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

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.

1 Introduction

The purpose of this paper is to derive the norm inequalities for the composite operator DG 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 DG 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 , where d is the exterior derivative, and d 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 [18] 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 R n , n2, throughout this paper. We use σB to express the ball with the same center as B and with diam(σB)=σdiam(B), σ>0. We do not distinguish the balls from cubes in this paper. We use |E| to denote the Lebesgue measure of a set E R n . We call w a weight if w L loc 1 ( R n ) and w>0 a.e. Let e 1 , e 2 ,, e n be the standard unit basis of R n , and let l = l ( R n ) be the linear space of l-vectors, which is spanned by the exterior products e I = e i 1 e i 2 e i l , corresponding to all ordered l-tuples I=( i 1 , i 2 ,, i l ), 1 i 1 < i 2 << i l n, l=0,1,,n. The Grassman algebra = l is a graded algebra with respect to the exterior products. For any α= α I e I and β= β I e I , the inner product in is defined by α,β= α I β I , with summation over all l-tuples I=( i 1 , i 2 ,, i l ) and all integers l=0,1,,n. The Hodge star operator : is defined by the rule 1= e 1 e 2 e n and αβ=βα=α,β(1) for all α, β. The norm of α is given by the formula | α | 2 =α,α=(αα) 0 =R. The Hodge star is an isometric isomorphism on Λ with : l n l and ( 1 ) l ( n l ) : l l .

A differential l-form ω on M is a de Rham current (see [[9], Chapter III]) on M with values in l ( R n ). Differential forms are extensions of functions. For example, in R n , the function u( x 1 , x 2 ,, x n ) is called a 0-form. Moreover, if u( x 1 , x 2 ,, x n ) is differentiable, then it is called a differential 0-form. The 1-form u(x) in R n can be written as u(x)= i = 1 n u i ( x 1 , x 2 ,, x n )d x i . If the coefficient functions u i ( x 1 , x 2 ,, x n ), i=1,2,,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 d x i 2 d x i k }, k=1,2,,n, that is, u(x)= I ω I (x)d x I = ω i 1 i 2 i k (x)d x i 1 d x i 2 d x i k , where I=( i 1 , i 2 ,, i k ), 1 i 1 < i 2 << i k n. Let D (M, l ) be the space of all differential l-forms on M, and let L p (M, l ) be the l-forms ω(x)= I ω I (x)d x I on M satisfying M | ω I | p < for all ordered l-tuples I, l=1,2,,n. We denote the exterior derivative by d: D (M, l ) D (M, l + 1 ) for l=0,1,,n1. The Hodge codifferential operator d : D (M, l + 1 ) D (M, l ) is given by d = ( 1 ) n l + 1 d on D (M, l + 1 ), l=0,1,,n1. The Dirac operator D involved in this paper is defined by D=d+ d . It is easy to check that D 2 =Δ, where Δ=d d + d d is the Laplace-Beltrami operator. Let l M be the l th exterior power of the cotangent bundle, C ( l M) be the space of smooth l-forms on M and W( l M)={u L loc 1 ( l M):u has generalized gradient}. The harmonic l-fields are defined by H( l M)={uW( l M):du= d u=0,u L p  for some 1<p<}. The orthogonal complement of in L 1 is defined by H ={u L 1 :u,h=0 for all hH}. Then the Green’s operator G is defined as G: C ( l M) H C ( l M) by assigning G(u) to be the unique element of H C ( l M) satisfying Poisson’s equation ΔG(u)=uH(u), where H is the harmonic projection operator that maps C ( l M) onto so that H(u) is the harmonic part of u. See [10] for more properties of these operators. We write u s , M = ( M | u | s ) 1 / s and u s , M , w = ( M | u | s w ( x ) d x ) 1 / s , where w(x) is a weight.

