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We establish the Poincar´e type inequalities for the composition of the maximal operator and the Green’s operator in John domains.. , n, are differentiable, then ux is called a differentia

Volume 2010, Article ID 723234, 12 pages

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,lingyi2001@hotmail.comand Hanson M Umoh,humoh@desu.edu

Received 31 December 2009; Accepted 12 March 2010

Academic Editor: Shusen Ding

Copyrightq 2010 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

with the same center as B and with diamσB  σ diamB, σ > 0 We do not distinguish the

u x1, x2, , x n  is called a 0-form Moreover, if ux1, x2, , x n is differentiable, then

i1u i x1, x2, , x n dx i If the coefficient functions ui x1, x2, , x n , i  1, 2, , n, are

differentiable, then ux is called a differential 1-form Similarly, a differential k-form ux

I

be the set of all l-forms inRn,

D

1.3 the space of all differential l-forms on Ω, and

L p

1.4

d : D

1.5

codifferential operator

d  : D

1.8

s,Ω



Ω|u| s dx

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

In this paper, we will focus on a class of differential forms satisfying the well-known

nonhomogeneous A-harmonic equation

where A :Ω × ∧lRn → ∧lRn  and B : Ω × ∧ lRn → ∧l−1Rn satisfy the conditions

p > 1 Then, A satisfies the required conditions and d  A x, du  0 becomes the p-harmonic

equation

d 

du |du| p−2

loc



The harmonic l-fields are defined by

Then, the Green’s operator G is defined as

G : C∞

1.16

is defined by

r>0

1

|Bx, r|



B x,r u

y s dy

where Bx, r is the ball of radius r, centered at x, 1 ≤ s < ∞ We write Mu  M1u if s  1.

r>0

1

|Bx, r|



B x,r u

y

− u B x,r s dy

u B

⎧

⎨

⎩



B

u

y

dy, l  0,

Differential forms, the Green’s operator, and maximal operators are widely used not

real applications, we often need to estimate the integrals with singular factors For example,

to deal with the singular integral if the potential function f 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

homotopy operator T and the projection operator H and established some inequalities for

same topic and derive global estimates for the singular integrals of these composite operators

in δ-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 the δ-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 ⊂ Rn is called a δ-John domain, δ > 0, if there exists a

that

Lemma 2.2 see 12

D a domain inRn Assume that u is a function in D such that φ |u| ∈ L1D, μ and μ{x ∈ D :

|u − c| > 0} > 0 for any constant c, where μ is a Radon measure defined by dμx  wxdx for a

weight w x Then, one has



D

φ a



D

for any positive constant a, where u D,μ  1/μDD udμ.



i

Q i∈V

χ√ 5/4Q

and some N > 1, and if Q i ∩Q j /  ∅, then there exists a cube R (this cube need not be a member of V) in

Q i ∩ Q j such that Q i ∪ Q j ⊂ NR Moreover, if Ω is δ-John, then there is a distinguished cube Q0∈ V

which can be connected with every cube Q ∈ V by a chain of cubes Q0  Q j0, Q j1, , Q j k  Q from

Lemma 2.4 see 14 1.10 in a domain D, σ > 10 <

s, and t < ∞ Then, there exists a constant C, independent of u, such that

for all balls B with σB ⊂ D, where σ > 1 is a constant.

Lemma 2.5 see 5 s be the Hardy-Littlewood maximal operator defined in1.17, G the

Green’s operator, and u ∈ L t Ω, ∧ l , l  1, 2, 3, , n, 1 ≤ s < t < ∞, a smooth differential form in a

bounded domain Ω Then,

for some constant C, independent of u.

Lemma 2.6 see 5 s Ω, ∧ l , l  1, 2, 3, , n, 1 ≤ s < ∞, be a smooth differential form

in a bounded domain Ω, M#

s the sharp maximal operator defined in1.18, and G the Green’s operator.

Then,



for some constant C, independent of u.

Lemma 2.7 Let u ∈ L t

A-harmonic equation1.10 in convex domain Ω, G the Green’s operator, and M s the Hardy-Littlewood

maximal operator defined in1.17 with 1 < s < t < ∞ Then, there exists a constant Cn, t, α, λ, ρ,

independent of u, such that



B

≤ Cn, t, α, λ, ρ

ρB

2.7

for all balls B with ρB ⊂ Ω and any real number α and λ with α > λ ≥ 0 and γ  λ − α/nt, where

x B is the center of the ball and ρ > 1 is a constant.

Proof Let ε ∈ 0, 1 be small enough such that εn < α − λ and B any ball with B ⊂ Ω, center x B

H ¨older’s inequality, we obtain



B



⎛

B

dx

⎞

⎠

1/t

≤



B

B

1

kt/ k−t

dx

⎞

⎠

k−t/kt

s k,B



B

,

2.8

k,B ≤ C2|B| m−k/mk m,ρB , 2.10

again, we find

m,ρB 



ρB



dx

≤



ρB



ρB



dx

t−m/mt

≤



ρB

t−m/mt

.

