Local Morrey and Campanato Spaces on Quasimetric Measure Spaces

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 ℝ^𝑛.

1. Introduction

A quasimetric on a nonempty set 𝑋 is a mapping 𝑑∶𝑋×𝑋→[0,∞) which satisfies the following conditions:(i)for every 𝑥,𝑦∈𝑋, 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(ii)for every 𝑥,𝑦∈𝑋, 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥);(iii)there is a constant 𝐾≥1 such that, for every 𝑥,𝑦,𝑧∈𝑋, 𝑑(𝑥,𝑧)≤𝐾𝑑𝑥,𝑦+𝑑𝑦,𝑧. (1)The pair (𝑋,𝑑) is then called a quasimetric space; if 𝐾=1, then 𝑑 is a metric and (𝑋,𝑑) is a metric space.

Given 𝑟>0 and 𝑥∈𝑋, let 𝐵(𝑥,𝑟)=𝑦∈𝑋∶𝑑𝑥,𝑦<𝑟 (2) be the “quasimetric” ball related to 𝑑 of radius 𝑟 and with center 𝑥. If (𝑋,𝑑) is a quasimetric space, then, 𝒯_𝑑, the topology in 𝑋 induced by 𝑑, is canonically defined by declaring 𝐺⊂𝑋 to be open, that is, 𝐺∈𝒯_𝑑, if and only if, for every 𝑥∈𝐺, there exists 𝑟>0 such that 𝐵(𝑥,𝑟)⊂𝐺 (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 𝑑 is a genuine metric. Moreover, the topology 𝒯_𝑑 is metrizable, see for instance [1] for references.

Two quasimetrics 𝑑 and 𝑑^′ on 𝑋 are said to be equivalent, if 𝑐⁻¹𝑑^′(𝑥,𝑦)≤𝑑(𝑥,𝑦)≤𝑐𝑑^′(𝑥,𝑦) with some 𝑐≥1 being independent of 𝑥,𝑦∈𝑋. It is clear that, for equivalent quasimetrics, induced topologies coincide. Moreover, for any 𝑎>0, 𝑑^𝑎 is a quasimetric as well and 𝒯_𝑑=𝒯_𝑑^𝑎. A quasimetric 𝑑 is called a 𝑞-metric, for 0<𝑞≤1, provided that 𝑑(𝑥,𝑧)≤𝑑𝑥,𝑦^𝑞+𝑑𝑦,𝑧^𝑞1/𝑞 (3) holds uniformly in 𝑥,𝑦,𝑧∈𝑋. It is easily checked that a 𝑞-metric enjoys the open ball property; that is, every ball related to 𝑑 is an open set in (𝑋,𝒯_𝑑). It is also known (see [1]) that, given 𝑑, for 𝑞 determined by the equality (2𝐾)^𝑞=2, 𝑑_𝑞 defined by 𝑑_𝑞𝑥,𝑦=inf^𝑛_𝑗=1𝑑𝑥_𝑗−1,𝑥_𝑗^𝑞∶𝑥=𝑥₀,𝑥₁,…,𝑥_𝑛=𝑦,𝑛≥1 (4) is a metric on 𝑋 which is equivalent to 𝑑^𝑞; more precisely, 𝑑_𝑞≤𝑑^𝑞≤4𝑑_𝑞. Consequently, 𝑑_(𝑞)∶=(𝑑_𝑞)^1/𝑞 is a 𝑞-metric equivalent with 𝑑; more precisely, 𝑑_(𝑞)≤𝑑≤4^1/𝑞𝑑_(𝑞). Thus, every quasimetric admits an equivalent 𝑞-metric that possesses the open ball property.

In what follows, if (𝑋,𝑑) is a given quasimetric space, then 𝑋 is considered as a topological space equipped with the (metrizable) topology 𝒯_𝑑. It may happen that a ball in 𝑋 is not a Borel set (i.e., it does not belong to the Borel 𝜎-algebra generated by 𝒯_𝑑), 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 𝑋 is additionally equipped with a Borel measure 𝜇 which is finite on bounded sets and nontrivial in the sense that 𝜇(𝑋)>0, we say that (𝑋,𝑑,𝜇) is a quasimetric measure space (we do not assume that 𝜇(𝐵)>0, for every ball 𝐵). In this paper, we additionally assume (similar to the assumption (1.3) made in [2]) that allballsin𝑋areopen; (5) taking into account what was mentioned above, this assumption does not narrow the generality of our considerations.