Let ω L loc 1 (M, l ), l=0,1,,n. We write ω locLip k (M, l ), 0k1, if

ω locLip k , M = sup σ Q M | Q | ( n + k ) / n ω ω Q 1 , Q <
(1.1)

for some σ1. Further, we write Lip k (M, l ) for those forms, whose coefficients are in the usual Lipschitz space with exponent k and write ω Lip k , M for this norm. Similarly, for ω L loc 1 (M, l ), l=0,1,,n, we write ωBMO(M, l ) if

ω , M = sup σ Q M | Q | 1 ω ω Q 1 , Q <
(1.2)

for some σ1. 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

In this section, we will develop Poincaré-type inequality with L s norm for the composite operator DG. This inequality will be used to prove other results in this paper. Using the same way in the proof of Propositions 5.15 and 5.17 in [12], we can prove that for any closed ball B ¯ =BB, we have

d d G ( u ) s , B ¯ + d d G ( u ) s , B ¯ + d G ( u ) s , B ¯ + d G ( u ) s , B ¯ + G ( u ) s , B ¯ C(s) u s , B ¯ .

Note that for any Lebesgue measurable function f defined on a Lebesgue measurable set E with |E|=0, we have E fdx=0. Thus, u s , B =0 and d d G ( u ) s , B + d d G ( u ) s , B + d G ( u ) s , B + d G ( u ) s , B + G ( u ) s , B =0 since |B|=0. Therefore, we obtain

d d G ( u ) s , B + d d G ( u ) s , B + d G ( u ) s , B + d G ( u ) s , M B + G ( u ) s , B = d d G ( u ) s , B ¯ + d d G ( u ) s , B ¯ + d G ( u ) s , B ¯ + d G ( u ) s , B ¯ + G ( u ) s , B ¯ C ( s ) u s , B ¯ = C ( s ) u s , B .

Hence, we have the following lemma.

Lemma 2.1Letube a smooth differential form defined inMand1<s<. Then there exists a positive constantC=C(s), independent ofu, such that

d d G ( u ) s , B + d d G ( u ) s , B + d G ( u ) s , B + d G ( u ) s , B + G ( u ) s , B C(s) u s , B

for any ballBM.

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

Lemma 2.2Letu L loc s (D, l ), l=1,2,,n, 1<s<, be a differential form in a bounded and convex domainD R n , and letTbe the homotopy operator defined on differential forms. Then there is also a decompositionu=Td(u)+dT(u)and

T u s , D C|D|diam(D) u s , D .

Using the notation above, we can define the l-form ω D D (D, l ) by

ω D = | D | 1 D ω(y)dy,l=0,and ω D =d(Tω),l=1,2,,n

for all ω L p (D, l ), 1p<.

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

Lemma 2.3Let0<α<, 0<β<and s 1 = α 1 + β 1 . Iffandgare measurable functions on R n , then

f g s , E f α , E g β , E

for anyE R n .

We now prove the following L s norm inequality for the composite operator DG of the Dirac operator D and Green’s operator G applied to differential forms.

Lemma 2.4Letu L loc s (M, l ), l=0,1,2,,n, 1<s<, be a differential form in a domainM, Dbe the Dirac operator andGbe Green’s operator. Then there exists a constantC, independent ofu, such that

D ( G ( u ) ) s , B C u s , B
(2.1)

for all ballsBM.

Proof Since the Dirac operator D can be expressed as D=d+ d , using Lemma 2.1, we have

D ( G ( u ) ) s , B = ( d + d ) G ( u ) s , B ( d G ( u ) s , B + d G ( u ) s , B C u s , B .
(2.2)

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 2.5Letu L loc s (M, l ), l=0,1,2,,n, 1<s<, be a differential form in a domainM, Dbe the Dirac operator andGbe Green’s operator. Then there exists a constantC, independent ofu, such that

D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B C|B|diam(B) u s , B
(2.3)

for all ballsBM.

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

D ( G ( u ) ) =d ( T ( D ( G ( u ) ) ) ) +T ( d ( D ( G ( u ) ) ) ) = ( D ( G ( u ) ) ) B +T ( d ( D ( G ( u ) ) ) ) ,
(2.4)

where T is the homotopy operator appearing in Lemma 2.2. From Lemma 2.2, for any differential form v, we have

T ( v ) s , B C 1 |B|diam(B) v s , B ,
(2.5)

where C 1 is a constant independent of v. Replacing v by d(D(G(u))) in (2.5) yields

T ( d ( D ( G ( u ) ) ) ) s , B C 2 |B|diam(B) d ( D ( G ( u ) ) ) s , B .
(2.6)

Noticing that ( D ( G ( u ) ) ) B =d(T(D(G(u)))) and using (2.6) and Lemma 2.1, we obtain

D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B = T ( d ( D ( G ( u ) ) ) ) s , B C 2 | B | diam ( B ) d ( D ( G ( u ) ) ) s , B C 2 | B | diam ( B ) d ( ( d + d ) G ( u ) ) s , B C 2 | B | diam ( B ) d d G ( u ) s , B C 3 | B | diam ( B ) u s , B ,
(2.7)

that is,

D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B C 3 |B|diam(B) u s , B .

