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This article is part of the series Inequalities in the A-Harmonic Equations and the Related Topics II.

Open Access Research

Orlicz norm inequalities for the composite operator and applications

Hui Bi12* and Shusen Ding3

Author Affiliations

1 Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, 150080, China

2 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

3 Department of Mathematics, Seattle University, Seattle, WA 98122, USA

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Journal of Inequalities and Applications 2011, 2011:69  doi:10.1186/1029-242X-2011-69


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2011/1/69


Received:8 March 2011
Accepted:24 September 2011
Published:24 September 2011

© 2011 Bi and Ding; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we first prove Orlicz norm inequalities for the composition of the homotopy operator and the projection operator acting on solutions of the nonhomogeneous A-harmonic equation. Then we develop these estimates to Lφ(µ)-averaging domains. Finally, we give some specific examples of Young functions and apply them to the norm inequality for the composite operator.

2000 Mathematics Subject Classification: Primary 26B10; Secondary 30C65, 31B10, 46E35.

Keywords:
Orlicz norm; the projection operator; the homotopy operator; Lφ(µ)-averaging domains

1. Introduction

Differential forms as the extensions of functions have been rapidly developed. In recent years, some important results have been widely used in PDEs, potential theory, nonlinear elasticity theory, and so forth; see [1-7] for details. However, the study on operator theory of differential forms just began in these several years and hence attracts the attention of many people. Therefore, it is necessary for further research to establish some norm inequalities for operators. The purpose of this article is to establish Orlicz norm inequalities for the composition of the homotopy operator T and the projection operator H.

Throughout this article, we always let E be an open subset of ℝn, n ≥ 2. The Lebesgue measure of a set E ⊂ ℝn is denoted by |E|. Assume that B ⊂ ℝn is a ball, and σB is the ball with the same center as B and with diam(σB) = σdiam(B). Let ∧k = ∧k(ℝn), k = 0, 1,..., n, be the linear space of all k-forms <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M1">View MathML</a>, where I = (i1, i2,...,ik), 1 ≤ i1 < i2 < < ik n. We use <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M2">View MathML</a> to denote the space of all differential k-forms in E. In fact, a differential k-form ω(x) is a Schwarz distribution in E with value in ∧k (ℝn). As usual, we still use ⋆ to denote the Hodge star operator, and use <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M3">View MathML</a> to denote the Hodge codifferential operator defined by d= (-1)nk+1 d⋆ on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M4">View MathML</a>. Here <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M5">View MathML</a> denotes the differential operator.

A weight w(x) is a nonnegative locally integrable function on ℝn. Lp(E, ∧k) is a Banach space equipped with norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M6">View MathML</a>. Let D be a bounded convex domain in ℝn, n ≥ 2, and C(∧kD) be the space of smooth k-forms on D, where ∧k D is the kth exterior power of the cotangent bundle. The harmonic k-field is defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M8">View MathML</a>. If we use <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M9">View MathML</a> to denote the orthogonal complement of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M10">View MathML</a> in L1, then the Green's operator G is defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M11">View MathML</a> by assigning G(ω) as the unique element of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M12">View MathML</a> satisfying ΔG(ω) = ω - H(ω), where H is the projection operator that maps C(∧kD) onto <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M10">View MathML</a> such that H(ω) is the harmonic part of ω; see [8] for more properties on the projection operator and Green's operator. The definition of the homotopy operator for differential forms was first introduced in [9]. Assume that D ⊂ ℝn is a bounded convex domain. To each y D, there corresponds a linear operator Ky : C(∧kD) → C(∧k-1D) satisfying that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M13">View MathML</a>. Then by averaging Ky over all points y in D, The homotopy operator T : C(∧kD) → C(∧k-1D) is defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M14">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M15">View MathML</a> is normalized so that ∫φ(y)dy = 1. In [9], those authors proved that there exists an operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M16">View MathML</a>, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M17">View MathML</a>

(1.1)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M18">View MathML</a>

(1.2)

for all differential forms ω Lp(D, ∧k) such that Lp(D, ∧k). Furthermore, we can define the k-form <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M19">View MathML</a> by the homotopy operator as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M20">View MathML</a>

(1.3)

for all ω Lp(D, ∧k), 1 ≤ p < ∞.

