R E S E A R C H
Open Access
Estimates for the composition of the
carathéodory and homotopy operators
Zhaoyang Tang
*and Jianmin Zhu
*Correspondence:
Department of Mathematics and System Science, National University of Defense Technology, Changsha, P.R. China
Abstract
In the present paper, we deal with the composition of carathéodory and homotopy operators for differential forms satisfying theA-harmonic equation in the bounded and convex domain. We obtain estimates for the composition and the form of inequalities with weights. Moreover, we also obtain the composition for the gradient, carathéodory, and homotopy operators. Then we obtain theW1,pnorm estimates for the composition operators.
Keywords: differential forms; composition operators; norm estimates
1 Introduction
The purpose of this paper is to establish the inequalities for the composition of the ho-motopy operatorT and the carathéodory operatorGapplied to differential forms inRn,
n≥. The homotopy operatorTis widely used in the decomposition and theLp-theory of
differential forms. And in [], we have extended the homotopy operator to the domain that is deformed to every point. In the meanwhile, the carathéodory operatorGform classic examples to discuss boundedness and continuity of nonlinear operators and play an im-portant part in advanced functional analysis, and in [] we have extended it to differential forms. In many situations, we need to estimate the various norms of the operators and their compositions.
Throughout this paper, we always assume thatis a bounded and convex domain and
Bis a ball inRn,n≥. LetσBbe the ball with the same center asBand withdiam(σB) =
σdiam(B),σ> . We do not distinguish the balls from cubes in this paper. For any subset
E⊂Rn, we use|E|to denote the Lebesgue measure ofE. In [], we have the estimate for
T(u):
T(u)p,F≤nσn–μ()(diam)up,F (.)
for allu∈Lploc(,∧l), whereF⊂is bounded and convex. And for carathéodory operator, we obtain
f(s,ω)≤a(s) +b|ω|p/p, s∈,ω∈Lp,∧l.
With these estimates, we can obtain the estimates for the composition of them. Finally, we obtain theW,pnorm estimates for the composition operator.
The main theorems are proved by reference to Chap. of [].
2 Some preliminaries about differential forms
The majority of notations and preliminaries used throughout this paper can be found in []. For the sake of convenience, we list briefly them in this section.
Lete,e, . . . ,endenote the standard orthogonal basis ofRn. Suppose thatl=l(Rn) is
the linear space ofl-covectors, generated by the exterior productseI=ei∧ei∧ · · · ∧eil,
corresponding to all orderedl-tuplesI= (i,i, . . . ,il), ≤i<i<· · ·<il≤n,l= , , . . . ,n.
The Grassmann algebra=nl=lis a graded algebra with respect to the exterior prod-ucts. Forα=αIeI ∈andβ=βIeI ∈, the inner product inis given byα,β =
αIβI with summation over alll-tuplesI= (i,i, . . . ,il) and all integralsl= , , . . . ,n. We
define the Hodge star operator :→by
ω=sign(π)αi,i,...,ik(x,x, . . . ,xn)dxj∧ · · · ∧dxjn–k,
whereω=αi,i,...,ik(x,x, . . . ,xn)dxi∧dxi∧· · ·∧dxikis ak-form,π= (i, . . . ,ik,j, . . . ,jn–k)
is a permutation of (, , . . . ,n) andsign(π) is the signature of the permutation. The norm ofα∈is given by the formula|α|=α,α = (α∧ α)∈=R.
A differentiall-formωis a Schwartz distribution onwith values inl(Rn). We use
D(,l) to denote the space of all differentiall-forms, andLp(,) to denote thel-forms
ω(x) =
I
ω(x)dxI=
ωi,i,...,il(x)dxi∧dxi∧ · · · ∧dxil
with all coefficientsωI∈Lp(,R). Thus,Lp(,l),p≥, is a Banach space with norm
ωp=ω(x)p,=
ω(x)p
/p
=
I
ωI(x)
p/
dx /p
.
