Journal of Inequalities and Applications Volume 2010, Article ID 723234,12pages doi:10.1155/2010/723234
Research Article
Global Estimates for Singular Integrals of
the Composition of the Maximal Operator and
the Green’s Operator
Yi Ling and Hanson M. Umoh
Department of Mathematical Sciences, Delaware State University, Dover, DE 19901, USA
Correspondence should be addressed to
Yi Ling,[email protected] Hanson M. Umoh,[email protected]
Received 31 December 2009; Accepted 12 March 2010
Academic Editor: Shusen Ding
Copyrightq2010 Y. Ling and H. M. Umoh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We establish the Poincar´e type inequalities for the composition of the maximal operator and the Green’s operator in John domains.
1. Introduction
LetΩbe a bounded, convex domain andBa ball inRn,n≥2. We useσBto denote the ball
with the same center asBand with diamσB σdiamB,σ >0. We do not distinguish the balls from cubes in this paper. We use|E|to denote then-dimensional Lebesgue measure of the setE⊆Rn. We say thatwis a weight ifw∈L1
locRnandw >0, a.e.
Differential forms are extensions of functions in Rn. For example, the function
ux1, x2, . . . , xn is called a 0-form. Moreover, if ux1, x2, . . . , xn is differentiable, then
it is called a differential 0-form. The 1-form ux in Rn can be written as ux
n
i1uix1, x2, . . . , xndxi. If the coefficient functions uix1, x2, . . . , xn, i 1,2, . . . , n, are
differentiable, then ux is called a differential 1-form. Similarly, a differentialk-formux
is generated by{dxi1∧dxi2∧ · · · ∧dxik},k1,2, . . . , n, that is,
ux
I
uIxdxI
ui1i2···ikxdxi1∧dxi2∧ · · · ∧dxik, 1.1
where∧is the Wedge Product,I i1, i2, . . . , ik, 1≤i1< i2<· · ·< ik≤n. Let
be the set of alll-forms inRn,
DΩ,∧l 1.3
the space of all differentiall-forms onΩ, and
LpΩ,∧l 1.4
thel-formsux IuIxdxI on Ωsatisfying
Ω|uI|pdx < ∞ for all ordered l-tuplesI,
l1,2, . . . , n. We denote the exterior derivative by
d:DΩ,∧l−→DΩ,∧l1 1.5
forl0,1, . . . , n−1, and define the Hodge star operator
:∧k−→ ∧n−k 1.6
as follows. IfuuIdxI,i1< i2<· · ·< ikis a differentialk-form, then
u −1IuIdxJ, 1.7
whereI i1, i2, . . . , ik,J {1,2, . . . , n} −I, andI kk1/2kj1ij. The Hodge
codifferential operator
d:DΩ,∧l1−→DΩ,∧l 1.8
is given byd −1nl1 donDΩ,∧l1,l0,1, . . . , n−1. We write
us,Ω
Ω|u|
sdx 1/s. 1.9
The differential forms can be used to describe various systems of PDEs and to express different geometric structures on manifolds. For instance, some kinds of differential forms are often utilized in studying deformations of elastic bodies, the related extrema for variational integrals, and certain geometric invariance. Differential forms have become invaluable tools for many fields of sciences and engineering; see 1,2for more details.
