Some Orlicz norms inequalities for the composite operator T ∘ d ∘ H

Abstract

In this article, we first establish the local inequality for the composite operator T ∘ d ∘ H with Orlicz norms. Then, we extend the local result to the global case in the L^φ (μ)-averaging domains.

1 Introduction

Recently as generalizations of the functions, differential forms have been widely used in many fields, such as potential theory, partial differential equations, quasiconformal mappings, and nonlinear analysis; see [1–4]. With the development of the theory of quasiconformal mappings and other relevant theories, a series of results about the solutions to different versions of the A-harmonic equation have been found; see [5–9]. Especially, the research on the inequalities of the various operators and their compositions applied to the solutions to different sorts of the A-harmonic equation has made great progress [5]. The inequalities equipped with the L^p -norm for differential forms have been very well studied. However, the inequalities with Orlicz norms have not been fully developed [9, 10]. Also, both L^p -norms and Orlicz norms of differential forms depend on the type of the integral domains. Since Staples introduced the L^s -averaging domains in 1989, several kinds of domains have been developed successively, including L^s (μ)-averaging domains, see [11–13]. In 2004, Ding [14] put forward the concept of the L^φ (μ)-averaging domains, which is considered as an extension of the other domains involved above and specified later.

The homotopy operator T, the exterior derivative operator d, and the projection operator H are three important operators in differential forms; for the first two operators play critical roles in the general decomposition of differential forms [15] while the latter in the Hodge decomposition [16]. This article contributes primarily to the Orlicz norm inequalities for the composite operator T ∘ d ∘ H applied to the solutions of the nonhomogeneous A-harmonic equation.

In this article, we first introduce some essential notation and definitions. Unless otherwise indicated, we always use Θ to denote a bounded convex domain in ℝ ⁿ (n ≥ 2), and let O be a ball in ℝ ⁿ . Let ρO denote the ball with the same center as O and diam(ρO) = ρdiam(O), ρ > 0. We say ν is a weight if $\nu \in L_{loc}^1\left({ℝ}^n\right)$ and ν > 0 a.e; see [17]. |D| is used to denote the Lebesgue measure of a set D ⊂ ℝ ⁿ , and the measure μ is defined by dμ = ν(x)dx. We use ||f|| _s,O for ${\left({\int }_O\mid f{\mid }^{s}dx\right)}^{\frac{1}{s}}$ and ||f||_s,O,νfor ${\left({\int }_O\mid f{\mid }^s\nu \left(x\right)dx\right)}^{\frac{1}{s}}$.

Let [5, 15]Λ^ℓ = Λ^ℓ (ℝ ⁿ ), ℓ = 0, 1,..., n, be the linear space of all ℓ-forms $\hslash \left(x\right)={\sum }_J{\hslash }_J\left(x\right)d{x}_J={\sum }_J{\hslash }_{j_1{j}_2\cdots j_{\ell }}\left(x\right)d{x}_{j_1}\wedge d{x}_{j_2}\cdots \wedge d{x}_{j_{\ell }}$ in ℝ ⁿ , where J = (j₁, j₂,..., j_ℓ), 1 ≤ j₁ < j₂ < ⋯ < j_ℓ ≤ n, ℓ = 0, 1,..., n, are the ordered ℓ-tuples. The Grassman algebra Λ^ℓ is a graded algebra with respect to the exterior products. For α = Σ _J α _J dx _J ∈ Λ^ℓ (ℝ ⁿ ) and β = Σ _J β _J dx _J ∈ Λ^ℓ (ℝ ⁿ ), the inner product in Λ^ℓ(ℝ ⁿ ) is given by 〈α, β〉 = Σ _J α _J β _J with summation over all ℓ-tuples J = (j₁, j₂, ..., j_ℓ), ℓ = 0, 1,..., n. Let C^∞ (Θ, ∧^ℓ) be the set of infinitely differentiable ℓ-forms on Θ ⊂ ℝ ⁿ , D'(Θ, Λ^ℓ) the space of all differential ℓ-forms in Θ and L^s (Θ, Λ^ℓ) the set of the ℓ-forms in Θ satisfying ${\displaystyle {\int }_{\Theta }}{({\Sigma }_J\mid {\omega }_J(x{)\mid }^2)}^{\frac{s}{2}}dx<\infty$ for all ordered ℓ-tuples J. The exterior derivative d: D'(Θ, Λ^ℓ) → D'(Θ, Λ^ℓ+1), ℓ = 0, 1,..., n - 1, is given by

$$d\hslash \left(x\right)=\sum _{i=1}^n\sum _J\frac{\partial {\omega }_{j_1{j}_2\cdots j_{\ell }}\left(x\right)}{\partial x_i}d{x}_i\wedge d{x}_{j_1}\wedge d{x}_{j_2}\cdots \wedge d{x}_{j_{\ell }}$$

(1.1)

for all ħ ∈ D'(Θ, Λ^ℓ), and the Hodge codifferential operator d^⋆ is defined as d^⋆ = (-1)^{n ℓ+1}⋆ d⋆ : D'(Θ, Λ^ℓ+1) → D'(Θ, Λ^ℓ), where ⋆ is the Hodge star operator.