The spaceLp(,l) is the subspace ofD(,l) with the condition
αLp ()=
n
i=
∂∂αxi
p/
dx /p
<∞.
The Sobolev spaceW,p(,l) ofl-forms isW,p(,l) =Lp(,)∩Lp(,l). The norms are given by
ωW,p(,∧l)= (diam)–ωp,+∇ωp,. (.)
We denote the exterior derivative byd:D(,l)→D(,l+) forl= , , . . . ,n– , which means
dω(x) =
n
k=
≤i<···<il≤n
∂ωi,i,...,il(x)
∂xk
dxk∧dxi∧dxi∧ · · · ∧dxil.
Its formal adjoint operator is defined by
In [], we define an operationKyfor anyy∈and we construct a homotopy operator
T:C∞(,∧l)→C∞(,∧l) by averagingK
yover all pointsy∈:
Tω=
ψ(y)Ky
φ*ωdy, (.)
whereψ inC∞ (U) is normalized so thatψ(y)dy= . We obtain the following decom-position for the operatorT:
ω=d(Tω) +T(dω). (.)
From [], we know that for any differential formu∈Lploc(,∧l),l= , , . . . ,n, <p<∞,
we have
∇(Tu)p,≤Cμ()up,, (.)
Tup,≤Cμ()diam(B)up,. (.)
whereμ(B) is flatness of(see []). See [–] for more details of differential forms and its applications.
Then we define the carathéodory conditions and carathéodory operator for differential forms (see []).
Definition . For a mappingf :× ∧l→ ∧l, whereis an open set inRn, we say that
f satisfies carathéodory conditions if
. For all mosts∈,f(s,ω)is continuous with respect toω, which means thatf can be expanded asf(s,ω) =JfJ(s,ω)dxJ, wherefJ:× ∧l→RandfJ(s,ω)is continuous
aboutωfor all mosts∈; and
. For any fixedω=IωdxI∈ ∧l,f(s,ω)is measurable abouts, which means that each
coefficient functionfJ(s,ω)is measurable aboutsfor any fixedω∈ ∧l.
Throughout this paper, we assume thatf(s,ω) satisfies the carathéodory condition (C -condition). Similarly, we can define the continuity off(s,ω) about (s,ω)∈× ∧l. Definition . Suppose that⊂Rnis a measurable set ( <mes≤+∞), andf :×
∧l→ ∧l. We define the carathéodory operatorG:∧l→ ∧lfor differential forms by
Gω(s) =fs,ω(s).
For the carathéodory operator, we have the similar result for differential forms as for the functions (see []).
Theorem . The carathéodory operator G maps continuously and boundedly Lp(,∧l)
into Lp(,∧l), if and only if, there exists b> , a(x)≥, a(x)∈Lp()satisfying the fol-lowing inequality:
f(x,ω)≤a(x) +b|ω|
p
p x∈,ω∈ ∧l. (.)
We define Muckenhoupt weights (see []).
Definition . A weightωsatisfies theAr()-condition in a subset⊂Rn, wherer> ,
and writeω∈Ar() when
sup
B
|B| B
ωdx
|B| B
ω/(–r)dx r–
<∞, (.)
where the supremum is over all ballsB⊂.
The following class of two-weight orAr,λ()-weights appeared in [] and [].
Definition . A pair of weights (ω,ω) satisfy theAr,λ()-condition in a setB⊂RN,
write (ω,ω)∈Ar,λ(B) for someλ≥ and <r<∞with /r+ /r= , if
sup
B⊂
|B| B
ωλdx /λr
|B| B
(/ω)/(r–)dx
λ(r–) <∞
for all ballsB⊂.
In the present paper, we deal with the A-harmonic equations formulated byd*A(x,du) =
B(x,du).
We also need the following weak reverse Hölder inequality (Lemma .. of []).
