A new weight class and Poincaré inequalities with the Radon measure

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.

1. Introduction

Let Ω be a domain in ℝⁿ, 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 ⊂ ℝⁿ. We say w is a weight if $w\in L_{loc}^1\left({ℝ}^n\right)$ 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 ⊂ ℝⁿ, where r > 1 , and write w ∈ A _r (E) when

$$\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_{B}wdx\right){\left(\frac{1}{\left|B\right|}{\int }_B{\left(\frac{1}{w}\right)}^{\frac{1}{r-1}}dx\right)}^{r-1}<\infty ,$$

(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 [2–11]. 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 $su{p}_B\left(\frac{1}{\left|B\right|}{\int }_B{w}^{\lambda }dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{\left(\frac{1}{w}\right)}^{1/\left(r-1\right)}dx\right)}^{r-1}<\infty$ 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 $A_r^{\lambda }\left(E\right)$-weights was introduced in [13]. We say that the weight w(x) > 0 satisfies the $A_r^{\lambda }\left(E\right)$-condition in E, r > 1 and λ > 0, and write $w\in A_r^{\lambda }\left(E\right)$, if $su{p}_B\left(\frac{1}{\left|B\right|}{\int }_{B}wdx\right){\left(\frac{1}{\left|B\right|}{\int }_B{w}^{1/\left(1-r\right)}dx\right)}^{\lambda \left(r-1\right)}<\infty$ for any ball B ⊂ E ⊂ ℝⁿ. Also, it is easy to see that $A_r^1\left(E\right)=A_r\left(E\right)$. Both A _r (λ, E) and $A_r^{\lambda }\left(E\right)$ have widely been used in the study of the weighted inequalities and integral estimates, see [4–6, 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 ℝⁿ, then || fg || _s,E ≤|| f ||_α,E⋅|| g || _β,E for any E ⊂ ℝⁿ.

We introduce the following class of functions which is an extension of the several existing classes of weights, such as $A_r^{\lambda }\left(E\right)$-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 ⊂ ℝⁿsatisfies 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

$$\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_B{g}^{\alpha }dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{g}^{-\beta }dx\right)}^{\gamma /\beta }<\infty ,$$

(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, $A_r^{\lambda }\left(E\right)=A\left(1,1/\left(r-1\right),\lambda ;E\right)$. 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 ⊂ ℝⁿ. Then, (i) There exists a constant α₀ > 1 such that A _r (E) ⊂ A(α₀,1/(r- 1),α₀; 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

$${\left(\frac{1}{\left|B\right|}{\int }_B{w}^{{\alpha }_0}dx\right)}^{1/{\alpha }_0}\le \frac{C_1}{\left|B\right|}{\int }_{B}wdx$$

(2.2)

for all balls B ⊂ E, i.e.,

$$\frac{1}{\left|B\right|}{\int }_B{w}^{{\alpha }_0}dx\le C_2{\left(\frac{1}{\left|B\right|}{\int }_{B}wdx\right)}^{{\alpha }_0}.$$

(2.3)

From (2.3) and (1.1), we obtain

$$\begin{array}{c}\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_B{w}^{{\alpha }_0}dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{w}^{-\frac{1}{r-1}}dx\right)}^{{\alpha }_0\left(r-1\right)}\hfill \\ \le C_2\underset{B}{\mathit{\text{sup}}}{\left(\frac{1}{\left|B\right|}{\int }_{B}wdx\right)}^{{\alpha }_0}{\left(\frac{1}{\left|B\right|}{\int }_B{w}^{-\frac{1}{r-1}}dx\right)}^{{\alpha }_0\left(r-1\right)}\hfill \\ \le C_2{\left(\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_{B}wdx\right){\left(\frac{1}{\left|B\right|}{\int }_B{\left(\frac{1}{w}\right)}^{\frac{1}{r-1}}dx\right)}^{r-1}\right)}^{{\alpha }_0}<\infty ,\hfill \end{array}$$

(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

$${\left({\int }_B{w}^{\alpha }dx\right)}^{1/\alpha }\le \left({\int }_{B}wdx\right){\left({\int }_B{1}^{\frac{\alpha }{1-\alpha }}dx\right)}^{\left(1-\alpha \right)/\alpha },$$