Consider the nonhomogeneous A-harmonic equation for differential forms

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M21">View MathML</a>

(1.4)

where A : E x ∧k(ℝn) → ∧k (ℝn) and B : E x ∧k (ℝn) → ∧k-1(ℝn) are two operators satisfying the conditions:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M22">View MathML</a>

(1.5)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M23">View MathML</a>

(1.6)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M24">View MathML</a>

(1.7)

for almost every x E and all ξ ∈ ∧k(ℝn). Here, a, b > 0 are some constants and 1 < p < ∞ is a fixed exponent associated with (1.4). A solution to (1.4) is an element of the Sobolev space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M25">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M26">View MathML</a>

(1.8)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M27">View MathML</a> with compact support.

2. Orlicz norm inequalities for the composite operator

In this section, we establish the weighted inequalities for the composite operator T H in terms of Orlicz norms. To state our results, we need some definitions and lemmas.

We call a continuously increasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 an Orlicz function. If the Orlicz function Φ is convex, then Φ is often called a Young function. The Orlicz space LΦ(E) consists of all measurable functions f on E such that ∫EΦ(|f |/λ)dx < ∞ for some λ = λ(f) > 0 with the nonlinear Luxemburg functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M28">View MathML</a>

(2.1)

Moreover, if Φ is a restrictively increasing Young function, then LΦ(E) is a Banach space and the corresponding norm || · || Φ,E is called Luxemburg norm or Orlicz Norm. The following definition appears in [10].

Definition 2.1. We say that an Orlicz function Φ lies in the class G(p, q, C), 1 ≤ p < q < and C ≥ 1, if (1) 1/C ≤ Φ(t1/p)/g(t) ≤ C and (2) 1/C ≤ Φ(t1/q)/h(t) ≤ C for all t > 0, where g(t) is a convex increasing function and h(t) is a concave increasing function on [0, ∞).

We note from [10] that each of Φ, g, and h mentioned in Definition 2.1 is doubling, from which it is easy to know that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M29">View MathML</a>

(2.2)

for all t > 0, where C1 and C2 are constants.

We also need the following lemma which appears in [1].

Lemma 2.2. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M30">View MathML</a>, k = 1, 2,..., n, 1 < s < ∞, be a smooth solution of the nonhomogeneous A-harmonic equation in a bounded convex domain D, H be the projection operator and T : C(∧kD) → C(∧k-1D) be the homotopy operator. Then there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M31">View MathML</a>

for all balls B with ρB D, where ρ > 1 is a constant.

The Ar weights, r > 1, were first introduced by Muckenhoupt [11] and play a crucial role in weighted norm inequalities for many operators. As an extension of Ar weights, the following class was introduced in [2].

Definition 2.3. We call that a measurable function w(x) defined on a subset E ⊂ ℝn satisfies the A(α, β, γ; E)-condition for some positive constants α, β, γ; write w(x) ∈ A(α,β, γ; E), if w(x) > 0 a.e. and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M32">View MathML</a>

where the supremum is over all balls B E.

We also need the following reverse Hölder inequality for the solutions of the nonhomogeneous A-harmonic equation, which appears in [3].

Lemma 2.4. Let u be a solution of the nonhomogeneous A-harmonic equation, σ > 1 and 0 < s, t < ∞. Then there exists a constant C, independent of u and B, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M33">View MathML</a>

for all balls B with σB E.

Theorem 2.5. Assume that u is a smooth solution of the nonhomogeneous A-harmonic equation in a bounded convex domain D, 1 < p, q < and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M34">View MathML</a>for some α > 1 and β > 0. Let H be the projection operator and T : C(∧kD) → C(∧k-1D), k = 1, 2,..., n, be the homotopy operator. Then there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M35">View MathML</a>

for all balls with σB D for some σ > 1.

Proof. Set s = αq and m = βp/(β + 1). From Lemma 2.2 and the reverse Hölder inequality, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M36">View MathML</a>

(2.3)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M37">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M38">View MathML</a>. Thus, using the Hölder inequality, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M39">View MathML</a>

(2.4)

Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M40">View MathML</a>. It is easy to find that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M41">View MathML</a>

(2.5)

Combining (2.3)-(2.5) immediately yields that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M42">View MathML</a>

This ends the proof of Theorem 2.5.

If we choose p = q in Theorem 2.5, we have the following corollary.

