Open Access

A new weight class and Poincaré inequalities with the Radon measure

Journal of Inequalities and Applications20122012:32

https://doi.org/10.1186/1029-242X-2012-32

Received: 11 May 2011

Accepted: 15 February 2012

Published: 15 February 2012

Abstract

We first introduce and study a new family of weights, the A(α, β, γ; E)-class which contains the well-known A r (E)-weight as a proper subset. Then, as applications of the A(α, β, γ ;E)-class, we prove the local and global Poincaré inequalities with the Radon measure for the solutions of the non-homogeneous A-harmonic equation which belongs to a kind of the nonlinear partial differential equations.

2000 Mathematics Subject Classification: Primary 26D10; Secondary 35J60; 31B05; 58A10; 46E35.

Keywords

weightsPoincaré-type inequalityharmonic equations and differential forms

1. Introduction

Let Ω be a domain in n , n ≥ 2 , B be a ball and σB be the ball with the same center and diam(σB) = σdiam(B), σ > 0. We use |E| to denote the Lebesgue measure of the set E n . We say w is a weight if w L l o c 1 ( n ) and w > 0 a.e. In 1972, Muckenhoupt [1] introduced the following A r (E)-weight in order to study the properties of the Hardy-Littlewood maximal operator. We say a weight w satisfies the A r (E) -condition in a subset E n , where r > 1 , and write w A r (E) when
sup B 1 | B | B w d x 1 | B | B 1 w 1 r - 1 d x r - 1 < ,
(1.1)

where the supremum is over all balls B E. Since then, the weight functions have been well studied and widely used in analysis and PDEs, particularly in areas of the measures and integrals, see [211]. In 1998, the following A r (λ, E)-weight class was introduced in [12]. We say that a weight w belongs to the A r (λ, E) class, 1 < r < ∞ and 0 < λ < ∞, or that w is an A r (λ, E)-weight, write w A r (λ, E) , if s u p B 1 B B w λ d x 1 B B 1 w 1 / ( r - 1 ) d x r - 1 < for all balls B E. Notice that if we choose λ = 1 , we find that A r (1, E) = A r (E). In 2000, the following class of A r λ ( E ) -weights was introduced in [13]. We say that the weight w(x) > 0 satisfies the A r λ ( E ) -condition in E, r > 1 and λ > 0, and write w A r λ ( E ) , if s u p B 1 B B w d x 1 B B w 1 / ( 1 - r ) d x λ ( r - 1 ) < for any ball B E n . Also, it is easy to see that A r 1 ( E ) = A r ( E ) . Both A r (λ, E) and A r λ ( E ) have widely been used in the study of the weighted inequalities and integral estimates, see [46, 12, 13] for example.

2. The A(α, β, γ; E)-class

In this section, we first introduce the A(α, β, γ; E)-class which is an extension of the A r (E)-weight. Then, we study the properties of this class. We will use the following Hölder inequality repeatedly in this article.

Lemma 2.1. Let 0 < α < ∞, 0 < β < ∞ and s-1 = α-1 + β-1. If f and g are measurable functions on n , then || fg || s,E ≤|| f ||α,E|| g || β,E for any E n .

We introduce the following class of functions which is an extension of the several existing classes of weights, such as A r λ ( E ) -weights, A r (λ, E)-weights, and A r (E)-weights.

Definition 2.2. We say that a measurable function g(x) defined on a subset E n satisfies the A(α, β, γ; E)-condition for some positive constants α, β, γ, write g(x) A(α, β, γ; E) if g(x) > 0 a.e., and
sup B 1 B B g α d x 1 B B g - β d x γ / β < ,
(2.1)

where the supremum is over all balls B E.

