Lipschitz and BMO norm inequalities for the composite operator on differential forms

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R E S E A R C H

Open Access

Lipschitz and BMO norm inequalities for

the composite operator on differential forms

Xuexin Li, Yong Wang and Yuming Xing

*

*_{Correspondence:}

[email protected]

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

Abstract

In this paper, we obtain Poincaré-type inequalities for the composite operator acting on diﬀerential forms and establish theLp_{, Lipschitz, and BMO norm estimates. We also} give the weighted versions of the comparison theorems for theLp_{, Lipschitz, and BMO} norms.

Keywords: norm inequality; diﬀerential form; Dirac operator

1 Introduction

Differential forms are a generalization of the traditional functions. In recent years, differ-ential forms have been widely used in physics systems, differdiffer-ential geometry, and PDEs. In this paper, we are interested in the properties of the composite operator acting on dif-ferential forms. Operator theory plays a critical role in investigating the properties of the solutions to partial differential equations. Many questions in partial differential equations involve estimating various norms of operators. The operator theory for functions has been very well developed in recent years. However, compared to the function cases, the opera-tor theory for differential forms is more complicated, so we need some advanced methods to deal with operators. This paper contributes to derive the properties of the composite operatorMs◦D◦Gon differential forms, whereMsis the general sharp maximal operator

deﬁned by

M

s(u) =M

su(x) =sup r>



|B(x,r)|

B(x,r)

u(t) –uB(x,r) s

dt

/s

for anyu(x)∈Lp₍_M_,l_{), where }_≤_s_≤_p_{. Here}_D_{is the Dirac operator proposed by the}

physicist Dirac. According to the needs of practical problems, different versions of the Dirac operators have been defined. The Dirac operator we are studying is the Hodge-Dirac operator defined byD=d+d∗. Heredis the exterior differential operator on differential forms, andd∗is the formal adjoint operator ofd. See [] for more details. The operatorG

is the well-known Green’s operator satisfying the equation

G(u) =u–H(u),

whereHis the harmonic projection operator. See [–] for more results and applications for the sharp maximal operator, the Dirac operator, and Green’s operator.

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In the following,Mstands for a bounded convex domain inRn_,_n_≥_{. The Lebesgue}

measure of a measurable setE⊆Rn_{is denoted by}_|_E_|_{. We use}_B_and_σ_B_{to denote}

con-centric balls such thatdiam(σB) =σdiam(B). Byl=l(Rn_{) we denote the linear space}

of alll-vectors spanned by the exterior productseI=ei∧ei ∧ · · · ∧eil for all ordered

l-tuplesI= (i,i, . . . ,il), ≤i<i<· · ·<il≤n. Thel-formu(x) =IuI(x)dxI is a linear

combination of the standard basisdxI₌_dxi

∧ · · · ∧dxil for all orderedl-tuplesI. If the coefficientuIis differential, we say thatuis a differentiall-form. ByD(M,l) we denote

the space of all diﬀerentiall-forms. Similarly, we writeLs(M,l) for thel-formu(x) onM

withuI satisfying_M|uI|s_<_∞_.

A diﬀerential l-form u∈D(M,l_{) is called a closed form if}_du_{=  in} _M_{. From the}

Poincaré lemmaddu=  we know thatduis a closed form. The module of a diﬀerential formuis given by|u|=∗(u∧ ∗u)∈D(M,).

A very important operator, the homotopy operatorT:C∞(M,l₎_→_C∞₍_M_,l–_{), is}

deﬁned by

Tu=

M

ϕ(y)Kyu dy

for diﬀerential formsu, whereϕ∈C∞_ (M) is normalized by_Mϕ(y)dy= , andKyis the

liner operator deﬁned by

(Kyu)(x;ξ, . . . ,ξl–) =





tl–u(tx+y–ty;x–y;ξ, . . . ,ξl–)dt.

For the homotopy operatorT, we have the following decomposition, which will be used repeatedly in this paper:

u=d(Tu) +T(du)

for any diﬀerential formu. A closed formuMis deﬁned byuM=d(Tu); in particular, when uis a -form,uM=|M|–_Mu(y)dy. In regard to Green’s operator, we need the following results in []:

dd∗G(u)s,B+d∗dG(u)s,B+dG(u)s,B+d∗G(u)s,B+G(u)s,B≤C(s)us,B,

d∗G(u)s,B=Gd∗(u)s,B, anddG(u)s,B=Gd(u)s,B

for any diﬀerential formuinMand  <s<∞. 2 Poincaré-type inequality

In this section, we give a Poincaré-type inequality for the composite operatorMs◦D◦G,

which will be used in the estimates for theLp_{, Lipschitz, and BMO norms. We will need}

the following lemmas.