In this paper, we will focus on a class of differential forms satisfying the well-known nonhomogeneousA-harmonic equation
whereA:Ω× ∧lRn → ∧lRnandB:Ω× ∧lRn → ∧l−1Rnsatisfy the conditions
|Ax, ξ| ≤a|ξ|p−1, Ax, ξ·ξ≥ |ξ|p, |Bx, ξ| ≤b|ξ|p−1 1.11
for almost everyx∈Ωand allξ∈ ∧lRn. Herea, b >0 are constants and 1< p <∞is a fixed
exponent associated with1.10. If the operatorB0,1.10becomesdAx, du 0, which
is called thehomogeneousA-harmonic equation. A solution to1.10is an element of the Sobolev spaceWloc1,pΩ,∧l−1such that
ΩAx, du·dϕBx, du·ϕ0 for allϕ∈Wloc1,pΩ,∧l−1
with compact support. Let A : Ω× ∧lRn → ∧lRnbe defined byAx, ξ ξ|ξ|p−2 with
p >1. Then,Asatisfies the required conditions anddAx, du 0 becomes thep-harmonic
equation
ddu|du|p−20 1.12
for differential forms. If u is a function 0-form, 1.12 reduces to the usual p-harmonic equation div∇u|∇u|p−2 0 for functions. A remarkable progress has been made recently in the study of different versions of the harmonic equations; see 3for more details.
LetC∞Ω,∧lbe the space of smoothl-forms onΩand
WΩ,∧lu∈L1locΩ,∧l:uhas generalized gradient. 1.13
The harmonicl-fields are defined by
HΩ,∧lu∈ WΩ,∧l:dudu0, u∈Lp for some 1< p <∞. 1.14
The orthogonal complement ofHinL1is defined by
H⊥u∈L1:< u, h >0 for allh∈ H. 1.15
Then, the Green’s operatorGis defined as
G:C∞Ω,∧l−→ H⊥∩C∞Ω,∧l 1.16
by assigningGuto be the unique element ofH⊥∩C∞Ω,∧lsatisfying Poisson’s equation
ΔGu u−Hu, whereHis the harmonic projection operator that mapsC∞Ω,∧lontoH
so thatHuis the harmonic part ofu. See 4for more properties of these operators. For any locallyLs-integrable formuy, the Hardy-Littlewood maximal operatorM
s
is defined by
Msu sup r>0
1
|Bx, r|
Bx,r
uysdy
1/s
whereBx, ris the ball of radiusr, centered atx, 1≤s <∞. We writeMu M1uifs1. Similarly, for a locallyLs-integrable formuy, we define the sharp maximal operatorM#
sby
M#
su sup r>0
1
|Bx, r|
Bx,r
uy−uBx,rsdy
1/s
, 1.18
where thel-formuB∈DB,∧lis defined by
uB
⎧ ⎨ ⎩
|B|−1
B
uydy, l0,
dTu, l1,2, . . . , n
1.19
for allu∈ LpB,∧l,1 ≤p < ∞, andT is the homotopy operator which can be found in 3.
Also, from 5, we know that bothMsuandM#suareLs-integrable 0-form.
Differential forms, the Green’s operator, and maximal operators are widely used not only in analysis and partial differential equations, but also in physics; see 2–4,6–9. Also, in real applications, we often need to estimate the integrals with singular factors. For example, when calculating an electric field, we will deal with the integralEr 1/4π0
Dρxr−
x/r−x3dx, where ρxis a charge density andxis the integral variable. The integral is singular if r ∈ D. When we consider the integral of the vector field F ∇f, we have to deal with the singular integral if the potential functionf contains a singular factor, such as the potential energy in physics. It is clear that the singular integrals are more interesting to us because of their wide applications in different fields of mathematics and physics. In recent paper 10, Ding and Liu investigated singular integrals for the composition of the homotopy operatorT and the projection operator Hand established some inequalities for these composite operators with singular factors. In paper 11, they keep working on the same topic and derive global estimates for the singular integrals of these composite operators inδ-John domains. The purpose of this paper is to estimate the Poincar´e type inequalities for the composition of the maximal operator and the Green’s operator over theδ-John domain.
2. Definitions and Lemmas
We first introduce the following definition and lemmas that will be used in this paper.
