Local Morrey and Campanato Spaces on Quasimetric Measure Spaces

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 R.

Given  > 0 and  ∈ , let  (, ) = { ∈  :  (, ) < } (2) be the "quasimetric" ball related to  of radius  and with center .If (, ) is a quasimetric space, then, T  , the topology in  induced by , is canonically defined by declaring  ⊂  to be open, that is,  ∈ T  , if and only if, for every  ∈ , there exists  > 0 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 T  is metrizable, see for instance [1] for references.Two quasimetrics  and   on  are said to be equivalent, if  −1   (, ) ≤ (, ) ≤   (, ) with some  ≥ 1 being independent of ,  ∈ .It is clear that, for equivalent quasimetrics, induced topologies coincide.Moreover, for any  > 0,   is a quasimetric as well and T  = T   .A quasimetric  is called a -metric, for 0 <  ≤ 1, provided that  (, ) ≤ ((, ) + (, )  ) 1/ (3) 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 (, T  ).It is also known (see [1]) that, given , for  determined by the equality (2)  = 2,   defined by is a metric on  which is equivalent to   ; more precisely,   ≤   ≤ 4  .Consequently,  () := (  ) 1/ is a -metric equivalent with ; more precisely,  () ≤  ≤ 4 1/  () .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 T  .It may happen that a ball in  is not a Borel set (i.e., it does not belong to the Borel algebra generated by T  ), 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 () > 0, we say that (, , ) is a quasimetric measure space (we do not assume that () > 0, for every ball ).In this paper, we additionally assume (similar to the assumption (1.3) made in [2]) that all balls in  are open; (5) 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  0 :  → [0, ∞) by setting  0 () = inf { :  ( (, )) > 0} ,  ∈ .
Given a function  :  → (0, ∞] such that  0 () < (), for every  ∈ , let B  () = B , () denote the family of balls (related to ) centered at  and with radius  satisfying  0 () <  < () (clearly balls with different radii but which coincide are identified as sets).Then we set Thus, B  denotes the family of all -local balls in  with positive measure.In case the lower estimate on the radius,  0 () < , is disregarded, we shall write B 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 B  ; thus,  ∈  1  loc, () :=  1 loc, (, , ) provided that ∫  || < ∞, for every ball  ∈ B  (and thus also for every  ∈ B ).Note that this notion of local integrability does not refer to compactness.Similarly, for 1 ≤  < ∞, we define   loc, () = { : ||  ∈  1  loc, ()}.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 B,  1 loc (),  , (), L , (), and so on (thus B 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 (R  ,  (∞) , ) 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 () = 0, for every ball , where  =  \ ; this happens, for instance, when () = (), where  ≥ 0 and  denotes Lebesgue measure on R  .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 () > 0, 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   () =   (, ), 1 ≤  < ∞, 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  > 0, means that  = (, ).For a function  ∈  1 loc, (), its average in a ball  = (, ) ∈ B  will be denoted by and similarly for any other Borel set , 0 < () < ∞, and any , whenever the integral makes sense.When the situation is specified to the Euclidean setting of R  , we shall consider either the metric  (2) induced by the norm ‖ ⋅ ‖ 2 or  (∞) induced by ‖ ⋅ ‖ ∞ .

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 B  .In practice,  will be usually defined on B, the family of all balls in .Then, a tempting alternative way of thinking about  is to treat it as a function  :  × R + → R + and then to define () = (, ), for  = (, ).There is, however, a pitfall connected with the fact that in general the mapping  × R + ∋ (, )  → (, ) ∈ B is not injective.Hence, we assume that  possesses the following property: (Thus, for instance, when  is bounded, i.e., diam() =   < ∞, the function  must obey the following rule: for every 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  ∈ R 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,  = R  ).We shall frequently test the constructed theory on these two functions.Finally, let us mention that it may happen that, for a constant  > 0 (playing the role of the dimension), we have uniformly in  ∈ B  .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 R = R ∪ {±∞} (when the real case is investigated).Given 1 ≤  < ∞, we define the generalized local fractional maximal operator  ,, acting on any admissible  by where the supremum is taken over all the balls from B  which contain , and its centered version by For  = 1, that is, when  ≡ 1, the maximal operators,  1,1, ,  # 1,1, , and M# 1,1, , and their centered counterparts were defined and investigated in [4] (in the setting of a metric measure space, in addition, satisfying () > 0, for every ball ).
Another property to be immediately noted is that holds, for 1 ≤ 1 ≤  < ∞, (see [9, p. 470], where its centered version is considered for  ≡ ∞ and  = 3).Also, the local fractional maximal operator where  is a Borel measure on  satisfying the upper growth condition for some 0 <  ≤ , with  playing the role of a dimension, uniformly in  > 0, and  ∈  (if  = R  ,  is Lebesgue measure, and  = , then  () is the classical fractional maximal operator) is covered by the presented general approach, since  () coincides with  1, ,−/ .Finally, a mixture of both, considered in [10] in the setting of (R  ,  (∞) ), coincides with  ,  , where   () =  ,, () = ( () 1−/  () ) 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.
