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`Journal of Function SpacesVolume 2014, Article ID 172486, 15 pageshttp://dx.doi.org/10.1155/2014/172486`
Research Article

## Local Morrey and Campanato Spaces on Quasimetric Measure Spaces

1Instytut Matematyki i Informatyki, Politechnika Wrocławska, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

2Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, Zhejiang 310023, China

Received 17 February 2014; Accepted 15 April 2014; Published 25 May 2014

Academic Editor: Dachun Yang

Copyright © 2014 Krzysztof Stempak and Xiangxing Tao. 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 define and investigate generalized local Morrey spaces and generalized local Campanato spaces, within a context of a general quasimetric measure space. The locality is manifested here by a restriction to a subfamily of involved balls. The structural properties of these spaces and the maximal operators associated to them are studied. In numerous remarks, we relate the developed theory, mostly in the “global” case, to the cases existing in the literature. We also suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets of .

#### 1. Introduction

A quasimetric on a nonempty set is a mapping which satisfies the following conditions:(i)for every , if and only if ;(ii)for every , ;(iii)there is a constant such that, for every , The pair is then called a quasimetric space; if , then is a metric and is a metric space.

Given and , let be the “quasimetric” ball related to of radius and with center . If is a quasimetric space, then, , the topology in induced by , is canonically defined by declaring to be open, that is, , if and only if, for every , there exists such that (at this point one easily checks directly that the topology axioms are satisfied for such a definition; note, however, that the balls themselves may not be open sets). Observe that this definition is consistent with the definition of metric topology in case when is a genuine metric. Moreover, the topology is metrizable, see for instance [1] for references.

Two quasimetrics and on are said to be equivalent, if with some being independent of . It is clear that, for equivalent quasimetrics, induced topologies coincide. Moreover, for any , is a quasimetric as well and . A quasimetric is called a -metric, for , provided that holds uniformly in . It is easily checked that a -metric enjoys the open ball property; that is, every ball related to is an open set in . It is also known (see [1]) that, given , for determined by the equality , defined by is a metric on which is equivalent to ; more precisely, . Consequently, is a -metric equivalent with ; more precisely, . Thus, every quasimetric admits an equivalent -metric that possesses the open ball property.

In what follows, if is a given quasimetric space, then is considered as a topological space equipped with the (metrizable) topology . It may happen that a ball in is not a Borel set (i.e., it does not belong to the Borel -algebra generated by ), see, for instance, [1] as an example. To avoid such pathological cases, the assumption that all balls are Borel sets must be made. Then, if is additionally equipped with a Borel measure which is finite on bounded sets and nontrivial in the sense that , we say that is a quasimetric measure space (we do not assume that , for every ball ). In this paper, we additionally assume (similar to the assumption (1.3) made in [2]) that taking into account what was mentioned above, this assumption does not narrow the generality of our considerations.

Let be a quasimetric measure space. Define the function by setting Observe that if , for some , then ; this is a consequence of the continuity property from below of the measure . The property “, for every ball ,” is equivalent with the statement that .

Given a function such that , for every , let denote the family of balls (related to ) centered at and with radius satisfying (clearly balls with different radii but which coincide are identified as sets). Then we set Thus, denotes the family of all -local balls in with positive measure. In case the lower estimate on the radius, , is disregarded, we shall write for the resulting family of balls.

By a -local integrability of a real or complex-valued function on , we mean its integrability with respect to the family of balls from ; thus, provided that , for every ball (and thus also for every ). Note that this notion of local integrability does not refer to compactness. Similarly, for , we define .

If , for some , then we will refer to as a locality function and to objects associated to as “local” objects. If identically, then we shall skip the subscript writing , , , , and so on (thus denotes the family of all balls in ) and refer to this setting as to the global one. Notice that the proofs of all results stated in the paper contain as a special case.

Parallel to the main theory, we shall also develop an alternative theory in the framework of closed balls . Note that, in the metric case, is indeed a closed set and, in general, if all balls are assumed to be Borel sets, then is Borel, too. The definitions of Morrey and Campanato spaces based on closed balls (in fact being closed cubes) in the framework of occur in the literature, compare, for instance, [3]. Clearly taking closed balls makes no difference with respect to the theory based on open balls, when has the property that , for every ball , where ; this happens, for instance, when , where and denotes Lebesgue measure on . In general, however, the two alternative ways may give different outcomes. Relevant comments indicating coincidences or differences of both theories will be given in several places.