We should notice that there are three parameters in the definition of the A(α, β, γ; E)-class. If we choose some special values for these parameters, we may obtain the existing weights. For example, if α = λ, β = 1/(r - 1) and γ = 1 in above definition, the A(α, β, γ; E) -class becomes A r (λ, E)-weight, that is A r (λ, E) = A(λ, 1/(r - 1),1;E). Similarly, A r λ ( E ) = A ( 1 , 1 / ( r - 1 ) , λ ; E ) . Also, it is easy to see that the A(α, β, γ; E)-class reduces to the usual A r (E)-weight if α = γ = 1 and β = 1/(r - 1). Moreover, we have the following theorem which establishes the relationship between the A r (E)-weight and the A(α, β, γ; E)-class.

Theorem 2.3. Let r > 1 be any constant and E n . Then, (i) There exists a constant α0 > 1 such that A r (E) A(α0,1/(r- 1)0; E). (ii) For any α with 0 < α < 1, A r (E) A(α,1/(r-1), α; E).

Proof. For w(x) A r (E), by the reverse Hölder inequality for the A r (E)-weight, there are constants α0 > 1 and C1 > 0 such that
1 B B w α 0 d x 1 / α 0 C 1 B B w d x
(2.2)
for all balls B E, i.e.,
1 B B w α 0 d x C 2 1 B B w d x α 0 .
(2.3)
From (2.3) and (1.1), we obtain
sup B 1 B B w α 0 d x 1 B B w - 1 r - 1 d x α 0 ( r - 1 ) C 2 sup B 1 B B w d x α 0 1 B B w - 1 r - 1 d x α 0 ( r - 1 ) C 2 sup B 1 B B w d x 1 B B 1 w 1 r - 1 d x r - 1 α 0 < ,
(2.4)
where the supremum is over all balls B E. Thus, w A(α0, 1/(r - 1), α0;E). Hence, A r (E) A(α0, 1/(r -1), α0; E). We have completed the proof of the first part of Theorem 2.3. Next, we prove the second part of the theorem. Let α (0,1) be any real number. Using the Hölder inequality with 1/α = 1 + (1 - α)/α, we have
B w α d x 1 / α B w d x B 1 α 1 - α d x ( 1 - α ) / α ,
(2.5)
that is
1 B B w α d x 1 / α 1 B B w d x
which can be written as
1 B B w α d x 1 B B w d x α .
(2.6)

Similar to inequality (2.4), using (2.6) and the definitions of the A r (E)-weight and the A(α, β,γ; E)-class, we obtain that A r (E) A(α, 1/(r-1), α; E) for any α with 0 < α < 1. The proof of Theorem 2.3 has been completed.

Example 2.4. Let Ω n be a bounded domain containing the origin and g(x) = |x| p , x Ω. We all know that g(x) = |x| p A r (Ω) for some r > 1 if and only if -n < p < n(r - 1). Now, we consider an example in 2, that is n = 2. Assume that D 2 is a bounded domain containing the origin and g(x) = |x|-3 is a function in D. Since p = -3 < -2 = -n, then g(x) = |x|-3 A r (D) for any r > 1. However, it is easy to check that g(x) = |x|-3 A(α, β, γ; D) for any positive numbers α, β, γ with 0 < α < 2/3.

Combining Theorem 2.3 and Example 2.4, we find that A r (E) is a proper subset of A(α, β, γ; E) for any positive constants α, β, γ and r with 0 < α < 2/ 3 and r > 1.

Theorem 2.5. If g1(x), g2(x) A(α, β, γ; E) for some α ≥ 1, β, γ > 0 and a subset E n , then g1(x) + g2(x) A(α, β, γ; E).