Let (𝑋,𝑑,𝜇) be a quasimetric measure space. Define the function 𝜌₀∶𝑋→[0,∞) by setting 𝜌₀(𝑥)=inf𝜀∶𝜇(𝐵(𝑥,𝜀))>0,𝑥∈𝑋. (6) Observe that if 𝜌₀(𝑥)>0, for some 𝑥∈𝑋, then 𝜇(𝐵(𝑥,𝜌₀(𝑥)))=0; this is a consequence of the continuity property from below of the measure 𝜇. The property “𝜇(𝐵)>0, for every ball 𝐵,” is equivalent with the statement that 𝜌₀≡0.

Given a function 𝜌∶𝑋→(0,∞] such that 𝜌₀(𝑥)<𝜌(𝑥), for every 𝑥∈𝑋, let ℬ_𝜌(𝑥)=ℬ_𝜌,𝑑(𝑥) denote the family of balls (related to 𝑑) centered at 𝑥 and with radius 𝑟 satisfying 𝜌₀(𝑥)<𝑟<𝜌(𝑥) (clearly balls with different radii but which coincide are identified as sets). Then we set ℬ_𝜌=ℬ_𝜌,𝑑=_𝑥∈𝑋ℬ_𝜌(𝑥). (7) Thus, ℬ_𝜌 denotes the family of all 𝜌-local balls in 𝑋 with positive measure. In case the lower estimate on the radius, 𝜌₀(𝑥)<𝑟, is disregarded, we shall write ℬ_𝜌 for the resulting family of balls.

By a 𝜌-local integrability of a real or complex-valued function on 𝑋, we mean its integrability with respect to the family of balls from ℬ_𝜌; thus, 𝑓∈𝐿¹_loc,𝜌(𝑋)∶=𝐿¹_loc,𝜌(𝑋,𝑑,𝜇) provided that ∫_𝐵|𝑓|𝑑𝜇<∞, for every ball 𝐵∈ℬ_𝜌 (and thus also for every 𝐵∈ℬ_𝜌). Note that this notion of local integrability does not refer to compactness. Similarly, for 1≤𝑝<∞, we define 𝐿^𝑝_loc,𝜌(𝑋)={𝑓∶|𝑓|^𝑝∈𝐿¹_loc,𝜌(𝑋)}.

If 𝜌(𝑥)≠∞, for some 𝑥∈𝑋, then we will refer to 𝜌 as a locality function and to objects associated to 𝜌 as “local” objects. If 𝜌≡∞ identically, then we shall skip the ∞ subscript writing ℬ, 𝐿¹_loc(𝑋), 𝑀_𝑝,𝜙(𝑋), ℒ_𝑝,𝜙(𝑋), and so on (thus ℬ denotes the family of all balls in 𝑋) and refer to this setting as to the global one. Notice that the proofs of all results stated in the paper contain 𝜌=∞ as a special case.