Definition 2.1. A proper subdomainΩ⊂Rnis called aδ-John domain,δ >0, if there exists a
pointx0∈Ωwhich can be joined with any other pointx∈Ωby a continuous curveγ⊂Ωso that
dξ, ∂Ω≥δ|x−ξ| 2.1
Lemma 2.2see 12. Letφbe a strictly increasing convex function on 0,∞withφ0 0and
Da domain inRn. Assume thatuis a function inDsuch thatφ|u| ∈L1D, μandμ{x ∈ D :
|u−c|>0}>0for any constantc, whereμis a Radon measure defined bydμx wxdxfor a weightwx. Then, one has
D
φa
2u−uD,μdμ≤
D
φa|u|dμ 2.2
for any positive constanta, whereuD,μ 1/μD
Dudμ.
Lemma 2.3see 13. EachΩhas a modified Whitney cover of cubesV{Qi}such that
i
Qi Ω,
Qi∈V
χ√5/4Q
i≤NχΩ 2.3
and someN >1, and ifQi∩Qj/∅, then there exists a cubeR(this cube need not be a member ofV) in
Qi∩Qjsuch thatQi∪Qj ⊂NR. Moreover, ifΩisδ-John, then there is a distinguished cubeQ0∈ V which can be connected with every cubeQ∈ Vby a chain of cubesQ0 Qj0, Qj1, . . . , Qjk Qfrom
Vand such thatQ⊂ρQji,i0,1,2, . . . , k, for someρρn, δ.
Lemma 2.4see 14. Letube a smooth differential form satisfying1.10in a domainD,σ >10< s, andt <∞. Then, there exists a constantC, independent ofu, such that
us,B≤C|B|t−s/stut,σB 2.4
for all balls B withσB⊂D, whereσ >1is a constant.
Lemma 2.5see 5. LetMsbe the Hardy-Littlewood maximal operator defined in1.17,Gthe
Green’s operator, andu∈LtΩ,∧l,l1,2,3, . . . , n,1≤s < t <∞, a smooth differential form in a
bounded domainΩ. Then,
MsGut,Ω≤Cut,Ω 2.5
for some constantC, independent of u.
Lemma 2.6see 5. Letu∈LsΩ,∧l,l1,2,3, . . . , n,1≤s <∞, be a smooth differential form
in a bounded domainΩ,M#
sthe sharp maximal operator defined in1.18, andGthe Green’s operator.
Then,
M#
sGus,Ω ≤C|Ω|1/sus,Ω 2.6
for some constantC, independent of u.
Lemma 2.7. Let u ∈ LtlocΩ,∧l, l 1,2, . . . , n, be a smooth differential form satisfying the A
maximal operator defined in1.17with1< s < t <∞. Then, there exists a constantCn, t, α, λ, ρ, independent ofu, such that
B
|MsGu|t
1
dx, ∂Ωαdx
1/t
≤Cn, t, α, λ, ρ|B|γ
ρB
|u|t 1
|x−xB|λ
dx 1/t
2.7
for all ballsBwithρB⊂Ωand any real numberαandλwithα > λ≥0andγ λ−α/nt, where
xBis the center of the ball andρ >1is a constant.
Proof. Letε∈0,1be small enough such thatεn < α−λandBany ball withB⊂Ω, centerxB
and radiusrB. Takingkt/1−ε,we see thatk > t. Note that 1/t1/k k−t/kt; using
H ¨older’s inequality, we obtain
B
|MsGu|t 1
dx, ∂Ωαdx
1/t ⎛ ⎝ B
|MsGu|
1
dx, ∂Ωα/t
t dx ⎞ ⎠ 1/t ≤ B
|MsGu|kdx
1/k⎛
⎝
B
1
dx, ∂Ωα/t
kt/k−t
dx ⎞ ⎠
k−t/kt
≤ MsGuk,B
B
dx, ∂Ω−αβdx
1/βt
,
2.8
whereβk/k−t. Sincek > t > s, usingLemma 2.5, we get
MsGuk,B ≤C1uk,B. 2.9
Letmntk/ntαk−λk, then 0< m < t < k. UsingLemma 2.4, we have
uk,B ≤C2|B|m−k/mkum,ρB, 2.10
whereρ >1 is a constant andρB ⊂Ω.By H ¨older’s inequality with 1/m1/t t−m/mt
again, we find
um,ρB
ρB
|u||x−xB|−λ/t|x−xB|λ/t
m dx 1/m ≤ ρB
|u||x−xB|−λ/t
t
dx
1/t
ρB
|x−xB|λ/t
mt/t−m
dx
t−m/mt
≤
ρB
|u|t|x−xB|−λdx
1/t
ρB
|x−xB|mλ/t−mdx
t−m/mt
.