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 DG.

Theorem 3.1Letu L loc s (M, l ), l=0,1,2,,n, 1<s<, be a differential form in a domainM, Dbe the Dirac operator andGbe Green’s operator. Then there exists a constantC, independent ofu, such that

D ( G ( u ) ) locLip k , M C u s , M ,
(3.1)

wherekis a constant with0k1.

Proof From Theorem 2.5, we find that

D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B C 1 |B|diam(B) u s , B
(3.2)

for all balls BM. Using the Hölder inequality with 1=1/s+(s1)/s, we find that

D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B = B | D ( G ( u ) ) ( D ( G ( u ) ) ) B | d x ( B | D ( G ( u ) ) ( D ( G ( u ) ) ) B | s d x ) 1 / s ( B 1 s / ( s 1 ) d x ) ( s 1 ) / s = | B | ( s 1 ) / s D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B = | B | 1 1 / s D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B | B | 1 1 / s ( C 1 | B | diam ( B ) u s , B ) C 2 | B | 2 1 / s + 1 / n u s , B .
(3.3)

Hence, using the definition of the Lipschitz norm, (3.3), and 21/s+1/n1k/n=11/s+1/nk/n>0, we have

D ( G ( u ) ) locLip k , M = sup σ B M | B | ( n + k ) / n D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B = sup σ B M | B | 1 k / n D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B sup σ B M | B | 1 k / n C 2 | B | 2 1 / s + 1 / n u s , B = sup σ B M C 2 | B | 1 1 / s + 1 / n k / n u s , B sup σ B M C 2 | M | 1 1 / s + 1 / n k / n u s , B C 3 sup σ B M u s , B C 3 u s , M .
(3.4)

The proof of Theorem 3.1 has been completed.

We have proved an estimate for the Lipschitz norm locLip k , M in Theorem 3.1. Now, we develop the estimates for the BMO norm , M . Let u locLip k (M, l ), l=0,1,,n, 0k1 and M be a bounded domain. Then from the definitions of the Lipschitz and BMO norms, we know that

u , M = sup σ B M | B | 1 u u B 1 , B = sup σ B M | B | k / n | B | ( n + k ) / n u u B 1 , B sup σ B M | M | k / n | B | ( n + k ) / n u u B 1 , B | M | k / n sup σ B M | B | ( n + k ) / n u u B 1 , B C 1 sup σ B M | B | ( n + k ) / n u u B 1 , B C 1 u locLip k , M ,

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

Lemma 3.2If a differential formu locLip k (M, l ), l=0,1,,n, 0k1, in a bounded domainM, thenuBMO(M, l )and

u , M C u locLip k , M ,
(3.5)

whereCis 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 3.3Letu L s (M, l ), l=0,1,2,,n, 1<s<, be a differential form in a domainM, Dbe the Dirac operator andGbe Green’s operator. Then there exists a constantC, independent ofu, such that

D ( G ( u ) ) , M C u s , M .
(3.6)

Proof Since inequality (3.5) holds for any differential form, we may replace u by D(G(u)) in inequality (3.5) and obtain

D ( G ( u ) ) , M C 1 D ( G ( u ) ) locLip k , M ,
(3.7)

where k is a constant with 0k1. On the other hand, from Theorem 3.1, we have

D ( G ( u ) ) locLip k , M C 2 u s , M .
(3.8)

Combining (3.7) and (3.8) gives D ( G ( u ) ) , M C 3 u s , M . The proof of Theorem 3.3 has been completed. □

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

Lemma 3.4Letφbe a strictly increasing convex function on[0,)withφ(0)=0, and letEbe a bounded domain in R n . Assume thatuis a smooth differential form inEsuch thatφ(k(|u|+| u E |)) L 1 (E;μ)for any real numberk>0andμ({xE:|u u E |>0})>0, whereμis a Radon measure defined bydμ=w(x)dxfor a weightw(x). Then for any positive constanta, we have

E φ ( a | u | ) dμC E φ ( 2 a | u u E | ) dμ,

whereCis a positive constant.