The following estimate for the homotopy operatorTappears in [].

Lemma . Let u∈Lt

loc,l= , , . . . ,n,  <t<∞,be a differential form in M,and T be the homotopy operator defined on differential forms.Then there exists a constant C, indepen-dent of u,such that

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We will use the generalized Hölder inequality repeatedly.

Lemma .[] Let <q<∞,  <p<∞,and s–₌_q–₊_p–_._{If f and g are measurable}

functions onRn_,_then

fgs,M≤ fq,Mgp,M

for any M⊂Rn_.

The following lemma appears in [].

Lemma . Letϕ: [, +∞)be a strictly increasing convex function such thatϕ() = .If

u(x)∈D(M,l)satisﬁesϕ(|u|)∈L₍_M_,_μ_),_{then for any a}_{> ,}_{we have}

M

ϕ

a

|u–uM|

dμ≤

M

ϕa|u|dμ.

First, we establish the boundedness for the composite operatorMs◦D◦G.

Lemma . Let u∈Lt(M,l),l= , , . . . ,n, ≤s<t<∞,be a diﬀerential form in a

do-main M.Then,there exists a constant C,independent of u,such that

_M

sDG(u) t,B≤C|B|diam(B)ut,B.

Proof For a ballBinM, using Lemma . for anyB(x,r)⊂Band the decomposition theo-rem, we have



|B(x,r)|

B(x,r)

DG(u) –DG(u)_B

(x,r) s

dt

/s

=|B(x,r)|–/s DG(u) –DG(u)_B

(x,r) s,B(x,r)

=|B(x,r)|–/s TdDG(u) _s,B

(x,r)

≤C|B(x,r)|–/s+/n dDG(u) _s,B

(x,r)

=C|B(x,r)|+/n–/s ddG(u) +dd∗G(u) _s,B₍_x_,_r₎

=C|B(x,r)|+/n–/s dd∗G(u) _s,B₍_x_,_r₎

≤C|B(x,r)|+/n–/sus,B(x,r). ()

Since  + /n– /s> , taking the supremum overr, we get

sup r>



|B(x,r)|

B(x,r)

DG(u) –DG(u)_B

(x,r) s

dt

/s

≤sup r>

C|B(x,r)|+/n–/sus,B(x,r)

≤sup r>

C|B|+/n–/sus,B

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Using the generalized Hölder inequality, we ﬁnd

us,B≤ ut,Bts/(t–s),B

=|B|(t–s)/tsut,B. ()

Combining () and (), we obtain

_M

sDG(u) t,B≤ C|B|

+/n–/s_u s,B _t,B

≤ C|B|+/n–/s+(t–s)/tsut,B _t,B

=C|B|+/n–/tut,Bt,B

=C|B|+/nut,B. ()

The proof of Lemma . is completed.

Theorem . Let u∈Lt₍_M_,l_),_l_{= , , . . . ,}_n_{, }_≤_s_<_t_<_∞_,_{be a diﬀerential form in a}

domain M.Then,

_M

sDG(u) –

M

sDG(u)

B t,B≤C|B|diam(B)ut,B, where C is a constant independent of u.

Proof Choosingϕ(t) =xt_,_a_{= , and}_ω₍_x₎_≡_{ in Lemma ., we have}

u–uBt,B≤Cut,B.

ReplacingubyMsDG(u) and using Lemma ., we get

_M

sDG(u) –

M

sDG(u)

B t,B≤C M

sDG(u) t,B

≤C|B|diam(B)ut,B. ()

The proof of Theorem . is completed.

3 Lipschitz and BMO norm inequalities

In this section, we compare theLp norm, Lipschitz norm, and BMO norm of the com-posite operatorMs◦D◦Gapplied to diﬀerential forms. Especially, when we estimate the

Lipschitz norm in terms of the BMO norm, we need the differential form to satisfy some versions of harmonic equations. We first introduce some definitions.

