-Weighted Inequalities with Lipschitz and BMO Norms

Absract

We first define a new kind of two-weight, then obtain some two-weight integral inequalities with Lipschitz norm and BMO norm for Green's operator applied to differential forms.

1. Introduction

Green's operator is often applied to study the solutions of various differential equations and to define Poisson's equation for differential forms. Green's operator has been playing an important role in the study of PDEs. In many situations, the process to study solutions of PDEs involves estimating the various norms of the operators. Hence, we are motivated to establish some Lipschitz norm inequalities and BMO norm inequalities for Green's operator in this paper.

In the meanwhile, there have been generally studied about -weighted [1, 2] and -weighted [3, 4] different inequalities and their properties. Results for more applications of the weight are given in [5, 6]. The purpose of this paper is to derive the new weighted inequalities with the Lipschitz norm and BMO norm for Green's operator applied to differential forms. We will introduce -weight, which can be considered as a further extension of the -weight.

We keep using the traditional notation.

Let be a connected open subset of , let be the standard unit basis of , and let be the linear space of -covectors, spanned by the exterior products , corresponding to all ordered -tuples , , . We let . The Grassman algebra is a graded algebra with respect to the exterior products. For and , the inner product in is given by with summation over all -tuples and all integers .

We define the Hodge star operator by the rule and for all . The norm of is given by the formula . The Hodge star is an isometric isomorphism on with and .

Balls are denoted by and is the ball with the same center as and with . We do not distinguish balls from cubes throughout this paper.

The -dimensional Lebesgue measure of a set is denoted by . We call a weight if and a.e. For and a weight , we denote the weighted -norm of a measurable function over by

(1.1)

where is a real number.

Differential forms are important generalizations of real functions and distributions. Specially, a differential -form on is a de Rham current [7, Chapter III] on with values in ; note that a 0-form is the usual function in . A differential -form on is a Schwartz distribution on with values in . We use to denote the space of all differential -forms = = . We write for the -forms with for all ordered -tuples . Thus is a Banach space with norm

(1.2)

For the vector-valued differential form

(1.3)

consists of differential forms

(1.4)

where the partial differentiations are applied to the coefficients of .

As usual, is used to denote the Sobolev space of -forms, which equals with norm

(1.5)

The notations and are self-explanatory. For and a weight , the weighted norm of over is denoted by

(1.6)

where is a real number.

We denote the exterior derivative by for . Its formal adjoint operator is given by on , . Let be the th exterior power of the cotangent bundle and let be the space of smooth -forms on . We set . The harmonic -fields are defined by = . The orthogonal complement of in is defined by = . Then, Green's operator is defined as by assigning to be 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 [8] for more properties of Green's operator.

The nonlinear elliptic partial differential equation = is called the homogeneous -harmonic equation or the -harmonic equation, and the differential equation

(1.7)

is called the nonhomogeneous -harmonic equation for differential forms, where and satisfy the following conditions:

(1.8)

for almost every and all . Here are constants and is a fixed exponent associated with (1.7). A solution to (1.7) is an element of the Sobolev space such that

(1.9)

for all with compact support.

Let be defined by with . Then, satisfies the required conditions and becomes the -harmonic equation

(1.10)

for differential forms. If is a function (a 0-form), (1.10) reduces to the usual -harmonic equation for functions. We should notice that if the operator equals in (1.7), then (1.7) reduces to the homogeneous -harmonic equation. Some results have been obtained in recent years about different versions of the -harmonic equation; see [9–11].

Let ,  . We write ,  , if

(1.11)

for some . Further, we write for those forms whose coefficients are in the usual Lipschitz space with exponent and write for this norm. Similarly, for , , we write if

(1.12)

for some . When is a 0-form, (1.12) reduces to the classical definition of .

Based on the above results, we discuss the weighted Lipschitz and BMO norms. For ,  , we write ,  , if

(1.13)

for some , where is a bounded domain, the Radon measure is defined by , is a weight and, is a real number. For convenience, we shall write the following simple notation for . Similarly, for ,  , we write if

(1.14)

for some , where the Radon measure is defined by , is a weight and is a real number. Again, we use to replace whenever it is clear that the integral is weighted.

2. Preliminary Knowledge and Lemmas

Definition 2.1.

We say that the weight satisfies the condition for some and ; let , if , a.e. and

(2.1)

for any ball .

If we choose and in Definition 2.1, we will obtain the usual -weight. If we choose , and in Definition 2.1, we will obtain the -weight [3]. If we choose , and in Definition 2.1, we will obtain the -weight [12].

Lemma 2.2 (see [1]).

If , then there exist constants and , independent of , such that

(2.2)

for all balls .

We need the following generalized Hölder inequality.

Lemma 2.3.

Let , and . If and are measurable functions on , then

(2.3)

for any .

The following version of weak reverse Hölder inequality appeared in [13].

Lemma 2.4.

Suppose that is a solution to the nonhomogeneous -harmonic equation (1.7) in , and . There exists a constant , depending only on and , such that

(2.4)

for all with .

Lemma 2.5 (see [14]).