Theorem . Let u be a solution of the nonhomogeneous A-harmonic equation in a do-mainand <s, t<∞. Then there exists a constant C, independent of u, such that
us,B≤C|B|(t–s)/stut,ρB (.)
for all balls B withρB⊂for someρ> .
ForAr-weightsω, we have the following reverse Hölder inequality (Lemma .. of []).
Theorem . Ifω∈Ar, r> , then there exist constantsβ> and C, independent ofω,
such that
ωβ,Q≤C|Q|(–β)/βω,Q (.)
for all balls Q⊂RN.
3 Main results and proofs
Theorem . Let u∈Lsloc(,∧l), l= , , . . . ,n, <s<∞, be a solution of the A-harmonic
equation in domainis bounded and convex and T:C∞(,∧l)→C∞(,∧l–)be the
homotopy operator. Assume thatρ> andω∈Ar()for some <r<∞. Then T(G(u))∈
Lsloc(,∧l). Moreover, there exists a constant C, independent of u, such that
TG(u)s,B,ωα≤C|B|diam(B)us,ρB,ωα (.)
Proof We only need to prove the inequality holds. With (.) and (.), we have
T
G(u)s,B≤Cμ()diamBG(u)s,B
≤Cμ()diamBa(x) +b|u|s,B ≤Cμ()diamBa(x)s,B+bus,B
≤Cμ()diamBus,B. (.)
Then just like the process of the proof for Theorem .. in [], we obtain the inequality. We discuss the inequality with <α< andα= separately. For <α< , first we set
t=s/( –α). With Hölder inequality, we obtain
TG(u)s,B,ωα =
B
TG(u)ωα/ssdx /s
≤TG(u)t,B
B
ωtα/(t–s)dx (t–s)/st
=TG(u)t,B
B
ωdx α/s
. (.)
By (.), we obtain
TG(u)t,B≤Cμ()diamBut,B. (.)
Letm=s/( +α(r– )), thenm<s. With (.) and (.) and using Theorem ., we have
TG(u)s,B,ωα ≤Cμ()diamBut,B B
ωdx α/s
≤Cμ()diamB|B|(m–t)/mtum,ρB B
ωdx α/s
. (.)
And using Hölder’s inequality again, we obtain
um,ρB =
ρB
|u|mdx /m
=
ρB
|u|ωα/sω–α/smdx /m
≤
ρB
|u|sωαdx /s
ρB
(/ω)/(r–i)dx α(r–)/s
(.)
for all ballsBwithρB⊂. With (.) and (.), we find that
TG(u)s,B,ωα ≤Cμ()diamB|B|(m–t)/mtus,ρB,ωα
×
ρB
(/ω)/(r–)dx α(r–)/s
B
ωdx α/s
Asω∈Ar(), we have
ρB
(/ω)/(r–)dx α(r–)/s
B
ωdx α/s
≤
ρB
ωdx ρB
(/ω)/(r–)dx
r–α/s
=
|ρB|r
|ρB| ρB
ωdx
|ρB| ρB
(/ω)/(r–)dx r–α/s
≤C|B|αr/s. (.)
With (.) and (.), we have
TG(u)s,B,ωα≤Cμ(B)diamBus,ρB,ωα (.)
for all ballsBwithρB⊂. This is just (.) with <α< . Then we prove the case ofα= . First, with Theorem ., we know
ωβ,B≤C|B|(–β)/βω,B, (.)
hereβ > andC> are all constants. Lett=sβ/(β– ), then we know <s<tand β=t/(t–s). With Hölder’s inequality (.), and (.), we obtain
T
G(u)s,B,ω =
B
T
G(u)ω/ssdx /s
≤
B
TG(u)tdx /t
B
ω/sst/(t–s)dx
(t–s)/st
=CT
G(u)t,Bω/β,sB
≤Cμ(B)diamBut,Bω/β,sB
≤Cμ(B)diamB|B|(–β)/βsω/β,sBut,B
≤Cμ(B)diamB|B|–/tω/β,sBut,B. (.)