The general notion of local maximal operators was introduced in [4] and some objects associated to them, mostly the BMO spaces, were investigated there in the setting of measure metric spaces. The present paper enhances investigation done in [4] in several directions. First, the broader context of quasimetric measure spaces is considered. Second, the condition , for every ball , is not assumed. Third, several variants of generalized maximal operators are admitted into our investigation. All this makes the developed theory more flexible in possible applications.

Throughout the paper, we use a standard notation. While writing estimates, we use the notation to indicate that with a positive constant independent of significant quantities. We shall write when simultaneously and ; for instance, means the equivalence of quasimetrics and , and so forth. By , , we shall denote the usual Lebesgue space on the measure space . Whenever we refer to a ball, we understand that its center and radius have been chosen (in general, these need not be uniquely determined by as a set). Then, writing , for a given ball and , means that . For a function , its average in a ball will be denoted by and similarly for any other Borel set , , and any , whenever the integral makes sense. When the situation is specified to the Euclidean setting of , we shall consider either the metric induced by the norm or induced by .

#### 2. Generalized Local Maximal Operators

By defining and investigating generalized local Morrey and Campanato spaces on quasimetric measure spaces, we adapt the general approach to these spaces presented by Nakai [2] (and follow the notation used there) and extend the concept of locality introduced in [4]. Also, we find it more convenient to work with relevant maximal operators when investigating the aforementioned spaces. An interesting concept of localization of Morrey and Campanato spaces on metric measure spaces recently appeared in [5]; this concept is, however, different from our concept. On the other hand, the concept of locality for Morrey and Campanato spaces on metric measure spaces that appeared in the recent paper [6] is consistent with the one we develop; see Remark 15 for further details.

Let be a positive function defined on . In practice, will be usually defined on , the family of all balls in . Then, a tempting alternative way of thinking about is to treat it as a function and then to define , for . There is, however, a pitfall connected with the fact that in general the mapping is not injective. Hence, we assume that possesses the following property: (Thus, for instance, when is bounded, i.e., , the function must obey the following rule: for every and , ).

Clearly, working with a general cannot lead to fully satisfactory results. Therefore, in what follows, we shall impose some additional mild (and natural) assumptions on in order to develop the theory. Frequently, in such assumptions, and will be interrelated. Of particular interest will be the functions where and denotes the radius of (the and stand for measure and radius, resp.). It is necessary to point out here that, for the second function, in fact, we consider a selector assigning to any , one of its possible radii (clearly this subtlety does not occur when, for instance, ). We shall frequently test the constructed theory on these two functions. Finally, let us mention that it may happen that, for a constant (playing the role of the dimension), we have uniformly in . Then,

Let the system be given. In what follows, by an admissible function on , we mean either a Borel measurable complex-valued function (when the complex case is considered) or a Borel measurable function with values in the extended real number system (when the real case is investigated). Given , we define the generalized local fractional maximal operator acting on any admissible by where the supremum is taken over all the balls from which contain , and its centered version by On the other hand, we define the generalized local sharp fractional maximal operator for any admissible by and similarly for its centered version . (If spaces of real functions are considered, then the infimum is taken over ; the analogous agreement applies in similar places.)

An alternative way of defining the local sharp maximal operator is but this makes sense only for . Similar comment applies to the analogous definition of . Clearly, uniformly in and . Observe an advantage of using instead of : is defined for only, while makes sense for much wider class of admissible functions.

For , that is, when , the maximal operators, , , and , and their centered counterparts were defined and investigated in [4] (in the setting of a metric measure space, in addition, satisfying , for every ball ).

Another property to be immediately noted is that holds, for , by an application of Hölder’s inequality; similar relation is valid for and and for the centered versions of the three operators.

Finally, in case of considering maximal operators based on closed balls, we shall use the notations , , and so forth. To be precise, the definition of is where denotes the family of all closed balls such that and similarly for other maximal operators considered above. Note that if , then ; this is a consequence of continuity property of the measure .