Note thatdx, ∂Ω≥ρ−1rBfor allx∈B, it follows that
dx, ∂Ω−αβ≤ρ−1rB
−αβ
. 2.12
Hence, we have
B
dx, ∂Ω−αβdx
1/βt
≤ρ−1rB
−α/t|B|1/βt
C3rB−α/t|B|1/βt.
2.13
Now, by the elementary integral calculation, we obtain
ρB
|x−xB|mλ/t−mdx
t−m/mt
≤C4
ρrB
λ/tnt−m/mt
. 2.14
Substituting2.9–2.14into2.8, we obtain
B
|MsGu|t
1
dx, ∂Ωαdx
1/t
< C5rB−α/tλ/tnt−m/mt|B|1/βtm−k/mk
ρB
|u|t|x−xB|−λdx
1/t
C5rBn/k−n/t|B|1/t−1/kλ−α/nt
ρB
|u|t|x−xB|−λdx
1/t
C6|B|1/k−1/t|B|1/t−1/kλ−α/nt
ρB
|u|t|x−xB|−λdx
1/t
C6|B|λ−α/nt
ρB
|u|t|x−xB|−λdx
1/t
Cn, t, α, λ, ρ|B|γ
ρB
|u|t|x−xB|−λdx
1/t
.
2.15
We have completed the proof.
Similarly, byLemma 2.6, we can prove the following lemma.
Lemma 2.8. Letu∈Ls
locΩ,∧l,1< s <∞,l 1,2, . . . , n, be a smooth differential form satisfying
the A-harmonic equation 1.10 in convex domain Ω, M#
s the sharp maximal operator defined in
1.18, and G Green’s operator. Then, there exists a constantCn, s, α, λ, ρ, independent ofu, such that
B
M#
sGu s 1
dx, ∂Ωαdx
1/s
≤Cn, s, α, λ, ρ|B|γ
ρB
|u|s 1
|x−xB|λ
dx 1/s
for all ballsBwithρB⊂Ωand any real numberαandλwithα > λ≥0andγ1/s−λ−α/ns, wherexBis the center of the ball andρ >1is a constant.
3. Main Results
Theorem 3.1. Letu ∈ LtlocΩ,∧l,l 1,2, . . . , n, be a smooth differential form satisfying the A -harmonic equation1.10,GGreen’s operator, andMsthe Hardy-Littlewood maximal operator defined
in1.17with1< s < t <∞. Then, there exists a constantCn, ρ, t, α, λ, N, Q0,Ω, independent of
u, such that
Ω
MsGu−MsGuQ0
t 1
dx, ∂Ωαdx
1/t
≤Cn, ρ, t, α, λ, N, Q0,Ω
Ω|u|
tgxdx 1/t
3.1
for any bounded and convexδ-John domainΩ⊂Rn, where
gx
i
χρQi
1
x−xQ
i
λ, 3.2
ρ > 1andα > λ ≥ 0are constants, the fixed cubeQ0 ⊂ Ω, the cubesQi ⊂ Ω, the constantN > 1
appeared inLemma 2.3, andxQi is the center ofQi.