(2.5)

that is

$${\left(\frac{1}{\left|B\right|}{\int }_B{w}^{\alpha }dx\right)}^{1/\alpha }\le \frac{1}{\left|B\right|}{\int }_{B}wdx$$

which can be written as

$$\frac{1}{\left|B\right|}{\int }_B{w^{\alpha }dx\le \left(\frac{1}{\left|B\right|}{\int }_{B}wdx\right)}^{\alpha }.$$

(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 ⊂ ℝⁿbe 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 ℝ², that is n = 2. Assume that D ⊂ ℝ² 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 ⊂ ℝⁿ, 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

$${\left({{\int }_B\left|g_1+g_2\right|}^{\alpha }dx\right)}^{\frac{1}{\alpha }}\le {\left({\int }_B{\left|g_1\right|}^{\alpha }dx\right)}^{\frac{1}{\alpha }}+{\left({\int }_B{\left|g_2\right|}^{\alpha }dx\right)}^{\frac{1}{\alpha }}.$$

(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

$$\begin{array}{ll}\hfill {\int }_B{\left(g_1+g_2\right)}^{\alpha }dx& \le {\left({\left({\int }_B{\left|g_1\right|}^{\alpha }dx\right)}^{\frac{1}{\alpha }}+{\left({\int }_B{\left|g_2\right|}^{\alpha }dx\right)}^{\frac{1}{\alpha }}\right)}^{\alpha }\\ \le 2^{\alpha }\left({\int }_B{\left|g_1\right|}^{\alpha }dx+{\int }_B{\left|g_2\right|}^{\alpha }dx\right).\end{array}$$

(2.8)

Note that g₁(x), g₂(x) ∈ A(α, β, γ; E). Using (2.8) , we obtain

$$\begin{array}{c}\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_B{\left(g_1+g_2\right)}^{\alpha }dx\right){\left(\frac{1}{\left|B\right|}{{\int }_B\left(g_1+g_2\right)}^{-\beta }dx\right)}^{\gamma /\beta }\hfill \\ \le \underset{B}{\mathit{\text{sup}}}{2}^{\alpha }\left(\frac{1}{\left|B\right|}{\int }_B{\left|g_1\right|}^{\alpha }dx+\frac{1}{\left|B\right|}{\int }_B{\left|g_2\right|}^{\alpha }dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{\left(g_1+g_2\right)}^{-\beta }dx\right)}^{\gamma /\beta }\hfill \\ \le \underset{B}{\mathit{\text{sup}}}{2}^{\alpha }\left(\frac{1}{\left|B\right|}{\int }_B{g}_1^{\alpha }dx{\left(\frac{1}{\left|B\right|}{\int }_B{g}_1^{-\beta }dx\right)}^{\gamma /\beta }+\frac{1}{\left|B\right|}{\int }_B{g}_2^{\alpha }dx{\left(\frac{1}{\left|B\right|}{\int }_B{g}_2^{-\beta }dx\right)}^{\gamma /\beta }\right)\hfill \\ <\infty .\hfill \end{array}$$

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 ⊂ ℝⁿ, where α _i , β _i > 0, i = 1,2, and $\frac{1}{\alpha }=\frac{1}{{\alpha }_1}+\frac{1}{{\alpha }_2},\frac{1}{\beta }=\frac{1}{{\beta }_1}+\frac{1}{{\beta }_2}$. Then, g₁(x)g₂(x) ∈ A(α, β, αγ; E).

Proof. Using Lemma 2.1 with $\frac{1}{\alpha }=\frac{1}{{\alpha }_1}+\frac{1}{{\alpha }_2}$ and $\frac{1}{\beta }=\frac{1}{{\beta }_1}+\frac{1}{{\beta }_2}$, respectively, we have

$${\left({\int }_B{\left(g_1{g}_2\right)}^{\alpha }dx\right)}^{1/\alpha }\le {\left({\int }_B{g}_1^{{\alpha }_1}dx\right)}^{1/{\alpha }_1}{\left({\int }_B{g}_2^{{\alpha }_2}dx\right)}^{1/{\alpha }_2},$$

(2.9)

$${\left({{\int }_B\left(g_1{g}_2\right)}^{-\beta }dx\right)}^{\gamma /\beta }\le {\left({\int }_B{g}_1^{-\beta }dx\right)}^{\gamma /{\beta }_1}{\left({\int }_B{g}_2^{-{\beta }_2}dx\right)}^{\gamma /{\beta }_2}.$$