Proof. Let g1(x), g2(x) A(α, β, γ; E). By Minkowski inequality, we find that
B g 1 + g 2 α d x 1 α B g 1 α d x 1 α + B g 2 α d x 1 α .
(2.7)
Since |a + b| s ≤ 2 s (|a| s + |b| s ) for any constants a, b, s with s > 0, from (2.7) , we have
B ( g 1 + g 2 ) α d x B g 1 α d x 1 α + B g 2 α d x 1 α α 2 α B g 1 α d x + B g 2 α d x .
(2.8)
Note that g1(x), g2(x) A(α, β, γ; E). Using (2.8) , we obtain
sup B 1 B B ( g 1 + g 2 ) α d x 1 B B ( g 1 + g 2 ) - β d x γ / β sup B 2 α 1 B B g 1 α d x + 1 B B g 2 α d x 1 B B ( g 1 + g 2 ) - β d x γ / β sup B 2 α 1 B B g 1 α d x 1 B B g 1 - β d x γ / β + 1 B B g 2 α d x 1 B B g 2 - β d x γ / β < .

Thus, g1(x) + g2(x) A(α, β, γ; E). The proof of Theorem 2.5 has been completed.

Theorem 2.6. Let g1(x) A(α1, β1, α1γ; E) and g2(x) A(α2, β2, α2γ; E) for some γ > 0 and any subset E n , where α i , β i > 0, i = 1,2, and 1 α = 1 α 1 + 1 α 2 , 1 β = 1 β 1 + 1 β 2 . Then, g1(x)g2(x) A(α, β, αγ; E).

Proof. Using Lemma 2.1 with 1 α = 1 α 1 + 1 α 2 and 1 β = 1 β 1 + 1 β 2 , respectively, we have
B ( g 1 g 2 ) α d x 1 / α B g 1 α 1 d x 1 / α 1 B g 2 α 2 d x 1 / α 2 ,
(2.9)
B ( g 1 g 2 ) - β d x γ / β B g 1 - β d x γ / β 1 B g 2 - β 2 d x γ / β 2 .
(2.10)
Combining (2.9) and (2.10) yields
B ( g 1 g 2 ) α d x 1 / α B ( g 1 g 2 ) - β d x γ / β B g 1 α 1 d x 1 / α 1 B g 1 - β 1 d x γ / β 1 B g 2 α 2 d x 1 / α 2 B g 2 - β 2 d x γ / β 2
(2.11)
which is equivalent to
B ( g 1 g 2 ) α d x B ( g 1 g 2 ) - β d x α γ / β 1 / α B g 1 α 1 d x B g 1 - β 1 d x α 1 γ / β 1 1 / α 1 B g 2 α 2 d x B g 2 - β 2 d x α 2 γ / β 2 1 / α 2 .
(2.12)
Noticing that g1(x) A(α1, β1, α1 γ; E) and g2(x) A(α2, β2, α2γ; E), we obtain
sup B 1 B B ( g 1 g 2 ) α d x 1 B B ( g 1 g 2 ) - β d x α γ / β sup B 1 B B g 1 α 1 d x 1 B B g 1 - β 1 d x α 1 γ β 1 α α 1 sup B 1 B B g 2 α 2 d x 1 B B g 2 - β 2 d x α 2 γ β 2 α α 2 < .
(2.13)

Thus, g1(x)g2(x) A(α, β, αγ; E). The proof of Theorem 2.6 has been completed.

Proposition 2.7. Let 0 < p < 1 and g(x) A(α, βp, γ; E). Then, g p (x) A(α, β, γ; E).

Proof. Using Lemma 2.1 with 1 α p = 1 α + 1 - p α p yields
B g α p d x 1 / α p B ( 1 - p ) / α p B g α d x 1 / α ,
that is
1 B B ( g p ) α d x 1 B B g α d x p .
(2.14)
Since g(x) A(α, βp, γ; E), using (2.14) , we find that
sup B 1 B B ( g p ) α d x 1 B B ( g p ) - β d x γ / β sup B 1 B B g α d x p 1 B B g - β p d x γ / β sup B 1 B B g α d x 1 B B g - β p d x γ / β p p sup B 1 B B g α d x 1 B B g - β p d x γ / β p p < .
(2.15)

Therefore, g p (x) A(α, β, γ; E). The proof of Proposition 2.7 has been completed.