Parallel to the main theory, we shall also develop an alternative theory in the framework of closed balls 𝐵(𝑥,𝑟)={𝑦∈𝑋∶𝑑(𝑥,𝑦)≤𝑟}. Note that, in the metric case, 𝐵(𝑥,𝑟) is indeed a closed set and, in general, if all balls are assumed to be Borel sets, then 𝐵(𝑥,𝑟) is Borel, too. The definitions of Morrey and Campanato spaces based on closed balls (in fact being closed cubes) in the framework of (ℝ^𝑛,𝑑^(∞),𝜇) 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 𝜇 has the property that 𝜇(𝜕𝐵)=0, for every ball 𝐵, where 𝜕𝐵=𝐵⧵𝐵; this happens, for instance, when 𝑑𝜇(𝑥)=𝑤(𝑥)𝑑𝑥, where 𝑤≥0 and 𝑑𝑥 denotes Lebesgue measure on ℝ^𝑛. 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 𝜇(𝐵)>0, for every ball 𝐵, 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 𝑆≲𝑇 to indicate that 𝑆≤𝐶𝑇 with a positive constant 𝐶 independent of significant quantities. We shall write 𝑆≃𝑇 when simultaneously 𝑆≲𝑇 and 𝑇≲𝑆; for instance, 𝑑≃𝑑^′ means the equivalence of quasimetrics 𝑑 and 𝑑^′, and so forth. By 𝐿^𝑝(𝑋)=𝐿^𝑝(𝑋,𝜇), 1≤𝑝<∞, we shall denote the usual Lebesgue 𝐿^𝑝 space on the measure space (𝑋,𝜇). 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 𝐵 as a set). Then, writing 𝜏𝐵, for a given ball 𝐵=𝐵(𝑥,𝑟) and 𝜏>0, means that 𝜏𝐵=𝐵(𝑥,𝜏𝑟). For a function 𝑓∈𝐿¹_loc,𝜌(𝑋), its average in a ball 𝐵=𝐵(𝑥,𝑟)∈ℬ_𝜌 will be denoted by 𝑓_𝐵=1𝜇(𝐵)_𝐵𝑓𝑑𝜇, (8) and similarly for any other Borel set 𝐴, 0<𝜇(𝐴)<∞, and any 𝑓, whenever the integral makes sense. When the situation is specified to the Euclidean setting of ℝ^𝑛, we shall consider either the metric 𝑑⁽²⁾ induced by the norm ‖⋅‖₂ or 𝑑^(∞) 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 𝜙 be a positive function defined on ℬ_𝜌. In practice, 𝜙 will be usually defined on ℬ, the family of all balls in 𝑋. Then, a tempting alternative way of thinking about 𝜙 is to treat it as a function 𝜙∶𝑋×ℝ₊→ℝ₊ and then to define 𝜙(𝐵)=𝜙(𝑥,𝑟), for 𝐵=𝐵(𝑥,𝑟). There is, however, a pitfall connected with the fact that in general the mapping 𝑋×ℝ₊∋(𝑥,𝑟)↦𝐵(𝑥,𝑟)∈ℬ is not injective. Hence, we assume that 𝜙 possesses the following property: 𝜙𝑥₁,𝑟₁=𝜙𝑥₂,𝑟₂whenever𝐵𝑥₁,𝑟₁=𝐵𝑥₂,𝑟₂. (9) (Thus, for instance, when 𝑋 is bounded, i.e., diam(𝑋)=𝑅_𝑋<∞, the function 𝜙 must obey the following rule: for every 𝑥₁,𝑥₂∈𝑋 and 𝑟₁,𝑟₂>𝑅_𝑋, 𝜙(𝑥₁,𝑟₁)=𝜙(𝑥₂,𝑟₂)).

Clearly, working with a general 𝜙 cannot lead to fully satisfactory results. Therefore, in what follows, we shall impose some additional mild (and natural) assumptions on 𝜙 in order to develop the theory. Frequently, in such assumptions, 𝜙 and 𝜇 will be interrelated. Of particular interest will be the functions 𝜙_𝑚,𝛼(𝐵)=𝜇(𝐵)^𝛼,𝜙_𝑟,𝛼(𝐵)=𝑟(𝐵)^𝛼, (10) where 𝛼∈ℝ and 𝑟(𝐵) denotes the radius of 𝐵 (the 𝑚 and 𝑟 stand for measure and radius, resp.). It is necessary to point out here that, for the second function, in fact, we consider a selector 𝐵↦𝑟(𝐵) assigning to any 𝐵, one of its possible radii (clearly this subtlety does not occur when, for instance, 𝑋=ℝ^𝑛). We shall frequently test the constructed theory on these two functions. Finally, let us mention that it may happen that, for a constant 𝑛>0 (playing the role of the dimension), we have 𝜇(𝐵)≃𝑟(𝐵)^𝑛, (11) uniformly in 𝐵∈ℬ_𝜌. Then, 𝜙_𝑚,𝛼(𝐵)≃𝜙_𝑟,𝛼(𝐵)^𝑛,𝐵∈ℬ_𝜌. (12)

Let the system (𝑋,𝑑,𝜇,𝜌,𝜙) be given. In what follows, by an admissible function on 𝑋, 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≤𝑝<∞, we define the generalized local fractional maximal operator 𝑀_{𝑝,𝜙,𝜌} acting on any admissible 𝑓 by 𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥)=sup_{𝑥∈𝐵∈ℬ𝜌}1𝜙(𝐵)1𝜇(𝐵)_𝐵𝑓^𝑝𝑑𝜇^1/𝑝,𝑥∈𝑋, (13) where the supremum is taken over all the balls from ℬ_𝜌 which contain 𝑥, and its centered version by 𝑀^𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥)=sup_{𝜌0(𝑥)(𝑥)<𝑟<𝜌}1𝜙(𝐵(×1𝑥,𝑟))𝜇(𝐵(𝑥,𝑟))_{𝐵(𝑥,𝑟)}𝑓^𝑝𝑑𝜇^1/𝑝,𝑥∈𝑋. (14) On the other hand, we define the generalized local sharp fractional maximal operator 𝑀^#_{𝑝,𝜙,𝜌} for any admissible 𝑓 by 𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥)=sup_{𝑥∈𝐵∈ℬ𝜌}inf_𝑧∈ℂ1𝜙(𝐵)×1𝜇(𝐵)_𝐵𝑓−𝑧^𝑝𝑑𝜇^1/𝑝,𝑥∈𝑋, (15) and similarly for its centered version 𝑀^#,𝑐_{𝑝,𝜙,𝜌}. (If spaces of real functions are considered, then the infimum is taken over 𝑧∈ℝ; the analogous agreement applies in similar places.)