Setm=s/r. With Theorem ., we have
ut,B≤C|B|(m–t)/mtum,ρB. (.)
And we use Hölder’s inequality again
um,ρB =
ρB
|u|ω/sω–/smdx /m
≤
ρB
|u|sωdx /s
ρB
(/ω)/(r–)dx
(r–)/s
. (.)
Withω∈Ar(), we have
ω/,Bs/ω//(sr–),ρB≤
ρB
ωdx ρB
=
|ρB|r
|ρB| ρB
ωdx
|ρB| ρB
(/ω)/(r–)dx r–/s
≤C|B|r/s. (.)
With (.)-(.), we have
TG(u)s,B,ω≤Cμ(B)diamB|B|–/tω/,Bs|B|(m–t)/mtum,ρB
≤Cμ(B)diamB|B|–/mω,/Bs/ω//(sr–),ρBus,ρB,ω
≤Cμ(B)diamBus,ρB,ω (.)
for all ballsBwithρB⊂. Thus, we complete the proof.
Actually by the method developed in [], for the two weight (ω,ω)∈Ar,λ(), we have
the following inequality.
Theorem . Let u∈Lsloc(,∧l), l= , , . . . ,n, <s<∞, be a solution of the A-harmonic
equation in domainis bounded and convex and T:C∞(,∧l)→C∞(,∧l–)be the
homotopy operator. Assume that ρ> and(ω,ω)∈Ar,λ()for some <r<∞,λ≥. Then, T(G(u))∈LSloc(,∧l). Moreover, there exists a constant C, independent of u, such
that
TG(u)s,B,ωα
≤C|B|diam(B)us,ρB,ω
α
(.)
for all balls B withρB⊂and any real numberαwith <α≤.
Proof Lett=λs/(λ–α). As s=t+(tst–s), with Hölder inequality, we have
B
T(Gu)ωαdx /s
=
B
T(Gu)ωα/ssdx /s
≤
B
T(Gu)tdx /t
B
ωα/sst/(t–s)dx (t–s)/st
≤T(Gu)t,B
B
ωλdx α/λs
(.)
for all ballsB⊂. Then, from (.), we obtain
T(Gu)t,B≤Cμ()diamBut,B. (.)
Letm=(λ+αλ(sr–)), then we knowm<s<t. With (.) and (.) and Theorem ., we have
B
T(Gu)sωαdx /s
≤C|B|μ()diamBut,B B
ωλdx α/λs
≤C|B||B|
m–t mtu
m,ρB B
ωλdx α/λs
Then by the generalized Hölder’s inequality, we have
um,ρB =
ρB|
u|mdx /m
=
ρB
|u|ωα/sω–α/smdx /m
≤
ρB
|u|sωαdx /s
ρB
(/ω)λ/(r–)dx
α(r–)/λs
(.)
for all ballsBwithρB⊂, where we use m =s +s–smm. Then with (.) and (.), we obtain
B
T(Gu)sωαdx /s
≤C|B||B|
m–t
mtμ()diamB ρB
|u|sωαdx /s
×
ρB
(/ω)λ/(r–)dx
α(r–)
λs
B
ωλdx α/λs
. (.)
Then, with (ω,ω)∈Ar,λ(), we have
B
ωλdx α/λs
ρB
(/ω)λ/(r–)dx
r–α/λs
≤
ρB
ωλdx α/λs
ρB
(/ω)λ/(r–)dx
r–α/λs
=|ρB|r
|ρB| ρB
ωλdx
|ρB| ρB
ωλ
/(r–)α/λs
≤C|B|αr/λs. (.)
With (.) and (.), we have
B
T(Gu)sωαdx /s
≤Cμ()diamB ρB
|u|sωαdx /s
(.)
for all ballsBwithρB⊂.
For the compositions of the gradient operator ∇, the homotopy operator T, the carathéodory operator G,∇ ◦T ◦G, we obtain the local Sobolev-Poincaréembedding theorem.