Corollary 2.6. Assume that u is a solution of the nonhomogeneous A-harmonic equation in a bounded convex domain D, 1 < q < and w(x) ∈ A(α, β, α; D) for some α > 1 and β > 0. Let H be the projection operator and T : C(∧kD) → C(∧k-1D), k = 1, 2,..., n, be the homotopy operator. Then there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M43">View MathML</a>

for all balls with σB D for some σ > 1.

Next, we prove the following inequality, which is a generalized version of the one given in Lemma 2.2. More precisely, the inequality in Lemma 2.2 is a special case of the following result when φ(t) = tp.

Theorem 2.7. Assume that φ is a Young function in the class G(p, q, C0), 1 < p < q < ∞, C0 ≥ 1 and D is a bounded convex domain. If u C(∧kD), k = 1, 2,..., n, is a solution of the nonhomogeneous A-harmonic equation in D, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M44">View MathML</a>and 1/p - 1/q ≤ 1/n, then there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M45">View MathML</a>

for all balls B with σB D, where σ > 1 is a constant.

Proof. From Lemma 2.2, we know that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M46">View MathML</a>

for 1 < s < ∞. Note that u is a solution of the nonhomogeneous A-harmonic equation. Hence, by the reverse Hölder inequality, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M47">View MathML</a>

(2.6)

where σ2 > σ1 > 1 are some constants. Thus, using that φ and g are increasing functions as well as Jensen's inequality for g, we deduce that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M48">View MathML</a>

(2.7)

Since 1/p - 1/q ≤ 1/n, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M49">View MathML</a>

(2.8)

Applying (2.7) and (2.8) and noting that g(t) ≤ C0φ(t1/p), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M50">View MathML</a>

(2.9)

It follows from (2.7) and (2.9) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M51">View MathML</a>

(2.10)

Applying Jensen's inequality once again to h-1 and considering that φ and h are doubling, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M52">View MathML</a>

This ends the proof of Theorem 2.7.

To establish the weighted version of the inequality obtained in the above Theorem 2.7, we need the following lemma which appears in [4].

Lemma 2.8. Let u be a solution of the nonhomogeneous A-harmonic equation in a domain E and 0 < p, q < ∞. Then, there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M53">View MathML</a>

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

Theorem 2.9. Assume that φ is a Young function in the class G(p, q, C0), 1 < p < q < ∞, C0 ≥ 1 and D is a bounded convex domain. Let dµ = w(x)dx, where w(x) ∈ A(α, β, α; D) for α > 1 and β > 0. If u C(∧kD), k = 1, 2,..., n, is a solution of the nonhomogeneous A-harmonic equation in D, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M91">View MathML</a>, then there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M54">View MathML</a>

for all balls B with σB D and |B| d0 > 0, where σ > 1 is a constant.

Proof. From Corollary 2.6 and Lemma 2.8, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M55">View MathML</a>

(2.11)

where σ2 > σ1 > 1 is some constant. Note that φ and g are increasing functions and g is convex in D. Hence by Jensen's inequality for g, we deduce that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M56">View MathML</a>

(2.12)

Set D1 = {x D : 0 < w(x) < 1} and D2 = {x D : w(x) ≥ 1}. Then D = D1 D2. We let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M57">View MathML</a>, if x D1 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M58">View MathML</a>, if x D2. It is easy to check that w(x) ∈ A(α, β, α; D) if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M59">View MathML</a>. Thus, we may always assume that w(x) ≥ 1 a.e. in D. Hence, we have µ(B) = ∫B w(x)dx |B| for all balls B D. Since p < q and |B| = d0 > 0, it is easy to find that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M60">View MathML</a>

(2.13)

It follows from (2.13) and g(t) ≤ C0φ(t1/p) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M61">View MathML</a>

(2.14)

Applying Jensen's inequality to h-1 and considering that φ and h are doubling, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M62">View MathML</a>

This ends the proof of Theorem 2.9.

Note that if we remove the restriction on balls B, then we can obtain a weighted inequality in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M63">View MathML</a>, for which the method of proof is analogous to the one in Theorem 2.9. We now give the statement as follows.