2.11

Note that dx, ∂Ω ≥ ρ − 1r B for all x ∈ B, it follows that

Hence, we have



B

2.13

Now, by the elementary integral calculation, we obtain



ρB

t−m/mt



ρr B

λ/t nt−m/mt



B

< C5r B−α/tλ/tnt−m/mt |B| 1/βtm−k/mk



ρB

 C5r Bn/k −n/t |B| 1/t−1/kλ−α/nt

ρB

 C6|B| 1/k−1/t |B| 1/t−1/kλ−α/nt

ρB

 C6|B| λ−α/nt

ρB

 Cn, t, α, λ, ρ



ρB

.

2.15

We have completed the proof

Lemma 2.8 Let u ∈ L s

the A-harmonic equation 1.10 in convex domain Ω, M# the sharp maximal operator defined in

that



B

#

≤ Cn, s, α, λ, ρ

ρB

2.16

for all balls B with ρB ⊂ Ω and any real number α and λ with α > λ ≥ 0 and γ  1/s − λ − α/ns,

where x B is the center of the ball and ρ > 1 is a constant.

3 Main Results

Theorem 3.1 Let u ∈ L t

A-harmonic equation1.10, G Green’s operator, and M s the Hardy-Littlewood maximal operator defined

u, such that



≤ Cn, ρ, t, α, λ, N, Q0,Ω

for any bounded and convex δ-John domainΩ ⊂ Rn , where

i

χ ρQ i

1

x − x Q i

ρ > 1 and α > λ ≥ 0 are constants, the fixed cube Q0 ⊂ Ω, the cubes Q i ⊂ Ω, the constant N > 1

appeared in Lemma 2.3, and x Q i is the center of Q i

Proof First, we use Lemma 2.3for the bounded and convex δ-John domain Ω There is a

Q i∈Vχ√ 5/4Q

i ≤







∪Q i

≤

⎛

Q i∈V

2t



Q i



Q

≤ C1t

⎛

⎜

⎛

Q i∈V



Q i

⎞

⎠

1/t



⎛

Q i∈V



Q i

⎞

⎠

1/t⎞

⎟

⎠.

3.3

Lemma 2.7:



Q i∈V



Q i

Q i∈V



Q i



n, ρ, t, α, λ,Ω 

Q i∈V

|Q i|γt



ρQ i

|u| t dμ i



n, ρ, t, α, λ,Ω|Ω|γt

Q i∈V



Ω



χ ρQ i



n, ρ, t, α, λ, N,Ω|Ω|γt





n, ρ, t, α, λ, N,Ω 

3.4

i0 s

Q j ∩ Q j

Note that



Q

1

≥



Q

1

3.7

μ

Q j i ∩ Q j i1





Q j i ∩ Q j i1





k i

1

Q j k



Q jk s



n, ρ, t, α, λ, N,Ωi1

k i

Q j k γt

Q j k



ρQ jk

|u| t dμ j k



n, ρ, t, α, λ, N,Ωi1

k i

Q j k

γt−1

ρQ jk

|u| t dμ j k



n, ρ, t, α, λ, N,Ωi1

k i

Ω



|u| t dμ j k

χ ρQ jk



n, ρ, t, α, λ, N,Ω 

Q i∈V



Ω



χ ρQ i



n, ρ, t, α, λ, N,Ω 

3.8

i1t i|s ≤ M s−1M

obtain



Q i∈V



Q i



n, ρ, t, α, λ, N,Ω 

Q i∈V



Q i



dμ



n, ρ, t, α, λ, N,Ω

⎛

Q∈V



Q i dμ

⎞

 C13



n, ρ, t, α, λ, N,Ω

Ωdμ





n, ρ, t, α, λ, N,ΩμΩ





n, ρ, t, α, λ, N,Ω 

3.9

theorem

Theorem 3.2 Let u ∈ L s

A-harmonic equation1.10, G Green’s operator, and M#the sharp maximal operator defined in1.18.

Then, there exists a constant C n, ρ, s, α, λ, N, Q0, Ω, independent of u, such that



Ω|M#

≤ Cn, ρ, s, α, λ, N, Q0,Ω

for any bounded and convex δ-John domainΩ ⊂ Rn , where

i

χ ρQ i

1

x − x Q i

ρ > 1 and α > λ ≥ 0 are constants, the fixed cube Q0 ⊂ Ω, the cubes Q i ⊂ Ω, the constant N > 1

appeared in Lemma 2.3, and x Q i is the center of Q i

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in δ-John domains The purpose of this paper is to estimate the Poincar´e type inequalities for the. .. the composition of the maximal operator and the Green’s operator over the δ-John domain.

2 Definitions and Lemmas

We first introduce the following definition and. ..

Differential forms, the Green’s operator, and maximal operators are widely used not

real applications, we often need to estimate the integrals with singular factors For example,

to