An alternative way of defining the local sharp maximal operator is 𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥)=sup_{𝑥∈𝐵∈ℬ𝜌}1𝜙(𝐵)×1𝜇(𝐵)_𝐵𝑓−𝑓_𝐵^𝑝𝑑𝜇^1/𝑝,𝑥∈𝑋, (16) but this makes sense only for 𝑓∈𝐿¹_loc,𝜌(𝑋). Similar comment applies to the analogous definition of 𝑀^#,𝑐_{𝑝,𝜙,𝜌}. Clearly, 𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥)≃𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥),𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥)≃𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥), (17) uniformly in 𝑓∈𝐿¹_loc,𝜌(𝑋) and 𝑥∈𝑋. Observe an advantage of using 𝑀^#_{𝑝,𝜙,𝜌} instead of 𝑀^#_{𝑝,𝜙,𝜌}: 𝑀^#_{𝑝,𝜙,𝜌}𝑓 is defined for 𝑓∈𝐿¹_loc,𝜌(𝑋) only, while 𝑀^#_{𝑝,𝜙,𝜌}𝑓 makes sense for much wider class of admissible functions.

For 𝜙=𝟏, that is, when 𝜙≡1, the maximal operators, 𝑀_1,𝟏,𝜌, 𝑀^#_1,𝟏,𝜌, and 𝑀^#_1,𝟏,𝜌, and their centered counterparts were defined and investigated in [4] (in the setting of a metric measure space, in addition, satisfying 𝜇(𝐵)>0, for every ball 𝐵).

Another property to be immediately noted is 𝑀_{𝑝1,𝜙,𝜌}𝑓(𝑥)≤𝑀_{𝑝2,𝜙,𝜌}𝑓(𝑥),𝑥∈𝑋, (18) that holds, for 1≤𝑝₁<𝑝₂<∞, by an application of Hölder’s inequality; similar relation is valid for 𝑀^#_{𝑝,𝜙,𝜌} and 𝑀^#_{𝑝,𝜙,𝜌} 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 𝑀_{𝑝,𝜙,𝜌}, 𝑀^#_{𝑝,𝜙,𝜌}, and so forth. To be precise, the definition of 𝑀_{𝑝,𝜙,𝜌} is 𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥)=sup_{𝑥∈𝐵∈ℬ𝜌}1𝜙𝐵1𝜇𝐵^𝐵𝑓^𝑝𝑑𝜇^1/𝑝,𝑥∈𝑋, (19) where ℬ_𝜌 denotes the family of all closed balls 𝐵=𝐵(𝑥,𝑟) such that 𝜌₀(𝑥)<𝑟<𝜌(𝑥) and similarly for other maximal operators considered above. Note that if 𝜌₀(𝑥)=inf{𝜀∶𝜇(𝐵(𝑥,𝜀))>0}, then 𝜌₀(𝑥)=𝜌₀(𝑥); this is a consequence of continuity property of the measure 𝜇.