Theorem . Let u∈Ls
loc(,∧l), l= , , . . . ,n, <s<∞, be a solution of the A-harmonic
equation in bounded and convex domain , T :C∞(,∧l)→C∞(,∧l–) be the
inde-pendent of u, such that
∇
TG(u)s,B≤Cμ()us,B (.)
and
TG(u)W,s(B)≤Cμ()us,B. (.)
Proof Actually, we only need to prove (.) and (.). From these two inequalities, the remaining part of the theorem follows. From (.), we obtain
∇
T(ω)p,≤Cμ()ωp, (.)
for anyω∈Lploc(∧lB). LetG(u) =ω, we have
∇
TG(u)s,B≤Cμ()G(u)s,B
≤Cμ()a(x)s,B+bus,B
≤Cμ()us,B. (.)
With the definition ofW,pnorm, (.), and (.), we have
TG(u)W,p(B)=diam(B)–T
G(u)s,B+∇TG(u)s,B
≤diam(B)–Cμ()diamBus,B+Cμ()us,B
≤Cμ()us,B. (.)
Thus, we obtain the inequality.
Using the same method as in the proof of Theorem ., we obtain the weighted inequality for∇(T(G(u)))s,B,ωα.
Corollary . Let u∈Ls
loc(,∧l), l= , , . . . ,n, <s<∞, be a solution of the A-harmonic
equation in bounded and convex domain, T:C∞(,∧l)→C∞(,∧l–)be the homotopy
operator,∇be the gradient operator and G be the carathéodory operator. Assume thatρ>
andω∈Ar()for some <r<∞. Then∇(T(G(u)))∈Lsloc(,∧l). Moreover, there exists a
constant C, independent of u, such that
∇
TG(u)s,B,ωα≤Cμ()us,B,ωα. (.)
ForG(T(u)), we also have the similar result.
Corollary . Let u∈Ls
loc(,∧l), l= , , . . . ,n, <s<∞, be a solution of the A-harmonic
equation in a bounded, convex domain and T:C∞(,∧l)→C∞(,∧l–)be the
ho-motopy operator. Assume thatρ> andω∈Ar()for some <r<∞. Then G(T(u))∈
Lsloc(,∧l). Moreover, there exists a constant C, independent of u, such that
GT(u)s,B,ωα≤C|B|diam(B)us,ρB,ωα (.)
Proof If (.) holds, thenG(T(u))∈Lsloc(,∧l) follows. Hence, we only need to prove
(.). From (.) and (.), we have
GT(u)s,B≤a(x)s,B+bT(u)s,B
≤Cμ()diam(B)us,B. (.)
Using the method in the proof of Theorem ., we obtain the inequality.
Actually for two weight (ω,ω)∈Ar,λ(), for someλ≤ and <r<∞, we have the
similar inequalities, with the method developed in the proof of Theorem ..
Corollary . Let u∈Lsloc(,∧l), l= , , . . . ,n, <s<∞, be a differential form
satisfy-ing A-harmonic equation in a bounded, convex domain⊂RN and T :C∞(,∧l)→
C∞(,∧l–)be the homotopy operator defined in (.). Assumeρ> and(ω
,ω)∈Ar,λ() for someλ≥and <r<∞. Then there exists a constant C, independent of u, such that
B
∇
T(u)sωαdx /s
≤C|B| ρB|
u|sωαdx /s
(.)
for all balls B withρB⊂and all real numberαwith <α<λ.
The above inequality is an extension of the usual inequality ofAr-weights. If we choose
ω(x) =ω(x) =ω(x) andλ= in the two weighted inequalities, we obtain theAr() weight
case.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Professor JZ gave the ideas. ZT gave the proofs and completed the manuscript.
Acknowledgement
The research of the first author was supported by the NSF of Hunan Province (No. 11JJ3004) and NSF of NUDT (No. JC10-02-02).
Received: 18 October 2011 Accepted: 6 June 2012 Published: 31 August 2012
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