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# Norm inequalities for composition of the Dirac and Green’s operators

- Shusen Ding
^{1}and - Bing Liu
^{2}Email author

**2013**:436

https://doi.org/10.1186/1029-242X-2013-436

© Ding and Liu; licensee Springer. 2013

**Received:**27 June 2013**Accepted:**27 August 2013**Published:**13 September 2013

## Abstract

We first prove a norm inequality for the composition of the Dirac operator and Green’s operator. Then, we estimate for the Lipschitz and BMO norms of the composite operator in terms of the ${L}^{s}$ norm of a differential form.

**MSC:**26B10, 30C65, 31B10, 46E35.

## Keywords

- Dirac operator
- Green’s operator
- differential forms and norm inequalities

## 1 Introduction

The purpose of this paper is to derive the norm inequalities for the composite operator $D\circ G$ of the Hodge-Dirac operator *D* and Green’s operator *G* on differential forms. Specifically, we will develop the upper bounds for norms of the composite operator $D\circ G$ applied to differential form *u* in terms of the norm of *u*. We all know that there are different versions of Dirac operators, such as the Hodge-Dirac operator associated to a Riemannian manifold and the euclidean Dirac operator arising in Clifford analysis. The Dirac operator studied in this paper is the Hodge-Dirac operator defined by $D=d+{d}^{\star}$, where *d* is the exterior derivative, and ${d}^{\star}$ is the Hodge codifferential, which is the formal adjoint to *d*. Both the Dirac operator *D* and Green’s operator *G* are widely studied and used in mathematics and physics. Since it was initiated by Paul Dirac in order to get a form of quantum theory compatible with special relativity, Dirac operators have been playing an important role in many fields of mathematics and physics, such as quantum mechanics, Clifford analysis and PDEs. Green’s operator is a key operator, which has been very well used in several areas of mathematics. In many situations, the process of studying solutions to PDEs involves estimating the various norms of the operators and their compositions. Hence, we are motivated to establish the upper bounds for the composite operators in this paper. See [1–8] for recent work on the Dirac operator, Green’s operator and their applications.

Let *M* be a bounded domain and *B* be a ball in ${\mathbb{R}}^{n}$, $n\ge 2$, throughout this paper. We use *σB* to express the ball with the same center as *B* and with $diam(\sigma B)=\sigma diam(B)$, $\sigma >0$. We do not distinguish the balls from cubes in this paper. We use $|E|$ to denote the Lebesgue measure of a set $E\subset {\mathbb{R}}^{n}$. We call *w* a weight if $w\in {L}_{\mathrm{loc}}^{1}({\mathbb{R}}^{n})$ and $w>0$ a.e. Let ${e}_{1},{e}_{2},\dots ,{e}_{n}$ be the standard unit basis of ${\mathbb{R}}^{n}$, and let ${\wedge}^{l}={\wedge}^{l}({\mathbb{R}}^{n})$ be the linear space of *l*-vectors, which is spanned by the exterior products ${e}_{I}={e}_{{i}_{1}}\wedge {e}_{{i}_{2}}\wedge \cdots \wedge {e}_{{i}_{l}}$, corresponding to all ordered *l*-tuples $I=({i}_{1},{i}_{2},\dots ,{i}_{l})$, $1\le {i}_{1}<{i}_{2}<\cdots <{i}_{l}\le n$, $l=0,1,\dots ,n$. The Grassman algebra $\wedge =\oplus {\wedge}^{l}$ is a graded algebra with respect to the exterior products. For any $\alpha =\sum {\alpha}^{I}{e}_{I}\in \wedge $ and $\beta =\sum {\beta}^{I}{e}_{I}\in \wedge $, the inner product in ∧ is defined by $\u3008\alpha ,\beta \u3009=\sum {\alpha}^{I}{\beta}^{I}$, with summation over all *l*-tuples $I=({i}_{1},{i}_{2},\dots ,{i}_{l})$ and all integers $l=0,1,\dots ,n$. The Hodge star operator ⋆: ∧→∧ is defined by the rule $\star 1={e}_{1}\wedge {e}_{2}\wedge \cdots \wedge {e}_{n}$ and $\alpha \wedge \star \beta =\beta \wedge \star \alpha =\u3008\alpha ,\beta \u3009(\star 1)$ for all *α*, $\beta \in \wedge $. The norm of $\alpha \in \wedge $ is given by the formula ${|\alpha |}^{2}=\u3008\alpha ,\alpha \u3009=\star (\alpha \wedge \star \alpha )\in {\wedge}^{0}=\mathbb{R}$. The Hodge star is an isometric isomorphism on Λ with $\star :{\wedge}^{l}\to {\wedge}^{n-l}$ and $\star \star {(-1)}^{l(n-l)}:{\wedge}^{l}\to {\wedge}^{l}$.

