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Abstract and Applied Analysis
Volume 2010, Article ID 242079, 23 pages
http://dx.doi.org/10.1155/2010/242079
Research Article

Some Remarks on Spaces of Morrey Type

Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy

Received 17 June 2010; Accepted 2 September 2010

Academic Editor: Boris Shekhtman

Copyright © 2010 Loredana Caso 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

We deepen the study of some Morrey type spaces, denoted by , defined on an unbounded open subset of . In particular, we construct decompositions for functions belonging to two different subspaces of , which allow us to prove a compactness result for an operator in Sobolev spaces. We also introduce a weighted Morrey type space, settled between the above-mentioned subspaces.

1. Introduction

Let be an unbounded open subset of . For and , we consider the space of the functions in such that where is the open ball with center and radius .

This space of Morrey type, defined by Transirico et al. in [1], is a generalization of the classical Morrey space and strictly contains when . Its introduction is related to the solvability of certain elliptic problems with discontinuous coefficients in the case of unbounded domains (see e.g., [13]).

In the first part of this work, we deepen the study of two subspaces of , denoted by and , that can be seen, respectively, as the closure of and in .We start proving some characterization lemmas that allow us to construct suitable decompositions of functions in and . This is done in the spirit of the classical decomposition , proved in [4] by Calderón and Zygmund for , where a given function in is decomposed, for any , in the sum of a part (whose norm can be controlled by and a remaining one . Analogous decompositions can be found also for different functional spaces (see e.g., [5, 6] for decompositions , , and ).

The idea of our decomposition, both for a in and , is the following: for any the function can be written as the sum of a “good” part , which is more regular, and of a “bad” part , whose norm can be controlled by means of a continuity modulus of the function itself.

Decompositions are useful in different contexts as the proof of interpolation results, norm inequalities and a priori estimates for solutions of boundary value problems.

For instance, in the study of several elliptic problems with solutions in Sobolev spaces, it is sometimes necessary to establish regularity results and a priori estimates for a fixed operator . These results often rely on the boundedness and possibly on the compactness of the multiplication operator which entails the estimate where depends on the regularity properties of and on the summability exponents, and is a given function in a normed space satisfying suitable conditions. In some particular cases, this cannot be done for the operator itself, but there is the need to introduce a suitable class of operators , whose coefficients, more regular, approximate the ones of . This “deviation” of the coefficients of from the ones of needs to be done controlling the norms of the approximating coefficients with the norms of the given ones. Hence, it is necessary to obtain estimates where the dependence on the coefficients is expressed just in terms of their norms. Decomposition results play an important role in this approximation process, providing estimates where the constants involved depend just on the norm of the given coefficients and on their moduli of continuity and do not depend on the considered decomposition.

In the framework of Morrey type spaces, in [1], the authors studied, for , the operator defined in (1.2), generalizing a well-known result proved by Fefferman in [7] (cf. also [8]). They established conditions for the boundedness and compactness of this operator. In [2], the boundedness result and the straightforward estimates have been extended to any

In view of the above considerations, the second part of this work is devoted to a further analysis of the multiplication operator defined in (1.2), for functions in . By means of our decomposition results, we are allowed to deduce a compactness result for the operator given in (1.2). The obtained estimates can be used in the study of elliptic problems to prove that the considered operators have closed range or are semi-Fredholm.

The deeper examination of the structure of and of its subspaces leads us to the definition of a new functional space, that is a weighted Morrey type space, denoted by .

In literature, several authors have considered different kinds of weighted spaces of Morrey type and their applications to the study of elliptic equations, both in the degenerate case and in the nondegenerate one (see e.g., [911]).

In this paper, given a weight in a class of measurable functions (see 6 for its definition), we prove that the corresponding weighted space is a space settled between and . In particular, we provide some conditions on that entail .

Taking into account the results of this paper, we are now in position to approach the study of some classes of elliptic problems with discontinuous coefficients belonging to the weighted Morrey type space .

2. Notation and Preliminary Results

Let be a Lebesgue measurable subset of and be the -algebra of all Lebesgue measurable subsets of . Given we denote by its Lebesgue measure and by its characteristic function. For every and every we set where is the open ball with center and radius and in particular, we put .

The class of restrictions to of functions with is denoted by and, for , is the class of all functions such that for any .

Let us recall the definition of the classical Morrey space .

For , and , is the set of the functions such that equipped with the norm defined by (2.1).

If is an unbounded open subset of and is fixed in , we can consider the space , which is larger than when . More precisely, is the set of all functions in such that endowed with the norm defined in (2.2).

We explicitly observe that a diadic decomposition gives for every the existence of , depending only on and , such that All the norms being equivalent, from now on, we consider the space

For the reader's convenience, we briefly recall some properties of functions in and needed in the sequel.

The first lemma is a particular case of a more general result proved in [12, Proposition ].

Lemma 2.1. Let be a sequence of mollifiers in . If and then

The second results concerns the zero extensions of functions in (see also [1, Remark ]).