With respect to the nonhomogeneous A-harmonic equation for differential forms, we indicate its general form as follows:

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

(1.2)

where A: Θ × Λ^ℓ(ℝ ⁿ ) → Λ^ℓ(ℝ ⁿ ) and B: Θ × Λ^ℓ(ℝ ⁿ ) → Λ^ℓ-1 (ℝ ⁿ ) satisfy the conditions: |A(x, η)| ≤ a|η|^s-1, A(x, η) · η ≥ |η| ^s , and |B(x, η)| ≤ b|η|^s-1for almost every x ∈ Θ and all η ∈ Λ^ℓ (ℝ ⁿ ). Here a, b > 0 are some constants, and 1 < s < ∞ is a fixed exponent associated with (1.2). A solution to (1.2) is an element of the Sobolev space $W_{loc}^{1,s}\left(\Theta ,{\Lambda }^{\ell -1}\right)$ such that

$${\int }_{\Theta }A\left(x,d\hslash \right)\cdot d\psi +B\left(x,d\hslash \right)\cdot \psi =0$$

(1.3)

for all $\psi \in W_{loc}^{1,s}\left(\Theta ,{\Lambda }^{\ell -1}\right)$ with compact support, where $W_{loc}^{1,s}\left(\Theta ,{\Lambda }^{\ell -1}\right)$ is the space of ℓ-forms whose coefficients are in the Sobolev space $W_{loc}^{1,s}\left(\Theta \right)$.

If the operator B = 0, (1.2) becomes

$$d^{*}A\left(x,d\hslash \right)=0,$$

(1.4)

which is called the (homogeneous) A-harmonic equation.

In [15], Iwaniec and Lutoborski gave the linear operator K _y : C^∞(Θ, Λ^ℓ) → C^∞(Θ, Λ^ℓ-1) as $\left(K_y\hslash \right)\left(x;{\theta }_1,\dots ,{\theta }_{\ell -1}\right)={\int }_0^1{t}^{\ell -1}\hslash \left(tx+y-ty;x-y,{\theta }_1,\dots ,{\theta }_{\ell -1}\right)dt$ for each y ∈ Θ. Then, the homotopy operator T: C^∞(Θ, Λ^ℓ) → C^∞(Θ, Λ^ℓ-1) is denoted by

$$T\hslash ={\int }_{\Theta }\upsilon \left(y\right)K_y\hslash dy,$$

(1.5)

where $\upsilon \in C_0^{\infty }\left(\Theta \right)$ is normalized so that ${\int }_{\Theta }\upsilon \left(y\right)dy=1$. The ℓ-form ħ _Θ ∈ D'(Θ, Λ^ℓ) is given by ${\hslash }_{\Theta }=\mid \Theta {\mid }^{-1}{\int }_{\Theta }\hslash \left(y\right)dy\left(\ell =0\right)$, ħ _Θ = d(Tħ)(ℓ = 1,..., n). In addition, we have the decomposition ħ = d(Tħ) + T(dħ) for each ħ ∈ L^s (Θ, Λ^ℓ), 1 ≤ s < ∞.

The definition of the H operator appeared in [16]. Let $L_{loc}^1\left(\Theta ,{\Lambda }^{\ell }\right)$ be the space of ℓ-forms whose coefficients are locally integrable, and $\mathcal{W}\left(\Theta ,{\Lambda }^{\ell }\right)$ the space of all $\Theta \in L_{loc}^1\left(\Theta ,{\Lambda }^{\ell }\right)$ that has generalized gradient. We define the harmonic ℓ-fields by $\mathcal{H}\left(\Theta ,{\Lambda }^{\ell }\right)=\left\{\Theta \in \mathcal{W}\left(\Theta ,{\Lambda }^{\ell }\right):d\hslash =d^{\star }\hslash =0,\hslash \in L^s\left(\Theta ,{\Lambda }^{\ell }\right)\mathsf{\text{for}}\mathsf{\text{some}}1<s<\infty \right\}$ and the orthogonal complement of $\mathcal{H}\left(\Theta ,{\Lambda }^{\ell }\right)$ in L¹(Θ, Λ^ℓ) as ${\mathcal{H}}^{\perp }=\left\{\omega \in L^1\left(\Omega ,{\Lambda }^{\ell }\right):<\omega ,h>=0forallh\in \mathcal{H}\left(\Theta ,{\Lambda }^{\ell }\right)\right\}$. Then, the H operator is defined by

$$H\left(\hslash \right)=\hslash -\Delta G\left(\hslash \right),$$

(1.6)

where ħ is in C^∞(Θ, Λ^ℓ), Δ = dd^⋆ + d^⋆d is the Laplace-Beltrami operator, and $G:C^{\infty }\left(\Theta ,{\Lambda }^{\ell }\right)\to {\mathcal{H}}^{\perp }\cap C^{\infty }\left(\Theta ,{\Lambda }^{\ell }\right)$ is the Green operator.

2 Main results

In this section, we first present some definitions of elementary conceptions, including Orlicz norms, the Young function, and the A(α, β, γ; Θ)-weight, then propose the local estimate for the composite operator of T ∘ d ∘ H with the Orlicz norm, and at last extend it to the global version in the L^φ (μ)-averaging domains. The proof of all the theorems in this section will be left in next section.

The Orlicz norm or Luxemburg norm differs from the traditional L^p -norm, whose definition is given as follows [18].

Definition 2.1. We call a continuously increasing function ϕ: [0, ∞) → [0, ∞) with ϕ(0) = 0 and ϕ(∞) = ∞ an Orlicz function, and a convex Orlicz function often denotes a Young function. Suppose that φ is a Young function, Θ is a domain with μ(Θ) < ∞, and f is a measurable function in Θ, then the Orlicz norm of f is denoted by

$$\parallel f{\parallel }_{\phi \left(\Theta ,\mu \right)}=inf\left\{\chi >0:\frac{1}{\mu \left(\Theta \right)}{\int }_{\Theta }\phi \left(\frac{\mid f\mid }{\chi }\right)d\mu \le 1\right\}.$$

(2.1)

The following class G(p, q, C) is introduced in [19], which is a special property of a Young function.