A differential *l*-form *ω* on *M* is a de Rham current (see [[9], Chapter III]) on *M* with values in ${\wedge}^{l}({\mathbb{R}}^{n})$. Differential forms are extensions of functions. For example, in ${\mathbb{R}}^{n}$, the function $u({x}_{1},{x}_{2},\dots ,{x}_{n})$ is called a 0-form. Moreover, if $u({x}_{1},{x}_{2},\dots ,{x}_{n})$ is differentiable, then it is called a differential 0-form. The 1-form $u(x)$ in ${\mathbb{R}}^{n}$ can be written as $u(x)={\sum}_{i=1}^{n}{u}_{i}({x}_{1},{x}_{2},\dots ,{x}_{n})\phantom{\rule{0.2em}{0ex}}d{x}_{i}$. If the coefficient functions ${u}_{i}({x}_{1},{x}_{2},\dots ,{x}_{n})$, $i=1,2,\dots ,n$, are differentiable, then $u(x)$ is called a differential 1-form. Similarly, a differential *k*-form $u(x)$ is generated by $\{d{x}_{{i}_{1}}\wedge d{x}_{{i}_{2}}\wedge \cdots \wedge d{x}_{{i}_{k}}\}$, $k=1,2,\dots ,n$, that is, $u(x)={\sum}_{I}{\omega}_{I}(x)\phantom{\rule{0.2em}{0ex}}d{x}_{I}=\sum {\omega}_{{i}_{1}{i}_{2}\cdots {i}_{k}}(x)\phantom{\rule{0.2em}{0ex}}d{x}_{{i}_{1}}\wedge d{x}_{{i}_{2}}\wedge \cdots \wedge d{x}_{{i}_{k}}$, where $I=({i}_{1},{i}_{2},\dots ,{i}_{k})$, $1\le {i}_{1}<{i}_{2}<\cdots <{i}_{k}\le n$. Let ${D}^{\prime}(M,{\wedge}^{l})$ be the space of all differential *l*-forms on *M*, and let ${L}^{p}(M,{\wedge}^{l})$ be the *l*-forms $\omega (x)={\sum}_{I}{\omega}_{I}(x)\phantom{\rule{0.2em}{0ex}}d{x}_{I}$ on *M* satisfying ${\int}_{M}{|{\omega}_{I}|}^{p}<\mathrm{\infty}$ for all ordered *l*-tuples *I*, $l=1,2,\dots ,n$. We denote the exterior derivative by $d:{D}^{\prime}(M,{\wedge}^{l})\to {D}^{\prime}(M,{\wedge}^{l+1})$ for $l=0,1,\dots ,n-1$. The Hodge codifferential operator ${d}^{\star}:{D}^{\prime}(M,{\wedge}^{l+1})\to {D}^{\prime}(M,{\wedge}^{l})$ is given by ${d}^{\star}={(-1)}^{nl+1}\star d\star $ on ${D}^{\prime}(M,{\wedge}^{l+1})$, $l=0,1,\dots ,n-1$. The Dirac operator *D* involved in this paper is defined by $D=d+{d}^{\star}$. It is easy to check that ${D}^{2}=\mathrm{\Delta}$, where $\mathrm{\Delta}=d{d}^{\star}+{d}^{\star}d$ is the Laplace-Beltrami operator. Let ${\wedge}^{l}M$ be the *l* th exterior power of the cotangent bundle, ${C}^{\mathrm{\infty}}({\wedge}^{l}M)$ be the space of smooth *l*-forms on *M* and $\mathcal{W}({\wedge}^{l}M)=\{u\in {L}_{\mathrm{loc}}^{1}({\wedge}^{l}M):u\text{has generalized gradient}\}$. The harmonic *l*-fields are defined by $\mathcal{H}({\wedge}^{l}M)=\{u\in \mathcal{W}({\wedge}^{l}M):du={d}^{\star}u=0,u\in {L}^{p}\text{for some}1p\mathrm{\infty}\}$. The orthogonal complement of ℋ in ${L}^{1}$ is defined by ${\mathcal{H}}^{\perp}=\{u\in {L}^{1}:\u3008u,h\u3009=0\text{for all}h\in \mathcal{H}\}$. Then the Green’s operator *G* is defined as $G:{C}^{\mathrm{\infty}}({\wedge}^{l}M)\to {\mathcal{H}}^{\perp}\cap {C}^{\mathrm{\infty}}({\wedge}^{l}M)$ by assigning $G(u)$ to be the unique element of ${\mathcal{H}}^{\perp}\cap {C}^{\mathrm{\infty}}({\wedge}^{l}M)$ satisfying Poisson’s equation $\mathrm{\Delta}G(u)=u-H(u)$, where *H* is the harmonic projection operator that maps ${C}^{\mathrm{\infty}}({\wedge}^{l}M)$ onto ℋ so that $H(u)$ is the harmonic part of *u*. See [10] for more properties of these operators. We write ${\parallel u\parallel}_{s,M}={({\int}_{M}{|u|}^{s})}^{1/s}$ and ${\parallel u\parallel}_{s,M,w}={({\int}_{M}{|u|}^{s}w(x)\phantom{\rule{0.2em}{0ex}}dx)}^{1/s}$, where $w(x)$ is a weight.