Remark 1. It may be worth mentioning that the following (local) variant of the Hardy-Littlewood maximal operator, 𝑀_𝜅,𝜌𝑓(𝑥)=sup_{𝑥∈𝐵∈ℬ𝜌}1𝜇(𝜅𝐵)_𝐵𝑓𝑑𝜇,𝑥∈𝑋, (20) and its centered version 𝑀^𝑐_𝜅,𝜌, where 𝜅>1 is given, both fall within the scheme presented here: 𝑀_𝜅,𝜌 coincides with 𝑀_{1,𝜙𝜅,𝜌}, where 𝜙_𝜅(𝐵)=𝜇(𝜅𝐵)/𝜇(𝐵). See [7, p. 493], where 𝜌≡∞ and 𝜅=5, [8, p. 126], where 𝜌≡∞, 𝜅>1, and 𝑀_𝜅 is considered in the setting of ℝ^𝑛 and closed cubes, and [9, p. 469], where 𝜌≡∞, 𝜅=3, and 𝑀^𝑐₃ is considered in the setting of ℝ^𝑛 and open (Euclidean) balls. This variant is an important substitute of the usual Hardy-Littlewood maximal operator (the limiting case of 𝜅=1) and is used frequently in the nondoubling case. Analogous comment concerns yet another variant of the Hardy-Littlewood maximal operator, 𝑀_{𝑝,𝜅,𝜌}𝑓(𝑥)∶=𝑀_𝜅,𝜌𝑓^𝑝(𝑥)^1/𝑝,𝑥∈𝑋, (21)1≤𝑝<∞, (see [9, p. 470], where its centered version is considered for 𝜌≡∞ and 𝜅=3). Also, the local fractional maximal operator 𝑀^(𝛼)𝑓(𝑥)=sup_{𝑥∈𝐵∈ℬ𝜌}1𝜇(𝐵)^{1−𝛼/𝜏𝐵}𝑓𝑑𝜇,𝑥∈𝑋, (22) where 𝜇 is a Borel measure on 𝑋 satisfying the upper growth condition 𝜇(𝐵(𝑥,𝑟))≲𝑟^𝜏, (23) for some 0<𝜏≤𝑛, with 𝑛 playing the role of a dimension, uniformly in 𝑟>0, and 𝑥∈𝑋 (if 𝑋=ℝ^𝑛, 𝜇 is Lebesgue measure, and 𝜏=𝑛, then 𝑀^(𝛼) is the classical fractional maximal operator) is covered by the presented general approach, since 𝑀^(𝛼) coincides with 𝑀_{1,𝜙𝑚,−𝛼/𝜏}. Finally, a mixture of both, 𝑀^(𝛼)_𝜅,𝑝𝑓(𝑥)=sup_{𝑥∈𝐵∈ℬ𝜌}1𝜇(𝜅𝐵)^{1−𝑝𝛼/𝜏𝐵}𝑓^𝑝𝑑𝜇^1/𝑝,𝑥∈𝑋, (24) considered in [10] in the setting of (ℝ^𝑛,𝑑^(∞)), coincides with 𝑀_{𝑝,𝜙𝜅}, where 𝜙_𝜅(𝐵)=𝜙_{𝜅,𝑝,𝛼}(𝐵)=𝜇(𝜅𝐵)^{1−𝑝𝛼/𝜏}𝜇(𝐵)^1/𝑝. (25)

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 𝜇, in addition satisfying the lower bound condition 𝜇(𝐵(𝑥,𝑟))≳𝑟^𝜏, 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 𝜌₀, 𝜌, and 𝜙. Namely, we assume that 𝜌₀ is an upper semicontinuous function (u.s.c. for short), 𝜌 is a lower semicontinuous function (l.s.c. for short), and 𝜙 satisfies ∀𝑟>0themapping𝑋∋𝑦⟼𝜙𝑦,𝑟isu.s.c. (26) It may be easily checked that in case 𝑑 is a genuine metric, 𝜌₀ is u.s.c. and 𝜙_𝑚,𝛼, 𝛼∈ℝ, satisfies (26).

Lemma 2. For any admissible 𝑓 and 1≤𝑝<∞, the functions 𝑀_{𝑝,𝜙,𝜌}𝑓, 𝑀^𝑐_{𝑝,𝜙,𝜌}𝑓, 𝑀^#_{𝑝,𝜙,𝜌}𝑓, and 𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓 are l.s.c., hence, Borel measurable, and the same is true for 𝑀^#_{𝑝,𝜙,𝜌}𝑓 and 𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓, when 𝑓∈𝐿¹_loc,𝜌(𝑋).

