Open Access

Global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norms

Journal of Inequalities and Applications20132013:526

https://doi.org/10.1186/1029-242X-2013-526

Received: 23 April 2013

Accepted: 8 October 2013

Published: 11 November 2013

Abstract

In this paper, we establish the global Poincaré-type inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norm.

Keywords

Poincaré-type inequalitiesOrlicz normsharp maximal operatorGreen’s operator

1 Introduction

The L p -theory of solutions of the homogeneous A-harmonic equation d A ( x , d u ) = 0 for differential forms u has been very well developed in recent years. Many L p -norm estimates and inequalities, including the Poincaré inequalities, for solution of the homogeneous A-harmonic equation have been established; see [1, 2]. The Poincaré inequalities for differential forms is an important tool in analysis and related fields, including partial differential equations and potential theory. However, the study of the nonhomogeneous A-harmonic equations d A ( x , d u ) = B ( x , d u ) has just begun [24]. In this paper, we focus on a class of differential forms satisfying the well-known nonhomogeneous A-harmonic equation d A ( x , d u ) = B ( x , d u ) .

Let us first introduce some necessary notation and terminology. Ω will refer to a bounded, convex domain in R n unless otherwise stated and B is a ball in R n , n 2 . We use σB to denote the ball with the same center as B and with diam ( σ B ) = σ diam ( B ) , σ > 0 . We do not distinguish the balls from cubes in this paper. We use | E | to denote the n-dimensional Lebesgue measure of the set E R n . We say w is a weight if w L loc 1 ( R n ) and w > 0 a.e. For a function u, we denote the average of u over B by
u B = 1 | B | B u d x ,
where | B | is the volume of B and the μ-average of u over B by
u B , μ = 1 μ ( B ) B u d μ .
Let l = l ( R n ) be the set of all l-forms in R n , let D ( Ω , l ) be the space of all differential l-forms on Ω, and let L p ( Ω , l ) be the l-forms u ( x ) = I u I ( x ) d x I on Ω satisfying Ω | u I | p d x < for all ordered l-tuples I, l = 1 , 2 , , n . We denote the exterior derivative by d : D ( Ω , l ) D ( Ω , l + 1 ) for l = 0 , 1 , , n 1 , and define the Hodge star operator : k n k as follows. If u = u I d x I , i 1 < i 2 < < i k , is a differential k-form, then u = ( 1 ) ( I ) u I d x J , where I = ( i 1 , i 2 , , i k ) , J = { 1 , 2 , , n } I , and ( I ) = k ( k + 1 ) 2 + j = 1 k i j . The Hodge codifferential operator
d : D ( Ω , l + 1 ) D ( Ω , l )
is given by d = ( 1 ) n l + 1 d on D ( Ω , l + 1 ) , l = 0 , 1 , , n 1 . We write
u s , Ω = ( Ω | u | s d x ) 1 / s .
The well-known nonhomogeneous A-harmonic equation is
d A ( x , d u ) = B ( x , d u ) ,
(1)
where A : Ω × l ( R n ) l ( R n ) and B : Ω × l ( R n ) l 1 ( R n ) satisfy the conditions:
| A ( x , ξ ) | a | ξ | p 1 , A ( x , ξ ) ξ | ξ | p , | B ( x , ξ ) | b | ξ | p 1
(2)
for almost every x Ω and all ξ l ( R n ) . Here, a , b > 0 are constants and 1 < p < is a fixed exponent associated with (1). If the operator B = 0 , equation (1) becomes d A ( x , d u ) = 0 , which is called the (homogeneous) A-harmonic equation. A solution to (1) is an element of the Sobolev space W loc 1 , p ( Ω , l 1 ) such that Ω A ( x , d u ) d φ + B ( x , d u ) φ = 0 for all φ W loc 1 , p ( Ω , l 1 ) with compact support. Let A : Ω × l ( R n ) l ( R n ) be defined by A ( x , ξ ) = ξ | ξ | p 2 with p > 1 . Then A satisfies the required conditions and d A ( x , d u ) = 0 becomes the p-harmonic equation
d ( d u | d u | p 2 ) = 0
(3)

for differential forms. If u is a function (0-form), equation (3) 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 [1] for more details.