Remark 1. It may be worth mentioning that the following (local) variant of the Hardy-Littlewood maximal operator, and its centered version , where is given, both fall within the scheme presented here: coincides with , where . See [7, p. 493], where and , [8, p. 126], where , , and is considered in the setting of and closed cubes, and [9, p. 469], where , , and is considered in the setting of and open (Euclidean) balls. This variant is an important substitute of the usual Hardy-Littlewood maximal operator (the limiting case of ) and is used frequently in the nondoubling case. Analogous comment concerns yet another variant of the Hardy-Littlewood maximal operator, , (see [9, p. 470], where its centered version is considered for and ). Also, the local fractional maximal operator where is a Borel measure on satisfying the upper growth condition for some , with playing the role of a dimension, uniformly in , and (if , is Lebesgue measure, and , then is the classical fractional maximal operator) is covered by the presented general approach, since coincides with . Finally, a mixture of both, considered in [10] in the setting of , coincides with , where

An interesting discussion of mapping properties of (global) fractional maximal operators in Sobolev and Campanato spaces in measure metric spaces equipped with a doubling measure , in addition satisfying the lower bound condition , is done by Heikkinen et al. in [11]. Investigation of local fractional maximal operators (from the point of view of their smoothing properties) defined in proper subdomains of the Euclidean spaces was given by Heikkinen et al. in [12]. See also comments at the end of Section 3.1.

The following lemma enhances [4, Lemmas 2.1 and 3.1]. By treating the centered case, we have to impose some assumptions on , , and . Namely, we assume that is an upper semicontinuous function (u.s.c. for short), is a lower semicontinuous function (l.s.c. for short), and satisfies It may be easily checked that in case is a genuine metric, is u.s.c. and , , satisfies (26).

Lemma 2. For any admissible and , the functions , , , and are l.s.c., hence, Borel measurable, and the same is true for and , when .

Proof. In the noncentered case no assumption on , , and is required. Indeed, fix , consider the level set , and take a point from this set. This means that there exists a ball such that and But the same ball , considered for any , also gives ; hence, , which shows that the level set is open. Exactly the same argument works for the level set except for the fact that, now, in (27), is replaced by . Finally, consider the level set and take a point from this set. There exists a ball and such that and, for every , we have . But the same ball is good enough, for any , in the sense that and, hence, , which shows that the level set is open.

In the centered case, we use the assumptions imposed on , , and . For , we write the level set as a union of open sets Each intersection on the right hand side is an open set. Indeed, is open, since, by assumption, is l.s.c. and is u.s.c. On the other hand, for every fixed , the function is l.s.c. as well. To show this, note that the limit of an increasing sequence of l.s.c. functions is a l.s.c. function, and, hence, it suffices to consider , . But then is l.s.c. as a product of three l.s.c. functions: is l.s.c. by continuity of from above, is l.s.c. by continuity of from below, and, finally, is l.s.c. as well, by the assumption (26) imposed on .

Exactly, the same argument works for the level set except for the fact that, now, in relevant places, has to be replaced by . Finally, for the level set , an argument similar to that given above combined with that used for does the job.

To relate maximal operators based on closed balls with these based on open balls, we must assume something more on the function . Namely, we assume that is defined on the union (rather than on only) and consider the following continuity condition: for every and , Note that , , satisfies (32) due to the continuity property of measure; in particular, satisfies (32).

We then have the following.

Lemma 3. Assume that (32) holds. Then, for , we have and the analogous identities for their centered counterparts. Consequently, for any , the functions , , , and are l.s.c. and, hence, Borel measurable.

Proof. For every and , we have To prove in (33), it is sufficient to check that, for any , such that , the following holds: Let and . Then, using the second part of (32), continuity of from below, and the monotone convergence theorem gives Similarly, to prove in (33), it suffices to check that, for any , such that , the following holds: Let and . Then, using the first part of (32), continuity of from above, and the dominated convergence theorem gives

The proof of (34) follows the line of the proof of (33) with the additional information that (note that ). Finally, the proofs of the centered versions go analogously.

Given , let be the “dilation constant” appearing in the version of the basic covering theorem for a quasimetric space with a constant in the quasitriangle inequality; see [13, Theorem 1.2]. It is easily seen that suffices (so that if is a metric, then and ). is called geometrically doubling provided that there exists such that every ball with radius can be covered by at most balls of radii . In the case when is such that , we say (cf. [4, p. 243]) that satisfies the -local -condition, , provided that In what follows, when the -local -condition is invoked, we tacitly assume that .

The following lemma enhances [4, Proposition 2.2].

Proposition 4. Suppose that and satisfy one of the following two assumptions: (i) and   satisfies  the  -local  -condition;(ii) uniformly  in  ,  and   is  geometrically  doubling.Then maps into boundedly and consequently; is bounded on , for any .

Proof. The assumption simply guarantees that , while the condition implies . To verify the weak type of both maximal operators in the latter replacement, note that, for , this is simply the conclusion of a version for quasimetric spaces of [4, Proposition 2.2], while, for , the result is essentially included in [7, Proposition 3.5] ( replaces 5 and the argument presented in the proof easily adapts to the local setting). Thus, each of the operators and is bounded on by applying Marcinkiewicz interpolation theorem, and, hence, the claim for follows.