(2.10)

Combining (2.9) and (2.10) yields

$$\begin{array}{c}{\left({\int }_B{\left(g_1{g}_2\right)}^{\alpha }dx\right)}^{1/\alpha }{\left({\int }_B{\left(g_1{g}_2\right)}^{-\beta }dx\right)}^{\gamma /\beta }\hfill \\ \le {\left({\int }_B{g}_1^{{\alpha }_1}dx\right)}^{1/{\alpha }_1}{\left({\int }_B{g}_1^{-{\beta }_1}dx\right)}^{\gamma /{\beta }_1}{\left({\int }_B{g}_2^{{\alpha }_2}dx\right)}^{1/{\alpha }_2}{\left({\int }_B{g}_2^{-{\beta }_2}dx\right)}^{\gamma /{\beta }_2}\hfill \end{array}$$

(2.11)

which is equivalent to

$$\begin{array}{c}{\left({{\int }_B{\left(g_1{g}_2\right)}^{\alpha }dx\left({\int }_B{\left(g_1{g}_2\right)}^{-\beta }dx\right)}^{\alpha \gamma /\beta }\right)}^{1/\alpha }\hfill \\ \le {\left({{\int }_B{g}_1^{{\alpha }_1}dx\left({\int }_B{g}_1^{-{\beta }_1}dx\right)}^{{\alpha }_1\gamma /{\beta }_1}\right)}^{1/{\alpha }_1}{\left({\int }_B{g}_2^{{\alpha }_2}dx{\left({\int }_B{g_2}^{-{\beta }_2}dx\right)}^{{\alpha }_2\gamma /{\beta }_2}\right)}^{1/{\alpha }_2}.\hfill \end{array}$$

(2.12)

Noticing that g₁(x) ∈ A(α₁, β₁, α₁ γ; E) and g₂(x) ∈ A(α₂, β₂, α₂γ; E), we obtain

$$\begin{array}{c}\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}\underset{B}{\int }{\left(g_1{g}_2\right)}^{\alpha }dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{\left(g_1{g}_2\right)}^{-\beta }dx\right)}^{\alpha \gamma /\beta }\hfill \\ \le {\left(\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_B{g}_1^{{\alpha }_1}dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{g}_1^{-{\beta }_1}dx\right)}^{\frac{{\alpha }_1\gamma }{{\beta }_1}}\right)}^{\frac{\alpha }{{\alpha }_1}}{\left(\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_B{g}_2^{{\alpha }_2}dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{g}_2^{-{\beta }_2}dx\right)}^{\frac{{\alpha }_2\gamma }{{\beta }_2}}\right)}^{\frac{\alpha }{{\alpha }_2}}\hfill \\ <\infty .\hfill \end{array}$$

(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 $\frac{1}{\alpha p}=\frac{1}{\alpha }+\frac{1-p}{\alpha p}$ yields

$${\left({\int }_B{g}^{\alpha p}dx\right)}^{1/\alpha p}\le {\left|B\right|}^{\left(1-p\right)/\alpha p}{\left({\int }_B{g}^{\alpha }dx\right)}^{1/\alpha },$$

that is

$$\frac{1}{\left|B\right|}{\int }_B{\left(g^p\right)}^{\alpha }dx\le {\left(\frac{1}{\left|B\right|}{\int }_B{g}^{\alpha }dx\right)}^p.$$

(2.14)

Since g(x) ∈ A(α, βp, γ; E), using (2.14) , we find that

$$\begin{array}{c}\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_B{\left(g^p\right)}^{\alpha }dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{\left(g^p\right)}^{-\beta }dx\right)}^{\gamma /\beta }\hfill \\ \le \underset{B}{\mathit{\text{sup}}}{\left(\frac{1}{\left|B\right|}{\int }_B{g}^{\alpha }dx\right)}^p{\left(\frac{1}{\left|B\right|}{\int }_B{g}^{-\beta p}dx\right)}^{\gamma /\beta }\hfill \\ \le \underset{B}{\mathit{\text{sup}}}{\left(\left(\frac{1}{\left|B\right|}{\int }_B{g}^{\alpha }dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{g}^{-\beta p}dx\right)}^{\gamma /\beta p}\right)}^p\hfill \\ \le {\left(\underset{B}{\mathit{\text{sup}}}\left(\frac{1}{\left|B\right|}{\int }_B{g}^{\alpha }dx\right){\left(\frac{1}{\left|B\right|}{\int }_B{g}^{-\beta p}dx\right)}^{\gamma /\beta p}\right)}^p\hfill \\ <\infty .\hfill \end{array}$$