Let C ( Ω , l ) be the space of smooth l-forms on Ω and
W ( Ω , l ) = { u L loc 1 ( Ω , l ) : u  has generalized gradient } .
The harmonic l-fields are defined by
H ( Ω , l ) = { u W ( Ω , l ) : d u = d u = 0 , u L p  for some  1 < p < } .
The orthogonal complement of in L 1 is defined by
H = { u L 1 : u , h = 0  for all  h H } .
Then Green’s operator G is defined as
G : C ( Ω , l ) H C ( Ω , l )

by assigning G ( u ) to be the unique element of H C ( Ω , l ) satisfying Poisson’s equation Δ G ( u ) = u H ( u ) , where H is the harmonic projection operator that maps C ( Ω , l ) onto so that H ( u ) is the harmonic part of u. See [5] for more properties of these operators.

In harmonic analysis, a fundamental operator is the Hardy-Littlewood maximal operator. The maximal function is a classical tool in harmonic analysis but recently it has been successfully used in studying Sobolev functions and partial differential equations. For any locally L s -integrable form u ( y ) , we define the Hardy-Littlewood maximal operator M s by
M s ( u ) = M s ( u ) ( x ) = sup r > 0 ( 1 | B ( x , r ) | B ( x , r ) | u ( y ) | s d y ) 1 s ,
(4)
where B ( x , r ) is the ball of radius r, centered at x, 1 s < . We write M ( u ) = M 1 ( u ) if s = 1 . Similarly, for a locally L s -integrable form u ( y ) , we define the sharp maximal operator M s # by
M s # ( u ) = M s # ( u ) ( x ) = sup r > 0 ( 1 | B ( x , r ) | B ( x , r ) | u ( y ) u B ( x , r ) | s d y ) 1 s .
(5)

Some interesting results about these operators have been established, see [3, 4] and [6] for more details.

The purpose of this paper is to estimate the global Poincaré-type inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norm.

2 Definitions and lemmas

We now introduce the following definition and lemmas that will be used in this paper.

Definition 1 We say the weight w ( x ) satisfies the A r ( Ω ) condition, r > 1 , write w A r ( Ω ) if w ( x ) > 1 a.e., and
sup B ( 1 | B | B w d x ) ( 1 | B | B ( 1 w ) 1 r 1 d x ) r 1 <
(6)

for any ball B Ω .

Definition 2 A proper subdomain Ω R n is called a δ-John domain, δ > 0 , if there exists a point x 0 Ω which can be joined with any other point x Ω by a continuous curve γ Ω so that
d ( ξ , Ω ) δ | x ξ |

for each ξ γ . Here d ( ξ , Ω ) is the Euclidean distance between ξ and Ω.

Lemma 1 [7]

Each Ω has a modified Whitney cover of cubes V = { Q i } such that
i Q i = Ω , Q i V χ 5 4 Q i N χ Ω

and some N > 1 , and if Q i Q j , then there exists a cube R (this cube need not be a member of ) in Q i Q j such that Q i Q j N R . Moreover, if Ω is δ-John, then there is a distinguished cube Q 0 V which can be connected with every cube Q V by a chain of cubes Q 0 = Q j 0 , Q j 1 , , Q j k = Q from and such that Q ρ Q j i , i = 0 , 1 , 2 , , k , for some ρ = ρ ( n , δ ) .

3 Poincaré inequalities

In this section, we prove the global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator with L p norm.

To get our result, we rewrite our Theorem 2 in [4] as follows.

Lemma 2 Let u be a smooth differential form satisfying A-harmonic equation (1) in a bounded domain Ω, let G be Green’s operator, and let M s be the sharp maximal operator defined in (4) with 1 < s p , q < . Then there exists a constant C, independent of u, such that
( B | M s ( G ( u ) ) M s ( G ( u ) ) B | q d μ ) 1 / q C ( δ , Ω ) | B | 1 + 1 n 1 p + 1 q ( σ B | u | p d μ ) 1 / p

for all balls B with σ B Ω , and a constant σ > 1 , where the measure μ is defined by d μ = w ( x ) d x and w ( x ) A r ( Ω ) with w δ > 0 for some r >1 and a constant δ.