Definition 2.2. Let f and g be correspondingly a convex increasing function and a concave increasing function on [0, ∞). Then, we call a Young function φ belongs to the class G(p, q, C), 1 ≤ p < q < ∞, C ≥ 1, if

$$\left(\mathsf{\text{i}}\right)\frac{1}{C}\le \frac{\phi \left(t^{\frac{1}{p}}\right)}{f\left(t\right)}\le C,\left(\mathsf{\text{ii}}\right)\frac{1}{C}\le \frac{\phi \left(t^{\frac{1}{q}}\right)}{g\left(t\right)}\le C$$

(2.2)

for all t > 0.

Remark. From [19], we assert that φ, f, g in above definition are doubling, namely, φ(2t) ≤ C₁φ(t) for all t > 0, and the completely similar property remains valid if φ is replaced correspondingly with f, g. Besides, we have

$$\left(\mathsf{\text{i}}\right)C_2{t}^q\le g^{-1}\left(\phi \left(t\right)\right)\le C_3{t}^q,\left(\mathsf{\text{ii}}\right)C_2{t}^p\le f^{-1}\left(\phi \left(t\right)\right)\le C_3{t}^p,$$

(2.3)

where C₁, C₂, and C₃ are some positive constants.

The following weight class appeared in [9].

Definition 2.3. Let ν(x) is a measurable function defined on a subset Θ ⊂ ℝ ⁿ. Then, we call ν(x) satisfies the A(α, β, γ; Θ)-condition for some positive constants α, β, γ, if ν(x) > 0 a.e. and

$$\underset{O}{sup}\left(\frac{1}{\mid O\mid }{\int }_O{\nu }^{\alpha }dx\right){\left(\frac{1}{\mid O\mid }{\int }_O{\left(\frac{1}{\nu }\right)}^{\beta }dx\right)}^{\frac{\gamma }{\beta }}<\infty ,$$

(2.4)

where the supremum is over all balls O with O ⊂ Θ. We write ν(x) ∈ A(α, β, γ; Θ).

Remark. Note that the A(α, β, γ; Θ)-class is an extension of some existing classes of weights, such as $A_r^{\lambda }\left(\Theta \right)$-weights, A _r (λ, Θ)-weights, and A _r (Θ)-weights. Taking the $A_r^{\lambda }\left(\Theta \right)$-weights for example, if $\alpha =1,\beta =\frac{1}{r-1}$, and γ = λ in the above definition, then the A(α, β, γ; Θ)-class reduces to the desired weights; see [9] for more details about these weights.

The main objective of this section is Theorem 2.4.

Theorem 2.4. Let v ∈ C^∞(Θ, Λ^ℓ), ℓ = 1, 2,..., n, be a solution of the nonhomogeneous A-harmonic equation (1.2) in a bounded convex domain Θ, T : C^∞(Θ, Λ^ℓ) → C^∞(Θ, Λ^ℓ-1) be the homotopy operator defined in (1.5), d be the exterior derivative defined in (1.1), and H be the projection operator defined in (1.6). Suppose that φ is a Young function in the class G(p, q, C₀), 1 ≤ p < q < ∞, C₀ ≥ 1, $\phi \left(\mid v\mid \right)\in L_{loc}^1\left(\Theta ;\mu \right)$, and dμ = ν(x)dx, where ν(x) ∈ A(α, β, α, Θ) for α > 1 and β > 0 with ν(x) ≥ ε > 0 for any × ∈ Θ. Then, there exists a constant C, independent of v, such that

$$\parallel T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O{\parallel }_{\phi \left(O,\mu \right)}\le C\parallel v{\parallel }_{\phi \left(\rho O,\mu \right)}$$

(2.5)

for all balls O with ρO ⊂ Θ, where ρ > 1 is a constant.

The proof of Theorem 2.4 depends upon the following two arguments, that is, Lemma 2.5 and Theorem 2.6.

In [9], Xing and Ding proved the following lemma, which is a weighted version of weak reverse inequality.

Lemma 2.5. Let v be a solution of the nonhomogeneous A-harmonic equation (1.2) in a domain Θ and 0 < s, t < ∞. Then, there exists a constant C, independent of v, such that

$${\left({\int }_O\mid v{\mid }^{s}d\mu \right)}^{\frac{1}{s}}\le C{\left(\mu \left(O\right)\right)}^{\frac{t-s}{st}}{\left({\int }_{\rho O}\mid v{\mid }^{t}d\mu \right)}^{\frac{1}{t}}$$

(2.6)

for all balls O with ρO ⊂ Θ for some ρ > 1, where the measure μ is defined as the preceding theorem.

Remark. We call attention to the fact that Lemma 2.5 contains a A(α, β, α; Θ)-weight, which makes the inequality be more flexible and more useful. For example, if let dμ = dx in Lemma 2.5, then it reduces to the common weak reverse inequality:

$$\parallel v{\parallel }_{s,O}\le C\mid O{\mid }^{\frac{t-s}{st}}\parallel v{\parallel }_{t,\rho O}.$$

(2.7)

For the composite operator T ∘ d ∘ H, we have the following inequality with A(α, β, α; Θ)-weight.

