Abstract and Applied Analysis

Volume 2013 (2013), Article ID 476309, 8 pages

http://dx.doi.org/10.1155/2013/476309

Research Article

A Generalization of Exponential Class and Its Applications

Hongya Gao, Chao Liu, and Hong Tian

College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

Received 9 April 2013; Accepted 26 May 2013

Academic Editor: Mieczysław Mastyło

Copyright © 2013 Hongya Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A function space, , , is defined. It is proved that is a Banach space which is a generalization of exponential class. An alternative definition of space is given. As an application, we obtain weak monotonicity property for very weak solutions of -harmonic equation with variable coefficients under some suitable conditions related to , which provides a generalization of a known result due to Moscariello. A weighted space is also defined, and the boundedness for the Hardy-Littlewood maximal operator and a Calderón-Zygmund operator with respect to is obtained.

1. Introduction

For and a bounded open subset , the grand Lebesgue space consists of all functions such that

where stands for the integral mean over . The grand Sobolev space consists of all functions such that

These two spaces, slightly larger than and , respectively, were introduced in the paper [1] by Iwaniec and Sbordone in 1992 where they studied the integrability of the Jacobian under minimal hypotheses. For in [2], imbedding theorems of Sobolev type were proved for functions . The small Lebesgue space was found by Fiorenza [3] in 2000 as the associate space of the grand Lebesgue space . Fiorenza and Karadzhov gave in [4] the following equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the nondecreasing rearrangement (provided that the underlying measure space has measure 1):

In [5], Greco et al. gave two more general definitions than (1) and (2) in order to derive existence and uniqueness results for -harmonic operators. For and , the grand space, denoted by , consists of functions such that

where

The grand Sobolev space consists of all functions belonging to and such that . That is,

Grand and small Lebesgue spaces are important tools in dealing with regularity properties for very weak solutions of -harmonic equation as well as weakly quasiregular mappings; see [6, 7].

The aim of the present paper is to provide a generalization , , of exponential class and prove that it is a Banach space. An alternative definition of is given in terms of weak Lebesgue spaces. As an application, we obtain weak monotonicity property for very weak solutions of -harmonic equation with variable coefficients under some suitable conditions related to . This paper also considers a weighted space and some boundedness result for classical operators with respect to this space.

In the sequel, the letter is used for various constants and may change from one occurrence to another.

2. A Generalization of Exponential Class

Recall that , the exponential class, consists of all measurable functions such that

for some . It is a Banach space under the norm

In this section, we define a space , , which is a generalization of , and prove that it is a Banach space.

Definition 1. For , the space is defined by

It is not difficult to see that

There are two special cases of that are worth mentioning since they coincide with two known spaces.

Case 1 (). In this case,

From the fact (see [8, page 12])

we get .

Case 2 (). The following proposition shows that can be regarded as a generalization of .

Proposition 2. .

Proof. In order to realize that a function in the space is in , it is sufficient to read the last lines of [2]. The vice versa is also true; see for example [9, Chap. VI, exercise no. 17].

It is clear that for any and any , we have the inclusions

The following theorem shows that if , then is slightly larger than .

Theorem 3. For , the space is a proper subspace of .

Proof. In the proof of Theorem 3 we always assume . Let , then there exists a constant , such that , a.e. . Thus,

which implies that .

The following example shows that is a proper subset. Since we have the inclusion (12), then it is no loss of generality to assume that . Consider the function defined in the open interval . It is obvious that . We now show that . In fact, for a positive integer, integration by parts yields

By L'Hospital's Law, one has

This equality together with (14) yields

By induction,

Recall that the function

is nondecreasing; thus (17) yields

where we have used the assumption and is the integer part of . The proof of Theorem 3 has been completed.

For functions , and , the addition and the multiplication are defined as usual.

Theorem 4. is a linear space on .

Proof. This theorem is easy to prove, so we omit the details.

For , we define

We drop the subscript from when there is no possibility of confusion.

Theorem 5. is a norm.

Proof. (1) It is obvious that and if and only if a.e. .

(2) For any , Minkowski’s inequality in yields

(3) For all and all , it is obvious that .

Theorem 6. is a Banach space.