Theorem 1 Let u L loc t ( Ω , l ) , l = 1 , 2 , , n , be a smooth differential form satisfying A-harmonic equation (1), let G be Green’s operator, and let M s be the sharp maximal operator defined in (4) with 1 < s < t < . Then there exists a constant C ( n , t , δ 0 , N , Ω ) , independent of u, such that
( Ω | M s ( G ( u ) ) ( M s ( G ( u ) ) ) Q 0 | t d μ ) 1 / t C ( n , t , δ 0 , N , Ω ) ( Ω | u | t d μ ) 1 / t
(7)

for any bounded and convex δ-John domain Ω R n , where the fixed cube Q 0 Ω , the constant N > 1 appeared in Lemma 1, and the measure μ is defined by d μ = w ( x ) d x and w ( x ) A r ( Ω ) with w δ 0 > 0 for some r >1 and a constant δ 0 .

Proof First, we use Lemma 1 for the bounded and convex δ-John domain Ω. There is a modified Whitney cover of cubes V = { Q i } for Ω such that Ω = Q i , and Q i V χ 5 4 Q i N χ Ω for some N > 1 . Moreover, there is a distinguished cube Q 0 V which can be connected with every cube Q V by a chain of cubes Q 0 = Q j 0 , Q j 1 , , Q j k = Q from and such that Q ρ Q j i , i = 0 , 1 , 2 , , k , for some ρ = ρ ( n , δ ) . Then, by the elementary inequality ( a + b ) t 2 t ( | a | t + | b | t ) , t 0 , we have
( Ω | M s ( G ( u ) ) ( M s ( G ( u ) ) ) Q 0 | t d μ ) 1 / t ( Q i V ( 2 t Q i | M s ( G ( u ) ) ( M s ( G ( u ) ) ) Q i | t d μ + 2 t Q i | ( M s ( G ( u ) ) ) Q i ( M s ( G ( u ) ) ) Q 0 | t d μ ) ) 1 / t C 1 ( t ) ( ( Q i V Q i | M s ( G ( u ) ) ( M s ( G ( u ) ) ) Q i | t d μ ) 1 / t + ( Q i V Q i | ( M s ( G ( u ) ) ) Q i ( M s ( G ( u ) ) ) Q 0 | t d μ ) 1 / t ) .
(8)
The first sum in (8) can be estimated by using Lemma 2.
Q i V Q i | M s ( G ( u ) ) ( M s ( G ( u ) ) ) Q i | t d μ C 2 ( n , t , δ 0 , Ω ) Q i V ρ i Q i | u | t d μ C 3 ( n , t , δ 0 , Ω ) Q i V Ω | u | t d μ C 4 ( n , t , N , δ 0 , Ω ) Ω | u | t d μ ,
(9)

where the measure μ is defined by d μ = w ( x ) d x and w ( x ) A r ( Ω ) with w δ 0 > 0 for some r >1 and a constant δ 0 .

To estimate the second sum in (8), we need to use the property of δ-John domain. Fix a cube Q i V and let Q 0 = Q j 0 , Q j 1 , , Q j k = Q i be the chain in Lemma 1. Then we have
| ( M s ( G ( u ) ) ) Q i ( M s ( G ( u ) ) ) Q 0 | i = 0 k 1 | ( M s ( G ( u ) ) ) Q j i ( M s ( G ( u ) ) ) Q j i + 1 | .
(10)
The chain { Q j i } also has the property that for each i, i = 0 , 1 , , k 1 , Q j i Q j i + 1 . Thus, there exists a cube D i such that D i Q j i Q j i + 1 and Q j i Q j i + 1 N D i , N > 1 . So,
max { | Q j i | , | Q j i + 1 | } | Q j i Q j i + 1 | max { | Q j i | , | Q j i + 1 | } | D i | N .
(11)
Note that
μ ( Q ) = Q d μ = Q w ( x ) d x Q δ 0 d x = δ 0 | Q | .
(12)
By (11), (12) and Lemma 2, we have
| ( M s ( G ( u ) ) ) Q j i ( M s ( G ( u ) ) ) Q j i + 1 | t = 1 μ ( Q j i Q j i + 1 ) Q j i Q j i + 1 | ( M s ( G ( u ) ) ) Q j i ( M s ( G ( u ) ) ) Q j i + 1 | t d μ 1 δ 0 | Q j i Q j i + 1 | Q j i Q j i + 1 | ( M s ( G ( u ) ) ) Q j i ( M s ( G ( u ) ) ) Q j i + 1 | t d μ N δ 0 max { | Q j i | , | Q j i + 1 | } Q j i Q j i + 1 | ( M s ( G ( u ) ) ) Q j i ( M s ( G ( u ) ) ) Q j i + 1 | t d μ C 5 ( n , t , δ 0 , N , Ω ) k = i i + 1 1 | Q j k | Q j k | M s ( G ( u ) ) ( M s ( G ( u ) ) ) Q j k | t d μ C 6 ( n , t , δ 0 , N , Ω ) k = i i + 1 | Q j k | 1 + 1 n | Q j k | σ j k Q j k | u | t d μ = C 6 ( n , t , δ 0 , N , Ω ) k = i i + 1 | Q j k | 1 n σ j k Q j k | u | t d μ C 7 ( n , t , δ 0 , N , Ω ) k = i i + 1 | Ω | 1 n Ω | u | t d μ C 8 ( n , t , δ 0 , N , Ω ) Q i V Ω | u | t d μ C 9 ( n , t , δ 0 , N , Ω ) Ω | u | t d μ .
(13)
Then, by (10), (13) and the elementary inequality | i = 1 M t i | s M s 1 i = 1 M | t i | s , we finally obtain
Q i V Q i | ( M s ( G ( u ) ) ) Q i ( M s ( G ( u ) ) ) Q 0 | t d μ C 10 ( n , t , δ 0 , N , Ω ) Q i V Q i ( Ω | u | t d μ ) d μ = C 10 ( n , t , δ 0 , N , Ω ) ( Q i V Q i d μ ) Ω | u | t d μ C 11 ( n , t , δ 0 , N , Ω ) ( Ω d μ ) Ω | u | t d μ = C 11 ( n , t , δ 0 , N , Ω ) μ ( Ω ) Ω | u | t d μ = C 12 ( n , t , δ 0 , N , Ω ) Ω | u | t d μ .
(14)