We call an equation a nonhomogeneousA-harmonic equation if

d_A₍_x_,_du_{) =}_B₍_x_,_du_), _()

where the operatorsA:M×l₍_Rn₎_→l₍_Rn_{) and}_B_:_M_×l₍_Rn₎_→l–₍_Rn_{) satisfy}

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for almost allx∈Mand allξ ∈l₍_Rn_{). Here}_p_{>  is a constant related to equation ()}

anda,b> . Now we give deﬁnitions of the BMO and Lipschitz norms. See [] for more details.

Foru∈L_loc(M,l),l= , , . . . ,n, we writeu∈loc Lip_k(M,l), ≤k≤, if

uloc Lipk,M= sup σB⊂M|

B|–(n+k)/nu–uB,B<∞

for someσ> .

Foru∈L_loc(M,l),l= , , . . . ,n, we writeu∈BMO(M,l) if

u∗,M= sup

σB⊂M|

B|–u–uB,B<∞

for someσ> . Similarly, we can deﬁne the weighted BMO norm and Lipschitz norm. Foru∈L_loc(M,l,w),l= , , , . . . ,n, we writeu∈loc Lip_k(M,l,w), ≤k≤, if

uloc Lip_k,M,w= sup

σB⊂M

μ(B)–(n+k)/nu–uB,B,w<∞

for someσ> , wherewis a weight, andμis the Radon measure deﬁned bydμ=w(x)dx. Foru∈L

loc(M,l,w),l= , , , . . . ,n, we writeu∈BMO(M,l,w) if

u∗,M,w= sup

σB⊂M

μ(B)–u–uB,B,w<∞

for someσ> , wherewis a weight, andμis the Radon measure deﬁned bydμ=w(x)dx. We also need the following inequality.

Lemma . [] Take ϕ be a strictly increasing convex function on [, +∞) such that

ϕ() = .If u(x)∈D(M,l₎_satisﬁes_ϕ₍_|_u_|₎_∈_L₍_M_,_μ₎_{and for any constant c}_,

μx∈M:|u–c|> > ,

whereμis the Radon measure deﬁned by dμ(x) =ω(x)dx with weightω(x),then for any a> ,we have

M

ϕa|u|dμ≤C

M

ϕa|u–uM|dμ,

where C is a constant independent of u.

Now, we estimate the Lipschitz norm ofMs◦D◦Gin terms of theLt-norm.

Theorem . Let u∈Lt₍_M_,l_),_l_{= , , . . . ,}_n_{, }_≤_s_<_t_<_∞_,_{be a diﬀerential form in M}_.

Then,there exists a constant C,independent of u,such that

_M

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Proof From Theorem . we obtain

_M

sDG(u) –

M

sDG(u)

B t,B≤C|B|diam(B)ut,B

for all ballsBwithB⊂M. Using the Hölder inequality, we have

_M

sDG(u) –

M

sDG(u)

B ,B

≤

B

_M

sDG(u) –

M

sDG(u)

B t

dx

/t

B

t/(t–)dx (t–)/t

≤ |B|(t–)/t M

sDG(u) –

M

sDG(u)

B t,B

=|B|–/t M_sDG(u) –M_sDG(u)_B _t,B

≤ |B|–/tC|B|diam(B)ut,B

≤C|B|–/t+/nut,B. ()

From the deﬁnition of the Lipschitz norm and () it follows that

_M

sDG(u) loc Lipk,M =_σsup

B⊂M

|B|–(n+k)/n M_sDG(u) –M_sDG(u)_B _,B = sup

σB⊂M|

B|––k/n M

sDG(u) –

M

sDG(u)

B ,B

≤ sup

σB⊂M|

B|––k/nC|B|–/t+/nut,B

= sup

σB⊂M

C|B|–k/n–/t+/nut,B

≤ sup

σB⊂M

C|M|–k/n–/t+/nut,B

≤C sup

σB⊂M ut,B

≤Cut,M. ()

This ends the proof of Theorem ..

From the deﬁnitions of the Lipschitz and BMO norms we can get a simple relationship.

Theorem . Let u∈Ls₍_M_,l_),_l_{= , , . . . ,}_n_{, }_≤_s_<_∞_,_{be a diﬀerential form in M}_._Then_,

_M

sDG(u) ∗,M≤C M

sDG(u) loc Lip_k,M,

where k is a constant with≤k≤,and C is a constant independent of u.