*k*and write ${\parallel \omega \parallel}_{{\mathrm{Lip}}_{k},M}$ for this norm. Similarly, for $\omega \in {L}_{\mathrm{loc}}^{1}(M,{\wedge}^{l})$, $l=0,1,\dots ,n$, we write $\omega \in BMO(M,{\wedge}^{l})$ if

for some $\sigma \ge 1$. When *ω* is a 0-form, (1.2) reduces to the classical definition of $BMO(M)$. The definitions of Lipschitz and BMO norms above appeared in [11].

## 2 ${L}^{s}$ norm inequalities

*f*defined on a Lebesgue measurable set

*E*with $|E|=0$, we have ${\int}_{E}f\phantom{\rule{0.2em}{0ex}}dx=0$. Thus, ${\parallel u\parallel}_{s,\partial B}=0$ and ${\parallel d{d}^{\ast}G(u)\parallel}_{s,\partial B}+{\parallel {d}^{\ast}dG(u)\parallel}_{s,\partial B}+{\parallel dG(u)\parallel}_{s,\partial B}+{\parallel {d}^{\ast}G(u)\parallel}_{s,\partial B}+{\parallel G(u)\parallel}_{s,\partial B}=0$ since $|\partial B|=0$. Therefore, we obtain

Hence, we have the following lemma.

**Lemma 2.1**

*Let*

*u*

*be a smooth differential form defined in*

*M*

*and*$1<s<\mathrm{\infty}$.

*Then there exists a positive constant*$C=C(s)$,

*independent of*

*u*,

*such that*

*for any ball*$B\subset M$.

The following results about the homotopy operator *T* can be found in [13].

**Lemma 2.2**

*Let*$u\in {L}_{\mathrm{loc}}^{s}(D,{\wedge}^{l})$, $l=1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form in a bounded and convex domain*$D\subset {\mathbb{R}}^{n}$,

*and let*

*T*

*be the homotopy operator defined on differential forms*.

*Then there is also a decomposition*$u=Td(u)+dT(u)$

*and*

*l*-form ${\omega}_{D}\in {D}^{\prime}(D,{\wedge}^{l})$ by

for all $\omega \in {L}^{p}(D,{\wedge}^{l})$, $1\le p<\mathrm{\infty}$.