Let α, β, γ > 0 be any constants. It is easy to prove that (i) 1 g ( x ) A ( α , β , γ ; E ) if and only if g(x) A(β, α, αβ/γ; E). (ii) g p (x) A(α, β, γ; E) if and only if g(x) A(αp, βp, γp; E) for any constant p > 0. Also, using the Hölder inequality and the definition of the A(α, β, γ; E)-class, we can prove the following monotone properties of the A(α, β, γ; E)-class.

Proposition 2.8. If α1 < α2, then A(α2, β, γ; E) A(α1, β, γ; E) for any β,γ > 0. If β1 < β2, then A(α, β2, γ; E) A(α, β1, γ; E) for any α, γ > 0.

From Theorem 2.3 and Proposition 2.8, we know that for every r > 1, there exists a constant α0 > 1 such that A r (E) A(α, 1/(r - 1),α; E) for any α with 0 < α < α0.

3. Local Poincaré inequalities

As applications of the A(α, β, γ; E)-class, we prove the local Poincaré inequalities with the Radon measure for the differential forms satisfying the non-homogeneous A-harmonic equation. Differential forms are extensions of functions in n . For example, the function u(x1, x2,...,x n ) is called a 0-form. The 1-form u(x) in n can be written as u ( x ) = i = 1 n u i ( x 1 , x 2 , . . . , x n ) d x i . If the coefficient functions u i (x1, x2,...,x n ), i = 1,2,...,n, are differentiable, then u(x) is called a differential 1-form. Similarly, a differential k-form u(x) is generated by d x i 1 d x i 2 d x i k , k = 1 , 2 , . . . , n , that is, u ( x ) = I u I ( x ) d x I = u i 1 i 2 i k ( x ) d x i 1 d x i 2 d x i k , where I = (i1, i2,...,i k ), 1 ≤ i1 < i2 < ... < i k n. Let l = l ( n ) be the set of all l-forms in n and L p (Ω, Λ l ) be the l-forms u(x) = Σ I u I (x) dx I in Ω satisfying ∫Ω |u I | p < ∞ for all ordered l-tuples I, l = 1,2,...,n. We denote the exterior derivative by d and the Hodge star operator by *. The Hodge codifferential operator d* is given by d* = (-1)nl+1*d*, l = 0,1,..., n - 1. We consider here the solutions to the nonlinear partial differential equation
d * A ( x , d u ) = B ( x , d u )
(3.1)
which is called non-homogeneous A -harmonic equation, where A : Ω × l ( n ) → l ( n ) and B : Ω × l ( n ) → l-1( n ) satisfy the conditions: |A(x, ξ)| ≤ a|ξ|p-1, A(x, ξ) ξ ≥ |ξ| p and |B(x, ξ)| ≤ b|ξ|p-1for almost every x Ω and all ξ l ( n ). Here a, b > 0 are constants and 1 < p < ∞ is a fixed exponent associated with (3.1). A solution to (3.1) is an element of the Sobolev space W l o c 1 , p ( Ω , l - 1 ) such that ∫Ω A(x, du) + B(x, du) φ = 0 for all φ W l o c 1 , p ( Ω , l - 1 ) with compact support. If u is a function (0-form) in n , the equation (3.1) reduces to
div A ( x , u ) = B ( x , u ) .
(3.2)
If the operator B = 0, Equation (3.1) becomes d*A(x, du) = 0, which is called the (homogeneous) A -harmonic equation. Let A : Ω × l ( n ) → l ( n ) be defined by A(x, ξ) = ξ|ξ|p- 2with p > 1. Then, A satisfies the required conditions and d*A(x, du) = 0 becomes the p-harmonic equation d*(du|du|p-2) = 0 for differential forms. See [5, 6, 916] for recent results on the solutions to the different versions of the A-harmonic equation. The operator K y with the case y = 0 was first introduced by Cartan [17]. Then, it was extended to the following version in [18]. Let D be a convex and bounded domain. To each y D there corresponds a linear operator K y : C(D, l ) → C(D, l-1) defined by ( K y u ) ( x ; ξ 1 , . . . , ξ l - 1 ) = 0 1 t l - 1 u ( t x + y - t y ; x - y , ξ 1 , . . . , ξ l - 1 ) d t . A homotopy operator T : C(D, l ) → C(D, l- 1) is defined by averaging K y over all points y D: Tu = ∫ D φ(y)K y udy, where ϕ C 0 ( D ) is normalized so that ∫ D φ(y)dy = 1. The l-form is defined by ω D = |D|-1 D ω(y) dy, l = 0, and ω D = d(T ω), l = 1,2,...,n for all ω L p (D, l ), 1 ≤ p ≤ ∞. For any differential form u L l o c s ( D , l ) , l = 1 , 2 , . . . , n , 1 < s < , we have
T u s , D C D d i a m ( D ) u s , D .
(3.3)