Proof From the deﬁnition of the BMO norms we obtain

_M

sDG(u) ∗,M = sup

σB⊂M

|B|– M_sDG(u) –M_sDG(u)_B _,B = sup

σB⊂M|

B|k/n|B|–(n+k)/n M

sDG(u) –

M

sDG(u)

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≤ sup

σB⊂M

|M|k/n|B|–(n+k)/n M_sDG(u) –M_sDG(u)_B _,B

≤ |M|k/n sup

σB⊂M|

B|–(n+k)/n M

sDG(u) –

M

sDG(u)

B ,B

≤C sup

σB⊂M

|B|–(n+k)/n M_sDG(u) –M_sDG(u)_B _,B

≤C M_sDG(u) _{loc Lip}

k,M. ()

This ends the proof of Theorem ..

Theorem . Let u∈Ls(M,l), l= , , . . . ,n,  <s<∞,satisfy equation()in M and

suppose thatμ{x∈M:|u–c|> }> .Then,

_M

sDG(u) loc Lip_k,M≤Cu∗,M,

where k is a constant with≤k≤,and C is a constant independent of u.

Proof From () we have

_M

sDG(u) –

M

sDG(u)

B ,B≤C|B|

–/s+/n_u s,B.

Using the reverse Hölder inequality and Lemma ., we get

us,B≤C|B|(–s)/su,σB≤C|B|(–s)/s u– (u)B _,_σ_B,

whereσ> . So, we have

_M

sDG(u) –

M

sDG(u)

B ,B≤C|B|

+/n _u_{– (}_u₎ B _,σB.

Lettingσ >σ, we have

_M

sDG(u) loc Lipk,M = sup σB⊂M

|B|–(n+k)/n M_sDG(u) –M_sDG(u)_B _,B

≤ sup

σB⊂M

C|B|+/n–k/n|B|– u– (u)B _,_σ_B

≤C sup

σB⊂M

C|B|– u– (u)B _,_σ_B

≤Cu∗,M. ()

This ends the proof of Theorem ..

By Theorem . and Theorem . we can easily estimate the BMO norm of the compos-ite operatorMs◦D◦G.

Corollary . Let u∈Lt₍_M_,l_),_l_{= , , . . . ,}_n_{, }_≤_s_<_t_<_∞_,_{be a diﬀerential form in M}_.

Then,

_M

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4 The weighted norm inequalities

In this section, we consider the weighted situation. The weight function we select is

A(α,β,γ,M)-weight, which contains the well-known Ar(M)-weight. We will use the Radon measure to deal with theA(α,β,γ,M)-weight in the proof.

Deﬁnition .[] We say that a measurable functionw(x) deﬁned on a subsetM⊂Rn

satisﬁes theA(α,β,γ,M)-condition for some positive constantsα,β,γifw(x) >  a.e. and

sup B⊂M



|B|

B wα_dx



|B|

B w–β_dx

γ/β <∞.

Now, we give estimates for the weighted Lipschitz and BMO norms.

Theorem . Let u∈Lq₍_M_,l_,_μ_),_l_{= , , . . . ,}_n_{, }_≤_s_<_q_<_∞_._{Assume that the Radon}

measureμis deﬁned by dμ=w(x)dx with w(x)∈A(α,β,γ,M)for someα> ,where <

p<∞,β=αq/(αp–p–αq),γ=αq/p,andαp–p–αq> .Then,

_M

sDG(u) loc Lipk,M,w≤Cup,M,w,

where C is a constant independent of u.

Proof Using the generalized Hölder inequality with exponents satisfying /q= /αq+ (α– )/αq, we get

B

_M

sDG(u) –

M

sDG(u)

B q

w(x)dx

/q

=

B

_M

sDG(u) –

M

sDG(u)

B

w(x)/qqdx

/q

≤

B

_M

sDG(u) –

M

sDG(u)

B

αq/(α–) dx

(α–)/αq

B

w(x)/qαqdx /αq

≤C|B|diam(B)

B|

u|αq/(α–)dx

(α–)/αq

B

w(x)αdx /αq

. ()