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

Definition 3.5 We say a differential form u l (E) belongs to the WRH( l ,E)-class and write uWRH( l ,E), l=0,1,2,,n, if for any constants 0<s,t<, the inequality

u s , B C | B | ( t s ) / s t u t , σ B
(3.9)

holds for any ball B with σBE, where σ>1 and C>0 are constants.

It is well known that any solutions of A-harmonic equations belong to WRH-class, see [1620] for example. Hence, the WRH-class is a large set of differential forms.

Theorem 3.6Letu L loc s (M, 1 ), l=0,1,2,,n, 1<s<, be a differential form such thatu u B WRH( l ,M)-class and the Lebesgue measure|{xB:|u u B |>0}|>0for any ballBM. Assume thatDis the Dirac operator, andGis Green’s operator. Then there exists a constantC, independent ofu, such that

D ( G ( u ) ) locLip k , M C u , M ,
(3.10)

wherekis a constant with0k1.

Proof Using Lemma 3.4 with φ(t)= t s , w(x)=1 over the ball B, we have

u s , B C 1 u u B s , B .
(3.11)

From Theorem 2.5 and (3.11), we obtain

D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B C 2 | B | diam ( B ) u s , B C 3 | B | diam ( B ) u u B s , B .
(3.12)

From the definition of the Lipschitz norm, the Hölder inequality with 1=1/s+(s1)/s and (3.12), for any ball B with BM, we find that

D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B = B | D ( G ( u ) ) ( D ( G ( u ) ) ) B | d x ( B | D ( G ( u ) ) ( D ( G ( u ) ) ) B | s d x ) 1 / s ( B 1 s s 1 d x ) ( s 1 ) / s = | B | ( s 1 ) / s D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B = | B | 1 1 / s D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B C 4 | B | 2 1 / s + 1 / n u u B s , B .
(3.13)

Next, since u u B WRH( l ,M)-class, we have

u u B s , B C 5 | B | ( 1 s ) / s u u B 1 , σ 1 B
(3.14)

for some constant σ 1 >1. Combination of (3.13) and (3.14) gives

D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B C 4 | B | 2 1 / s + 1 / n u u B s , B C 6 | B | 1 + 1 / n u u B 1 , σ 1 B .
(3.15)

Hence, we obtain

| B | ( n + k ) / n D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B C 6 | B | 1 / n k / n u u B 1 , σ 1 B = C 6 | B | 1 + 1 / n k / n | B | 1 u u B 1 , σ 1 B C 7 | B | 1 + 1 / n k / n | σ 1 B | 1 u u B 1 , σ 1 B C 7 | M | 1 + 1 / n k / n | σ 1 B | 1 u u B 1 , σ 1 B C 8 | σ 1 B | 1 u u B 1 , σ 1 B .
(3.16)

Thus, taking the supremum on both sides of (3.16) over all balls σ 2 BM with σ 2 > σ 1 and using the definitions of the Lipschitz and BMO norms, we find that

D ( G ( u ) ) locLip k , M = sup σ 2 B M | B | ( n + k ) / n D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B C 7 sup σ 2 B M | σ 1 B | 1 u u B 1 , σ 1 B C 7 u , M ,
(3.17)

that is,

D ( G ( u ) ) locLip k , M C u , M .
(3.18)

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 3.7Letu L s (M, l ), l=0,1,2,,n, 1<s<, be a differential form in a domainM, Dbe the Dirac operator andGbe Green’s operator. Then there exists a constantC, independent ofu, such that

D ( G ( u ) ) , M C D ( G ( u ) ) locLip k , M .
(3.19)

4 Weighted inequalities

In this section, we establish the weighted norm comparison inequalities for the composition of the Dirac operator and Green’s operator applied to differential form defined in a domain M R n . For ω L loc 1 (M, l , w α ), l=0,1,,n, we write ω locLip k (M, l , w α ), 0k1 if

