BMO and Lipschitz norm estimates for the composition of Green’s operator and the potential operator

R E S E A R C H

Open Access

BMO and Lipschitz norm estimates for the

composition of Green’s operator and the

potential operator

Zhimin Dai

_{, Yuming Xing}

_{and Shusen Ding}

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Harbin Institute of Technology, Harbin, China

Full list of author information is available at the end of the article

Abstract

In this paper, we establish BMO and Lipschitz norm inequalities for the composition of Green’s operator and the potential operator. We also investigate the relationship among the Lipschitz norm, the BMO norm and theLp-norm. Finally, we display some examples for applications.

MSC: Primary 35J60; secondary 31B05; 58A10; 46E35

Keywords: differential forms; Lipschitz norm; BMO norm; Green’s operator; potential operator

1 Introduction

Differential forms are extensions of functions and can be used to describe various systems in partial differential equations (or PDEs), physics, theory of elasticity, quasiconformal analysis,etc. Differential forms have become invaluable tools for many fields of sciences and engineering; see [, ] for more details.

Now we introduce some notations and definitions. Letbe an open subset ofRn_(n≥)

andObe a ball inRn_{. LetρOdenote the ball with the same center asOanddiam(ρO) =}

ρdiam(O),ρ> . A weightw(x) is a nonnegative locally integrable function inRn_.|D|is

used to denote the Lebesgue measure of a setD⊂Rn_{. Let∧=∧(R}n_{),= , , . . . ,n, be}

the linear space of all-forms(x) =_JJ(x)dxJ=

Jjj···j(x)dxj∧dxj· · ·∧dxjinR n_,

whereJ= (j,j, . . . ,j), ≤j<j<· · ·<j≤n, are the ordered-tuples. Moreover, if each

of the coefficientJ(x) of(x) is differential on, then we call(x) a differential-form

onand useD(,∧_{) to denote the space of all differential-forms on.C}∞_(,∧)

de-notes the space of smooth-forms on. We denote the exterior derivative byd, and the Hodge codifferential operatord_{is defined asd= (–)}n+_d:D(,+_)→D(,),

where is the Hodge star operator. For ≤p<∞, Lp_{(,∧) is a Banach space with}

the norm p,= (

|(x)|

p_dx)/p_{= (} (

J|J(x)|)p/dx)/p<∞. For a weightw(x), we

write p,,w= (

||pw(x)dx)/p. Similarly, the notationsL p

loc(,∧

_{) andW},p

loc(,∧

_{) are}

self-explanatory.

From [], ifis a differential form in a bounded convex domain, then there is a de-composition

=d(T) +T(d), (.)

whereTis called a homotopy operator. For the homotopy operator, we know that

T p,O≤C|O|diam(O) p,O (.)

holds for any differential form∈Lp_loc(O,∧_{),= , , . . . ,n,  <p<∞. Furthermore, we}

can define the-form∈D(,∧) by

=

⎧ ⎨ ⎩

||–

(y)dy, = ;

d(T), = , . . . ,n (.)

for all∈Lp_{(,∧), ≤p<∞.}

In this paper, we focus on a class of differential forms satisfying the well-known nonho-mogeneous A-harmonic equation

dA(x,d) =B(x,d), (.)

whereA:× ∧(Rn_{)→ ∧(R}n_{) andB:× ∧(R}n_{)→ ∧}–_(Rn_{) satisfy the conditions:} |A(x,η)| ≤a|η|s–_{,A(x,η)·η≥ |η|}s_{and|B(x,η)| ≤b|η|}s–_{for almost everyx∈and all} η∈ ∧_(Rn_{). Herea,b>  are some constants and  <s<∞is a fixed exponent associated}

with (.). A solution to (.) is an element of the Sobolev spaceW_loc,s(,∧–_{) such that}

A(x,d)·dψ+B(x,d)·ψ=  (.)

for all ψ ∈W_loc,s(,∧–_{) with compact support. The various deformations of (.) are}

shown in [].

Recently, Bi extended the definition of a potential operator to the set of all differential forms in []. For any differential -form(x) =_JJ(x)dxJ, the potential operatorPis

defined by

P(x) =P J

J(x)dxJ

=

J

PJ(x)

dxJ= J

K(x,y)J(y)dy dxJ, (.)

where the kernelK(x,y) is a non-negative measurable function defined forx=y,J(x) is

defined on⊂Rn and the summation is over all ordered-tuplesJ. For more results related to the potential operatorP, see [–].