Substituting (9) and (14) in (8), we have completed the proof of Theorem 1. □

4 Poincaré inequality with Orlicz norm

In this section, we give a global Poincaré inequality with Orlicz norm for the composition of the sharp maximal operator and Green’s operator.

Definition 3 Let φ be a continuously increasing convex function on [ 0 , ) with φ ( 0 ) = 0 , and let Λ be a domain with μ ( Λ ) < . If u is a measurable function in Λ, then we define the Orlicz norm of u by
u L ( φ , Λ , μ ) = inf { k > 0 : 1 μ ( Λ ) Λ φ ( | u ( x ) | k ) d μ 1 } .
(15)

A continuously increasing function ψ : [ 0 , ) [ 0 , ) with φ ( 0 ) = 0 is called an Orlicz function. A convex Orlicz function φ is often called a Young function.

In [8], Buckley and Koskela gave the following class of functions.

Definition 4 We say a Young function φ lies in the class G ( p , q , C ) , 1 p < q < , C 1 , if (i) 1 / C φ ( t 1 / p ) / g ( t ) C and (ii) 1 / C φ ( t 1 / q ) / h ( t ) C for all t > 0 , where g is a convex increasing function and h is a concave increasing function on [ 0 , ) .

From [8] and [9], we know that the class G ( p , q , C ) contains some very interesting functions, such as φ ( t ) = t p and φ ( t ) = t p log + α ( t ) , p 1 , α R , and each of φ, g and h is doubling in the sense that its values at t and 2t are uniformly comparable for all t > 0 , and the consequent fact that
C 1 t q h 1 ( φ ( t ) ) C 2 t q , C 1 t p g 1 ( φ ( t ) ) C 2 t p ,
(16)

where C 1 and C 2 are constants.

Now, we are ready to give our another global Poincaré inequality with Orlicz norm.

Theorem 2 Let φ be a Young function in the class G ( p , q , C 0 ) , 1 p < q < , C 0 1 , let u L loc t ( Ω , l ) , l = 1 , 2 , , n , be a smooth differential form satisfying A-harmonic equation (1) in Ω, let G be Green’s operator, and let M s be the sharp maximal operator defined in (4) with 1 < s t < . Then there exists a constant C, independent of u, such that
M s ( G ( u ) ) M s ( G ( u ) ) Q 0 L ( φ , Ω , μ ) C u L ( φ , Ω , μ )

for any bounded and convex δ-John domain Ω R n with μ ( Ω ) < , where the fixed cube Q 0 Ω appeared in Lemma 1, and the measure μ is defined by d μ = w ( x ) d x and w ( x ) A r ( Ω ) with w δ 0 > 0 for some r >1 and a constant δ 0 .