ω locLip k , M , w α = sup σ Q M ( μ ( Q ) ) ( n + k ) / n ω ω Q 1 , Q , w α <
(4.1)

for some σ>1, where M is a bounded domain, the measure μ is defined by dμ=w(x)dx, w is a weight. For convenience, we use the following simple notation locLip k (M, l ) for locLip k (M, l ,w). Similarly, for ω L loc 1 (M, l ,w), l=0,1,,n, we will write ωBMO(M, l ,w) if

ω , M , w = sup σ Q M ( μ ( Q ) ) 1 ω ω Q 1 , Q , w <
(4.2)

for some σ>1, where the measure μ is defined by dμ=w(x)dx, w is a weight. Again, we write BMO(Ω, l ) to replace BMO(M, 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 A r (M), if w(x)>0 a.e., and

sup B ( 1 | B | B w d x ) ( 1 | B | B ( 1 w ) 1 / ( r 1 ) d x ) ( r 1 ) <

for any ball BM.

For uWRH( l ,M), using the Hölder inequality, we extend inequality (2.3) into the following weighted version

D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B , w C 3 |B|diam(B) u s , σ B , w
(4.3)

for all balls B with σBM, where σ>1 is a constant.

Theorem 4.2Letu L s (M, 1 ,μ), l=0,1,2,,n, 1<s<, be a differential form in a bounded domainMsuch thatuWRH( l ,M)-class. Assume thatDis the Dirac operator, andGis Green’s operator, where the measureμis defined bydμ=wdxandw A r (M)for somer>1withw(x)ε>0for anyxM. Then there exists a constantC, independent ofu, such that

D ( G ( u ) ) locLip k , M , w C u s , M , w ,
(4.4)

wherekis a constant with0k1.

Proof Since w(x)ε>0, we have

μ(B)= B wdx B εdx= C 1 |B|,

which gives

1 μ ( B ) C 2 | B |
(4.5)

for any ball B. Using (4.3) and the Hölder inequality with 1=1/s+(s1)/s, we find that

D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B , w = B | D ( G ( u ) ) ( D ( G ( u ) ) ) B | d μ ( B | D ( G ( u ) ) ( D ( G ( u ) ) ) B | s d μ ) 1 / s ( B 1 s / ( s 1 ) d μ ) ( s 1 ) / s = ( μ ( B ) ) ( s 1 ) / s D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B , w = ( μ ( B ) ) 1 1 / s D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B , w ( μ ( B ) ) 1 1 / s ( C 3 | B | diam ( B ) u s , σ B , w ) C 4 ( μ ( B ) ) 1 1 / s | B | 1 + 1 / n u s , σ B , w .
(4.6)

From the definition of the weighted Lipschitz norm, (4.5) and (4.6), we obtain

D ( G ( u ) ) locLip k , M , w = sup σ B M ( μ ( B ) ) ( n + k ) / n D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B , w = sup σ B M ( μ ( B ) ) 1 k / n D ( G ( u ) ) ( D ( G ( u ) ) ) B 1 , B , w C 5 sup σ B M ( μ ( B ) ) 1 / s k / n | B | 1 + 1 / n u s , σ B , w C 6 sup σ B M | B | 1 / s k / n + 1 + 1 / n u s , σ B , w C 6 sup σ B M | M | 1 / s k / n + 1 + 1 / n u s , σ B , w C 6 | M | 1 / s k / n + 1 + 1 / n sup σ B M u s , σ B , w C 7 u s , M , w
(4.7)

since 1/sk/n+1+1/n>0 and |M|<. We have completed the proof of Theorem 4.2.

Next, we estimate the BMO norm in terms of the L s norm. Let u locLip k (M, l ), l=0,1,,n, 0k1, in a bounded domain M. From the definitions of the weighted Lipschitz and the weighted BMO norms, we have

u , M , w = sup σ B M ( μ ( B ) ) 1 u u B 1 , B , w = sup σ B M ( μ ( B ) ) k / n ( μ ( B ) ) ( n + k ) / n u u B 1 , B , w sup σ B M ( μ ( M ) ) k / n ( μ ( B ) ) ( n + k ) / n u u B 1 , B , w ( μ ( M ) ) k / n sup σ B M ( μ ( B ) ) ( n + k ) / n u u B 1 , B , w C 1 sup σ B M ( μ ( B ) ) ( n + k ) / n u u B 1 , B , w C 1 u locLip k , M , w ,
(4.8)

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

Theorem 4.3Letu locLip k (M, l ,μ), l=0,1,,n, 0k1, be any differential form in a bounded domainM, where the measureμis defined bydμ=wdxandw A r (M)for somer>1. ThenuBMO(Ω, l ,w)and

u , Ω , w C u locLip k , Ω , w ,
(4.9)

whereCis a constant.