Theorem 2.6. Let us assume, in addition to the definitions of the homotopy operator T, the exterior derivative d, the projection operator H, and the measure μ in Theorem 2.4, that q is any integer satisfying 1 < q < ∞, v ∈ C^∞(Θ, Λ^ℓ), ℓ = 1, 2,..., n, be a solution of the nonhomogeneous A-harmonic equation (1.2) in a bounded convex domain Θ and $\mid v\mid \in L_{loc}^q\left(\Theta ;\mu \right)$. Then, there exists a constant C, independent of ν, such that

$${\left({\int }_O\mid T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O{\mid }^{q}d\mu \right)}^{\frac{1}{q}}\le Cdiam\left(O\right)\mid O\mid {\left({\int }_{\rho O}\mid v{\mid }^{q}d\mu \right)}^{\frac{1}{q}}$$

(2.8)

for all balls O with ρO ⊂ Θ for some ρ > 1.

For the purpose of Theorem 2.6, we will need the following Lemmas 2.7 (the general Hölder inequality) and 2.8 that were proved in [5].

Lemma 2.7. Let f and g are two measurable functions on ℝ ⁿ, α, β, γ are any three positive constants with γ^-1 = α^-1 + β^-1. Then, there exists the inequality such that

$$\parallel fg{\parallel }_{\gamma ,\Theta }\le \parallel f{\parallel }_{\alpha ,\Theta }\parallel g{\parallel }_{\beta ,\Theta }$$

(2.9)

for any Θ ⊂ ℝ ⁿ .

Lemma 2.8. Let us assume, in addition to the definitions of the homotopy operator T, the exterior derivative d, and the projection operator H in Theorem 2.4, that ν ∈ C^∞(Θ, Λ^ℓ), ℓ = 1, 2,..., n, be a solution of the nonhomogeneous A-harmonic equation (1.2) in a bounded convex domain Θ and $\mid v\mid \in L_{loc}^s\left(\Theta \right)$. Then, there exists a constant C, independent of v, such that

$$\parallel T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O{\parallel }_{s,O}\le C\mid O\mid diam\left(O\right)\parallel v{\parallel }_{s,\rho O}$$

(2.10)

for all balls O with ρO ⊂ Θ, where ρ > 1 is a constant.

Remark. Note that in Theorem 2.4, φ may be any Young function, provided it lies in the class G(p, q, C₀), 1 ≤ p < q < ∞, C₀ ≥ 1. From [19], we know that the function $\phi \left(t\right)=t^p{log}_{+}^{\alpha }t$ belongs to G(p₁, p₂, C), 1 ≤ p₁ < p < p₂, t > 0, and α ∈ ℝ. Here log₊t is a cutoff function such that log₊t = 1 for t ≤ e otherwise log₊t = log t. Moreover, if α = 0, one verifies easily that φ(t) = t^p is as well in the class G(p₁, p₂, C), 1 ≤ p₁ < p₂ < ∞. Therefore, fixing the function $\phi \left(t\right)=t^p{log}_{+}^{\alpha }t$, α ∈ ℝ in Theorem 2.4, we get the following result.

Corollary 2.9. Let us assume, in addition to the definitions of the homotopy operator T, the exterior derivative d, the projection operator H, and the measure μ in Theorem 2.4, that$\phi \left(t\right)=t^p{log}_{+}^{\alpha }t$, p > 1, t > 0, α ∈ ℝ, ν ∈ C^∞(Θ, Λ^ℓ), ℓ = 1, 2,..., n, be a solution of the nonhomogeneous A-harmonic equation (1.2) in a bounded convex domain Θ and $\phi \left(\mid v\mid \right)\in L_{loc}^1\left(\Theta ;\mu \right)$. Then, there exists a constant C, independent of v, such that

$$\begin{array}{c}{\int }_O\mid T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O{\mid }^p{log}_{+}^{\alpha }\left(\mid T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\mid \right)d\mu \\ \le C{\int }_{\rho O}\mid v{\mid }^p{log}_{+}^{\alpha }\mid v\mid d\mu \end{array}$$

(2.11)

for all balls O with ρO ⊂ Θ for some ρ > 1. The following definition of the L^φ (μ)-averaging domains can be found in [5, 14].

Definition 2.10. Let φ be a Young function on [0, +∞) with φ(0) = 0. We call a proper subdomain Θ ⊂ ℝ ⁿan L^φ (μ)-averaging domains, if μ (Θ) < ∞ and there exists a constant C such that

$${\int }_{\Theta }\phi \left(\tau \mid \hslash -{\hslash }_{\Theta }\mid \right)d\mu \le C\underset{4O\subset \Theta }{sup}{\int }_O\phi \left(\sigma \mid \hslash -{\hslash }_O\mid \right)d\mu$$

(2.12)

for all Θ such that$\phi \left(\mid \Theta \mid \right)\in L_{loc}^1\left(\Theta ;\mu \right)$, where the measure μ is defined by dμ = ν(x)dx, ν(x) is a weight, and τ, σ are constants with 0 < τ, σ ≤ 1, and the supremum is over all balls O with 4O ⊂ Θ.

By Definition 2.10, we arrive at the following global case of Theorem 2.4.

Theorem 2.11. Let us assume, in addition to the definitions of the homotopy operator T, the exterior derivative d, the projection operator H, the measure μ, and the Young function φ in Theorem 2.4, that ν ∈ C^∞(Θ, Λ^k ), k = 1, 2,..., n, be a solution of the nonhomogeneous A-harmonic equation (1.2) in a bounded L^φ (μ)-averaging domains Θ and φ(|ν|) ∈ L¹(Θ; μ). Then, there is a constant C, independent of ν, such that

$$\parallel T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_{\Theta }{\parallel }_{\phi \left(\Theta ,\mu \right)}\le C\parallel v\mid {\mid }_{\phi \left(\Theta ,\mu \right)}.$$

(2.13)

Since John domains are very special L^φ (μ)-averaging domains, the preceding theorem immediately yields the following corollary.