Proof Let g, h be the functions in the G ( p , q , C 0 ) condition. Note that φ is an increasing function. Using Theorem 1, (i) in Definition 4, and Jensen’s inequality, we obtain
φ ( 1 k ( Ω | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | t d μ ) 1 / t ) φ ( 1 k C 1 ( Ω | u | t d μ ) 1 / t ) = φ ( ( 1 k t C 1 t Ω | u | t d μ ) 1 / t ) C 0 g ( 1 k t C 1 t Ω | u | t d μ ) = C 0 g ( Ω 1 k t C 1 t | u | t d μ ) C 0 Ω g ( 1 k t C 1 t | u | t ) d μ .
(17)
Again, from (i) in Definition 4, we have
g ( x ) C 0 φ ( x 1 t ) .
Thus, we obtain
Ω g ( 1 k t C 1 t | u | t ) d μ C 0 Ω φ ( 1 k C 1 | u | ) d μ .
(18)
Combining (17) and (18) yields
φ ( 1 k ( Ω | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | t d μ ) 1 / t ) C 0 2 Ω φ ( 1 k C 1 | u | ) d μ = C 2 Ω φ ( 1 k C 1 | u | ) d μ .
(19)
Now, using Jensen’s inequality for h 1 , (16) and (ii) in Definition 4, and noticing that φ is doubling, we see
Ω φ ( | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | k ) d μ = h ( h 1 ( Ω φ ( | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | k ) d μ ) ) h ( Ω h 1 ( φ ( | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | k ) ) d μ ) h ( C 3 Ω ( | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | k ) t d μ ) C 0 φ ( ( C 3 Ω ( | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | k ) t d μ ) 1 t ) = C 0 φ ( 1 k ( C 3 Ω ( | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | ) t d μ ) 1 t ) C 4 φ ( 1 k ( Ω ( | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | ) t d μ ) 1 t ) .
(20)
Substituting (19) into (20) and using the fact that φ is doubling, we get
Ω φ ( | M s ( G ( u ) ) M s ( G ( u ) ) Q 0 | k ) d μ
(21)
C 5 Ω φ ( 1 k C 1 | u | ) d μ C 6 Ω φ ( 1 k | u | ) d μ .
(22)
Therefore, from Definition 3, we have
M s ( G ( u ) ) M s ( G ( u ) ) Q 0 L ( φ , Ω , μ ) C 6 u L ( φ , Ω , μ ) .

 □

Declarations

Acknowledgements

The first author was supported by NSF of P.R. China (No. 11071048).

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology
(2)
Department of Mathematical Science, Delaware State University

References

  1. Agarwal RP, Ding S, Nolder CA: Inequalities for Differential Forms. Springer, Berlin; 2009.View ArticleGoogle Scholar
  2. Agarwal RP, Ding S: Global Caccioppoli-type and Poincaré inequalities with Orlicz norms. J. Inequal. Appl. 2010., 2010: Article ID 727954Google Scholar
  3. Ling Y, Umoh HM: Global estimates for singular integrals of the composition of the maximal operator and the Green’s operator. J. Inequal. Appl. 2010., 2010: Article ID 723234Google Scholar
  4. Ling Y, Gejun B: Some local Poincaré inequalities for the composition of the sharp maximal operator and the Green’s operator. Comput. Math. Appl. 2012, 63: 720–727. 10.1016/j.camwa.2011.11.036MathSciNetView ArticleGoogle Scholar
  5. Warner FW: Foundations of Differentiable Manifolds and Lie Groups. Springer, New York; 1983.View ArticleGoogle Scholar
  6. Ding S: Norm estimate for the maximal operator and Green’s operator. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2009, 16: 72–78. Differential Equations and Dynamical Systems, suppl. S1MathSciNetGoogle Scholar
  7. Nolder CA: Hardy-Littlewood theorems for A -harmonic tensors. Ill. J. Math. 1999, 43: 613–631.MathSciNetGoogle Scholar
  8. Buckley SM, Koskela P: Orlicz-Hardy inequalities. Ill. J. Math. 2004, 48: 787–802.MathSciNetGoogle Scholar
  9. Ding S: L ( φ , μ ) -averaging domains and Poincaré inequalities with Orlicz norm. Nonlinear Anal. 2010, 73: 256–265. 10.1016/j.na.2010.03.018MathSciNetView ArticleGoogle Scholar

Copyright

© Gejun and Ling; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.