Proof. In the noncentered case no assumption on 𝜌₀, 𝜌, and 𝜙 is required. Indeed, fix 𝑓, consider the level set 𝐹_𝜆=𝐹_𝜆(𝑓)={𝑥∈𝑋∶𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥)>𝜆}, and take a point 𝑥₀ from this set. This means that there exists a ball 𝐵∈ℬ_𝜌 such that 𝑥₀∈𝐵 and 1𝜙(𝐵)1𝜇(𝐵)_𝐵𝑓^𝑝𝑑𝜇^1/𝑝>𝜆. (27) But the same ball 𝐵, considered for any 𝑦∈𝐵, also gives 𝑀_{𝑝,𝜙,𝜌}𝑓(𝑦)>𝜆; hence, 𝐵⊂𝐹_𝜆, which shows that the level set is open. Exactly the same argument works for the level set {𝑥∈𝑋∶𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥)>𝜆} except for the fact that, now, in (27), 𝑓 is replaced by 𝑓−⟨𝑓⟩_𝐵. Finally, consider the level set 𝐹^#_𝜆={𝑥∈𝑋∶𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥)>𝜆} and take a point 𝑥₀ from this set. There exists a ball 𝐵∈ℬ_𝜌 and 𝜀>0 such that 𝑥₀∈𝐵 and, for every 𝑧∈ℂ, we have (1/𝜙(𝐵))(⟨|𝑓−𝑧|^𝑝⟩_𝐵)^1/𝑝>𝜆+𝜀. But the same ball 𝐵 is good enough, for any 𝑦∈𝐵, in the sense that 𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑦)>𝜆 and, hence, 𝐵⊂𝐹^#_𝜆, which shows that the level set is open.
In the centered case, we use the assumptions imposed on 𝜌₀, 𝜌, and 𝜙. For 𝑀^𝑐_{𝑝,𝜙,𝜌}, we write the level set 𝐹^𝑐_𝜆=𝐹^𝑐_𝜆(𝑓)={𝑥∈𝑋∶𝑀^𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥)>𝜆} as a union of open sets 𝐹^𝑐_𝜆=_𝑟>01𝑦∈𝑋∶𝜙𝐵𝑓𝑦,𝑟^{𝑝𝐵(𝑦,𝑟)1/𝑝}∩𝑦>𝜆𝑦∈𝑋∶𝜌>𝑟>𝜌₀𝑦. (28) Each intersection on the right hand side is an open set. Indeed, 𝑦∈𝑋∶𝜌𝑦>𝑟>𝜌₀𝑦=𝑦∩𝑦∈𝑋∶𝜌>𝑟𝑦∈𝑋∶𝜌₀𝑦<𝑟 (29) is open, since, by assumption, 𝜌 is l.s.c. and 𝜌₀ is u.s.c. On the other hand, for every fixed 𝑟>0, the function 1𝑋∋𝑦⟼𝜙𝐵𝑓𝑦,𝑟^{𝑝𝐵(𝑦,𝑟)1/𝑝} (30) 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 𝑓=𝜒_𝐴, 𝜇(𝐴)<∞. But then 1𝑋∋𝑦⟼𝜙𝐵𝜇𝑦,𝑟𝐴∩𝐵𝑦,𝑟^1/𝑝𝜇𝐵𝑦,𝑟 (31) is l.s.c. as a product of three l.s.c. functions: 𝑋∋𝑦↦𝜇(𝐴∩𝐵(𝑦,𝑟))^1/𝑝 is l.s.c. by continuity of 𝜇 from above, 𝑋∋𝑦↦(1/𝜇(𝐵(𝑦,𝑟))) is l.s.c. by continuity of 𝜇 from below, and, finally, 𝑋∋𝑦↦𝜙(𝐵(𝑦,𝑟))⁻¹ is l.s.c. as well, by the assumption (26) imposed on 𝜙.
Exactly, the same argument works for the level set 𝐹_𝜆^#,𝑐={𝑥∈𝑋∶𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥)>𝜆} except for the fact that, now, in relevant places, 𝑓 has to be replaced by 𝑓−⟨𝑓⟩_𝐵. Finally, for the level set 𝐹_𝜆^#,𝑐={𝑥∈𝑋∶𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥)>𝜆}, an argument similar to that given above combined with that used for 𝑀^#_{𝑝,𝜙,𝜌} 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 𝜙. Namely, we assume that 𝜙 is defined on the union ℬ_𝜌∪ℬ_𝜌 (rather than on ℬ_𝜌 only) and consider the following continuity condition: for every 𝑦₀∈𝑋 and 𝜌₀(𝑦₀)<𝑟₀<𝜌(𝑦₀), 𝜙𝐵𝑦₀,𝑟₀=lim_{𝑟→𝑟+0}𝜙𝐵𝑦₀,𝜙𝐵𝑦,𝑟₀,𝑟₀=lim_{𝑟→𝑟−0}𝜙𝐵𝑦₀,𝑟. (32) Note that 𝜙_𝑚,𝛼, 𝛼∈ℝ, satisfies (32) due to the continuity property of measure; in particular, 𝜙≡1 satisfies (32).

We then have the following.

Lemma 3. Assume that (32) holds. Then, for 1≤𝑝<∞, we have 𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥)=𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥),𝑓∈𝐿^𝑝_{𝑙𝑜𝑐,𝜌}(𝑋),𝑥∈𝑋,𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥)=𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥),𝑓∈𝐿^𝑝_{𝑙𝑜𝑐,𝜌}(𝑋),𝑥∈𝑋, (33)(34) and the analogous identities for their centered counterparts. Consequently, for any 𝑓∈𝐿^𝑝_{𝑙𝑜𝑐,𝜌}(𝑋), the functions 𝑀_{𝑝,𝜙,𝜌}𝑓, 𝑀^𝑐_{𝑝,𝜙,𝜌}𝑓, 𝑀^#_{𝑝,𝜙,𝜌}𝑓, and 𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓 are l.s.c. and, hence, Borel measurable.