Proof. Suppose that , and for any positive integer ,

Since is -finite, then with . It is no loss of generality to assume that the s are disjoint. Equation (17) implies that for any positive integer ,

Thus, by the completeness of , there exists , such that

Hence for any positive integer , there exists a subsequence of , , such that

If we let

then

It is no loss of generality to assume that the subsequence of is itself; thus

We now prove and , . In fact, by (22), for any , there exists , such that if , then

Let ; one has

Hence , and , . This completes the proof of Theorem 6.

Definition 7. The grand Sobolev space consists of all functions belonging to and such that . That is,

This definition will be used in Section 4.

3. An Alternative Definition of

In this section, we give an alternative definition of in terms of weak Lebesgue spaces. Let us first recall the definition of weak spaces or the Marcinkiewicz spaces, ; see [10, Chapter 1, Section 2], [11, Chapter 2, Section 5], or [12, Chapter 2, Section 18].

Definition 8. Let . We say that if and only if there exists a positive constant such that

for every , where is the -dimensional Lebesgue measure of and denotes the distribution function of .

For , we recall that if , then for every , and if and only if for every measurable set , the following inequality holds

for some constant .

Equation (32) is equivalent to

Recall also that

Definition 9. For , the weak space is defined by

The following theorem shows that ; thus, can be regarded as an alternative definition of the space .

Theorem 10.

Proof. We divided the proof into two steps.

Step 1 . If , for each , one can split the integral in the right-hand side of (35) to obtain

The second integral has been estimated by the inequality , which is a direct consequence of the definition of the constant (see (34)). Setting , we arrive at

This implies that

Therefore

here we have used (40) and the definition of .

Step 2 . Since for any ,

then

This implies that

Thus

The proof of Theorem 10 has been completed.

4. An Application

In this section, we give an application of the space to monotonicity property of very weak solutions of the -harmonic equation

where is a mapping satisfying the following assumptions:

(1)the mapping is measurable for all ,(2)the mapping is continuous for a.e. , for all , and a.e. ,(3)

(4)

where , , a.e. .

Conditions (1) and (2) insure that the composed mapping is measurable whenever is measurable. The degenerate ellipticity of the equation is described by condition (3). Finally, condition (4) guarantees that for any and any , can be integrated for against functions in with compact support.

Definition 11. A function , , is called a very weak solution of (46) if

for all .

A fruitful idea in dealing with the continuity properties of Sobolev functions is the notion of monotonicity. In one dimension a function is monotone if it satisfies both a maximum and minimum principle on every subinterval. Equivalently, we have the oscillation bounds for every interval . The definition of monotonicity in higher dimensions closely follows this observation.

A continuous function defined in a domain is monotone if

for every ball . This definition in fact goes back to Lebesgue [13] in 1973 where he first showed the relevance of the notion of monotonicity in the study of elliptic PDEs in the plane. In order to handle very weak solutions of -harmonic equation, we need to extend this concept, dropping the assumption of continuity. The following definition can be found in [14]; see also [6, 7].

Definition 12. A real-valued function is said to be weakly monotone if for every ball and all constants such that

we have

for almost every .

For continuous functions (51) holds if and only if on . Then (52) says we want the same condition in , that is, the maximum and minimum principles.

Manfredi's paper [14] should be mentioned as the beginning of the systematic study of weakly monotone functions. Koskela et al. obtained in [15] that -harmonic functions are weakly monotone. In [16], the first author obtained a result which states that very weak solutions of the -harmonic equation are weakly monotone provided that is small enough. The objective of this section is to extend the operator to spaces slightly larger than .

Theorem 13. Let , a.e. , . If is a very weak solution to (46), then it is weakly monotone in provided that .

Proof. For any ball and , let

It is obvious that

Consider the Hodge decomposition (see [6]),

The following estimate holds

Definition 11 with acting as a test function yields

Hölder's inequality together with the conditions (3) and (4) and equations (56) and (57) yields

The condition implies that

Since , then

By , we have

Combining (58)–(61) and taking into account the assumption , a.e. , we arrive at , a.e. . This implies that vanishes a.e. in , and thus must be the zero function in , completing the proof of Theorem 13.

Remark 14. We remark that the result in Theorem 13 is a generalization of a result due to Moscariello; see [17, Corollary 4.1].

5. A Weighted Version

A weight is a locally integrable function on which takes values in almost everywhere. For a weight and a measurable set , we define and the Lebesgue measure of by . The weighted Lebesgue spaces with respect to the measure are denoted by with . Given a weight , we say that satisfies the doubling condition if there exists a constant such that for any cube , we have , where denotes the cube with the same center as whose side length is 2 times that of . When satisfies this condition, we denote , for short.

