Journal of Function Spaces

Volume 2014 (2014), Article ID 172486, 15 pages

http://dx.doi.org/10.1155/2014/172486

Research Article

Local Morrey and Campanato Spaces on Quasimetric Measure Spaces

Krzysztof Stempak¹ and Xiangxing Tao²

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 . Then consists of balls of the form , , , where denotes the space of all sequences on , and , for any and , and, hence, and , for . In addition, every -component is of the form , , and, hence, . Similarly, for any , we have , , and, hence, and .

3.1. Morrey and Campanato Spaces on Open Proper Subsets of

In this subsection we suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets of .

As it was mentioned in [4, p. 259], indecisions accompany choosing a suitable definition of for a general open proper subset of equipped with Lebesgue measure. The “right” way seems to be the following: consists of those functions such that

where the supremum is taken over all closed balls (or closed cubes, if one prefers; then the character should be replaced by ), entirely contained in ; see [20]. Throughout this section stands for the Lebesgue measure of , a measurable subset of . Note that such a definition has a local flavor: the locality function entering the scene is

where the distance from to is given by

and or .

Similar indecisions accompany the process of choosing a suitable definition of Morrey and Campanato spaces for a general open proper subset . The spaces , , and , determined by

where (see also [21]), were originally introduced by Morrey [22] (with a restriction to open and bounded subsets). For a definition of (nowadays called after Campanato, the Campanato space), also with a restriction to open and bounded subsets, see [23].

An alternative way of defining generalized Morrey and Campanato spaces on open proper (not necessarily bounded) subset is by using our general approach with the locality function given above. To fix the attention let us assume, for a moment, that . Thus, for a given function , we define and . Explicitely, this means that, for ,

by the definition of , is the family of all closed balls entirely contained in .

Given a parameter , we now define the locality function as

so that . Then, for a function as above, we define and . The structure of the above definition of and reveals that if is not connected, then the defined spaces are isometrically isomorphic to the direct sums of the corresponding spaces built on the connected components of with norm for the direct sum of the given spaces. Indeed, if, for instance, , ( is finite or countable), where each is a connected component of , and denotes the restriction of to , then

Thus, without loss of generality, we can assume (and we do this) that is connected.

The analogous definitions (and comments associated to them) obey . To distinguish between the two cases corresponding to the choice of or , when necessary, we shall write and , and , and so forth. Also, the family of balls related to will be denoted by , while the family of cubes related to will be denoted by .

In what follows, rather than considering a general , we limit ourselves to the specific case of . Clearly, satisfies (32) and, hence, distinguishing between open or closed balls (or open or closed cubes) is not necessary. We write in place of and similarly in other occurences. Our goal is to prove that the definitions on Morrey and Campanato spaces do not depend on choosing balls or cubes; this is contained in Theorem 20.

The following propositions partially contain [24, Theorems 3.5 and 3.9] as special cases.

Proposition 18. Let and be given. The spaces are independent of the choice of the scale parameter with equivalence of the corresponding norms. The analogous statement is valid for the spaces .

Proof. Let . We shall prove the inequalities

which give the inclusions and . The inequalities opposite to (80) and (81) (with ) are obvious, and thus the opposite inclusions follow.

Consider first the case of (80). There exists such that bisecting any cube    times results in obtaining a family of congruent subcubes of each of them in . Thus

and the result follows.

Considering (81), we shall apply the procedure similar to that used in the proof of [24, Theorem 3.5]. Take and . Then

where is the quotient space and is the quotient norm. Since the dual to is identified with , where is the exponent conjugate to , , and denotes the subspace of consisting of functions with , therefore

According to [24, Lemma 3.1], there exist constants and such that, for every and every function , there exist subcubes of and functions such that and , for , and . Take satisfying , where is the cube chosen earlier, and select subcubes and functions with properties as above. Then

Hence,

and, consequently, .

Proposition 19. Let and be given. The spaces are independent of the choice of the scale parameter with equivalence of the corresponding norms. The analogous statement is valid for the spaces .

Proof. The present proof mimics the one of Proposition 18, since essentially it suffices to replace the character by and, assuming are given, to use the following geometrical properties of Euclidean balls. The first one says that there exists such that every ball may be covered by a family of balls each of them in and with radii smaller than that of . The second one (more sophisticated) is contained in [24, Lemma 3.8] and says that there exist constants and such that, for every and every function , there exist balls with radii smaller than that of and functions such that and , for , and . (The fact that radii of are smaller than the radius of is not directly indicated in the statement of [24, Lemma 3.8] but it is implicitly contained in the construction included in the proof of that lemma.)

For the sake of completeness, we include an outline of the proof of the first aforementioned property. We shall use the following simple geometrical fact: given and , there exists such that, for any sphere , one can find points on that sphere such that

(if , then we set ).