Green’s operator and the potential operator are of quite importance in the study of po-tential theory and nonlinear elasticity; see [, , , –] for more properties of these two operators. In many situations, the process of studying solutions of PDEs involves estimat-ing the various norms of the operators. However, the study on the composition of the potential operator and other operators is yet to be fully developed. Hence, we are mo-tivated to establish some norm inequalities for the composite operatorG◦Papplied to differential forms.

Let∈L loc(,∧

_{),= , , . . . ,n. We write∈locLip}

k(,∧), ≤k≤, if locLipk,= sup

ρO⊂|

O|–(n+k)/n –O ,O<∞ (.)

for someρ≥. Further, we writeLip_k(,∧_{) for those forms whose coefficients are in}

the usual Lipschitz space with exponentkand write Lip_k,for this norm. Similarly, for ∈L

loc(,∧

_{),= , , . . . ,n, we write∈BMO(,∧) if} ,= sup

ρO⊂

|O|– –O ,O<∞ (.)

for someρ≥. Whenis a -form, equation (.) reduces to the classical definition of

BMO(). As to the definitions of the weighted Lipschitz and BMO norms, we will present them in Section .

The purpose of this paper is to derive the Lipschitz and BMO norm inequalities for the composition of Green’s operator Gand the potential operatorP applied to differential forms.

2 Estimates for Lipschitz and BMO norms

In this section, we establish the estimates for Lipschitz and BMO norms for the composite operatorG◦P. We need the following lemmas and definition.

The following inequality is the well-known Hölder inequality and gets proved with the Cauchy-Schwarz inequality in [].

Lemma . Let (,μ) be a measure space and Lp_{(μ) = L}p_{(,μ) ={f : ⊂ R}n _→

C; f p,,μ<∞}be a Lebesgue space with the Lp-norm

f p,,μ=

⎧ ⎨ ⎩

( f p_dμ)/p_{, ≤p<∞;} ess sup_x∈|f(x)|, p=∞.

(.)

If p,q≥with/p+ /q= ,and if f ∈Lp_{(μ)and g∈L}q_{(μ),then fg∈L}_(μ)and

fg ,,μ≤ f p,,μ g q,,μ. (.)

Remark Ifμis a Lebesgue measure, that is,dμ=dx, then (.) reduces to the inequality

fg ,≤ f p, g q,. (.)

Lemma .[] Let∈D(,∧_{)be a solution to the nonhomogeneous A-harmonic}

equa-tion(.)onandρ> be a constant.Then there exists a constant C,independent of,

such that

d p,O≤Cdiam(O)– –c p,ρO (.)

Lemma .[] Letbe a solution of the nonhomogeneous A-harmonic equation(.)in a domainand <s,t<∞.Then there exists a constant C,independent of,such that

s,O≤C|O|(t–s)/ts t,ρO (.)

for all balls O withρO⊂,whereρ> is a constant. The following definition is introduced in [].

Definition . A kernelKonRn_×Rn_{(n≥) is said to satisfy the standard estimates if}

there existα,  <α≤, and a constantCsuch that for all distinct pointsxandyinRnand allzwith|x–z|<

|x–y|,

() K(x,y)≤C|x–y|–n; () K(x,y) –K(z,y)≤Cx–z

x–y

α|x–y|–n;

() K(z,y) –K(x,y)≤Cx–z x–y

α|x–y|–n. (.)

The followingLp_{-norm and Lipschitz norm inequalities for the compositionG◦P of}

Green’s operator and the potential operator appear in [].

Lemma . Let∈Lp_{(,∧),= , , . . . ,n,  <p<∞,be a differential form in a bounded} convex domain⊂Rn,P be the potential operator defined in(.)with the kernel K(x,y)

satisfying the condition()of the standard estimates(.)and G be Green’s operator.Then there exists a constant C,independent of,such that

GP()–GP()_Op,O≤C|O|diam(O) p,O (.)

for all balls O with O⊂.

Lemma . Let∈Lp(,∧_{),= , , . . . ,n,  <p<∞,be a differential form in a bounded} domain,P be the potential operator defined in(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.)and G be Green’s operator.Then there exists a constant C,independent of,such that

GP()_locLip

k,≤C p,, (.)

where k is a constant with≤k≤.

Lemma .[] Letϕbe a strictly increasing convex function on[,∞)withϕ() =  (that is,ϕis a Young function),and D be a bounded domain inRn.Assume thatis a smooth differential form in D such thatϕ(k(||+|D|))∈L(D;μ)for any real number k> and

μ({x∈D:|–D|> }) > ,whereμis a Radon measure defined by dμ=w(x)dx for a weight w(x).Then,for any positive constant a,we have

D

ϕa||dμ≤C

D

ϕa|–D|

dμ, (.)