Proof. First, we use Lemma 2.3for the bounded and convex δ-John domain Ω. There is a modified Whitney cover of cubesV {Qi}forΩsuch thatΩ ∪Qi, and
Qi∈Vχ√5/4Qi ≤
NχΩfor someN >1. Moreover, there is a distinguished cubeQ0∈ Vwhich can be connected with every cubeQ ∈ V by a chain of cubesQ0 Qj0, Qj1, . . . , Qjk Q from Vsuch that
Q⊂ρQji,i0,1,2, . . . , k, for someρρn, δ. Then, by the elementary inequalityab
s ≤
2s|a|s|b|s,s≥0, we have
Ω
MsGu−MsGuQ0
t 1
dx, ∂Ωαdx
1/t
∪Qi
MsGu−MsGuQ0
tdμ
1/t
≤
⎛
⎝
Qi∈V
2t
Qi
MsGu−MsGuQi
tdμ
2t
Qi
MsGuQi−MsGuQ0
tdμ
≤C1t ⎛ ⎜ ⎝ ⎛ ⎝ Qi∈V Qi
MsGu−MsGuQi
tdμ ⎞ ⎠ 1/t ⎛ ⎝ Qi∈V Qi
MsGuQi−MsGuQ0
tdμ
⎞ ⎠
1/t⎞
⎟ ⎠.
3.3
The first sum in 3.3 can be estimated by using Lemma 2.2 with ϕ xt, a 2, and
Lemma 2.7:
Qi∈V
Qi
MsGu−MsGuQi
tdμ ≤ Qi∈V Qi
2t|MsGu|tdμ
≤C2
n, ρ, t, α, λ,Ω
Qi∈V
|Qi|γt
ρQi
|u|tdμi
≤C3
n, ρ, t, α, λ,Ω|Ω|γt
Qi∈V
Ω
|u|tdμi
χρQi
C4
n, ρ, t, α, λ, N,Ω|Ω|γt
Ω|u|
tgxdx
C5
n, ρ, t, α, λ, N,Ω
Ω|u|
tgxdx,
3.4
whereμxandμixare the Radon measures defined bydμ 1/dx, ∂Ωαdxanddμix
1/|x−xQi|
λdx, respectively.
To estimate the second sum in3.3, we need to use the property ofδ-John domain. Fix a cubeQi ∈ Vand letQ0Qj0, Qj1, . . . , Qjk Qibe the chain inLemma 2.3. Then we have
MsGuQi−MsGuQ0
≤k−1
i0
MsGuQji −MsGuQji1. 3.5
The chain{Qji}also has property that for eachi,i0,1, . . . , k−1,Qji∩Qji1/∅. Thus, there
exists a cubeDisuch thatDi⊂Qji∩Qji1andQji∪Qji1 ⊂NDi,N >1, so,
maxQji,Qji1
Qji∩Qji1
≤ maxQji,Qji1 |Di| ≤
Note that
μQ
Q
1
dx, ∂Ωαdx
≥
Q
1
diamΩαdx
C7n, α,Ω|Q|,
3.7
whereC7n, α,Ωis a positive constant. By3.6,3.7, andLemma 2.7, we have
MsGuQji−MsGuQji1 t
1
μQji∩Qji1
Qji∩Qji1
MsGuQji−MsGuQji1 t dx
dx, ∂Ωα
≤ C8n, α,Ω
Qji∩Qji1
Qji∩Qji1
MsGuQji−MsGuQji1 t dx
dx, ∂Ωα
≤ C8n, α,ΩC6N
maxQji,Qji1
Qji∩Qji1
MsGuQji−MsGuQji1 t
dμ
≤C9n, t, α, N,Ω
i1
ki
1
Qjk
Qjk
MsGu−MsGuQjk t
dμ
≤C10
n, ρ, t, α, λ, N,Ω
i1
ki
Qjk
γt
Qj
k
ρQjk
|u|tdμjk
C10
n, ρ, t, α, λ, N,Ω
i1
ki
Qj
k
γt−1
ρQjk
|u|tdμjk
≤C11
n, ρ, t, α, λ, N,Ω
i1
ki
|Ω|γt−1 Ω
|u|tdμjk
χρQjk
≤C12
n, ρ, t, α, λ, N,Ω
Qi∈V
Ω
|u|tdμi
χρQi
C12
n, ρ, t, α, λ, N,Ω
Ω|u|
tgxdx.