We will use the following generalized Hölder’s inequality repeatedly in this paper.

**Lemma 2.3**

*Let*$0<\alpha <\mathrm{\infty}$, $0<\beta <\mathrm{\infty}$

*and*${s}^{-1}={\alpha}^{-1}+{\beta}^{-1}$.

*If*

*f*

*and*

*g*

*are measurable functions on*${\mathbb{R}}^{n}$,

*then*

*for any*$E\subset {\mathbb{R}}^{n}$.

We now prove the following ${L}^{s}$ norm inequality for the composite operator $D\circ G$ of the Dirac operator *D* and Green’s operator *G* applied to differential forms.

**Lemma 2.4**

*Let*$u\in {L}_{\mathrm{loc}}^{s}(M,{\wedge}^{l})$, $l=0,1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form in a domain*

*M*,

*D*

*be the Dirac operator and*

*G*

*be Green’s operator*.

*Then there exists a constant*

*C*,

*independent of*

*u*,

*such that*

*for all balls*$B\subset M$.

*Proof*Since the Dirac operator

*D*can be expressed as $D=d+{d}^{\star}$, using Lemma 2.1, we have

We have completed the proof of Lemma 2.4.

Next, we prove the Poincaré-type inequality for the composition of the Dirac operator and Green’s operator, which forms the foundation of this paper. □

**Theorem 2.5**

*Let*$u\in {L}_{\mathrm{loc}}^{s}(M,{\wedge}^{l})$, $l=0,1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form in a domain*

*M*,

*D*

*be the Dirac operator and*

*G*

*be Green’s operator*.

*Then there exists a constant*

*C*,

*independent of*

*u*,

*such that*

*for all balls*$B\subset M$.

*Proof*Applying the decomposition of differential forms described in Lemma 2.2 to the form $D(G(u))$ yields

*T*is the homotopy operator appearing in Lemma 2.2. From Lemma 2.2, for any differential form

*v*, we have

*v*. Replacing

*v*by $d(D(G(u)))$ in (2.5) yields

We have completed the proof of Theorem 2.5. □

## 3 Upper bounds for Lipschitz and BMO norms

In this section, we establish the upper bounds for Lipschitz norms and BMO norms in terms of ${L}^{s}$ norms. Using Theorem 2.5, we now obtain the upper bounds for Lipschitz norm of the composite operator $D\circ G$.

**Theorem 3.1**

*Let*$u\in {L}_{\mathrm{loc}}^{s}(M,{\wedge}^{l})$, $l=0,1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form in a domain*

*M*,

*D*

*be the Dirac operator and*

*G*

*be Green’s operator*.

*Then there exists a constant*

*C*,

*independent of*

*u*,

*such that*

*where**k**is a constant with*$0\le k\le 1$.

*Proof*From Theorem 2.5, we find that

The proof of Theorem 3.1 has been completed.

*M*be a bounded domain. Then from the definitions of the Lipschitz and BMO norms, we know that

where ${C}_{1}$ is a positive constant. Hence, we have the following inequality between the Lipschitz norm and the BMO norm. □

**Lemma 3.2**

*If a differential form*$u\in {\mathrm{locLip}}_{k}(M,{\wedge}^{l})$, $l=0,1,\dots ,n$, $0\le k\le 1$,

*in a bounded domain*

*M*,

*then*$u\in BMO(M,{\wedge}^{l})$

*and*

*where**C**is a constant*.

Combining Theorems 3.1 and Lemma 3.2, we obtain the following inequality between the BMO norm and the ${L}^{s}$ norm.

**Theorem 3.3**

*Let*$u\in {L}^{s}(M,{\wedge}^{l})$, $l=0,1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form in a domain*

*M*,

*D*

*be the Dirac operator and*

*G*

*be Green’s operator*.

*Then there exists a constant*

*C*,

*independent of*

*u*,

*such that*

*Proof*Since inequality (3.5) holds for any differential form, we may replace

*u*by $D(G(u))$ in inequality (3.5) and obtain

*k*is a constant with $0\le k\le 1$. On the other hand, from Theorem 3.1, we have

Combining (3.7) and (3.8) gives ${\parallel D(G(u))\parallel}_{\star ,M}\le {C}_{3}{\parallel u\parallel}_{s,M}$. The proof of Theorem 3.3 has been completed. □

We will need the following lemma that appeared in [14].