(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) $\frac{1}{g\left(x\right)}\in A\left(\alpha ,\beta ,\gamma ;E\right)$ 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 ℝⁿ. For example, the function u(x₁, x₂,...,x _n ) is called a 0-form. The 1-form u(x) in ℝⁿcan be written as $u\left(x\right)={\sum }_{i=1}^n{u}_i\left(x_1,x_2,...,x_n\right)d{x}_i$. 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 $\left\{d{x}_{i_1}\wedge d{x}_{i_2}\wedge \cdots \wedge d{x}_{i_k}\right\},k=1,2,...,n$, that is, $u\left(x\right)={\sum }_I{u}_I\left(x\right)d{x}_I=\sum u_{i_1{i}_2\cdots i_k}\left(x\right)d{x}_{i_1}\wedge d{x}_{i_2}\wedge \cdots \wedge d{x}_{i_k}$, where I = (i₁, i₂,...,i _k ), 1 ≤ i₁ < i₂ < ... < i _k ≤ n. Let ∧^l= ∧^l(ℝⁿ) be the set of all l-forms in ℝⁿand 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

$$d^{*}A\left(x,du\right)=B\left(x,du\right)$$

(3.1)

which is called non-homogeneous A -harmonic equation, where A : Ω × ∧^l(ℝⁿ) → ∧^l(ℝⁿ) and B : Ω × ∧^l(ℝⁿ) → ∧^l-1(ℝⁿ) satisfy the conditions: |A(x, ξ)| ≤ a|ξ|^p-1, A(x, ξ) ⋅ ξ ≥ |ξ|^pand |B(x, ξ)| ≤ b|ξ|^p-1for almost every x ∈ Ω and all ξ ∈ ∧^l(ℝⁿ). 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 $W_{loc}^{1,p}\left(\mathrm{\Omega },{\wedge }^{l-1}\right)$ such that ∫_Ω A(x, du) ⋅ dφ + B(x, du) ⋅ φ = 0 for all $\phi \in W_{loc}^{1,p}\left(\mathrm{\Omega },{\wedge }^{l-1}\right)$ with compact support. If u is a function (0-form) in ℝⁿ, the equation (3.1) reduces to

$$\mathsf{\text{div}}A\left(x,\nabla u\right)=B\left(x,\nabla u\right).$$

(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(ℝⁿ) → ∧^l(ℝⁿ) 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*(du|du|^p-2) = 0 for differential forms. See [5, 6, 9–16] 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 $\left(K_{y}u\right)\left(x;{\xi }_1,...,{\xi }_{l-1}\right)={\int }_0^1{t}^{l-1}u\left(tx+y-ty;x-y,{\xi }_1,...,{\xi }_{l-1}\right)dt$. 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 $\varphi \in C_0^{\infty }\left(D\right)$ 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 $u\in L_{loc}^s\left(D,{\wedge }^l\right),l=1,2,...,n,1<s<\infty$, we have

$${∥Tu∥}_{s,D}\le C\left|D\right|diam\left(D\right){∥u∥}_{s,D}.$$

(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, B}≤ C|B|^(t-s)/st||du|| _t,σB for all balls or cubes B with σB ⊂ Ω.

Theorem 3.2. Let $u\in L_{loc}^s\left(\mathrm{\Omega },{\wedge }^l\right)$ be a solution of the non-homogeneous A-harmonic equation (3.1) in a domain $\mathrm{\Omega },du\in L_{loc}^s\left(\mathrm{\Omega },{\wedge }^{l+1}\right),l=0,1,...,n-1$ and 1 < s < ∞. Then, there exists a constant C, independent of u, such that

$${\left({\int }_B{\left|u-u_B\right|}^{s}d\mu \right)}^{1/s}\le C\left|B\right|diam\left(B\right){\left({\int }_{\sigma B}{\left|du\right|}^{s}d\mu \right)}^{1/s}$$

(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

$${∥u-u_B∥}_{t,B}={∥T\left(du\right)∥}_{t,B}\le C_1\left|B\right|diam\left(B\right){∥du∥}_{t,B}$$