A weight function is in the Muckenhoupt class with if there exists such that for any cube

where . We define .

Let be a weight. The Hardy-Littlewood maximal operator with respect to the measure is defined by

We say that is a Calderón-Zygmund operator if there exists a function which satisfies the following conditions:

For a weight and , we define the space as follows:

where

The following lemma comes from [18].

Lemma 15. If and , then the operator is bounded on .

Theorem 16. The operator is bounded on for and .

Proof. By Lemma 15, since for and , the operator is bounded on , then

This implies that

completing the proof of Theorem 16.

The following lemma can be found in [19].

Lemma 17. If , then there exists such that .

The following lemma can be found in [20, 21].

Lemma 18. If and , then a Calderón-Zygmund operator is bounded on .

Theorem 19. A Calderón-Zygmund operator is bounded on for and .

Proof. By and Lemma 17, one has for some . For , Hölder's inequality yields

Thus,

Lemma 18 yields

as desired.

Acknowledgment

This study was funded by NSF of Hebei Province (A2011201011).

References

T. Iwaniec and C. Sbordone, “On the integrability of the Jacobian under minimal hypotheses,” Archive for Rational Mechanics and Analysis, vol. 119, no. 2, pp. 129–143, 1992.

N. Fusco, P.-L. Lions, and C. Sbordone, “Sobolev imbedding theorems in borderline cases,” Proceedings of the American Mathematical Society, vol. 124, no. 2, pp. 561–565, 1996.

A. Fiorenza, “Duality and reflexivity in grand Lebesgue spaces,” Collectanea Mathematica, vol. 51, no. 2, pp. 131–148, 2000.

A. Fiorenza and G. E. Karadzhov, “Grand and small Lebesgue spaces and their analogs,” Journal for Analysis and its Applications, vol. 23, no. 4, pp. 657–681, 2004.

L. Greco, T. Iwaniec, and C. Sbordone, “Inverting the p-harmonic operator,” Manuscripta Mathematica, vol. 92, no. 2, pp. 249–258, 1997.

T. Iwaniec and G. Martin, Geometric Function Theorey and Nonlinear Analysis, The Clarendon Press, Oxford, UK, 2001.

H. Y. Gao, H. M. Wang, and G. Z. Gu, “Weak monotonicity for the component functions of weak solutions of the Beltrami system,” Acta Mathematica Scientia A, vol. 29, no. 3, pp. 651–655, 2009 (Chinese).

M. D. Cheng, D. G. Deng, and R. L. Long, Real Analysis, Higher Education Press, Beijing, China, 1993.

J. B. Garnett, Bounded Analytic Functions, Springer, New York, NY, USA, 1st edition, 2007.

S. Campanato, Sistemi ellittici in forma di divergenza, Quaderni Scuola Normale Superiore Pisa, Pisa, Italy, 1980.

E. Giusti, Metodi diretti nel calcolo delle variazioni, U.M.I., Bologna, Italy, 1994.

A. Kufner, O. John, and S. Fučík, Function Spaces, Noordhoff International, Leyden, Mass, USA, 1977.

H. Lebesgue, Oeuvres Scientifiques, Volume I-V, F. Chatelet and G. Choquet, Ed., Institue de Mathematiques de l'Universite, Geneva, Switzerland, 1973.

J. J. Manfredi, “Weakly monotone functions,” The Journal of Geometric Analysis, vol. 4, no. 3, pp. 393–402, 1994.

P. Koskela, J. J. Manfredi, and E. Villamor, “Regularity theory and traces of -harmonic functions,” Transactions of the American Mathematical Society, vol. 348, no. 2, pp. 755–766, 1996.

H. Y. Gao, “Some properties of weakly quasiregular mappings,” Acta Mathematica Sinica. Chinese Series, vol. 5, no. 1, pp. 191–196, 2002 (Chinese).

G. Moscariello, “On the integrability of “finite energy” solutions for p-harmonic equations,” Nonlinear Differential Equations and Applications, vol. 11, no. 3, pp. 393–406, 2004.

J. García-Cuerva and J. L. R. de Francia, Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherland, 1985.

M. Q. Zhou, Lectures on Harmonic Analysis, Beijing University Press, Beijing, China, 1999, (Chinese).

J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2000.

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, NJ, USA, 2004.