**Lemma 3.4**

*Let*

*φ*

*be a strictly increasing convex function on*$[0,\mathrm{\infty})$

*with*$\phi (0)=0$,

*and let*

*E*

*be a bounded domain in*${\mathbb{R}}^{n}$.

*Assume that*

*u*

*is a smooth differential form in*

*E*

*such that*$\phi (k(|u|+|{u}_{E}|))\in {L}^{1}(E;\mu )$

*for any real number*$k>0$

*and*$\mu (\{x\in E:|u-{u}_{E}|>0\})>0$,

*where*

*μ*

*is a Radon measure defined by*$d\mu =w(x)\phantom{\rule{0.2em}{0ex}}dx$

*for a weight*$w(x)$.

*Then for any positive constant*

*a*,

*we have*

*where**C**is a positive constant*.

The following $WRH$-class of differential forms was introduced in [15].

**Definition 3.5**We say a differential form $u\in {\wedge}^{l}(E)$ belongs to the $WRH({\wedge}^{l},E)$-class and write $u\in WRH({\wedge}^{l},E)$, $l=0,1,2,\dots ,n$, if for any constants $0<s,t<\mathrm{\infty}$, the inequality

holds for any ball *B* with $\sigma B\subset E$, where $\sigma >1$ and $C>0$ are constants.

It is well known that any solutions of *A*-harmonic equations belong to $WRH$-class, see [16–20] for example. Hence, the $WRH$-class is a large set of differential forms.

**Theorem 3.6**

*Let*$u\in {L}_{\mathrm{loc}}^{s}(M,{\wedge}^{1})$, $l=0,1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form such that*$u-{u}_{B}\in WRH({\wedge}^{l},M)$-

*class and the Lebesgue measure*$|\{x\in B:|u-{u}_{B}|>0\}|>0$

*for any ball*$B\subset M$.

*Assume that*

*D*

*is the Dirac operator*,

*and*

*G*

*is Green’s operator*.

*Then there exists a constant*

*C*,

*independent of*

*u*,

*such that*

*where**k**is a constant with*$0\le k\le 1$.

*Proof*Using Lemma 3.4 with $\phi (t)={t}^{s}$, $w(x)=1$ over the ball

*B*, we have

*B*with $B\subset M$, we find that

The proof of Theorem 3.6 has been completed. □

Replacing *u* by $D(G(u))$ in Lemma 3.2, we obtain the following comparison inequality between the Lipschitz norm and the BMO norm.

**Corollary 3.7**

*Let*$u\in {L}^{s}(M,{\wedge}^{l})$, $l=0,1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form in a domain*

*M*,

*D*

*be the Dirac operator and*

*G*

*be Green’s operator*.

*Then there exists a constant*

*C*,

*independent of*

*u*,

*such that*

## 4 Weighted inequalities

*M*is a bounded domain, the measure

*μ*is defined by $d\mu =w(x)\phantom{\rule{0.2em}{0ex}}dx$,

*w*is a weight. For convenience, we use the following simple notation ${\mathrm{locLip}}_{k}(M,{\wedge}^{l})$ for ${\mathrm{locLip}}_{k}(M,{\wedge}^{l},w)$. Similarly, for $\omega \in {L}_{\mathrm{loc}}^{1}(M,{\wedge}^{l},w)$, $l=0,1,\dots ,n$, we will write $\omega \in BMO(M,{\wedge}^{l},w)$ if

for some $\sigma >1$, where the measure *μ* is defined by $d\mu =w(x)\phantom{\rule{0.2em}{0ex}}dx$, *w* is a weight. Again, we write $BMO(\mathrm{\Omega},{\wedge}^{l})$ to replace $BMO(M,{\wedge}^{l},w)$ when it is clear that the integral is weighted.