Corollary 2.12. Let us assume, in addition to the definitions of the homotopy operator T, the exterior derivative d, the projection operator H, the measure μ, and the Young function φ in Theorem 2.4, that ν ∈ C^∞(Θ, Λ^k ), k = 1, 2,..., n, be a solution of the nonhomogeneous A-harmonic equation (1.2) in a bounded John domains Θ and φ(|ν| ∈ L¹(Θ; μ). Then, there is a constant C, independent of u, such that

$$\parallel T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_{\Theta }{\parallel }_{\phi \left(\Theta ,\mu \right)}\le C\parallel v{\parallel }_{\phi \left(\Theta ,\mu \right)}.$$

(2.14)

Remark. Note that the L^s -averaging domains and L^s (μ)-averaging domains are also special L^φ (μ)-averaging domains. Thus, Theorem 2.11 also holds for the L^s -averaging domains and L^s (μ)-averaging domains, respectively.

3 The proof of main results

In this section, we will give the proof of several theorems mentioned in the previous section.

Proof of Theorem 2.6. Let $t=\frac{\alpha q}{\alpha -1}$ and $r=\frac{\beta q}{\beta +1}$, then r < q < t. From Lemma 2.7 with $\frac{1}{q}=\frac{1}{t}+\frac{t-q}{tq}$, Lemma 2.8 and (2.6), we have

$$\begin{array}{c}{\left({{\int }_O\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\right|}^q\nu \left(x\right)dx\right)}^{\frac{1}{q}}\\ ={\left({\int }_O(\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\right|\nu {\left(x\right)}^{\frac{1}{q}}{)}^{q}dx\right)}^{\frac{1}{q}}\\ \le {\left({{\int }_O\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\right|}^{t}dx\right)}^{\frac{1}{t}}{\left({\int }_O{\left(\nu \left(x\right)\right)}^{\frac{t}{t-q}}dx\right)}^{\frac{t-q}{tq}}\\ \le C_{1}diam\left(O\right)\mid O\parallel \mid v\mid {\mid }_{t,{\rho }_{1}O}{\left({\int }_O{\left(\nu \left(x\right)\right)}^{\alpha }dx\right)}^{\frac{1}{\alpha q}}\\ \le C_{2}diam\left(O\right)\mid O{\mid }^{1+\frac{r-t}{rt}}{∥v∥}_{r,{\rho }_{2}O}{\left({\int }_O{\left(\nu \left(x\right)\right)}^{\alpha }dx\right)}^{\frac{1}{\alpha q}},\end{array}$$

(3.1)

where ρ₂, ρ₁ are two constants satisfying ρ₂ > ρ₁ > 1.

By virtue of Lemma 2.7 with $\frac{1}{r}=\frac{1}{q}+\frac{q-r}{rq}$, we obtain that

$$\begin{array}{l}\parallel v{\parallel }_{r,{\rho }_{2}O}\\ ={\left({\displaystyle {\int }_{{\rho }_{2}O}}\mid v{\mid }^{r}dx\right)}^{\frac{1}{r}}\\ ={\left({\displaystyle {\int }_{{\rho }_{2}O}}{(\mid v\mid (\nu (x{))}^{\frac{1}{q}}\cdot {(\nu (x))}^{\frac{-1}{q}})}^{r}dx\right)}^{\frac{1}{r}}\\ \le {\left({\displaystyle {\int }_{{\rho }_{2}O}}\mid v{\mid }^q\nu (x)dx\right)}^{\frac{1}{q}}{\left({\displaystyle {\int }_{{\rho }_{2}O}}{(\nu (x))}^{\frac{-r}{q-r}}dx\right)}^{\frac{q-r}{rq}}\\ ={\left({\displaystyle {\int }_{{\rho }_{2}O}}\mid v{\mid }^{q}d\mu \right)}^{\frac{1}{q}}{\left({\displaystyle {\int }_{{\rho }_{2}O}}{(\nu (x))}^{-\beta }dx\right)}^{\frac{1}{\beta q}}\mathrm{.}\end{array}$$

(3.2)

Observe that v(x) ∈ A(α, β, α, Θ), hence

$$\begin{array}{c}{\left({\int }_O{\left(\nu \left(x\right)\right)}^{\alpha }dx\right)}^{\frac{1}{\alpha q}}{\left({\int }_{{\rho }_{2}O}{\left(\nu \left(x\right)\right)}^{-\beta }dx\right)}^{\frac{1}{\beta q}}\\ \le {\left(\left({\int }_{{\rho }_{2}O}{\left(\nu \left(x\right)\right)}^{\alpha }dx\right){\left({\int }_{{\rho }_{2}O}{\left(\nu \left(x\right)\right)}^{-\beta }dx\right)}^{\frac{\alpha }{\beta }}\right)}^{\frac{1}{\alpha q}}\\ ={\left(\mid {\rho }_{2}O{\mid }^{1+\frac{\alpha }{\beta }}\left(\frac{1}{\mid {\rho }_{2}O\mid }{\int }_{{\rho }_{2}O}{\left(\nu \left(x\right)\right)}^{\alpha }dx\right){\left(\frac{1}{\mid {\rho }_{2}O\mid }{\int }_{{\rho }_{2}O}{\left(\nu \left(x\right)\right)}^{-\beta }dx\right)}^{\frac{\alpha }{\beta }}\right)}^{\frac{1}{\alpha q}}\\ \le C_3\mid {\rho }_{2}O{\mid }^{\frac{1}{\alpha q}+\frac{1}{\beta q}}.\end{array}$$