Using Lemma . withϕ(t) =tp_{andw(x) =  over the ballO, we obtain}

p,O≤C –O p,O, (.)

whereCis a constant.

Theorem . Let∈Lp_{(,∧),= , , . . . ,n,  <p<∞,be a solution of the} nonhomo-geneous A-harmonic equation (.)in a bounded convex domain, P be the potential operator defined in(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.)and G be Green’s operator.Then there exists a constant C,independent of,

such that

GP()_locLip

k,≤C ,, (.)

where k is a constant with≤k≤.

Proof From Lemma . and (.), we obtain

GP()–GP()_Op,O≤C|O|diam(O) p,O

≤C|O|diam(O) –O p,O. (.)

From the decomposition (.), (.) and (.), we have

–O p,O= Td p,O≤C|O|diam(O) d p,O≤C|O||O|/n d p,O. (.)

Combining (.) with (.) yields

GP()–GP()_Op,O≤C|O|diam(O) –O p,O

≤C|O|diam(O)

C|O||O|/n d p,O

≤C|O|+/ndiam(O) d p,O

≤C|O|+/n d p,O. (.)

Using the definition of the Lipschitz norm, (.) with  = /p+ (p– )/pand (.), for any ballOwithO⊂, it follows that

GP()–GP()_O,O=

O

GP()–GP()_Odx

≤

O

_G

P()–GP()_Opdx

/p

O



p p–_dx

(p–)/p

=|O|(p–)/pGP()–GP()_Op,O

≤ |O|–/pC|O|+/n d p,O

From Lemma ., we have

d p,O≤Cdiam(O)– –c p,ρO≤C|O|–/n –c p,ρO (.)

for any closed formcand any ballOwithρO⊂, whereρ>  is a constant.

Sinceis a solution of equation (.) andcis a closed form,–cis also a solution of equation (.). By Lemma ., we obtain

–c p,ρO≤C|O|(–p)/p –c ,ρO (.)

for some constantρ>ρ>  withρO⊂.

Combining (.), (.) and (.), we obtain

GP()–GP()_O,O≤C|O|–/p+/n d p,O ≤C|O|–/p+/n

C|O|–/n –c p,ρO

≤C|O|–/p+/n –c p,ρO ≤C|O|–/p+/n

C|O|(–p)/p –c ,ρO

≤C|O|–/p+/n+/p– –c ,ρO

=C|O|+/n –c ,ρO (.)

for any closed formc.

Sincecis any closed form in (.), we may choosec=ρOin (.). By the definitions

of the Lipschitz norms, and noticing ≤k≤, we find that

GP()_locLip

k,=_ρOsup_⊂|O|

–(n+k)/n_GP()–GP() O,O

≤ sup

ρO⊂

|O|––k/nC|O|+/n –ρO ,ρO

=C sup

ρO⊂|

O|+/n–k/n –ρO ,ρO ≤C sup

ρO⊂

|O|+/n–k/n|ρO|– –ρO ,ρO ≤C sup

ρO⊂|

|+/n–k/n|ρO|– –ρO ,ρO ≤C||+/n–k/n sup

ρO⊂

|ρO|– –ρO ,ρO

≤C ,, (.)

whereρ>ρ>ρwithρO⊂.

The proof of Theorem . has been completed. We have developed some estimates for the Lipschitz norm · locLipk,. Now, we establish

Lemma .[] If a differential form∈locLip_k(,∧_{), = , , . . . ,n, ≤k≤,in a} bounded convex domain,then∈BMO(,∧_)and

,≤C locLipk,, (.)

where C is a constant.

SinceG(P()) is a differential form whenis a differential form, we have the following theorem.

Theorem . If a differential form G(P())∈locLip_k(,∧_{),= , . . . ,n, ≤k≤,in} a bounded convex domain,then there exists a constant C,independent of,such that G(P())∈BMO(,∧_)and

GP()_,≤CGP()_locLipk,, (.)

where the definitions of G and P are the same as in the preceding theorem.

Based on the above results, we estimate the BMO norm · ,of compositionG◦Pin

terms ofLpnorm.

Theorem . Let∈Lp(,∧_{),  <p<∞,be a differential form in a bounded convex} domain,G be Green’s operator and P be the potential operator defined in equation(.)

with the kernel K(x,y)satisfying the condition()of the standard estimates(.).Then there exists a constant C,independent of,such that

GP()_,≤C p,. (.)

Proof From Lemma ., we have

GP()_locLip

k,≤C p,. (.)