**Definition 4.1**We say the weight $w(x)$ satisfies the ${A}_{r}(M)$ condition, $r>1$, write $w\in {A}_{r}(M)$, if $w(x)>0$ a.e., and

for any ball $B\subset M$.

for all balls *B* with $\sigma B\subset M$, where $\sigma >1$ is a constant.

**Theorem 4.2**

*Let*$u\in {L}^{s}(M,{\wedge}^{1},\mu )$, $l=0,1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form in a bounded domain*

*M*

*such that*$u\in WRH({\wedge}^{l},M)$-

*class*.

*Assume that*

*D*

*is the Dirac operator*,

*and*

*G*

*is Green’s operator*,

*where the measure*

*μ*

*is defined by*$d\mu =w\phantom{\rule{0.2em}{0ex}}dx$

*and*$w\in {A}_{r}(M)$

*for some*$r>1$

*with*$w(x)\ge \epsilon >0$

*for any*$x\in M$.

*Then there exists a constant*

*C*,

*independent of*

*u*,

*such that*

*where**k**is a constant with*$0\le k\le 1$.

*Proof*Since $w(x)\ge \epsilon >0$, we have

*B*. Using (4.3) and the Hölder inequality with $1=1/s+(s-1)/s$, we find that

since $-1/s-k/n+1+1/n>0$ and $|M|<\mathrm{\infty}$. We have completed the proof of Theorem 4.2.

*M*. From the definitions of the weighted Lipschitz and the weighted BMO norms, we have

where ${C}_{1}$ is a positive constant. Thus, we have obtained the following result. □

**Theorem 4.3**

*Let*$u\in {\mathrm{locLip}}_{k}(M,{\wedge}^{l},\mu )$, $l=0,1,\dots ,n$, $0\le k\le 1$,

*be any differential form in a bounded domain*

*M*,

*where the measure*

*μ*

*is defined by*$d\mu =w\phantom{\rule{0.2em}{0ex}}dx$

*and*$w\in {A}_{r}(M)$

*for some*$r>1$.

*Then*$u\in BMO(\mathrm{\Omega},{\wedge}^{l},w)$

*and*

*where**C**is a constant*.

**Theorem 4.4**

*Let*$u\in {L}^{s}(M,{\wedge}^{1},\mu )$, $l=0,1,2,\dots ,n$, $1<s<\mathrm{\infty}$,

*be a differential form in a bounded domain*

*M*

*such that*$u\in WRH({\wedge}^{l},M)$-

*class*.

*Assume that*

*D*

*is the Dirac operator*,

*and*

*G*

*is Green’s operator*,

*where the measure*

*μ*

*is defined by*$d\mu =w\phantom{\rule{0.2em}{0ex}}dx$

*and*$w\in {A}_{r}(M)$

*for some*$r>1$

*with*$w(x)\ge \epsilon >0$

*for any*$x\in M$.

*Then there exists a constant*

*C*,

*independent of*

*u*,

*such that*

*holds for any bounded domain**M*.

*Proof*Replacing

*u*by $D(G(u))$ in Theorem 4.4, we have

*k*is a constant with $0\le k\le 1$. Using Theorem 4.3, we obtain

This ends the proof of Theorem 4.4. □

## 5 Applications

As applications of the results proved in this paper, we consider the following examples.

**Example 5.1**Let

*u*be a differential 1-form in ${\mathbb{R}}^{3}-\{(0,0,0)\}$ defined by

*r*such that $(0,0,0)\notin \overline{B}$. It is easy to see that

*u*be the 1-form discussed in Example 5.1, using Theorem 3.3, we obtain an upper bound for ${\parallel D(G(u))\parallel}_{\star ,M}$ as follows

In fact, it would be very hard to estimate ${\parallel D(G(u))\parallel}_{\star ,M}$ directly from calculation of the operator norm.

**Example 5.2**Let $u(x,y,z)$ be a 1-form defined in ${\mathbb{R}}^{3}$ by

**Remark** (i) The Poincaré-type inequalities for the composition of the Dirac operator and Green’s operator presented in (2.3) and (4.3) can be extended into the global case. (ii) It should be noticed that the domains involved in this paper are general bounded domains, which largely increases the flexibility and applicability of our results.

## Declarations

## Authors’ Affiliations

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