Theorem 2.10. Assume that φ is a Young function in the class G(p, q, C0), 1 < p < q < ∞, C0 ≥ 1 and D is a bounded convex domain. Let dµ = w(x)dx, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M64">View MathML</a>for α > 1 and β > 0. If u C(∧kD), k = 1, 2,..., n, is a solution of the nonhomogeneous A-harmonic equation in D, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M91">View MathML</a>and 1/p - 1/q ≤ 1/n, then there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M65">View MathML</a>

for all balls B with σB D, where σ > 1 is a constant.

Directly from the proof of Theorem 2.7, if we replace |T(H(u))-(T(H(u)))B| by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M66">View MathML</a>, then we immediately have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M67">View MathML</a>

(2.15)

for all balls B with σB D and λ > 0. Furthermore, from the definition of the Orlicz norm and (2.15), the following Orlicz norm inequality holds.

Corollary 2.11. Assume that φ is a Young function in the class G(p, q, C0), 1 < p < q < ∞, C0 ≥ 1 and D is a bounded convex domain. If u C(∧kD), k = 1, 2,..., n, is a solution of the nonhomogeneous A-harmonic equation in D, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M68">View MathML</a>and 1/p - 1/q ≤ 1/n, then there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M69">View MathML</a>

(2.16)

for all balls B with σB D, where σ > 1 is a constant.

Next, we extend the local Orlicz norm inequality for the composite operator to the global version in the Lφ(µ)-averaging domains.

In [12], Staples introduced Ls-averaging domains in terms of Lebesgue measure. Then, Ding and Nolder [6] developed Ls-averaging domains to weighted versions and obtained a similar characterization. At the same time, they also established a global norm inequality for conjugate A-harmonic tensors in Ls(µ)-averaging domains. In the following year, Ding [5] further generalized Ls-averaging domains to Lφ(µ)-averaging domains, for which Ls(µ)-averaging domains are special cases when φ(t) = ts. The following definition appears.

Definition 2.12. Let φ be an increasing convex function defined on [0, ∞) with φ(0) = 0. We say a proper subdomain Ω ⊂ ℝn an Lφ(µ)-averaging domain, if µ(Ω) < and there exists a constant C such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M70">View MathML</a>

for some balls B0 ⊂ Ω and all u such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M71">View MathML</a>, where 0 < τ, σ < are constants and the supremum is over all balls B ⊂ Ω.

Theorem 2.13. Let φ be a Young function in the class G(p, q, C0), 1 < p < q < ∞, C0 ≥ 1 and D is a bounded convex Lφ(dx)-averaging domain. Suppose that φ(|u|) ∈ L1(D, dx), u C(∧1D) is a solution of the nonhomogeneous A-harmonic equation in D and 1/p - 1/q ≤ 1/n. Then there exists a constant C, independent of u, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M72">View MathML</a>

(2.17)

where B0 D is a fixed ball.

Proof. Since D is an Lφ(dx)-averaging domain and φ is doubling, from Theorem 2.7, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M73">View MathML</a>

We have completed the proof of Theorem 2.13.

Clearly, (2.17) implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M74">View MathML</a>

(2.18)

Similarly, we also can develop the inequalities established in Theorems 2.9 and 2.10 to Lφ(µ)-averaging domains, for which = w(x)dx and w(x) ∈ A(α, β, α; D) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M63">View MathML</a>, respectively.

3. Applications

The homotopy operator provides a decomposition to differential forms ω Lp(D, ∧k) such that Lp(D, ∧k+1). Sometimes, however, the expression of T(H(u)) or (TH(u))B may be quite complicated. However, using the estimates in the previous section, we can obtain the upper bound for the Orlicz norms of T(H(u)) or (TH(u))B. In this section, we give some specific estimates for the solutions of the nonhomogeneous A-harmonic equation. Meantime, we also give several Young functions that lie in the class G(p, q, C) and then establish some corresponding norm inequalities for the composite operator.

In fact, the nonhomogeneous A-harmonic equation is an extension of many familiar equations. Let B = 0 and u be a 0-form in the nonhomogeneous A-harmonic equation (1.4). Thus, (1.4) reduces to the usual A-harmonic equation:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M75">View MathML</a>

(3.1)

In particular, if we take the operator A(x, ξ) = ξ|ξ|p-2, then Equation 3.1 further reduces to the p-harmonic equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M76">View MathML</a>

(3.2)

It is easy to verify that the famous Laplace equation Δu = 0 is a special case of p = 2 to the p-harmonic equation.