3.8
Then, by 3.5,3.8, and the elementary inequality |Mi1ti|s ≤ Ms−1Mi1|ti|s, we finally
obtain Qi∈V Qi
MsGuQi−MsGuQ0
tdμ
≤C13
n, ρ, t, α, λ, N,Ω
Qi∈V
Qi
Ω|u|
tgxdx dμ
C13
n, ρ, t, α, λ, N,Ω
⎛ ⎝ Qi∈V Qi dμ ⎞ ⎠
Ω|u|
C13
n, ρ, t, α, λ, N,Ω
Ωdμ
Ω|u|
tgxdx
C14
n, ρ, t, α, λ, N,ΩμΩ
Ω|u|
tgxdx
C15
n, ρ, t, α, λ, N,Ω
Ω|u|
tgxdx.
3.9
Substituting3.4and3.9in3.3, we have completed the proof ofTheorem 3.1.
Using the proof method for Theorem 3.1 and Lemma 2.8, we get the following theorem.
Theorem 3.2. Letu ∈ LslocΩ,∧l,l 1,2, . . . , n, be a smooth differential form satisfying the A
-harmonic equation1.10,GGreen’s operator, andM#
sthe sharp maximal operator defined in1.18.
Then, there exists a constantCn, ρ, s, α, λ, N, Q0,Ω, independent ofu, such that
Ω|M #
sGu−M#sGuQ0|
s 1
dx, ∂Ωαdx
1/s
≤Cn, ρ, s, α, λ, N, Q0,Ω
Ω|u|
sgxdx 1/s
3.10
for any bounded and convexδ-John domainΩ⊂Rn, where
gx
i
χρQi
1
x−xQi
λ, 3.11
ρ > 1andα > λ ≥ 0are constants, the fixed cubeQ0 ⊂ Ω, the cubesQi ⊂ Ω, the constantN > 1
appeared inLemma 2.3, andxQi is the center ofQi.
References
1 R. K. Sachs and H. H. Wu,General Relativity for Mathematicians, Springer, New York, NY, USA, 1977. 2 C. von Westenholz, Differential Forms in Mathematical Physics, North-Holland, Amsterdam, The
Netherlands, 1978.
3 R. P. Agarwal, S. Ding, and C. Nolder,Inequalities for Differential Forms, Springer, New York, NY, USA, 2009.
4 F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, vol. 94 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1983.
5 S. Ding, “Norm estimates for the maximal operator and Green’s operator,”Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 16, supplement S1, pp. 72–78, 2009.
6 A. Banaszuk and J. Hauser, “Approximate feedback linearization: a homotopy operator approach,” SIAM Journal on Control and Optimization, vol. 34, no. 5, pp. 1533–1554, 1996.
7 H. Cartan,Differential Forms, Houghton Mifflin, Boston, Mass, USA, 1970.
8 D. P. Knobles, “Projection operator method for solving coupled-mode equations: an application to separating forward and backward scattered acoustic fields,” inTheoretical and Computational Acoustics, pp. 221–232, World Scientific, River Edge, NJ, USA, 2004.
10 S. Ding and B. Liu, “A singular integral of the composite operator,”Applied Mathematics Letters, vol. 22, no. 8, pp. 1271–1275, 2009.
11 S. Ding and B. Liu, “A singular integral of the composite operator,”Applied Mathematics Letters, vol. 22, no. 8, pp. 1271–1275, 2009.
12 S. Ding, “Lϕμ-averaging domains and the quasi-hyperbolic metric,”Computers & Mathematics with
Applications, vol. 47, no. 10-11, pp. 1611–1618, 2004.
13 C. A. Nolder, “Hardy-Littlewood theorems forA-harmonic tensors,”Illinois Journal of Mathematics, vol. 43, no. 4, pp. 613–632, 1999.