpha}\right\|}_{L^{r}(\Omega)}=\frac{\left|{{\left\langle{\alpha,\beta}\right\rangle}}\right|}{{\left\|{\beta}\right\|}_{L^{r^{\prime}}(\Omega)}}.$

A fortiori for any choice of $\beta$:

$\displaystyle{\left\|{\alpha}\right\|}_{L^{r}(\Omega)}\leq\sup_{\beta\in L^{r^{\prime}}(\Omega)}\frac{\left|{{\left\langle{\alpha,\beta}\right\rangle}}\right|}{{\left\|{\beta}\right\|}_{L^{r^{\prime}}(\Omega)}}.$

To prove the other direction, we use the Hölder inequalities, Lemma 6.1:

$\displaystyle\forall\beta\in L_{p,q}^{r^{\prime}}(\Omega),\ \frac{\left|{{\left\langle{\alpha,\beta}\right\rangle}}\right|}{{\left\|{\beta}\right\|}_{L^{r^{\prime}}(\Omega)}}\leq{\left\|{\alpha}\right\|}_{L^{r}(\Omega)}.$

The proof is complete. $\hfill\blacksquare$

Now we are in position to state:

Lemma 6.3**.**

The dual space of the Banach space $\displaystyle L_{p,q}^{r}(\Omega)$ is $\displaystyle L_{n-p,n-q}^{r^{\prime}}(\Omega).$

Proof.

Suppose first that $u\in L_{n-p,n-q}^{r^{\prime}}(\Omega).$ Then consider:

$\displaystyle\forall\alpha\in L_{p,q}^{r}(\Omega),\ {\mathcal{L}}(\alpha):=\int_{\Omega}{\alpha\wedge u}={\left\langle{\alpha,{\overline{*u}}}\right\rangle}.$

This is a linear form on $\displaystyle L_{p,q}^{r}(\Omega)$ and its norm, by definition, is

$\displaystyle{\left\|{{\mathcal{L}}}\right\|}=\sup_{\alpha\in L^{r}(\Omega)}\frac{\left|{{\left\langle{\alpha,{\overline{*u}}}\right\rangle}}\right|}{{\left\|{\alpha}\right\|}_{L^{r}(\Omega)}}.$

By use of Lemma 6.2 we get

$\displaystyle{\left\|{{\mathcal{L}}}\right\|}={\left\|{{\overline{*u}}}\right\|}_{L^{r^{\prime}}_{p,q}(\Omega)}={\left\|{u}\right\|}_{L^{r^{\prime}}_{n-p,n-q}(\Omega)}.$

So we have ${\left({L_{p,q}^{r}(\Omega)}\right)}^{*}\supset L_{n-p,n-q}^{r^{\prime}}(\Omega)$ with the same norm.

Conversely take a continuous linear form ${\mathcal{L}}$ on $\displaystyle L_{p,q}^{r}(\Omega).$ We have, again by definition, that:

$\displaystyle{\left\|{{\mathcal{L}}}\right\|}=\sup_{\alpha\in L^{r}(\Omega)}\frac{\left|{{\mathcal{L}}(\alpha)}\right|}{{\left\|{\alpha}\right\|}_{L^{r}(\Omega)}}.$

Because ${\mathcal{D}}_{p,q}(\Omega)\subset L_{p,q}^{r}(\Omega),$ ${\mathcal{L}}$ is a continuous linear form on $\displaystyle{\mathcal{D}}_{p,q}(\Omega),$ hence, by definition, ${\mathcal{L}}$ can be represented by a $(n-p,n-q)$-current $u.$ So we have:

$\displaystyle\forall\alpha\in{\mathcal{D}}_{p,q}(\Omega),\ {\mathcal{L}}(\alpha):=\int_{\Omega}{\alpha\wedge u}={\left\langle{\alpha,{\overline{*u}}}\right\rangle}.$

Moreover we have, by Lemma 6.2,

$\displaystyle{\left\|{{\mathcal{L}}}\right\|}=\sup_{\alpha\in{\mathcal{D}}_{p,q}(\Omega)}\frac{\left|{{\left\langle{\alpha,*\bar{u}}\right\rangle}}\right|}{{\left\|{\alpha}\right\|}_{L^{r}(\Omega)}}={\left\|{*u}\right\|}_{L^{r^{\prime}}(\Omega)}$

because $\displaystyle{\mathcal{D}}_{p,q}(\Omega)$ is dense in $\displaystyle L_{p,q}^{r}(\Omega).$ So we proved

$\displaystyle{\left({L_{p,q}^{r}(\Omega)}\right)}^{*}\subset L_{n-p,n-q}^{r^{\prime}}(\Omega)$ with the same norm.

The proof is complete. $\hfill\blacksquare$