Weighted Inequalities for Potential Operators with Lipschitz and BMO Norms

Abstract

Some Lipschitz norm and BMO norm inequalities for potential operator to the versions of differential forms are obtained, and some properties of a new kind of weight are derived.

1. Introduction

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 potential operator to the versions of differential forms.

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 that is, . For and a weight , we denote the weighted -norm of a measurable function over by

(11)

where is a real number.

Differential forms are important generalizations of real functions and distributions; 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

(12)

For , the vector-valued differential form consists of differential forms , 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

(13)

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

(14)

where is a real number. We denote the exterior derivative by  :  for . Its formal adjoint operator  :  is given by on .

Let . We write if

(15)

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

(16)

for some . When is a 0-form, (1.6) reduces to the classical definition of .

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

(17)

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

(18)

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.

From [1], if is a differential form defined in a bounded, convex domain , then there is a decomposition

(19)

where is called a homotopy operator. Furthermore, we can define the -form by

(110)

for all .

For any differential -form , we define the potential operator by

(111)

where the kernel is a nonnegative measurable function defined for , and the summation is over all ordered -tuples . It is easy to find that the case reduces to the usual potential operator. That is,

(112)

where is a function defined on . Associated with , the functional is defined as

(113)

where is some sufficiently small constant and is a ball with radius . Throughout this paper, we always suppose that satisfies the following conditions: there exists such that

(114)

and there exists such that

(115)

On the potential operator and the functional , see [2] for details.

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

(116)

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

(117)

for almost every and all . Here are constants and is a fixed exponent associated with (1.16). A solution to (1.16) is an element of the Sobolev space such that

(118)

for all with compact support. When is a 0-form, that is, is a function, (1.16) is equivalent to

(119)

Lots of results have been obtained in recent years about different versions of the -harmonic equation, see [3–5].

2. The Estimate for Potential Operators with Lipschitz Norm and BMO Norm

In this section, we give the estimate for potential operators with Lipschitz norm and BMO norm applied to differential forms. The following strong type inequality for potential operators appears in [6].

Lemma 2.1 (see [6]).

Let , be a differential form defined in a bounded, convex domain , and let be coefficient of with for all ordered -tuples . Assume that and is the potential operator with for any , then there exists a constant , independent of , such that

(21)

We will establish the following estimate for potential operators.

Theorem 2.2.

Let , be a differential form defined in a bounded, convex domain , and let be coefficient of with for all ordered -tuples . Assume that and is the potential operator with for any , then there exists a constant , independent of , such that

(22)

Proof.

By the definition of the Lipschitz norm, (2.1), and hölder's inequality with , we have

(23)

since and , where is a constant and .

By the definition of the BMO norm, we have

(24)

We have completed the proof of Theorem 2.2.

3. The Weight

In this section, we introduce the weight appeared in [7].

Definition 3.1.

Let be two locally integrable nonnegative functions in and assume that almost everywhere. We say that belongs to the class, and , or that is an weight, write or when it will not cause any confusion, if

(31)

for all balls .

The following results show that the weights have the properties similar to those of the weights.

Theorem 3.2.

If , then .

Proof.

Let . Since , by Hölder's inequality,

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so that

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Thus, we find that

(34)

for all balls since . Therefore, , and hence .

Theorem 3.3.

If and the measures are defined by , then

(35)

where is a ball in and is a measurable subset of .

Proof.

By Hölder's inequality, we have

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This implies

(37)

Note that , by Hölder's inequality again, we have

(38)

so that

(39)

Hence, we obtain

(310)

Since , there exists a constant such that

(311)

Combining (3.7), (3.10), and (3.11), we deduce that

(312)

Hence,

(313)

The desired result is obtained.

If we choose and in Theorem 3.3, we will obtain

(314)

which is called the strong doubling property of weights; see [8].

4. The Weighted Inequality for Potential Operators

In this section, we are devoted to develop some two-weight norm inequalities for potential operator to the versions of differential forms. We need the following lemmas.

Lemma 4.1 (see [9]).

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

(41)

for all balls .

Lemma 4.2.

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

(42)

for any .

Lemma 4.3 (see [10]).

Let be a solution of the nonhomogeneous A-harmonic equation in and , then there exists a constant , independent of , such that

(43)

for all with .

Theorem 4.4.

Let , be a solution of the nonhomogeneous -harmonic equation (1.16) in a bounded domain and is the potential operator with for any , 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

(44)

where is a constant with .

Proof.

Since , using Lemma 4.1, there exist constants and , such that

(45)

for any ball .

Since , by Lemma 4.2, we have

(46)

Choose where , , then and . Since , by Lemma 4.2 and (4.5), we have

(47)

From Lemma 2.1, we have

(48)

Applying Lemma 4.3 (the weak reverse Hölder inequality for the solutions of the nonhomogeneous -harmonic equation), we obtain

(49)

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

(410)

Since , then

(411)

Since , combining with (4.6)–(4.11), we have

(412)

From the definition of the BMO norm, we obtain

(413)

for all balls with and . We have completed the proof of Theorem 4.4.