(3.3)

Combining (3.1)-(3.3), we obtain that

$$\begin{array}{c}{\left({\int }_O\mid T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O{\mid }^q\nu \left(x\right)dx\right)}^{\frac{1}{q}}\\ \le C_{4}diam\left(O\right)\mid O{\mid }^{1+\frac{r-t}{rt}}\mid {\rho }_{2}O{\mid }^{\frac{1}{\alpha q}+\frac{1}{\beta q}}{\left({\int }_{{\rho }_{2}O}\mid v{\mid }^q\nu \left(x\right)dx\right)}^{\frac{1}{q}}\\ \le C_{5}diam\left(O\right)\mid O\mid {\left({\int }_{{\rho }_{2}O}\mid v{\mid }^{q}d\mu \right)}^{\frac{1}{q}}.\end{array}$$

(3.4)

Therefore, we have completed the proof of Theorem 2.6.

By Lemma 2.5 and Theorem 2.6, we obtain the proof of Theorem 2.4.

Proof of Theorem 2.4. First, we observe that $\mu \left(O\right)={\int }_O\nu \left(x\right)dx\ge {\int }_O\epsilon dx=C_1\mid O\mid$, thereby

$$\frac{1}{\mu \left(O\right)}\le \frac{C_2}{\mid O\mid }$$

(3.5)

for all balls O ⊂ Θ.

We obtain from Theorem 2.6 and Lemma 2.5 that

$$\begin{array}{c}{\left({\int }_O\mid T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O{\mid }^{q}d\mu \right)}^{\frac{1}{q}}\\ \le C_{1}diam\left(O\right)\mid O\mid {\left({\int }_{{\rho }_{1}O}\mid v{\mid }^{q}d\mu \right)}^{\frac{1}{q}}\\ \le C_{2}diam\left(O\right)\mid O\mid (\mu \left({\rho }_{1}O\right){)}^{\frac{p-q}{pq}}{\left({\int }_{{\rho }_{2}O}\mid v{\mid }^{p}d\mu \right)}^{\frac{1}{p}},\end{array}$$

(3.6)

where ρ₂, ρ₁ with ρ₂ > ρ₁ > 1 are two constants. Note that φ is an increasing function, and f is an increasing convex function in [0, ∞), by Jensen's inequality for f, we obtain that

$$\begin{array}{c}\phi \left(\frac{1}{\chi }{\left({\int }_O{\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\right|}^{q}d\mu \right)}^{\frac{1}{q}}\right)\\ \le \phi \left(\frac{1}{\chi }{C}_2\left|O\right|diam\left(O\right){\left(\mu \left({\rho }_{1}O\right)\right)}^{\frac{\left(p-q\right)}{pq}}{\left({\int }_{{\rho }_{2}O}{\left|v\right|}^{p}d\mu \right)}^{\frac{1}{p}}\right)\\ =\phi \left({\left(\frac{1}{{\chi }^p}{C}_{{}_2}^p{\left|O\right|}^p{\left(diam\left(O\right)\right)}^p{\left(\mu \left({\rho }_{1}O\right)\right)}^{\frac{\left(p-q\right)}{q}}{\int }_{{\rho }_{2}O}{\left|v\right|}^{p}d\mu \right)}^{\frac{1}{p}}\right)\\ \le C_{3}f\left(\frac{1}{{\chi }^p}{C}_{{}_2}^p{\left|O\right|}^p{\left(diam\left(O\right)\right)}^p{\left(\mu \left({\rho }_{1}O\right)\right)}^{\frac{\left(p-q\right)}{q}}{\int }_{{\rho }_{2}O}{\left|v\right|}^{p}d\mu \right)\\ =C_{3}f\left({\int }_{{\rho }_{2}O}\frac{1}{{\chi }^p}{C}_{{}_2}^p{\left|O\right|}^p{\left(diam\left(O\right)\right)}^p{\left(\mu \left({\rho }_{1}O\right)\right)}^{\frac{\left(p-q\right)}{q}}{\left|v\right|}^{p}d\mu \right)\\ \le C_3{\int }_{{\rho }_{2}O}f\left(\frac{1}{{\chi }^p}{C}_{{}_2}^p{\left|O\right|}^p{\left(diam\left(O\right)\right)}^p{\left(\mu \left({\rho }_{1}O\right)\right)}^{\frac{\left(p-q\right)}{q}}{\left|v\right|}^p\right)d\mu .\end{array}$$

(3.7)

Since 1 ≤ p < q < ∞, we have $1+\frac{p-q}{pq}=1+\frac{1}{q}-\frac{1}{p}>0$, which yields

$$\begin{array}{c}diam\left(O\right)\mid O\mid \mu {\left({\rho }_{1}O\right)}^{\frac{p-q}{pq}}\\ \le C_{4}diam\left(\Theta \right)\mid O\parallel {\rho }_{1}O{\mid }^{{}^{\frac{p-q}{pq}}}\\ \le C_{5}diam\left(\Theta \right)\mid O{\mid }^{1+\frac{p-q}{pq}}\\ \le C_{6}diam\left(\Theta \right)\mid \Theta {\mid }^{1+\frac{p-q}{pq}}\le C_7.\end{array}$$

(3.8)

