A new weight class and Poincaré inequalities with the Radon measure

Let Ω be a domain in ℝ 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 ⊂ ℝ ⁿ . We say w is a weight if w ∈ L l o c 1 ( ℝ 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 ⊂ ℝ ⁿ , where r > 1 , and write w ∈ A _r (E) when

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 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 ⊂ ℝ 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 [4‐6, 12, 13] for example.

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 ℝ ⁿ , then || fg || _s,E ≤|| f ||_α,E⋅|| g || _β,E for any E ⊂ ℝ 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 ⊂ ℝ 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

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 ⊂ ℝ ⁿ . 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

for all balls B ⊂ E, i.e.,

From (2.3) and (1.1), we obtain

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

that is

which can be written as

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 ⊂ ℝ 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 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 ⊂ ℝ 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

Since |a + b| ^s ≤ 2 ^s (|a| ^s + |b| ^s ) for any constants a, b, s with s > 0, from (2.7) , we have

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

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 ⊂ ℝ 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

Combining (2.9) and (2.10) yields

which is equivalent to

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

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

that is

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

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.

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 ℝ ⁿ . For example, the function u(x₁, x₂,...,x _n ) is called a 0-form. The 1-form u(x) in ℝ 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 (ℝ ⁿ ) be the set of all l-forms in ℝ 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

which is called non-homogeneous A -harmonic equation, 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 and |B(x, ξ)| ≤ b|ξ|^p-1for 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 (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) ⋅ dφ + 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 ℝ ⁿ , the equation (3.1) reduces to

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 (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*(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 ( 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

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 ∈ 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

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

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

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

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

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

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

that is

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

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.

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 ⊂ ℝ ⁿ is called an L ^s -averaging domain, s ≥ 1, if there exists a constant C such that

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

for some ball B₀ ⊂ Ω and all u ∈ L l o c s ( Ω ; μ ), 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

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

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,

with G(x) ≥ 1. Hence, we may suppose that g(x) ≥ 1 a.e. in Ω. Thus, for any D ⊂ Ω, we have

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

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

that is

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

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

where Ω is an L ^s (μ)-averaging domain Ω ⊂ ℝ ⁿ with 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.