Proof. For every 𝑦₀∈𝑋 and 𝑟₀>0, we have 𝐵𝑦₀,𝑟₀=_𝑟>𝑟0𝐵𝑦₀,𝑟,𝐵𝑦₀,𝑟₀=_0<𝑟<𝑟0𝐵𝑦₀,𝑟. (35) To prove ≥ in (33), it is sufficient to check that, for any 𝐵₀=𝐵(𝑦₀,𝑟₀)∈ℬ_𝜌, such that 𝑥₀∈𝐵₀, the following holds: 1𝜙𝐵₀1𝜇𝐵_0𝐵0𝑓^𝑝𝑑𝜇^1/𝑝≤sup_{𝑥∈𝐵∈ℬ𝜌}1𝜙𝐵1𝜇𝐵^𝐵𝑓^𝑝𝑑𝜇^1/𝑝. (36) Let 𝑟_𝑛→𝑟⁻₀ and 𝑟_𝑛>𝜌₀(𝑦₀). Then, using the second part of (32), continuity of 𝜇 from below, and the monotone convergence theorem gives 1𝜙𝐵𝑦₀,𝑟_𝑛1𝜇𝐵𝑦₀,𝑟_𝑛^𝐵(𝑦₀^,𝑟_𝑛⁾𝑓^𝑝𝑑𝜇^1/𝑝⟶1𝜙𝐵₀1𝜇𝐵_0𝐵0𝑓^𝑝𝑑𝜇^1/𝑝. (37) Similarly, to prove ≤ in (33), it suffices to check that, for any 𝐵₀=𝐵(𝑦₀,𝑟₀)∈ℬ_𝜌, such that 𝑥₀∈𝐵₀, the following holds: 1𝜙𝐵₀1𝜇𝐵₀^𝐵₀𝑓^𝑝𝑑𝜇^1/𝑝≤sup_{𝑥∈𝐵∈ℬ𝜌}1𝜙(𝐵)1𝜇(𝐵)_𝐵𝑓^𝑝𝑑𝜇^1/𝑝. (38) Let 𝑟_𝑛→𝑟⁺₀ and 𝑟_𝑛<𝜌(𝑦₀). Then, using the first part of (32), continuity of 𝜇 from above, and the dominated convergence theorem gives 1𝜙𝐵𝑦₀,𝑟_𝑛1𝜇𝐵𝑦₀,𝑟_{𝑛𝐵(𝑦0,𝑟𝑛)}𝑓^𝑝𝑑𝜇^1/𝑝⟶1𝜙𝐵₀1𝜇𝐵₀^𝐵₀𝑓^𝑝𝑑𝜇^1/𝑝. (39)
The proof of (34) follows the line of the proof of (33) with the additional information that 𝑓_{𝐵(𝑦0,𝑟0)}=lim_𝑛→∞𝑓_{𝐵(𝑦0,𝑟𝑛)},𝑓_{𝐵(𝑦0,𝑟0)}=lim_𝑛→∞𝑓_{𝐵(𝑦0,𝑟𝑛)} (40) (note that 𝐿^𝑝_loc,𝜌(𝑋)⊂𝐿¹_loc,𝜌(𝑋)). Finally, the proofs of the centered versions go analogously.

Given (𝑋,𝑑), let 𝜅_𝐾 be the “dilation constant” appearing in the version of the basic covering theorem for a quasimetric space with a constant 𝐾 in the quasitriangle inequality; see [13, Theorem 1.2]. It is easily seen that 𝜅_𝐾=𝐾(3𝐾+2) suffices (so that if 𝑑 is a metric, then 𝐾=1 and 𝜅_𝐾=5). (𝑋,𝑑) is called geometrically doubling provided that there exists 𝑁∈ℕ such that every ball with radius 𝑟 can be covered by at most 𝑁 balls of radii (1/2)𝑟. In the case when (𝑋,𝑑,𝜇) is such that 𝜌₀≡0, we say (cf. [4, p. 243]) that 𝜇 satisfies the 𝜌-local 𝜏-condition, 𝜏>1, provided that 𝜇(𝜏𝐵)≲𝜇(𝐵),𝐵∈ℬ_𝜌. (41) In what follows, when the 𝜌-local 𝜏-condition is invoked, we tacitly assume that 𝜌₀≡0.

The following lemma enhances [4, Proposition 2.2].