It follows from (i) in Definition 2.2 that $f\left(t\right)\le C_8\phi \left(t^{\frac{1}{p}}\right)$. Thus,

$$\begin{array}{l}{\displaystyle {\int }_{{\rho }_{2}O}}f\left(\frac{1}{{\chi }^p}{C}_2^p\mid O{\mid }^p{(diam(O))}^p{(\mu ({\rho }_{1}O))}^{\frac{p-q}{q}}\mid v{\mid }^p\right)d\mu \\ \le C_8{\displaystyle {\int }_{{\rho }_{2}O}}\phi \left(\frac{1}{\chi }{C}_2\mid O\mid (diam(O))(\mu ({\rho }_{1}O{))}^{\frac{p-q}{q}}\mid v\mid \right)d\mu \\ \le C_8{\displaystyle {\int }_{{\rho }_{2}O}}\phi \left(\frac{1}{\chi }{C}_9\mid v\mid \right)d\mu \\ \le C_{10}{\displaystyle {\int }_{{\rho }_{2}O}}\phi \left(\frac{1}{\chi }\mid v\mid \right)d\mu \mathrm{.}\end{array}$$

(3.9)

Combining (3.7) and (3.9), we obtain that

$$\begin{array}{c}\phi \left(\frac{1}{\chi }{\left({\int }_O\mid T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O{\mid }^{q}d\mu \right)}^{\frac{1}{q}}\right)\\ \le C_3{\int }_{{\rho }_{2}O}f\left(\frac{1}{{\chi }^p}{C}_2^p\mid O{\mid }^p{\left(diam\left(O\right)\right)}^p\left(\mu \left({\rho }_{1}O\right)\right)\frac{\left(p-q\right)}{q}\mid v{\mid }^p\right)d\mu \\ \le C_{11}{\int }_{{\rho }_{2}O}\phi \left(\frac{1}{\chi }\mid v\mid \right)d\mu .\end{array}$$

(3.10)

Applying Jensen's inequality to g^-1 and considering that φ and g are doubling, we obtain that

$$\begin{array}{l}{\displaystyle {\int }_O\phi \left(\frac{\left|T(d(H(v)))-{(T(d(H(v))))}_O\right|}{\chi }\right)d\mu }\\ =g\left(g^{-1}\left({\displaystyle {\int }_O\phi \left(\frac{\left|T(d(H(v)))-{(T(d(H(v))))}_O\right|}{\chi }\right)d\mu }\right)\right)\\ \le g\left({\displaystyle {\int }_O{g}^{-1}\left(\phi \left(\frac{\left|T(d(H(v)))-{(T(d(H(v))))}_O\right|}{\chi }\right)\right)d\mu }\right)\\ \le g\left(C_{12}{\displaystyle {\int }_O{\left(\frac{\left|T(d(H(v)))-{(T(d(H(v))))}_O\right|}{\chi }\right)}^{q}d\mu }\right)\\ \le C_{13}\phi \left({\left(C_{12}{\displaystyle {\int }_O{\left(\frac{\left|T(d(H(v)))-{(T(d(H(v))))}_O\right|}{\chi }\right)}^{q}d\mu }\right)}^{\frac{1}{q}}\right)\\ \le C_{14}\phi \left(\frac{1}{\chi }{\left({\displaystyle {\int }_O{|T(d(H(v)))-{(T(d(H(v))))}_O|}^{q}d\mu }\right)}^{\frac{1}{q}}\right)\\ \le C_{15}{\displaystyle {\int }_{{\rho }_{2}O}\phi }\left(\frac{\mid v\mid }{\chi }\right)d\mu .\end{array}$$

(3.11)

Therefore,

$$\begin{array}{c}\frac{1}{\mu \left(O\right)}{\int }_O\phi \left(\frac{\mid T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\mid }{\chi }\right)d\mu \\ \le \frac{1}{\mu \left(O\right)}{C}_{15}{\int }_{{\rho }_{2}O}\phi \left(\frac{\mid v\mid }{\chi }\right)d\mu \\ \le \frac{1}{\mu \left({\rho }_{2}O\right)}{C}_{16}{\int }_{{\rho }_{2}O}\phi \left(\frac{\mid v\mid }{\chi }\right)d\mu .\end{array}$$

(3.12)

By Definition 2.1 and (3.12), we achieve the desired result

$$\parallel T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\mid {\mid }_{\phi \left(O,\mu \right)}\le C\parallel v\mid {\mid }_{\phi \left(\rho O,\mu \right)}.$$

(3.13)

With the aid of Definition 2.10, We proceed now to derive Theorem 2.11.

Proof of Theorem 2.11. Note that Θ is a L^φ (μ)-averaging domains, and φ is doubling, from Definition 2.10 and (3.12), we have

$$\begin{array}{c}\frac{1}{\mu \left(\Theta \right)}{\int }_{\Theta }\phi \left(\frac{\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_{\Theta }\right|}{\chi }\right)d\mu \\ \le C_1\frac{1}{\mu \left(\Theta \right)}\underset{4O\subset \Theta }{sup}{\int }_O\phi \left(\frac{\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\right|}{\chi }\right)d\mu \\ \le C_1\frac{1}{\mu \left(\Theta \right)}\underset{4O\subset \Theta }{sup}\left(C_2{\int }_{\rho O}\phi \left(\frac{\left|v\right|}{\chi }\right)d\mu \right)\\ \le C_3\frac{1}{\mu \left(\Theta \right)}\underset{4O\subset \Theta }{sup}{\int }_{\Theta }\phi \left(\frac{\left|v\right|}{\chi }\right)d\mu \\ \le C_3\frac{1}{\mu \left(\Theta \right)}{\int }_{\Theta }\phi \left(\frac{\left|v\right|}{\chi }\right)d\mu .\end{array}$$

(3.14)