(3.5)

for any t > 1. Now, choose t = αs/(α - 1), then, t > s. Using the Hölder inequality and (3.5), we obtain

$$\begin{array}{cc}\hfill {\left({\int }_B{\left|u-u_B\right|}^{s}d\mu \right)}^{1/s}& ={\left({\int }_B{\left|u-u_B\right|}^{s}g\left(x\right)dx\right)}^{1/s}\hfill \\ ={\left({\int }_B{\left(\left|u-u_B\right|g^{1/s}\left(x\right)\right)}^{s}dx\right)}^{1/s}\hfill \\ \le {\left({\int }_B{\left|u-u_B\right|}^{t}dx\right)}^{1/t}{\left({\int }_B{g}^{t/\left(t-s\right)}\left(x\right)dx\right)}^{\left(t-s\right)/st}\hfill \\ \le C_2\left|B\right|diam\left(B\right){∥du∥}_{t,B}{\left({\int }_B{g}^{\alpha }\left(x\right)dx\right)}^{1/\alpha s}.\hfill \end{array}$$

(3.6)

Let m = βs/(1 + β), then 0 < m < s. From Lemma 3.1, we have

$${∥du∥}_{t,B}\le C_3\left|B\right|\frac{m-t}{mt}∥du∥m,{\sigma }_{1}B,$$

(3.7)

where σ₁ > 1 is a constant. Using the Hölder inequality again, we find that

$$\begin{array}{cc}\hfill {∥du∥}_{m,{\sigma }_{1}B}& ={\left({\int }_{{\sigma }_{1}B}{\left(\left|du\right|{\left(g\left(x\right)\right)}^{1/s}{\left(g\left(x\right)\right)}^{-1/s}\right)}^{m}dx\right)}^{1/m}\hfill \\ \le {\left({\int }_{{\sigma }_{1}B}{\left|du\right|}^{s}g\left(x\right)dx\right)}^{1/s}{\left({\int }_{{\sigma }_{1}B}{\left(g^{-1/s}\left(x\right)\right)}^{\frac{ms}{s-m}}dx\right)}^{\frac{s-m}{ms}}\hfill \\ \le {\left({\int }_{{\sigma }_{1}B}{\left|du\right|}^{s}g\left(x\right)dx\right)}^{1/s}{\left({\int }_{{\sigma }_{1}B}{\left(g\left(x\right)\right)}^{\frac{-m}{s-m}}dx\right)}^{\frac{s-m}{ms}}\hfill \\ \le {\left({\int }_{{\sigma }_{1}B}{\left|du\right|}^{s}d\mu \right)}^{1/s}{\left({\int }_{{\sigma }_{1}B}{g}^{-\beta }\left(x\right)dx\right)}^{1/\beta s}.\hfill \end{array}$$

(3.8)

Since g ∈ A(α, β, α; Ω), it follows that

$$\begin{array}{c}{\left({\int }_B{g}^{\alpha }\left(x\right)dx\right)}^{1/\alpha s}{\left({\int }_{{\sigma }_{1}B}{g}^{-\beta }\left(x\right)dx\right)}^{1/\beta s}\hfill \\ \le {\left(\left({\int }_{{\sigma }_{1}B}{g}^{\alpha }\left(x\right)dx\right){\left({\int }_{{\sigma }_{1}B}{g}^{-\beta }\left(x\right)dx\right)}^{\alpha /\beta }\right)}^{1/\alpha s}\hfill \\ ={\left({\left|{\sigma }_{1}B\right|}^{1+\frac{\alpha }{\beta }}\left(\frac{1}{\left|{\sigma }_{1}B\right|}{\int }_{{\sigma }_{1}B}{g}^{\alpha }\left(x\right)dx\right){\left(\frac{1}{\left|{\sigma }_{1}B\right|}{\int }_{{\sigma }_{1}B}{g}^{-\beta }\left(x\right)dx\right)}^{\alpha /\beta }\right)}^{1/\alpha s}\hfill \\ \le C_4{\left|B\right|}^{1/\alpha s+1/\beta s}.\hfill \end{array}$$

(3.9)