Remark 5. It is probably worth pointing out that in the setting of , closed cubes, and an arbitrary Borel measure on which is finite on bounded sets, the maximal operator is of weak type () with respect to and thus is bounded on , for any (since ; the same is true for ). The details are given in [8, p. 127]. The same is valid for open (Euclidean) balls; see [14, Theorem 1.6]. In [14], Sawano also proved that, for an arbitrary separable locally compact metric space equipped with a Borel measure which is finite on bounded sets (every such a measure is Radon), for every , the associated centered maximal operator is of weak type () with respect to , and the result is sharp with respect to . See also Terasawa [15], where the same result, except for the sharpness, is proved without the assumption on separability of a metric space but with an additional assumption on the involved measure.

#### 3. Local Morrey and Campanato Spaces

The generalized local Morrey and Campanato spaces in the setting of the given system , , are defined by the requirements respectively. Note that the identities hold for any admissible . Therefore, using the centered versions of the operators and in (43) and (44) does not affect the spaces and the norms. Also, and, hence, using either or in place of in (44) does not affect the spaces, and, due to (17), the norms remain equivalent. It is also worth noting that, in the definitions of the spaces and , a priori we do not require to belong to but, a posteriori, indeed and .

Other properties to be observed are the inequality which holds, for any admissible , and gives and the continuous embeddings for , that follow from (18) and its version for .

When and is a metric, for and , the space coincides with the local BMO space defined and investigated in [4] in the setting of a metric measure space satisfying , for every ball .

Since is merely a seminorm, a genuine norm is generated by considering the quotient space , where the subspace is Unlikely to the case of , may be bigger than the space of constant functions. As it was explained in [4, p. 249], coincides with the space of functions which are constant -a.e. on each of -components of , where -components are obtained by means of the equivalence relation and provided that there exist balls such that , , and , .

In what follows we shall abuse slightly the language (in fact, we already did it) using in several places the term norm instead of (the proper term) seminorm.

The definition of the generalized local Morrey and Campanato spaces based on closed balls requires using in (43) and (44) the operators and , respectively. The resulting spaces are then denoted by and , respectively. Lemma 3 immediately leads to Corollary 6.

Corollary 6. Assume that (32) holds. Then, for , we have with identity of the corresponding norms in the first case and equivalence of norms in the second case.

Remark 7. Consider the global case; that is, . In the setting of equipped with the Euclidean distance and Lebesgue measure, the classical Morrey and Campanato spaces and (in the notation from [16]) correspond to the choice of (up to a multiplicative constant), where , , and , and are explicitely given by

If , then clearly . It is also known (see [16] for references) that, for , ; for , and ; and for , with . Here, denotes the space of all constant functions on .

Recall that a quasimetric measure space is called a space of homogeneous type provided that is doubling; that is, it satisfies uniformly in and ; clearly, the doubling condition implies that .

In the framework of a space of homogeneous type , a systematic treatment of generalized Campanato, Morrey, and Hölder spaces was presented by Nakai [2]. We refer to this paper for a discussion (among other things) of the relations between these spaces. In the nondoubling case, that is, in the setting of and a Borel measure that satisfies the growth condition (23), a theory of Morrey spaces was developed by Sawano and Tanaka [3] and Sawano [17]; for details, see Remarks 13 and 14.

Remark 8. Of course it may happen that is trivial in the sense that it contains only the null function. The triviality of is equivalent with the statement that, for every nonnull function , there exists such that . For instance, if with Lebesgue measure, , , and , with , then , for every nonnull and every (so that , for every ). Similarly, it may happen that is trivial in the sense that it consists of functions from only. This time, the triviality of is equivalent with the statement that, for every function , there exists such that . For instance, if with Lebesgue measure, , , and , with , then , for every and every (so that , for every ; in particular, (48) then implies that , for ).

See also [18] for further remarks on triviality of (the global case; equipped with the Euclidean metric and Lebesgue measure). In the same place, [18], the following interesting observation is made. Let be a function, , for , , and let be given. If , for every , then is decreasing and with equivalency of norms. Similarly, if , for every , then for , is increasing and with equivalency of norms.

In the Euclidean setting of with Lebesgue measure, the definition of the classical Morrey and Campanato spaces by using either the Euclidean balls or the Euclidean cubes (with sides parallel to the axes) gives the same outcome. Choosing balls or cubes means using either the metric or . In the general setting, we consider two equivalent quasimetrics on and possibly different and functions.