By Definition 2.1 and (3.14), we conclude that

$$\parallel T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_{\Theta }{\parallel }_{\phi \left(\Theta ,\mu \right)}\le C\parallel v{\parallel }_{\phi \left(\Theta ,\mu \right)}.$$

(3.15)

4 Applications

If we choose A to be a special operator, for example, A(x, dħ) = dħ|dħ|^s-2, then (1.4) reduces to the following s-harmonic equation:

$$d^{\star }\left(d\hslash \mid d\hslash {\mid }^{s-2}\right)=0.$$

(4.1)

In particular, we may let s = 2, if ħ is a function (0-form), then Equation 4.1 is equivalent to the well-known Laplace's equation Δħ = 0. The function ħ satisfying Laplace's equation is referred to as the harmonic function as well as one of the solutions of Equation 4.1. Therefore, all the results in Section 2 still hold for the ħ. As to the harmonic function, one finds broaden applications in the elliptic partial differential equations, see [20] for more related information.

We may make use of the following two specific examples to conform the convenience of the main inequality (3.11) in evaluating the upper bound for the L^φ -norm of |T(d(H(v))) - (T(d(H(v)))) _O |. Obviously, we may take advantages of (3.11) to make this estimating process easily, without calculating T(d(H(v))) and (T(d(H(v)))) _O complicatedly.

Example 4.1. Let ε, r be two distinct constants satisfying$\frac{1}{e}<\epsilon <r<1$, y = (y₁, y₂,..., y _n ) be a fixed point in ℝ ⁿ (n > 2), φ(t) = t^p log₊t, p > 1, $v=({\sum }_{i=1}^n{\left(x_i-y_i\right)}^2{)}^{\frac{2-n}{2}}$and O = {x = (x₁,..., x _n )| : ε² ≤ (x₁ - y₁)² + ⋯ + (x _n - y _n ) ≤ r²}.

First, by simple computation, we have

$$v_{x_i}=\left(2-n\right)\left(x_i-y_i\right){\left(\sum _{i=1}^n{\left(x_i-y_i\right)}^2\right)}^{\frac{-n}{2}},$$

(4.2)

$$v_{x_i{x}_i}=\left(2-n\right){\left(\sum _{i=1}^n{\left(x_i-y_i\right)}^2\right)}^{\frac{-\left(n+2\right)}{2}}\left(\sum _{i=1}^n{\left(x_i-y_i\right)}^2-n{\left(x_i-y_i\right)}^2\right),$$

(4.3)

then we get

$$\Delta v=\sum _{i=1}^n{v}_{x_i{x}_i}=0,$$

(4.4)

so the harmonic property of v is confirmed.

Observe that |O| = σ _n rⁿ , where σ _n denotes the volume of a unit ball in ℝ ⁿ (n > 2), and $1<\frac{1}{r^{n-2}}\le \mid v\mid =\mid ({\sum }_{i=1}^n{\left(x_i-y_i\right)}^2{)}^{\frac{2-n}{2}}\mid \le \frac{1}{{\epsilon }^{n-2}}$, applying (3.11) with χ = 1, dμ = dx, we obtain

$$\begin{array}{c}{\int }_O\phi \left(\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\right|\right)dx\\ ={\int }_O{\left(\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\right|\right)}^p{log}_{+}\left(\left|T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\right|\right)dx\\ \le C\left({\int }_{\rho O}{\left|v\right|}^p{log}_{+}\left|v\right|dx\right)\\ \le C\left({\left(\frac{1}{{\epsilon }^{\left(n-2\right)}}\right)}^{p}log\frac{1}{{\epsilon }^{\left(n-2\right)}}\left|\rho O\right|\right)\\ =\left(\frac{1}{{\epsilon }^{\left(n-2\right)p}}\left({\sigma }_n{\rho }^n{r}^n\right)\right)log\frac{1}{{\epsilon }^{\left(n-2\right)}}\\ =\frac{C{\rho }^n{\sigma }_n{r}^n}{{\epsilon }^{\left(n-2\right)p}}log\frac{1}{{\epsilon }^{\left(n-2\right)}}.\end{array}$$

(4.5)

Example 4.2. Let us assume, in addition to the definitions of ε, r, φ of Example 4.1, that y = (y₁, y₂) be a fixed point in ℝ², $v=log{\left({\sum }_{i=1}^2{\left(x_i-y_i\right)}^2\right)}^{\frac{1}{2}}$and O = {x = (x₁, x₂)| : ε² ≤ (x₁ - y₁)² + (x₂ - y₂) ≤ r²}.

Similarly, we observe to begin with that

$$v_{x_i}=\frac{x_i-y_i}{{\sum }_{i=1}^2{\left(x_i-y_i\right)}^2},$$

(4.6)

$$v_{x_i{x}_i}=\frac{{\displaystyle {\sum }_{i=1}^2{(x_i-y_i)}^2}-2{(x_i-y_i)}^2}{{{({\displaystyle {\sum }_{i=1}^2(x_i}-y_i)}^2)}^2}\mathrm{.}$$

(4.7)

Thus,

$$\Delta v=\sum _{i=1}^2{u}_{x_i{x}_i}=0,$$

(4.8)

which implies the function v is harmonic.

With respect to the estimation of ${\int }_O\phi \left(\mid T\left(d\left(H\left(v\right)\right)\right)-{\left(T\left(d\left(H\left(v\right)\right)\right)\right)}_O\mid \right)dx$, Example 4.2 proceeds in much the same way after replacing |O| = σ _n rⁿ and $1<\mid v\mid \le \frac{1}{{\epsilon }^{n-2}}$ with |O| = πr² and |log ε| < |v| ≤ |log r| < 1, respectively. Here we omit the reminder process.