Using Theorem . and (.), it follows that

_G

P()_,≤CG

P()_locLip

k,≤C p,. (.)

The proof of Theorem . has been completed. Similar to the proof of Theorem ., using Theorems . and ., we can prove the following theorem.

Theorem . Let ∈ Lp_{(,∧),  <p <∞, be a solution of the nonhomogeneous}

A-harmonic equation (.)in a bounded convex domain,G be Green’s operator and P be the potential operator defined in equation(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.).Then there exists a constant C,independent of,such that

3 Two weight estimates

In this section, we discuss the weighted Lipschitz and BMO norms []. For∈L loc(,

∧_,wα_{),= , , . . . ,n, we write∈locLip}

k(,∧,wα), ≤k≤, if locLipk,,wα= sup

ρO⊂

μ(O)–(n+k)/n –O ,O,wα<∞ (.)

for someρ> , whereis a bounded domain. The measureμis defined bydμ=w(x)α_dx, wis a weight andαis a real number. For convenience, we will write the simple notation

locLip_k(,∧_{) for locLip}

k(,∧,wα). Similarly, for∈Lloc(,∧

_,wα_{),= , , . . . ,n, we}

write∈BMO(,∧_,wα_{) if} ,,wα= sup

ρO⊂

μ(O)– –O ,O,wα<∞ (.)

for someρ> , where the measureμis defined bydμ=w(x)α_{dx,wis a weight andαis a}

real number. Again, we will writeBMO(,∧_{) to replaceBMO(,∧,w}α_{) when it is clear}

that the integral is weighted.

Definition .[] A pair of weights (w(x),w(x)) satisfies theAr,λ()-condition in a set

⊂Rn. Write (w(x),w(x))∈Ar,λ() for someλ≥ and  <r<∞with _r+_r =  if

sup O⊂



|O|

O wλ

dx 

λr _

|O|

O



w λr

r

dx

 λr

<∞. (.)

Lemma .[] Let∈Lp_{(,∧,ν),= , . . . ,n,  <p<∞,be a solution of the} nonho-mogeneous A-harmonic equation(.)in a bounded convex domain,P be the potential operator defined in(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.)and G be Green’s operator.Assume that(w(x),w(x))∈Ar,λ()for some

λ≥and <r<∞.Then there exists a constant C,independent of,such that

GP()–GP()_Op,O,wα

 ≤C|O|diam(O) p,ρO,w α

 (.)

for all balls O withρO⊂,whereρ> andαare two constants with≤α<λ.

Theorem . Let∈Lp_{(,∧,ν),= , . . . ,n,  <p<∞,be a solution of the} nonhomo-geneous A-harmonic equation (.)in a bounded convex domain, P be the potential operator defined in(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.)and G be Green’s operator.The measuresμandνare defined by dμ=wα

dx,

dν=wα

dx,and(w(x),w(x))∈Ar,λ()for someλ≥and <r<∞with w(x)≥> for

any x∈.Then there exists a constant C,independent of,such that

GP()_locLip

k,,wα ≤C p,,w α

, (.)

where k andαare constants with≤k≤and <α≤.

Proof Sinceμ(O) =_Owα

dx≥

O

α_dx=C

|O|, we have

for any ballO. Using (.) and Lemma . with  = /p+ (p– )/p, it follows that

GP()–GP()_O,O,wα 

=

O

GP()–GP()_Odμ

≤

O

GP()–GP()_Opdμ /p

O

p/(p–)dμ (p–)/p

=μ(O)(p–)/pGP()–GP()_Op,O,wα 

=μ(O)–/pGP()–GP()_Op,O,wα 

≤μ(O)–/pC|O|diam(O) p,ρO,wα

≤C

μ(O)–/p|O|+/n p,ρO,wα

. (.)

Notice that  – /p+ /n–k/n>  and||<∞, from (.), (.) and (.), we obtain

GP()_locLip

k,,wα = sup ρO⊂

μ(O)–(n+k)/nGP()–GP()_O,O,wα 

≤ sup

ρO⊂

μ(O)––k/nC

μ(O)–/p|O|+/n p,ρO,wα

=C sup

ρO⊂

μ(O)–/p–k/n|O|+/n p,ρO,wα_

≤C sup

ρO⊂

C|O|– /p+k/n

|O|+/n p,ρO,wα_

≤C sup

ρO⊂

|O|–/p+/n–k/n p,ρO,wα 

≤C sup

ρO⊂|

|–/p+/n–k/n p,ρO,wα_

≤C||–/p+/n–k/n sup

ρO⊂

p,ρO,wα 

≤C p,,wα_ (.)