Remark 2.2. Let . If we denote by the zero extension of outside , then and for every where is a constant independent of , and .
Furthermore, if , then and where is a constant independent of and .

For a general survey on Morrey and Morrey type spaces, we refer to [1, 2, 13, 14].

3. The Spaces and

This section is devoted to the study of two subspaces of , denoted by and . Here, we point out the peculiar characteristics of functions belonging to these sets by means of two characterization lemmas.

Let us put, for and ,

Lemma 3.1. Let , and . The following properties are equivalent:

We denote by the subspace of made up of functions verifying one of the above properties.

Proof of Lemma 3.1. The equivalence between (3.2) and (3.3) is proved in of [1, Lemma ]. Let us show that (3.2) entails (3.4) and vice versa.
Fix in the closure of in , then for each there exists a function such that Fixed , from (3.5), it easily follows that On the other hand Therefore, if we set from(3.7), we deduce that, if , then Putting together (3.6) and (3.9), we get (3.4).
Conversely, if we take a function satisfying (3.4), for any there exists such that if with , then .
For each we set Observe that Therefore, if we put from (3.11), it follows that and then To end the proof, we define the function Indeed, by construction and by (3.14), one gets that .

Remark 3.2. It is easily seen (see also [1]) that if , then

Now, we introduce two classes of applications needed in the sequel.

For we denote by a function of class such that To define the second class, we first fix in satisfying and equivalent to dist (for more details on the existence of such an see for instance [15]). Hence, for we put It is easy to prove that belongs to , for any . Moreover, where

Lemma 3.3. Let , and . The following properties are equivalent:

The subspace of of the functions satisfying one of the above properties will be denoted by .

Proof of Lemma 3.3. The equivalence between (3.21) and (3.22) is a consequence of (3.3) and of [1, Lemmas and ]. The one between (3.21) and (3.24) follows from of [1, Remark ]. Always in [1], see Lemma and Remark , it is proved that (3.21) entails (3.25) and vice versa. Let us show that (3.21) and (3.23) are equivalent too.
Let us firstly assume that belongs to the closure of in .
Clearly, this entails that is in the closure of in , thus by Lemma 3.1, one has that It remains to show that To this aim, observe that fixed there exists such that On the other hand, if we consider the sets defined in (3.20), one has Therefore, since has a compact support, there exists Then, since one has that supp hence for all
The above considerations together with (3.28) give, for any , that is, (3.27).
Conversely, assume that and that (3.23) holds.
First of all, we observe that denoted by the zero extension of to , by (2.7) of Remark 2.2, there exists a positive constant , independent of , and of , such that Furthermore, by (3.23), we get that fixed there exists such that Therefore, Set by construction Hence, taking into account (2.8) of Remark 2.2, one has that On the other hand, (3.23) together with Lemma 3.1 give that , then from Remark 3.2, we get So, using (2.7) of Remark 2.2, we have that and Arguing as in [16, Lemma ], from (3.36)–(3.39), we conclude that We are now in the hypotheses of Lemma 2.1. Hence, denoted by a sequence of mollifiers in , we can find a positive integer such that Set , one has . Furthermore, using (3.34) and (3.41), we get this concludes the proof.

4. Decompositions of Functions in and

The characterizations of the spaces and naturally lead us to the introduction of the following moduli of continuity.

Let be a function in . A modulus of continuity of in is a map such that If belongs to , a modulus of continuity of in is an application such that Let us show now the decomposition results.

Lemma 4.1. Let , and . For any , one has with and

Proof. Given and , we introduce the set Observe that Set In view of (4.6), this gives the first inequality in (4.4), the second one easily follows from (4.5).

Lemma 4.2. Let , and . For any , one has with

Proof. To prove this second decomposition result, we exploit again the definition of the set introduced in (4.5) and inequality (4.6).
In this case, for any , we define To obtain the first inequality in (4.10), we observe that (4.6) gives The second one is a consequence of (4.5).

5. A Compactness Result

In this section, as application, we use the previous results to prove the compactness of a multiplication operator on Sobolev spaces.

To this aim, let us recall an imbedding theorem proved in [2, Theorem ].

Let us specify the assumptions: is an open subset of having the cone property with cone , the parameters satisfy one of the following conditions: with when and , and with when

Theorem 5.1. Under hypothesis and if or holds, for any and for any one has . Moreover, there exists a constant , depending on and , such that

Putting together Lemma 4.1 and Theorem 5.1, we easily have the following result.

Corollary 5.2. Under hypothesis and if or holds, for any and for any one has for each , where is the constant of (5.1).

If is in the previous estimate can be improved as showed in the corollary below.

Corollary 5.3. Under hypothesis and if or holds, for any and for any there exists an open set with the cone property, such that for each , where is the constant of (5.1).

Proof. Fix and . In view of Lemma 4.2 and Theorem 5.1, for any we have Using again Lemma 4.2, we obtain Putting together (5.4) and (5.5), we get (5.3), with obtained as follows: fixed and , the set is union of the open cones with opening , height and such that .