Lemma 3.1. [14] Let u be a differential form satisfying the non-homogeneous A-harmonic equation (3.1) in Ω, σ > 1 and 0 < s, t < ∞. Then, there exists a constant C, independent of u, such that ||du||s, BC|B|(t-s)/st||du|| t,σB for all balls or cubes B with σB Ω.

Theorem 3.2. Let u L l o c s ( Ω , l ) be a solution of the non-homogeneous A-harmonic equation (3.1) in a domain Ω , d u L l o c s ( Ω , l + 1 ) , l = 0 , 1 , . . . , n - 1 and 1 < s < ∞. Then, there exists a constant C, independent of u, such that
B u - u B s d μ 1 / s C B d i a m ( B ) σ B d u s d μ 1 / s
(3.4)

for all balls B with σB Ω, where the Radon measure μ is defined by dμ = g(x)dx and g A(α, β, α; Ω), α > 1, β > 0.

Proof. By the decomposition theorem of differential forms, we have u = d(Tu) + T(du) = u B + T(du), where d is the exterior differential operator and T is the homotopy operator.

From (3.3), we obtain
u - u B t , B = T ( d u ) t , B C 1 B d i a m ( B ) d u t , B
(3.5)
for any t > 1. Now, choose t = αs/(α - 1), then, t > s. Using the Hölder inequality and (3.5), we obtain
B u - u B s d μ 1 / s = B u - u B s g ( x ) d x 1 / s = B u - u B g 1 / s ( x ) s d x 1 / s B u - u B t d x 1 / t B g t / ( t - s ) ( x ) d x ( t - s ) / s t C 2 B d i a m ( B ) d u t , B B g α ( x ) d x 1 / α s .
(3.6)
Let m = βs/(1 + β), then 0 < m < s. From Lemma 3.1, we have
d u t , B C 3 B m - t m t d u m , σ 1 B ,
(3.7)
where σ1 > 1 is a constant. Using the Hölder inequality again, we find that
d u m , σ 1 B = σ 1 B d u ( g ( x ) ) 1 / s ( g ( x ) ) - 1 / s m d x 1 / m σ 1 B d u s g ( x ) d x 1 / s σ 1 B g - 1 / s ( x ) m s s - m d x s - m m s σ 1 B d u s g ( x ) d x 1 / s σ 1 B ( g ( x ) ) - m s - m d x s - m m s σ 1 B d u s d μ 1 / s σ 1 B g - β ( x ) d x 1 / β s .
(3.8)
Since g A(α, β, α; Ω), it follows that
B g α ( x ) d x 1 / α s σ 1 B g - β ( x ) d x 1 / β s σ 1 B g α ( x ) d x σ 1 B g - β ( x ) d x α / β 1 / α s = σ 1 B 1 + α β 1 σ 1 B σ 1 B g α ( x ) d x 1 σ 1 B σ 1 B g - β ( x ) d x α / β 1 / α s C 4 B 1 / α s + 1 / β s .
(3.9)
Combining (3.6), (3.7), and (3.8) and using (3.9), we have
B u - u B s d μ 1 / s C 5 B d i a m ( B ) B m - t m t σ 1 B d u s d μ 1 / s B g α ( x ) d x 1 / α s σ 1 B g - β ( x ) d x 1 / β s C 5 d i a m ( B ) B 1 + 1 t - 1 m σ 1 B d u s d μ 1 / s B g α ( x ) d x σ 1 B g - β ( x ) d x α / β 1 / α s C 6 B d i a m ( B ) σ 1 B d u s d μ 1 / s ,
that is
B u - u B s d μ 1 / s C 6 B d i a m ( B ) σ 1 B d u s d μ 1 / s .