Using the generalized Hölder inequality with exponents satisfying (α– )/αq= /p+ (αp–

p–αq)/αqp, we get

B

|u|αq/(α–)_dx (α–)/αq

=

B

uw(x)/pw(x)–/pαq/(α–)dx (α–)/αq

≤

B

uw(x)/ppdx

/p

×

B

w(x)–/pαqp/(αp–p–αq)dx

(αp–p–αq)/αqp

=

B

|u|pw(x)dx

/p

×

B

w(x)–αq/(αp–p–αq)dx

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Sincew(x)∈A(α,₍_α_p–p–αq_α_q),α_pq,M), we have

B

w(x)αdx

/αq

B

=|B|/αq



|B|

w(x)αdx



|B|

B

(αp–p–αq)/p/αq

≤C. ()

Combining (), (), and (), we obtain

B

_M

sDG(u) –

M

sDG(u)

B q

w(x)dx

/q

≤C|B|diam(B)

B

|u|pw(x)dx

/p

.

It follows that

_M

sDG(u) –

M

sDG(u)

B ,B,w

=

B

_M

sDG(u) –

M

sDG(u)

Bdμ

≤

B

_M

sDG(u) –

M

sDG(u)

B q

w(x)dx

/q

B

q/(q–)dμ

(q–)/q

=μ(B)(q–)/q M_sDG(u) –M_sDG(u)_B _q,B,w

≤Cμ(B)(q–)/q|B|diam(B)up,B,w. ()

Finally, we get

_M

sDG(u) loc Lip_k,M,w

= sup

σB⊂M

μ(B)–(n+k)/n M_sDG(u) –M_sDG(u)_B _,B,w

≤C sup

σB⊂M

μ(B)–(n+k)/n+(q–)/q++/nup,B,w

≤C sup

σB⊂M

μ(M)–(n+k)/n+(q–)/q++/nup,B,w

≤Cup,M,w. ()

This ends the proof of Theorem ..

Similarly to the proof of Theorem ., we have the following corollary.

Corollary . Let u∈Ls₍_M_,l_,_μ_),_l_{= , , . . . ,}_n_{, }_≤_s_<_∞_,_{be a diﬀerential form in M}_,_μ

and w(x)be the same as in Theorem..Then,

_M

sDG(u) ∗,M,w≤C M

sDG(u) loc Lip_k,M,w,

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The following corollary can be obtained by combining Theorem . and Corollary ..

Corollary . Let u,μ,w(x),and p be as in Theorem..Then,

_M

sDG(u) ∗,M,w≤Cup,M,w, where C is a constant independent of u.

5 Applications

In this section, we apply our results to some diﬀerential forms.

Example . LetM={(x,y,z) :x_+x_+· · ·+x

n≤} ⊂Rnandu(x, . . . ,xn) be deﬁned in

Rn_by

u(x, . . . ,xn) =

n

i=

e+x+···+xn xi  + (x_+· · ·+x

n) dxi.

Sou(x, . . . ,xn) is a diﬀerential form inM. Now we estimateMsDG(u) – (MsDG(u))Mt,M.

By simple calculation we obtain

ut,M =

M

|u|tdx

/t

≤

M

e+x+···+xn_x

+· · ·+xn

t/

dx

/t

≤

M

et/dx

/t

=e|M|/t.

Using Theorem ., we have

_M

sDG(u) –

M

sDG(u)

M t,≤C|M|diam(M)e

_|_M_|/t

= eC

πn/ ( +n/)

+/t

.

Example . Letu(x, . . . ,xn) be deﬁned inRnby

u(x, . . . ,xn) =

n

i=



 +x

+· · ·+xn dxi.

For a ball B⊂Rn with radiusr, it is diﬃcult to estimate the upper bound directly for

M

sDG(u)loc Lip_k,B, but by Theorem . we have

_M

sDG(u) loc Lip_k,B≤Cut,B

≤C

B

x_+· · ·+x_n/ +x_+· · ·+x_nt/dx

(11)

≤C|B|/t

=C

πn/_rn

( +n/)

/t

.

Similarly, we also obtain an upper bound for the BMO norm of the composite operator M

s◦D◦G.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to the main results. XL drafted the manuscript. YW and YX improved the ﬁnal version. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The third author was supported in part by the Foundation of Education Department of Heilongjiang Province (#12541133).

Received: 14 August 2015 Accepted: 13 November 2015

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