We are now in position to prove the compactness result.

Corollary 5.4. Suppose that condition is satisfied, that or holds, and fix . Then, the operator is compact.

Proof. Observe that if is a bounded open set with the cone property, the operator is compact.
Indeed, if is a bounded open set, the operator is linear and bounded. Moreover, since has the cone property, the Rellich-Kondrachov Theorem (see e.g., [17]) applies and gives that the operator is compact.
Let us consider now a sequence bounded in and let be such that . According to the above considerations, fixed there exist a subsequence and such that On the other hand, given and , in view of Corollary 5.3, there exists a constant and an open set with the cone property, independent of , such that From (5.11) and (5.10) written for and , for one has By (5.12) and (4.2), we conclude that is a Cauchy sequence in , which gives the compactness of the operator defined in (5.6).

6. The Space

In this section, we introduce some weighted spaces of Morrey type settled between and . To this aim, given , we consider the set defined in [18] as the class of measurable weight functions such that It is easy to show that if and only if there exists , independent on and , such that Furthermore, We put For and we denote by the Banach space made up of measurable functions such that equipped with the norm It can be proved that the space is dense in (see e.g., [18, 19]).

From now on, we consider and we denote by the positive real number such that .

Lemma 6.1. Let , and . The following properties are equivalent:

We denote by the set of functions satisfying one of the above properties.

Proof of Lemma 6.1. We start proving the equivalence between (6.6) and (6.7). This proof is in the spirit of the one of Lemma 3.1. For the reader's convenience, we write down just few lines pointing out the main differences.
If (6.6) holds, fixed there exists a function such that From (6.9), we get that for any , Furthermore, in view of the equivalence of the spaces and given by (2.3) and taking into account (6.2), where depends only on and . Hence, set from (6.11) we deduce that if , then Putting together (6.10) and (6.13), we obtain (6.7).
Now, assume that is a function in and that (6.7) holds. Then, for any there exists such that if with then For each we define the set Using again (2.3), there exists depending on the same parameters as such that Therefore, if we put from (6.16), we obtain and then We conclude setting . Indeed, by (6.15), and (6.19) gives that .
Arguing similarly, we prove also that (6.6) entails (6.8) and vice versa. Indeed, if and (6.6) holds, we can obtain as before (6.10) and (6.11).
On the other hand, there exists a constant such that Putting together (6.11) and (6.20), we obtain where . Now, set from (6.21), we deduce that if then From (6.10) and (6.23), we obtain (6.8).
Conversely, assume that (6.8) holds. We consider again the sets introduced in (6.15). From (6.16), we get Therefore, if we put from (6.24), we obtain and then, (6.8) being verified, We conclude the proof setting . Indeed, clearly and (6.27) gives .

Arguing in the spirit of Section 4, we want to obtain a decomposition result also for functions in . To this aim, we put for and In view of the previous lemma, we can define a modulus of continuity of a function in as a map such that

Lemma 6.2. Let , and . For any , one has with and where is a positive constant only depending on and and where is that of (6.2).

Proof. Fix , for any we set where The thesis followed by (6.2) and (2.3) arguing as in the proof of Lemma 4.1.

Let us show the following inclusion.

Lemma 6.3. Let and . Then,

Proof. For , clearly and then (6.6) holds. On the other hand, for we can show that if , then (6.7) holds. Indeed, observe that by (2.3), there exists a constant such that for any Moreover, there exists a constant such that Hence, fixed and set we deduce that, if then

Now, we can prove a further characterization of .

Lemma 6.4. LetThen, is the closure of in

Proof. Clearly, if by (6.6), one has also that is in the closure of in .
Conversely, let us prove that if belongs to the closure of in then (6.8) holds. Indeed, given there exists a function , for an , such that Hence, given Now, observe that since by Lemma 6.3, we get and therefore, using (6.8) of Lemma 6.1, we obtain that if then This, together with (6.38), ends the proof.

A straightforward consequence of the definitions (3.21) of Lemma 3.3, (6.6) of Lemma 6.1, and (3.2) of Lemma 3.1 is given by the following result.

Lemma 6.5. Let and . Then,

Let us show that if vanishes at infinity, the first inclusion stated in the lemma above becomes an identity.

Lemma 6.6. Let and . If is such that then

Proof. We show the inclusion , the converse being stated in Lemma 6.5. In view of Lemma 6.4, it is enough to verify that if (6.40) holds, then , for any .
To this aim, given , we fix and we prove that (3.25) is satisfied. Observe that by Lemmas 6.3 and 6.5. Moreover, for any and if there exists a constant such that On the other hand, if , clearly one has We can treat the first term on the right-hand side of this last equality as done in (6.41) obtaining the constant being the one of (6.41).
Concerning the second one, observe that for any and we have the inclusion , where denotes an n-dimensional cube of center and edge . Now, there exists a positive integer such that we can decompose the cube in