Let be a smooth form and let be Green's operator, , and . Then, there exists a constant , independent of , such that

(2.5)

for all balls .

We need the following Lemma 2.6 (Caccioppoli inequality) that was proved in [8].

Lemma 2.6 (see [8]).

Let be a solution to the nonhomogeneous -harmonic equation (1.7) in and let be a constant. Then, there exists a constant , independent of , such that

(2.6)

for all balls or cubes with and all closed forms . Here .

Lemma 2.7 (see [14]).

Let , , , be a smooth form in a domain . Then, there exists a constant , independent of , such that

(2.7)

where is a constant with .

Lemma 2.8 (see [14]).

Let , , , be a smooth form in a bounded domain and let be Green's operator. Then, there exists a constant , independent of , such that

(2.8)

3. Main Results and Proofs

Theorem 3.1.

Let , , , be a solution of the nonhomogeneous A-harmonic equation (1.7) in a bounded domain and let be Green's operator, where the Radon measures and are defined by , . Assume that for some , . Then, there exists a constant , independent of , such that

(3.1)

where is a constant with , and is a constant with .

Proof.

Choose where ; then and . Since , by Lemmas 2.3 and 2.5, we have

(3.2)

for all ball . Choosing , then . From Lemma 2.4, we have

(3.3)

where and . Using Hölder inequality with , we have

(3.4)

Since , then

(3.5)

Since , combining with (3.2), (3.3), (3.4), and (3.5), we have

(3.6)

Notice that , ; from (3.6) and the Hölder inequality with , we find that

(3.7)

We have completed the proof of Theorem 3.1.

Remark 3.

Specially, choosing and in Theorem 3.1, we have

(3.8)

Next, we will establish the following weighted norm comparison theorem between the Lipschitz and the BMO norms.

Theorem 3.2.

Let , , , be a solution of the nonhomogeneous A-harmonic equation (1.7) in a bounded domain and let be Green's operator, where the Radon measures and are defined by , . Assume that for some , with for any . Then, there exists a constant , independent of , such that

(3.9)

where is a constant with , and is a constant with .

Proof.

Choose where ; then and . Since , by Lemma 2.3, we have

(3.10)

for any ball and some constant with . Choosing in Lemma 2.6, we find that

(3.11)

where is a constant and . Combining (3.8), (3.10), and (3.11), it follows that

(3.12)

Choosing , then . Applying the weak reverse Hölder inequality for the solutions of the nonhomogeneous -harmonic equation, we obtain

(3.13)

where is a constant and . Substituting (3.13) into (3.12), we have

(3.14)

Using Hölder inequality with , we have

(3.15)

Since , then

(3.16)

Since , combining with (3.14), (3.15), and (3.16), we have

(3.17)

Since , we have

(3.18)

for all ball . Notice that and ; from (3.17), we have

(3.19)

where is a constant and . We have completed the proof of Theorem 3.2.

Now, we will prove the following weighted inequality between the BMO norm and the Lipschitz norm for Green's operator.

Theorem 3.3.

Let , , , be a solution of the nonhomogeneous A-harmonic equation (1.7) in a bounded domain and let be Green's operator, where the Radon measures and are defined by , . Assume that and for some , with for any . Then, there exists a constant , independent of , such that

(3.20)

where is a constant with .

Proof.

Since , using Lemma 2.2, there exist constants and , such that

(3.21)

for any ball .

Since , by Lemma 2.3, we have

(3.22)

Choose where , ; then and . Since , by Lemma 2.3 and (3.21), we have

(3.23)

From Lemmas 2.5 and 2.6 with , we have

(3.24)

where is a constant and . Applying the weak reverse Hölder inequality for the solutions of the nonhomogeneous -harmonic equation, we obtain

(3.25)

where is a constant and . Choosing , then . Using Hölder inequality with , we have

(3.26)

Since and , combining with (3.22), (3.23), (3.24), and (3.25), we have

(3.27)

From the definitions of the Lipschitz and BMO norms, we obtain

(3.28)

for all balls with and . Substituting (3.27) into (3.28), we have

(3.29)

We have completed the proof of Theorem 3.3.

Using the same methods, and by Lemmas 2.7 and 2.8, we can estimate Lipschitz norm and BMO norm of Green's operator in terms of norm.

Theorem 3.4.

Let , , , be a solution of the nonhomogeneous A-harmonic equation (1.7) in a bounded domain and let be Green's operator, where the Radon measures and are defined by , . Assume that and for some , with for any . Then, there exists a constant , independent of , such that

(3.30)

where is a constant with , and is a constant with .

Theorem 3.5.

Let , , , be a solution of the nonhomogeneous A-harmonic equation (1.7) in a bounded domain and let be Green's operator, where the Radon measures and are defined by , . Assume that and for some , with for any . Then, there exists a constant , independent of , such that

(3.31)

where is a constant with .

Remark 3.

Note that the differentiable functions are special differential forms (0-forms). Hence, the usual -harmonic equation for functions is the special case of the -harmonic equation for differential forms. Therefore, all results that we have proved for solutions of the -harmonic equation in this paper are still true for -harmonic functions.