Proof.In the noncentered case no assumption on  0 , , and  is required.Indeed, fix , consider the level set   =   () = { ∈  :  ,, () > }, and take a point  0 from this set.This means that there exists a ball  ∈ B  such that  0 ∈  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 { ∈  : M# ,, () > } except for the fact that, now, in (27),  is replaced by  − ⟨⟩  .Finally, consider the level set  #  = { ∈  :  # ,, () > } and take a point  0 from this set.There exists a ball  ∈ B  and  > 0 such that  0 ∈  and, for every  ∈ C, we have (1/())(⟨| − |  ⟩  ) 1/ > +.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  0 , , 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  0 is u.s.c.On the other hand, for every fixed  > 0, 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:  ∋   → ( ∩ (, )) Exactly, the same argument works for the level set F#,  = { ∈  : M#, ,, () > } 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 B  ∪ B  (rather than on B  only) and consider the following continuity condition: for every  0 ∈  and  0 ( 0 ) <  0 < ( 0 ), Note that  , ,  ∈ R, satisfies (32) due to the continuity property of measure; in particular,  ≡ 1 satisfies (32).
We then have the following.
Lemma 3. Assume that (32) holds.Then, for 1 ≤  < ∞, we have and the analogous identities for their centered counterparts.Consequently, for any  ∈  Proof.For every  0 ∈  and  0 > 0, we have To prove ≥ in (33), it is sufficient to check that, for any  0 = ( 0 ,  0 ) ∈ B  , such that  0 ∈  0 , the following holds: Let   →  − 0 and   >  0 ( 0 ).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  0 = ( 0 ,  0 ) ∈ B  , such that  0 ∈  0 , the following holds: Let   →  + 0 and   < ( 0 ).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   loc, () ⊂  1 loc, ()).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   = (3+2) suffices (so that if  is a metric, then  = 1 and   = 5).(, ) is called geometrically doubling provided that there exists  ∈ N such that every ball with radius  can be covered by at most  balls of radii (1/2).In the case when (, , ) is such that  0 ≡ 0, we say (cf.[4, p. 243]) that  satisfies the -local -condition,  > 1, provided that In what follows, when the -local -condition is invoked, we tacitly assume that  0 ≡ 0. .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  ≥ 2, the associated centered maximal operator M  is of weak type (1, 1) 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.
Since ‖ ⋅ ‖ L ,, is merely a seminorm, a genuine norm is generated by considering the quotient space L ,, ()/ 0, , where the subspace  0, is 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.
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  = R  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.
In the Euclidean setting of R  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  (2) 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 1 ≤  < ∞ and the system (, , , , ) be given, and suppose that the triple (  ,   ,   ) is different from (, , ).Assume also that there exists  0 ∈ N such that, for any ball  ∈ B , , there exists a covering {  1 , . . .,    0 } of  consisting of balls from B  ,  such that   (   ) (   ) 1/ ≲  () () where in the second sum summation goes only over these 's for which (   ) > 0. 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  ∈ L ,  ,  ,  () and  ∈ B , , and consider   ,  ⊂ 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  ∈ B  .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  0 = 1 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  ∈ R, let  =  ()  , and   = 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 (R  ,  (∞) , ), where  is a Borel measure on R  finite on bounded sets (recall that every such measure is automatically a Radon measure) which may be nondoubling.
For a parameter  > 1 and 1 ≤  ≤  < ∞, the Morrey space M   (, ) (in the notation of [3] but with the roles of  and  switched) is the space of functions on R  satisfying sup where the supremum is taken over all (closed) cubes with the property () > 0. The space M   (, ) coincides with our space  , (R  ,  (∞) , ) (i.e.,  ≡ ∞), where It was proved in [3, Proposition 1.1] (the growth condition (23) did not intervene there) that M   (, ) does not depend on the choice of  > 1.This corresponds to the situation of  =   =  (∞) ,  =   ≡ ∞,  as above and   () = () −1/ (  ) 1/−1/ ,   > 1, in Corollary 10 since, as it can be easily observed, for 1 <  <   < ∞ say, we have  ≤   and, on the other hand, the assumption of Proposition 9 is satisfied due to simple geometrical properties of cubes in R  (see [3, p. 1536] for details).