Combining (3.6), (3.7), and (3.8) and using (3.9), we have

$$\begin{array}{c}{\left({\int }_B{\left|u-uB\right|}^{s}d\mu \right)}^{1/s}\hfill \\ \le C_5\left|B\right|diam\left(B\right){\left|B\right|}^{\frac{m-t}{mt}}{\left({\int }_{{\sigma }_{1}B}{\left|du\right|}^{s}d\mu \right)}^{1/s}{\left({\int }_B{g}^{\alpha }\left(x\right)dx\right)}^{1/\alpha s}{\left({\int }_{{\sigma }_{1}B}{g}^{-\beta }\left(x\right)dx\right)}^{1/\beta s}\hfill \\ \le C_{5}diam\left(B\right){\left|B\right|}^{1+\frac{1}{t}-\frac{1}{m}}{\left({\int }_{{\sigma }_{1}B}{\left|du\right|}^{s}d\mu \right)}^{1/s}{\left(\left({\int }_B{g}^{\alpha }\left(x\right)dx\right){\left({\int }_{{\sigma }_{1}B}{g}^{-\beta }\left(x\right)dx\right)}^{\alpha /\beta }\right)}^{1/\alpha s}\hfill \\ \le C_6\left|B\right|diam\left(B\right){\left({\int }_{{\sigma }_{1}B}{\left|du\right|}^{s}d\mu \right)}^{1/s},\hfill \end{array}$$

that is

$${\left({\int }_B{\left|u-u_B\right|}^{s}d\mu \right)}^{1/s}\le C_6\left|B\right|diam\left(B\right){\left({\int }_{{\sigma }_{1}B}{\left(du\right)}^{s}d\mu \right)}^{1/s}.$$

We have completed the proof of Theorem 3.2.

Let $g\left(x\right)=\frac{1}{{\left|x-x_B\right|}^{\lambda }}$, where x _B be the center of the ball B ⊂ Ω and $0<\lambda <\frac{n}{\alpha },\alpha >1$. Then, g(x) ∈ A (α, β, α; Ω). From Theorem 3.2, we have the following corollary.

Corollary 3.3. Let $u\in L_{loc}^s\left(\mathrm{\Omega },{\wedge }^l\right)$ be a solution of the non-homogeneous A-harmonic equation (3.1) in a domain $\mathrm{\Omega },du\in L_{loc}^s\left(\mathrm{\Omega },{\wedge }^{l+1}\right),l=0,1,...,n-1$ and 1 < s < ∞. Then, there exists a constant C, independent of u, such that

$${\left({\int }_B{\left|u-u_B\right|}^{s}d\mu \right)}^{1/s}\le C\left|B\right|diam\left(B\right){\left({\int }_{\sigma B}{\left|du\right|}^{s}d\mu \right)}^{1/s}$$

(3.10)

for all balls B with σB ⊂ Ω, where the Radon measure μ is defined by $d\mu =\frac{1}{{\left|x-x_B\right|}^{\lambda }}dx$, x _B is the center of the ball $B\subset \mathrm{\Omega },0<\lambda <\frac{n}{\alpha }$ 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 ⊂ ℝⁿis called an L^s-averaging domain, s ≥ 1, if there exists a constant C such that

$${\left(\frac{1}{\left|\mathrm{\Omega }\right|}{\int }_{\mathrm{\Omega }}{\left|u-u_{\mathrm{\Omega }}\right|}^{s}dx\right)}^{1/s}\le C\underset{B\subset \mathrm{\Omega }}{\mathit{\text{sup}}}{\left(\frac{1}{\left|B\right|}{\int }_B{\left|u-u_B\right|}^{s}dx\right)}^{1/s}$$

for all $u\in L_{loc}^s\left(\mathrm{\Omega }\right)$.

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 ⊂ ℝⁿan L^s(μ)-averaging domain, s ≥ 1, if there exists a constant C such that

$${\left(\frac{1}{\mu \left(\mathrm{\Omega }\right)}{\int }_{\mathrm{\Omega }}{\left|u-u_{B_0}\right|}^{s}d\mu \right)}^{1/s}\le C\underset{B\subset \mathrm{\Omega }}{\mathit{\text{sup}}}{\left(\frac{1}{\mu \left(B\right)}{\int }_B{\left|u-u_B\right|}^{s}dx\right)}^{1/s}$$

for some ball B₀ ⊂ Ω and all $u\in L_{loc}^s\left(\mathrm{\Omega };\mu \right)$, where the Radon measure μ(x) is defined by dμ = 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

$${\left({\int }_{\mathrm{\Omega }}{\left|u-u_{B_0}\right|}^{s}d\mu \right)}^{1/s}\le C{\left(\mu \left(\mathrm{\Omega }\right)\right)}^{1+1/n}{\left({\int }_{\mathrm{\Omega }}{\left|du\right|}^{s}d\mu \right)}^{1/s}$$