In ℝ3, consider that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M77">View MathML</a>

(3.3)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M78">View MathML</a>. It is easy to check that = 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M79">View MathML</a>|. Hence, ω is a solution of the nonhomogeneous A-harmonic equation. Let B be a ball with the origin O σB, where σ > 1 is a constant. Usually the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M80">View MathML</a> is not easy to estimate due to the complexity of the operators T and H as well as the function φ. However, by Theorem 2.7, we can give an upper bound of Orlicz norm. Specially, if the Young function φ is not very complicated, sometimes it is possible to obtain a specific upper bound. For instance, take φ(t) = tplog+t, where log+ t = 1 if t e and log+ t = log t if t > e. It is easy to verify that φ(t) = tplog+t is a Young function and belongs to G(p1, p2, C) for some constant C = C(p1, p2, p). Let 0 < M < ∞ be the upper bound of |ω| in σB. Thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M81">View MathML</a>

where σ > 1 is some constant. Also, if we let φ(t) = tplog+t in Theorem 2.13, we can obtain a global estimate in a bounded convex Lφ(dx)-averaging domain D without the origin. That is

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M82">View MathML</a>

where B0 D is a fixed ball and N is the upper bound of |ω| in D.

Next we give some examples of Young functions that lie in G(p, q, C) and then apply them to Theorem 2.9.

Consider the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M83">View MathML</a>, 1 < p < ∞, α ∈ ℝ. Obviously, if we take α = 1, then Ψ (t) reduces to φ(t) = tplog+ t mentioned above. It is easy to check that for all 1 ≤ p1 < p < p2 < ∞ and α ∈ ℝ, the function Ψ(t) ∈ G(p1, p2, C), where C is dependent on p, p1, p2 and α. However, Ψ(t) is not always a Young function. More precisely, Ψ (t) cannot guarantee to be both increasing and convex. However, note that for Ψ (t), we can always find K > 1 depending on p and α such that the function Ψ (t) is increasing and convex on both [0, 1] and [K, ∞). Furthermore, if let ΨK(t) = Ψ(t) on [0, 1] ∪ [K, ∞) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M84">View MathML</a> in (1, K), then ΨK(t) still lies in G(p1, p2, C) for some C = C(p, α, p1, p2). It is worth noting that after such modification ΨK(t) is convex in the entire interval [0, ∞), in the sense that ΨK(t) is a Young function that lies in the class G(p, q, C); see [10] for more details on ΨK(t). Thus, we have the following result.

Corollary 3.1. Assume that u C(∧kD), k = 1, 2,..., n, is a solution of the nonhomogeneous A-harmonic equation in D, where D is a bounded convex domain. Let dµ = w(x)dx and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M85">View MathML</a>, where w(x) ∈ A(α, β, α; D) for α > 1 and β > 0. Then, for the composition of the homotopy operator T and the projection operator H, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M86">View MathML</a>

for all balls B with σB D and |B| d0 > 0. Here σ and C are constants and C is independent of u.

For the other example consider the function Φ(t) = tp sin t, on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M87">View MathML</a> and Φ(t) = tp, in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M88">View MathML</a>, 3 < p < ∞. It is easy to check that Φ(t) is a Young function and for all 0 < p1 < p + 1 < p2 < ∞, Φ(t) ∈ G(p1, p2,C), where C = C(p, p1, p2) ≥ 1 is some constant. Thus, Theorem 2.9 holds for Φ(t) and we have the following corollary.

Corollary 3.2. Assume that u C(∧kD), k = 1, 2,..., n, is a solution of the nonhomogeneous A-harmonic equation in D, where D is a bounded convex domain. Let dμ = w(x)dx and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M89">View MathML</a>, where w(x) ∈ A(α, β, α; D) for α > 1 and β > 0. Then, for the composition of the homotopy operator T and the projection operator H, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/69/mathml/M90">View MathML</a>

for all balls B with σB D and |B| d0 > 0. Here σ and C are constants and C is independent of u.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

HB and SD jointly contributed to the main results and HB wrote the paper. All authors read and approved the final manuscript.

Acknowledgements

The authors express their sincere thanks to the referee for his/her thoughtful suggestions. H.B. was supported by the Foundation of Education Department of Heilongjiang Province in 2011 (#12511111) and by the Youth Foundation at the Harbin University of Science and Technology (# 2009YF033).

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