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, B  , is defined with an aid of an admissible function  :  → (0, ∞) and a parameter  ∈ (0, ∞) and agrees with our class B  for the locality function () = () (in [6], an assumption of geometrical nature is imposed on ).Then, the Morrey-type space M , B  (), 1 ≤  ≤  < ∞, was defined as the space of functions on  satisfying sup The investigations in the general setting were next specified in [6] to the important example of the Gauss measure space (R  ,  (2) ,   ), where   denotes the Gauss measure   () =  −/2 exp(−‖‖ 2 2 ).The importance of this example lies in the fact that the measure space (R  ,   ) is the natural environment for the Ornstein-Uhlenbeck operator −(1/2)Δ+  ⋅ ∇.In the context of (R  ,  (2) ,   ), the Campanato-type space E , B  (  ) was also defined as the space of functions on  satisfying (the additional summand ‖‖  1 (  ) was added due to the specific character of the involved measure space).
In the final example of this section, we analyse a specific case that shows that, in general, things may occur unexpected.
Consider first  ≡ ∞.Then, for any  = (()) ∈N ∈ ℓ  (N, ),  # ,1  and  #, ,1  are constant functions: where 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 Ω; see [20].Throughout this section || stands for the Lebesgue measure of , a measurable subset of Ω.Note that such a definition has a local flavor: the locality function entering the scene is where the distance from  ∈ Ω to Ω is given by and  =  (2) or  =  (∞) .Similar indecisions accompany the process of choosing a suitable definition of Morrey and Campanato spaces for a general open proper subset Ω ⊂ R  .The spaces  , (Ω), 1 ≤  < ∞, and 0 ≤  ≤ , determined by where  0 = diam Ω (see also [21]), were originally introduced by Morrey [22] (with a restriction to open and bounded subsets).For a definition of L , (Ω) (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 Ω ⊂ R  is by using our general approach with the locality function  Ω given above.To fix the attention let us assume, for a moment, that  =  (∞) .Thus, for a given function  : B  (Ω) → (0,∞), we define  , (Ω) :=  ,, Ω (Ω) and L , (Ω) := L ,, Ω (Ω).Explicitely, this means that, for  :=  Ω , by the definition of , B  (Ω) is the family of all closed balls entirely contained in Ω.
Given a parameter 0 <  ≤ 1, we now define the locality function   =  ,Ω as so that  1 =  Ω .Then, for a function  as above, we define  ,, (Ω) :=  ,,  (Ω) and L ,, (Ω) := L ,,  (Ω).The structure of the above definition of  ,, (Ω) and L ,, (Ω) reveals that if Ω is not connected, then the defined spaces are isometrically isomorphic to the direct sums of the corresponding spaces built on the connected components of Ω with ℓ ∞ norm for the direct sum of the given spaces.Indeed, if, for instance,  ∈  ,, (Ω), Ω = ⋃ ∈ Ω  ( is finite or countable), where each Ω  is a connected component of Ω, and   denotes the restriction of  to Ω  , then Thus, without loss of generality, we can assume (and we do this) that Ω is connected.The analogous definitions (and comments associated to them) obey  =  (2) .To distinguish between the two cases corresponding to the choice of  (2) or  (∞) , when necessary, we shall write  (2)   and  (∞)  ,  B ,, (Ω) and  Q ,, (Ω), and so forth.Also, the family of balls related to  (2)   will be denoted by B Ω  , 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  Q ,, (Ω) in place of  Q , , , (Ω) 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 1 ≤  < ∞ and  ≥ −1/ be given.The spaces  Q ,, (Ω) are independent of the choice of the scale parameter  ∈ (0, 1) with equivalence of the corresponding norms.The analogous statement is valid for the spaces L Q ,, (Ω).
Proof.Let 0 <  <  < 1.We shall prove the inequalities which give the inclusions The inequalities opposite to (80) and (81) (with  = 1) are obvious, and thus the opposite inclusions follow.
Consider first the case of (80).There exists  = (, , ) such that bisecting any cube  ∈ Q Ω   times results in obtaining a family {  } of 2  congruent subcubes of  each of them in Q Ω  .Thus and the result follows.
The results of Propositions 18 and 19 allow us to define (the choice of  = 1/2 being "random") and similarly for  B , (Ω), L Q , (Ω), L B , (Ω), and the corresponding norms.The following theorem partially contains [24, Theorem 4.2] as a special case.
Theorem 20.Let 1 ≤  < ∞ and  ≥ −1/ 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 Ω) .The opposite inclusion and inequality are proved in an analogous way.
Clearly, the concept of Morrey and Campanato spaces on open proper subsets of R  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 Ω ⊂ R  with the locality function Ω ∋   → dist(, R  \ Ω).
The assumption (92), with  being Lebesgue measure and the metric being  (∞) is obviously satisfied (clearly the usual Hardy-Littlewood operator is also bounded on   (R  )).
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 0 <  < , we define the fractional integral operator   by letting For functions ,  : (0, ∞) → (0, ∞), we shall consider the following conditions (compare them with the assumptions imposed in [32]): The proof is complete.