Theorem 4.4Letu L s (M, 1 ,μ), l=0,1,2,,n, 1<s<, be a differential form in a bounded domainMsuch thatuWRH( l ,M)-class. Assume thatDis the Dirac operator, andGis Green’s operator, where the measureμis defined bydμ=wdxandw A r (M)for somer>1withw(x)ε>0for anyxM. Then there exists a constantC, independent ofu, such that

D ( G ( u ) ) , M , w C u s , M , w
(4.10)

holds for any bounded domainM.

Proof Replacing u by D(G(u)) in Theorem 4.4, we have

D ( G ( u ) ) , M , w C 1 D ( G ( u ) ) locLip k , M , w ,
(4.11)

where k is a constant with 0k1. Using Theorem 4.3, we obtain

D ( G ( u ) ) locLip k , M , w C 2 u s , M , w .
(4.12)

Substituting (4.12) into (4.11), we obtain

D ( G ( u ) ) , M , w C 3 u s , M , w .

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 R 3 {(0,0,0)} defined by

u(x,y,z)= x d x x 2 + y 2 + z 2 + y d y x 2 + y 2 + z 2 + z d z x 2 + y 2 + z 2 ,
(5.1)

which can be considered as a vector field in R 3 . Let B R 3 be a ball with radius r such that (0,0,0) B ¯ . It is easy to see that

u s , B = ( B | u | s d x d y d z ) 1 / s = ( B 1 d x d y d z ) 1 / s = | B | 1 / s .

Using Theorem 2.5, we obtain an upper bound C | B | 1 + 1 / s diam(B) for the complicated operator norm D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B . Specifically, we have

D ( G ( u ) ) ( D ( G ( u ) ) ) B s , B C | B | 1 + 1 / s diam(B)=2rC ( 4 3 π r 3 ) 1 + 1 / s .

Choosing M=B and u be the 1-form discussed in Example 5.1, using Theorem 3.3, we obtain an upper bound for D ( G ( u ) ) , M as follows

D ( G ( u ) ) , M C ( 4 3 π r 3 ) 1 / s .

In fact, it would be very hard to estimate D ( G ( u ) ) , M directly from calculation of the operator norm.

Example 5.2 Let u(x,y,z) be a 1-form defined in R 3 by

u(x,y,z)= 2 3 π ( arctan y x 1 d x + arctan y x + 1 d y + arctan z x 2 + 1 d z ) .
(5.2)

Let r>0 be a constant, ( x 0 , y 0 , z 0 ) be a fixed point with x 0 >2r, y 0 >2r, z 0 >2r and

M= { ( x , y , z ) : ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 r 2 } .

It would be very complicated for us to obtain the upper bound for the Lipschitz norm D ( G ( u ) ) locLip k , M if we evaluated

D ( G ( u ) ) locLip k , M = sup σ Q M | Q | ( n + k ) / n D ( G ( u ) ) ( D ( G ( u ) ) ) Q 1 , Q

directly. However, using Theorem 3.1, we can easily obtain the upper bound of the norm D ( G ( u ) ) locLip k , M as follows. First, we know that |M|= 4 3 π r 3 and

| u ( x , y , z ) | 2 3 π ( | arctan y x 1 | + | arctan y x + 1 | + | arctan z x 2 + 1 | ) 2 3 π ( π 2 + π 2 + π 2 ) = 1 .
(5.3)

Applying (3.1), we have

D ( G ( u ) ) locLip k , M C u s , M = C ( M | u ( x , y , z ) | s d x d y d z ) 1 / s C ( M 1 d x d y d z ) 1 / s = C | M | 1 / s = C ( 4 3 π r 3 ) 1 / s .

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.

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Correspondence to Bing Liu.

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DS had the first drift of the paper and BL added some contents and re-organized the paper.

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Ding, S., Liu, B. Norm inequalities for composition of the Dirac and Green’s operators. J Inequal Appl 2013, 436 (2013). https://doi.org/10.1186/1029-242X-2013-436

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Keywords

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