Now, take any . In fact, we shall prove the aforementioned property for the “maximal” ball with . Let , , where is large enough (to be determined in the last step of the argument). Using the above geometrical fact, on each sphere , , we choose finite number of points such that the balls centered at these points and with radii equal covering the annulus . More precisely, given , we apply the geometrical fact with and (so that ) and (so that ). It is clear that the union of all chosen balls covers and there is of them. To verify that each of these balls is in , take with center lying on the sphere (this is the worst case). Since, for , we have and , it is clear that

Hence, if is chosen to be the least positive integer with the property , then

and the required property follows (note that depends on and , and, hence, depends on , , and , as claimed).

The results of Propositions 18 and 19 allow us to define and (the choice of being “random”) and similarly for , , , and the corresponding norms. The following theorem partially contains [24, Theorem 4.2] as a special case.

Theorem 20. Let and be given. Then, we have

with equivalence of the corresponding norms.

Proof. We focus on proving the statement concerning the Campanato spaces; the argument for the Morrey spaces is analogous (and slightly simpler). Given a cube or a ball , by or , we will denote the ball circumscribed on or the cube circumscribed on , respectively. By the inequality , it is clear that, for , , if , and , if . Moreover, and , where and , depend on the dimension only.

Fix and take . For any and defined above,

Consequently, which also shows that . The results of Propositions 18 and 19 now give and . The opposite inclusion and inequality are proved in an analogous way.

Clearly, the concept of Morrey and Campanato spaces on open proper subsets of may be generalized to open proper subsets of a general quasimetric space . See [4, Section 5], where the concept of local maximal operators in such framework was mentioned. Finally, we mention that the presented concept of locality for open proper subdomains in the Euclidean spaces is rather common. See, for instance, the recent paper [25] where the regularity of the local Hardy-Littlewood operator was studied and the paper [12] where notions of local fractional operators were introduced and studied, in both cases in the setting of with the locality function .

4. Boundedness of Operators on Local Morrey Spaces

Boundedness of classical operators of harmonic analysis on Morrey spaces was investigated in a vast number of papers; see, for instance, [3, 5, 17, 26–30] and references cited there.

In this section, we assume the system to be fixed. We begin with a result on the boundedness of local Hardy-Littlewood maximal operator between local Morrey spaces. For the notational convenience, let , where is the constant from the quasitriangle inequality (if is a metric, then and ). Observe that the assumptions we impose in Proposition 21 are satisfied, for instance, when , for some , uniformly in , , , and satisfies the -local -condition. Recall also that when it comes to the boundedness of on , we have the conclusion of Proposition 4 to our disposal.

Proposition 21. Let and be a nonincreasing function such that satisfies

uniformly in . If is bounded on , then, it is also bounded from to .

Proof. For the notational convention, let ; that is,

Take , fix a ball , and consider the splitting adjusted to in the sense that . Then, for any ,

where the supremum is taken over all balls such that and . If and is one of such balls, then the fact that is nonincreasing gives

Consequently,

This estimate, subadditivity of , and the assumption that is bounded on give

This shows the required estimate .

Remark 22. Consider the global case, . To rediscover the classical result of Chiarenza and Frasca, [31, Theorem 1], which is the boundedness of the usual Hardy-Littlewood operator on the space , and (see Remark 7), take which is decreasing. The assumption (92), with being Lebesgue measure and the metric being is obviously satisfied (clearly the usual Hardy-Littlewood operator is also bounded on ).

Similarly, if is a space of homogeneous type and , , so that , then condition (92) is satisfied and, hence, , the Hardy-Littlewood operator associated to , maps boundedly into itself.

In the literature, several variants of fractional integrals over quasimetric measure spaces are considered. Here, we shall consider a variant in the setting of a quasimetric measure spaces with satisfying the upper growth condition (23) with . For any appropriate function and , we define the fractional integral operator by letting

For functions , we shall consider the following conditions (compare them with the assumptions imposed in [32]):

Proposition 23. Let and be functions satisfying (99). In addition, assume that and satisfy

uniformly in . If is bounded from to , then, for any , it is also bounded from to .

Proof. Since , it is sufficient to consider the case . The estimate to be proved is

uniformly in and . Take and and consider the decomposition , , and . It suffices to verify (101) with replaced by , , on the left hand side of this estimate. For , using the assumption on the boundedness of and (100), we write

For , note that, for any , we have ( denotes the completion of in ), and, therefore,

With this pointwise estimate, it follows that

The proof is complete.

Remark 24. García-Cuerva and Gatto proved that [33, Corollary 3.3], for a metric measure space , satisfying (23), is bounded from to provided that and . The assumption that is a metric may be relaxed; see [1], and in fact we can assume to be a quasi-metric measure space satisfying (23). Let . Then, and satisfy (99). In addition, if we assume that satisfies , uniformly in and , then (100) holds with constraints on as above. Therefore, with all these assumptions, is bounded from to .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was initiated when Krzysztof Stempak visited the Department of Mathematics of Zhejiang University of Science and Technology, China, in April 2013. He is thankful for the warm hospitality he received. The authors would like to thank the referees for their careful comments. The research of Krzysztof Stempak is supported by NCN of Poland under Grant 2013/09/B/ST1/02057. The research of Xiangxing Tao is supported by NNSF of China under Grants nos. 11171306 and 11071065.

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