The proof of Theorem . has been completed. We now estimate the · ,,wα_ norm in terms of theLp-norm.

Theorem . Let∈Lp_{(,∧,ν),= , . . . ,n,  <p<∞,be a solution of the} nonhomo-geneous A-harmonic equation (.)in a bounded convex domain, P be the potential operator defined in(.)with the kernel K(x,y)satisfying the condition()of the standard estimates(.)and G be Green’s operator.The measuresμandνare defined by dμ=wα_dx,

dν=wα

dx,and(w(x),w(x))∈Ar,λ()for someλ≥and <r<∞with w(x)≥> for

any x∈.Then there exists a constant C,independent of,such that

GP()_,,wα

 ≤C p,,w α

, (.)

Proof From (.) and (.), it follows that

,,wα  = sup

ρO⊂

μ(O)– –O ,O,wα 

= sup ρO⊂

μ(O)k/nμ(O)–(n+k)/n –O ,O,wα

≤ sup

ρO⊂

μ()k/nμ(O)–(n+k)/n –O ,O,wα_ ≤μ()k/n sup

ρO⊂

μ(O)–(n+k)/n –O ,O,wα 

≤C sup

ρO⊂

μ(O)–(n+k)/n –O ,O,wα ≤C locLip_k,,wα

, (.)

whereCis a positive constant. ReplacingbyG(P()) in (.), we find that GP()_,,wα

 ≤C

GP()_locLipk,,wα

, (.)

wherekis a constant with ≤k≤. From Theorem ., we obtain

GP()_locLip

k,,wα ≤C p,,w α

, (.)

Substituting (.) into (.), we have

GP()_,,wα

 ≤C p,,w α

. (.)

We have completed the proof of Theorem ..

4 Applications

If we chooseA,Bto be a special operator, for example,A(x,d) =d|d|s–,B= , then (.) reduces to the followings-harmonic equation:

dd|d|s–= . (.)

In particular, we may lets= , then (.) reduces to

d(d) = . (.)

We may make use of the following two specific examples to conform the convenience of the inequality (.) in evaluating the upper bound for the BMO norm ofG(P()). Ob-viously, we may take advantage of (.) to make this estimating process easy, without calculating G(P()) ,in a complicated way.

Example . Let = (x_ +x_ + xx+ xx +xx)dx + (x +x+ xx + xx+

xx)dx+ (x+x+ xx+ xx+ xx)dx,={x= (x,x,x) :|≤x+x+x< },

P be the potential operator defined in (.) with the kernelK(x,y) satisfying the condi-tion () of the standard estimates (.) andGbe Green’s operator.

First, by simple computation, we have

d=xdx∧dx+ xdx∧dx+xdx∧dx, (.) (d) =xdx– xdx+xdx,d

(d)= . (.)

Sinced_{= (–)}×+_{d= –d,}

d(d) = –d(d)= . (.)

This implies thatsatisfies (.). Observe that

p, =

||p_dx

/p

=

x_+x_+ xx+ xx+xx 

+x_+x_+ xx+ xx+ xx 

+ x_+x_+ xx+ xx+ xx p/

dx

/p

≤

( +  +  +  + )

+ ( +  +  +  + ) + ( +  +  +  + )p/dx

/p

≤√||/p

=√

π



/p

. (.)

Applying (.), we obtain

GP()_,≤C p,≤C √



π



/p

. (.)

Example . Let us assume, in addition to the definitions ofGandPof Example .,

=ex_sinx

Similarly, to begin with, we observe that

∂ ∂x

=∂



∂x 

=exsinx, (.) ∂

∂x

=excosx,

∂ ∂x_ = –e

x_sinx

. (.)

Thus,

= , (.)

which implies the functionis harmonic. Observe that

p,=

||p_dx

/p

=

π



dθ 



ercosθ_sin(rsinθ)p_{r dr}

/p

≤e(π·/)/p

=π/pe. (.)

Applying (.), we obtain

_G

P()_,≤C p,≤Cπ/pe. (.)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ZD finished the proof and the writing work. YX gave ZD some excellent advice on the proof and writing. SD gave ZD lots of help in revising the paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Harbin Institute of Technology, Harbin, China.}2_{Department of Mathematics, Seattle}

University, Seattle, WA 98122, USA.

Acknowledgements

The authors wish to thank the anonymous referees for their time and thoughtful suggestions.

Received: 23 October 2012 Accepted: 8 January 2013 Published: 22 January 2013

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doi:10.1186/1029-242X-2013-26