The result that follows compares generalized local Morrey and Campanato spaces for the given system with these of under convenient and, in some sense, natural assumptions.

Proposition 9. Let and the system be given, and suppose that the triple is different from . Assume also that there exists such that, for any ball , there exists a covering of consisting of balls from such that Then, and consequently . Similarly, if, for any ball , there exists a ball such that and then and, hence, .

Proof. To prove the first claim, take and , and consider a covering of , , consisting of balls from and satisfying (54). We have where in the second sum summation goes only over these 's for which . Taking the supremum over the relevant balls on the left hand side shows the required estimate and, hence, the inclusion. To prove the second claim, take and , and consider , , satisfying (56). Then Taking again the supremum over the relevant balls on the left hand side shows the second required estimate and, hence, the second inclusion.

Corollary 10. Under the assumptions of Proposition 9 and the analogous assumptions but with the roles of and switched, we have with equivalency of the corresponding norms.

Remark 11. In the case when, in the system , only is replaced by , it may happen that uniformly in . Then, the conclusion of Proposition 9 is obvious but, at the same moment, this is the simplest case of the assumption made in Proposition 9, with and the covering of consisting of .

The following example generalizes the situation of equivalency of theories based on the Euclidean balls or cubes mentioned above.

Example 12. Let be a space of homogeneous type. Assume that is a quasimetric equivalent with and . Given , let and . Then, for and , we have uniformly in and , and consequently with equivalency of the corresponding norms. Indeed, assuming that , for a , we have (in what follows means a ball related to ) and, hence, we take as a covering of . The doubling property of then implies and, therefore, (54) follows with and declared as above. The “dual” estimate follows analogously.

Remark 13. Sawano and Tanaka [3] defined and investigated Morrey spaces in the setting of , where is a Borel measure on finite on bounded sets (recall that every such measure is automatically a Radon measure) which may be nondoubling.

For a parameter and , the Morrey space (in the notation of [3] but with the roles of and switched) is the space of functions on satisfying where the supremum is taken over all (closed) cubes with the property . The space coincides with our space (i.e., ), where It was proved in [3, Proposition 1.1] (the growth condition (23) did not intervene there) that does not depend on the choice of . This corresponds to the situation of , , as above and , , in Corollary 10 since, as it can be easily observed, for say, we have and, on the other hand, the assumption of Proposition 9 is satisfied due to simple geometrical properties of cubes in (see [3, p. 1536] for details).

Remark 14. Sawano [17] defined and investigated the so-called generalized Morrey spaces in the same setting of (with closed cubes). For a parameter and a nondecreasing function , the space was defined as the space of functions on satisfying The space coincides with our space , where (Note that, for , , we also have ). It was proved in [1, Proposition 1.2] (again the growth assumption did not intervene there) that is independent of with equivalency of norms. This result may also be seen as a consequence of Corollary 10. Indeed, by taking in this corollary, for , we have . On the other hand, the assumption of Proposition 9 is satisfied by the argument already mentioned in Remark 13 (geometrical properties of cubes in ).

Remark 15. Recently, Liu et al. [6] defined and investigated the local Morrey spaces in the setting of a locally doubling metric measure space . The latter means that the measure possesses the doubling and the reverse doubling properties only on a class of admissible balls. This class, , is defined with an aid of an admissible function and a parameter and agrees with our class for the locality function (in [6], an assumption of geometrical nature is imposed on ). Then, the Morrey-type space , , was defined as the space of functions on satisfying The investigations in the general setting were next specified in [6] to the important example of the Gauss measure space , where denotes the Gauss measure . The importance of this example lies in the fact that the measure space is the natural environment for the Ornstein-Uhlenbeck operator . In the context of , the Campanato-type space was also defined as the space of functions on satisfying (the additional summand was added due to the specific character of the involved measure space).

Remark 16. In [19, Theorems 4 and 5] an example of Borel measure in was provided ( being absolutely continuous with respect to Lebesgue measure) such that () and () differ.

In the final example of this section, we analyse a specific case that shows that, in general, things may occur unexpected.

Example 17. Take , to be the metric on , and to be the measure on such that , where and (so that ). Note that is nondoubling; it is not even locally doubling and if is a ball, then either , for some , or , and, hence, , for every ball . Then , for , and . For simplicity, we now treat the case only.

Consider first . Then, for any , and are constant functions: where . Hence, , , , and is identified with , where denotes the space of constant sequences. Similarly, for any , and, hence, with identity of norms.

Consider now the case of