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

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.

1 Introduction

The L p -theory of solutions of the homogeneous A-harmonic equation d A(x,du)=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,du)=B(x,du) 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,du)=B(x,du).

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 , n2. We use σB to denote the ball with the same center as B and with diam(σB)=σdiam(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 udx,

where |B| is the volume of B and the μ-average of u over B by

u B , μ = 1 μ ( B ) B udμ.

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 dx< 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,,n1, 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,,n1. We write

u s , Ω = ( Ω | u | s d x ) 1 / s .

The well-known nonhomogeneous A-harmonic equation is

d A(x,du)=B(x,du),
(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 xΩ 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,du)=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,du)dφ+B(x,du)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,du)=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 ΔG(u)=uH(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, 1s<. 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 BΩ.

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 xΩ by a continuous curve γΩ so that

d(ξ,Ω)δ|xξ|

for each ξγ. 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 NR. Moreover, if Ω is δ-John, then there is a distinguished cube Q 0 V which can be connected with every cube QV by a chain of cubes Q 0 = Q j 0 , Q j 1 ,, Q j k =Q from and such that Qρ 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<sp,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 σBΩ, and a constant σ>1, where the measure μ is defined by dμ=w(x)dx and w(x) A r (Ω) with wδ>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 dμ=w(x)dx 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 QV by a chain of cubes Q 0 = Q j 0 , Q j 1 ,, Q j k =Q from and such that Qρ 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 ), t0, 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 dμ=w(x)dx 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,,k1, 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 dμ= Q w(x)dx Q δ 0 dx= 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), 1p<q<, C1, if (i) 1/Cφ( t 1 / p )/g(t)C and (ii) 1/Cφ( 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), p1, αR, 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 ), 1p<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<st<. 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 dμ=w(x)dx 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 ) dμ C 0 Ω φ ( 1 k C 1 | u | ) dμ.
(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 ) dμ
(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 ) .

 □

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Acknowledgements

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

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Gejun, B., Ling, Y. Global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norms. J Inequal Appl 2013, 526 (2013). https://doi.org/10.1186/1029-242X-2013-526

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Keywords

  • Poincaré-type inequalities
  • Orlicz norm
  • sharp maximal operator
  • Green’s operator