We have completed the proof of Theorem 3.2.

Let g ( x ) = 1 x - x B λ , where x B be the center of the ball B Ω and 0 < λ < n α , α > 1 . Then, g(x) A (α, β, α; Ω). From Theorem 3.2, we have the following corollary.

Corollary 3.3. Let u L l o c s ( Ω , l ) be a solution of the non-homogeneous A-harmonic equation (3.1) in a domain Ω , d u L l o c s ( Ω , l + 1 ) , l = 0 , 1 , . . . , n - 1 and 1 < s < ∞. Then, there exists a constant C, independent of u, such that
B u - u B s d μ 1 / s C B d i a m ( B ) σ B d u s d μ 1 / s
(3.10)

for all balls B with σB Ω, where the Radon measure μ is defined by d μ = 1 x - x B λ d x , x B is the center of the ball B Ω , 0 < λ < n α and α > 1 is a constant.

4. Global Poincaré inequalities

In this section, we will prove the global Poincaré inequalities with the Radon measure for solutions of the nonhomogeneous A-harmonic equation in L s (μ)-averaging domains. In 1989, Staples [19] introduced the following L s -averaging domains.

Definition 4.1. A proper subdomain Ω n is called an L s -averaging domain, s ≥ 1, if there exists a constant C such that
1 Ω Ω u - u Ω s d x 1 / s C sup B Ω 1 B B u - u B s d x 1 / s

for all u L l o c s ( Ω ) .

Also, in [19], the L s -averaging domain is characterized in terms of the quasi-hyperbolic metric. Particularly, Staples proved that any John domain is L s -averaging domain, see [20] for more results on the averaging domains. In [15], the L s -averaging domains were extended to the following L s (μ)-averaging domains.

Definition 4.2. We call a proper subdomain Ω n an L s (μ)-averaging domain, s ≥ 1, if there exists a constant C such that
1 μ Ω Ω u - u B 0 s d μ 1 / s C sup B Ω 1 μ B B u - u B s d x 1 / s

for some ball B0 Ω and all u L l o c s ( Ω ; μ ) , where the Radon measure μ(x) is defined by = w(x)dx and w(x) is a weight. Here, the supremum is over all balls B with B Ω.

Theorem 4.3. Let u L s (Ω, 0) be a solution of the non-homogeneous A -harmonic equation (3.2) in a domain Ω, du L s (Ω, 1), 1 < s < ∞. Then, there exists a constant C, independent of u, such that
Ω u - u B 0 s d μ 1 / s C μ ( Ω ) 1 + 1 / n Ω d u s d μ 1 / s
(4.1)

for any L s (μ)-averaging domain Ω n with μ (Ω) < ∞, where B0 is some ball appearing in Definition 4.2 and the Radon measure μ is defined by dμ = g(x)dx, g(x) A(α, β, α; Ω), α >1, β > 0.

Proof. We may assume g(x) ≥ 1 a.e. in Ω. Otherwise, let Ω1 = Ω {x Ω : 0 < g(x) < 1} and Ω2 = Ω {x Ω : g(x) ≥ 1}. Then, Ω = Ω1 Ω2. We define G(x) by
G ( x ) = 1 , x Ω 1 g ( x ) , x Ω 2 .

