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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 476309, 8 pages
A Generalization of Exponential Class and Its Applications
College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
Received 9 April 2013; Accepted 26 May 2013
Academic Editor: Mieczys####^~^~^~^~^~^####x142;aw Masty####^~^~^~^~^~^####x142;o
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.
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 Calder####^~^~^~^~^~^####xf3;n-Zygmund operator with respect to is obtained.
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  by Iwaniec and Sbordone in 1992 where they studied the integrability of the Jacobian under minimal hypotheses. For in , imbedding theorems of Sobolev type were proved for functions . The small Lebesgue space was found by Fiorenza  in 2000 as the associate space of the grand Lebesgue space . Fiorenza and Karadzhov gave in  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 , 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,
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. .
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####^~^~^~^~^~^####x2019;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 .
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  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 ; 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 .
Manfredi's paper  should be mentioned as the beginning of the systematic study of weakly monotone functions. Koskela et al. obtained in  that -harmonic functions are weakly monotone. In , 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 ), The following estimate holds Definition 11 with acting as a test function yields H####^~^~^~^~^~^####xf6;lder'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)####^~^~^~^~^~^####x2013;(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.
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 Calder####^~^~^~^~^~^####xf3;n-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 .
Lemma 15. If and , then the operator is bounded on .
Theorem 16. The operator is bounded on for and .
The following lemma can be found in .
Lemma 17. If , then there exists such that .
Lemma 18. If and , then a Calder####^~^~^~^~^~^####xf3;n-Zygmund operator is bounded on .
Theorem 19. A Calder####^~^~^~^~^~^####xf3;n-Zygmund operator is bounded on for and .
This study was funded by NSF of Hebei Province (A2011201011).
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