Poincaré Inequalities for Composition Operators with 𝐿^𝜑-Norm

Abstract

We establish the Poincaré-type inequalities for the composition of the homotopy operator, exterior derivative operator, and the projection operator with norm applied to the nonhomogeneous -harmonic equation in -averaging domains.

1. Introduction

The purpose of the paper is to develop the Poincaré-type inequalities for the composition of the homotopy operator , exterior derivative operator , and the projection operator with -norm. These operators play critical roles in investigating the properties of the solutions to PDEs and in controlling oscillatory behavior of the solutions in domains [1–6]. We first establish the local Poincaré inequalities for the composition in -averaging domains. Then, we prove the global Poincaré inequalities for the composition of in -averaging domains.

In this paper, we assume is a bounded and convex domain in and is a ball that is centred at with as its radius. For any , we use to denote the ball with centred at with radius . We do not distinguish the balls from the cubes in this paper. We use to denote the Lebesgue measure of a set . We call a weight if and a.e. For a function , we denote the average of over by . Differential forms are extensions of functions in . For example, the function is called a -form. Moreover, if is differentiable, it is called a differential -form. The -form in can be written as . If the coefficient functions , are differentiable, is called a differential -form. Similarly, a differential -form is generated by , , that is, = , where , . Let be the set of all -forms in , be the space of all differential -forms on and be the -forms on satisfying for all ordered -tuples , . We denote the exterior derivative by for , and define the Hodge star operator as follows: if , , is a differential -form, then , where   , and . The Hodge codifferential operator is given by on .

We use to denote a bounded and convex domain on . Let be the th exterior power of the cotangent bundle, let be the space of smooth -forms on , and . The harmonic -fields are defined by . The orthogonal complement of in is defined by . Then, the Green's operator is defined as by assigning as the unique element of satisfying Poisson's equation , where is the harmonic projection operator that maps onto so that is the harmonic part of . See [7, 8] for more properties of these operators. The differential forms can be used to describe various systems of PDEs and to express different geometric structures on manifolds. See [9, 10].

The operator with the case was first introduced by Cartan in [11]. Then, it was extended to the following version in [12]. To each there corresponds a linear operator defined by − and the decomposition . A homotopy operator is defined by averaging over all points , where is normalized so that .

We are particularly interested in a class of differential forms satisfying the well-known nonhomogeneous -harmonic equation where and satisfy the conditions for almost every and all . Here and are constants and is a fixed exponent associated with (1). A solution to (1) is an element of the Sobolev space such that for all with compact support. If is a function (form) in , (1) reduces to If the operator , (1) becomes which is called the homogeneous -harmonic equation. Let be defined by with . Then, satisfies the required conditions and becomes the -harmonic equation for differential forms. Some results have been obtained in recent years about different versions of the -harmonic equation; see [1, 2, 8, 9, 13–15].

2. Main Results and Proofs

Definition 1. Let be a continuously increasing convex function on with , and let be a domain with . If is a measurable function in , then we define the Orlicz norm of by A continuously increasing function with is called an Orlicz function, and a convex Orlicz function is often called a Young function.

From Definition 1, it is easy to see that for any domain if is finite.

Definition 2. Let be an increasing convex function on with . We call a proper subdomain an Orlicz space , if and there exists a constant such that for some ball and all integrable functions in , where the supremum is over all balls with .

Definition 3 (see [15]). We say that a Young function lies in the class , if and for all , where is a convex increasing function and is a concave increasing function on .

From [15], we know that the class contains some very interesting functions, such as and , and each of and is doubling in the sense that its values at and are uniformly comparable for all , and the consequent fact that where and are constants. We will need the following reverse Hlder inequality.

Lemma 4 (see [4]). Let be a solution of the nonhomogeneous -harmonic equation (1) in a bounded and convex domain and . Then, there exists a constant , independent of , such that for all balls with for some .

Lemma 5 (see [1]). Let be a solution of the nonhomogeneous -harmonic equation (1) in a bounded and convex domain . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that for all balls with .

Lemma 6 (see [1]). Let be a solution of the nonhomogeneous -harmonic equation (1) in a bounded and convex domain . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that for all balls with , where is a constant.

Theorem 7. Let be a Young function in the class , , , and let be a bounded convex domain. Assume that and is a solution of the nonhomogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that for some and all balls with .

Proof. For any constant , from Lemmas 4 and 5, (i) in Definition 3, using the fact that is an increasing function, Jensen's inequality, and noticing that and are doubling, we have Since , then,. Hence, we have . From (i) in Definition 3, we find that . Thus, Combining (13) and (14) yields Using Jensen's inequality for , (8), and noticing that and are doubling, we obtain Substituting (15) into (16) and noticing that is doubling, we have From Definition 2 and (17), we have the following version of Poincaré inequality with the Orlicz norm: We have completed the proof of Theorem 7.

Theorem 8. Let be a Young function in the class , and let be a bounded convex domain. Assume that and is a solution of the non-homogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that for some and all balls with .

Proof. For any constant , from Lemma 5, (i) in Definition 3, using the fact that is an increasing function, Jensen's inequality, and noticing that and are doubling, we have Since , then . From (i) in Definition 3, we find that . Thus, Combining (20) and (21) yields Using Jensen's inequality for , (8), and noticing that and are doubling, we obtain Substituting (22) into (23) and noticing that is doubling, we have From Definition 2 and (24), we have the following version of Poincaré inequality with the Orlicz norm: We have completed the proof of Theorem 8.

Using a similar method to the proof of Theorem 8, we can establish the following version of Poincaré inequality with the Orlicz norm.

Theorem 9. Let be a Young function in the class , and let be a bounded convex domain. Assume that and is a solution of the non-homogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that for some and all balls with .

Theorem 10. Let be a Young function in the class , and let be a bounded convex domain. Assume that and is a solution of the non-homogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that where is some fixed ball.

Proof. From definition of the and (12), we have We have completed the proof of Theorem 10.

Using a similar method to the proof of Theorem 8, we obtain Theorem 11.

Theorem 11. Let be a Young function in the class , and let be a bounded convex domain. Assume that and is a solution of the non-homogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that where is some fixed ball.

It has been proved in [5] that any John domain is special -averaging domain. Hence, we have the following results.

Corollary 12. Let be a Young function in the class , and let be a bounded John domain. Assume that and is a solution of the non-homogeneous -harmonic (3) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that where is some fixed ball.

For some special convex function, we have the following theorems.

Theorem 13. Let or   a Young function, and a bounded convex domain. Assume that and is a solution of the nonhomogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that for some and all balls with .

Theorem 14. Let or ,, a Young function, and a bounded convex domain. Assume that and is a solution of the nonhomogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that where is some fixed ball.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of the paper.

Acknowledgment

The author would like to thank Professor Shusen Ding for his precious and thoughtful suggestions.