Then, G(x) ≥ g(x) and it is easy to check that g(x) A(α, β, α; Ω) if and only if G(x) A(α, β, α; Ω).

Thus,
Ω u - u B 0 s d μ 1 / s = Ω u - u B 0 s g ( x ) d x 1 / s Ω u - u B 0 s G ( x ) d x 1 / s
(4.2)
with G(x) ≥ 1. Hence, we may suppose that g(x) ≥ 1 a.e. in Ω. Thus, for any D Ω, we have
μ ( D ) = D d μ = D g ( x ) d x D d x = D .
(4.3)
Note that diam(B) = C1|B|1/n. From Theorem 3.2, we obtain
1 B B u - u B s d μ 1 / s C 2 B 1 + 1 / n - 1 / s σ B d u s d μ 1 / s .
(4.4)
By definition of the L s (μ) -averaging domain, (4.3) , (4.4) and noticing that 1 + 1/n - 1/s > 0, we find that
1 μ ( Ω ) Ω u - u B 0 s d μ 1 / s C 3 sup B Ω 1 μ ( B ) B u - u B s d μ 1 / s C 3 sup B Ω 1 B B u - u B s d μ 1 / s C 4 sup B Ω B 1 + 1 / n - 1 / s σ B d u s d μ 1 / s C 4 Ω 1 + 1 / n - 1 / s sup B Ω σ B d u s d μ 1 / s C 4 Ω 1 + 1 / n - 1 / s Ω d u s d μ 1 / s C 4 ( μ ( Ω ) ) 1 + 1 / n - 1 / s Ω d u s d μ 1 / s ,
that is
Ω u - u B 0 s d μ 1 / s C μ ( Ω ) 1 + 1 / n Ω d u s d μ 1 / s .

The proof of Theorem 4.3 has been completed.

In [15], it has been proved that any John domain is an L s (μ)-averaging domain. Hence, we have the following corollary.

Corollary 4.4. Let u L s (Ω, 0) be a solution of the non-homogeneous A-harmonic equation (3.2) in a John domain Ω with μ(Ω) < ∞, du L s (Ω, 1), 1 < s < ∞. Then, there exists a constant C, independent of u, such that
Ω u - u B 0 s d μ 1 / s C Ω d u s d μ 1 / s ,
(4.5)

where B0 is some ball appearing in Definition 4.2 and the Radon measure μ is defined by dμ = g(x)dx and g(x) A(α, β, α; Ω), α > 1, β > 0.

Example 4.5. Since the usual p-harmonic equation div (u|u|p-2) = 0 and the A-harmonic equation div A (x, u) = 0 for functions are the special cases of the non-homogeneous A- harmonic equation, all results proved in Sections 3 and 4 are still true for p-harmonic functions and A-harmonic functions.

Remark. (i) Since an L s -averaging domain is a special L s (μ)-averaging domain, then the inequality (4.1) still holds in any L s -averaging domain. (ii) Since μ(Ω) < ∞, the inequality (4.1) can be written as
Ω u - u B 0 s d μ 1 / s C Ω d u s d μ 1 / s ,

where Ω is an L s (μ)-averaging domain Ω n with μ(Ω) < ∞ and B0 is some ball appearing in Definition 4.2, and the Radon measure μ is defined by = g(x)dx and g(x) A(α, β, α; Ω), α > 1, β > 0. (iii) The inequalities obtained in this article are extensions of the usual A r (E)-weighted inequalities since the A r (E) is a proper subset of the A(α, β, α; E)-class which can be used to extend many results with the A r (E)-weight into the A(α, β, α; E)-weight.

Declarations

Acknowledgements

I would like to thank Professor Shusen Ding for his precious and thoughtful suggestions which greatly improved the presentation of the article. This work was supported by the Foundation of Education Department of Heilongjiang Province in 2011 (#12511111).

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology

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© Xing; licensee Springer. 2012

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