(4.1)

for any L^s(μ)-averaging domain Ω ⊂ ℝⁿwith 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

$$G\left(x\right)=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill x\in {\mathrm{\Omega }}_1\hfill \\ \hfill g\left(x\right),\hfill & \hfill x\in {\mathrm{\Omega }}_2.\hfill \end{array}\right.$$

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,

$$\begin{array}{ll}\hfill {\left({\int }_{\Omega }{\left|u-u_{B_0}\right|}^{s}d\mu \right)}^{1/s}& ={\left({\int }_{\Omega }{\left|u-u_{B_0}\right|}^{s}g\left(x\right)dx\right)}^{1/s}\\ \le {\left({\int }_{\Omega }{\left|u-u_{B_0}\right|}^{s}G\left(x\right)dx\right)}^{1/s}\end{array}$$

(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

$$\mu \left(D\right)={\int }_{D}d\mu ={\int }_{D}g\left(x\right)dx\ge {\int }_{D}dx=\left|D\right|.$$

(4.3)

Note that diam(B) = C₁|B|^1/n. From Theorem 3.2, we obtain

$${\left(\frac{1}{\left|B\right|}{\int }_B{\left|u-u_B\right|}^{s}d\mu \right)}^{1/s}\le C_2{\left|B\right|}^{1+1/n-1/s}{\left({\int }_{\sigma B}{\left|du\right|}^{s}d\mu \right)}^{1/s}.$$

(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

$$\begin{array}{cc}\hfill {\left(\frac{1}{\mu \left(\Omega \right)}{\int }_{\Omega }{\left|u-u_{B_0}\right|}^{s}d\mu \right)}^{1/s}& \le C_3\underset{B\subset \Omega }{sup}{\left(\frac{1}{\mu \left(B\right)}{\int }_B{\left|u-u_B\right|}^{s}d\mu \right)}^{1/s}\hfill \\ \le C_3\underset{B\subset \Omega }{sup}{\left(\frac{1}{\left|B\right|}{\int }_B{\left|u-u_B\right|}^{s}d\mu \right)}^{1/s}\hfill \\ \le C_4\underset{B\subset \Omega }{sup}{\left|B\right|}^{1+1/n-1/s}{\left({\int }_{\sigma B}{\left|du\right|}^{s}d\mu \right)}^{1/s}\hfill \\ \le C_4{\left|\Omega \right|}^{1+1/n-1/s}\underset{B\subset \Omega }{sup}{\left({\int }_{\sigma B}{\left|du\right|}^{s}d\mu \right)}^{1/s}\hfill \\ \le C_4{\left|\Omega \right|}^{1+1/n-1/s}{\left({\int }_{\Omega }{\left|du\right|}^{s}d\mu \right)}^{1/s}\hfill \\ \le C_4{\left(\mu \left(\Omega \right)\right)}^{1+1/n-1/s}{\left({{\int }_{\Omega }\left|du\right|}^{s}d\mu \right)}^{1/s},\hfill \end{array}$$

that is

$${\left({\int }_{\mathrm{\Omega }}{\left|u-u_{B_0}\right|}^{s}d\mu \right)}^{1/s}\le C{\left(\mu \left(\mathrm{\Omega }\right)\right)}^{1+1/n}{\left({\int }_{\mathrm{\Omega }}{\left|du\right|}^{s}d\mu \right)}^{1/s}.$$

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

$${\left({\int }_{\mathrm{\Omega }}{\left|u-u_{B_0}\right|}^{s}d\mu \right)}^{1/s}\le C{\left({\int }_{\mathrm{\Omega }}{\left|du\right|}^{s}d\mu \right)}^{1/s},$$

(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

$${\left({\int }_{\mathrm{\Omega }}{\left|u-u_{B_0}\right|}^{s}d\mu \right)}^{1/s}\le C{\left({\int }_{\mathrm{\Omega }}{\left|du\right|}^{s}d\mu \right)}^{1/s},$$

where Ω is an L^s(μ)-averaging domain Ω ⊂ ℝⁿwith μ(Ω) < ∞ and 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. (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.