Proposition 4. Suppose that 𝜙 and 𝜇 satisfy one of the following two assumptions: (i)1≲𝜙  and  𝜇  satisfies  the  𝜌-local  𝜅_𝐾-condition;(ii)𝜇(𝜅_𝐾𝐵)/𝜇(𝐵)≲𝜙(𝐵)  uniformly  in  𝐵∈ℬ_𝜌,  and  (𝑋,𝑑)  is  geometrically  doubling.Then 𝑀_1,𝜙,𝜌 maps 𝐿¹(𝑋,𝜇) into 𝐿^1,∞(𝑋,𝜇) boundedly and consequently; 𝑀_1,𝜙,𝜌 is bounded on 𝐿^𝑝(𝑋,𝜇), for any 1<𝑝<∞.

Proof. The assumption 1≲𝜙 simply guarantees that 𝑀_1,𝜙,𝜌≲𝑀_1,𝟏,𝜌, while the condition 𝜇(𝜅_𝐾𝐵)/𝜇(𝐵)≲𝜙(𝐵) implies 𝑀_1,𝜙,𝜌≲𝑀_{1,𝜅𝐾,𝜌}. To verify the weak type (1,1) of both maximal operators in the latter replacement, note that, for 𝑀_1,𝟏,𝜌, this is simply the conclusion of a version for quasimetric spaces of [4, Proposition 2.2], while, for 𝑀_{1,𝜅𝐾,𝜌}, the result is essentially included in [7, Proposition 3.5] (𝜅_𝐾 replaces 5 and the argument presented in the proof easily adapts to the local setting). Thus, each of the operators 𝑀_1,𝟏,𝜌 and 𝑀_{1,𝜅𝐾,𝜌} is bounded on 𝐿^𝑝(𝑋,𝜇) by applying Marcinkiewicz interpolation theorem, and, hence, the claim for 𝑀_1,𝜙,𝜌 follows.

Remark 5. It is probably worth pointing out that in the setting of ℝ^𝑛, closed cubes, and an arbitrary Borel measure 𝜇 on ℝ^𝑛 which is finite on bounded sets, the maximal operator 𝑀_𝜅 is of weak type (1,1) with respect to 𝜇 and thus is bounded on 𝐿^𝑝(𝜇), for any 𝜅>1 (since 𝑀_𝜅,𝜌≤𝑀_𝜅; the same is true for 𝑀_𝜅,𝜌). 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 𝜅≥2, the associated centered maximal operator 𝑀^𝑐_𝜅 is of weak type (1,1) with respect to 𝜇, and the result is sharp with respect to 𝜅. 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 (𝑋,𝑑,𝜇,𝜌,𝜙), 𝐿_{𝑝,𝜙,𝜌}(𝑋)∶=𝐿_{𝑝,𝜙,𝜌},ℒ𝑋,𝑑,𝜇_{𝑝,𝜙,𝜌}(𝑋)∶=ℒ_{𝑝,𝜙,𝜌}𝑋,𝑑,𝜇, (42)1≤𝑝<∞, are defined by the requirements 𝑓_{𝐿𝑝,𝜙,𝜌}∶=sup_𝑥∈𝑋𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥)<∞,𝑓_{ℒ𝑝,𝜙,𝜌}∶=sup_𝑥∈𝑋𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥)<∞, (43)(44) respectively. Note that the identities sup_𝑥∈𝑋𝑀^𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥)=sup_𝑥∈𝑋𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥),sup_𝑥∈𝑋𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥)=sup_𝑥∈𝑋𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥), (45) hold for any admissible 𝑓. Therefore, using the centered versions of the operators 𝑀_{𝑝,𝜙,𝜌} and 𝑀^#_{𝑝,𝜙,𝜌} in (43) and (44) does not affect the spaces and the norms. Also, sup_𝑥∈𝑋𝑀^#,𝑐_{𝑝,𝜙,𝜌}𝑓(𝑥)=sup_𝑥∈𝑋𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥), (46) and, hence, using either 𝑀^#,𝑐_{𝑝,𝜙,𝜌} or 𝑀^#_{𝑝,𝜙,𝜌} in place of 𝑀^#_{𝑝,𝜙,𝜌} 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 𝐿_{𝑝,𝜙,𝜌}(𝑋) and ℒ_{𝑝,𝜙,𝜌}(𝑋), a priori we do not require 𝑓 to belong to 𝐿^𝑝_loc,𝜌(𝑋) but, a posteriori, indeed 𝐿_{𝑝,𝜙,𝜌}(𝑋)⊂𝐿^𝑝_loc,𝜌(𝑋) and ℒ_{𝑝,𝜙,𝜌}(𝑋)⊂𝐿^𝑝_loc,𝜌(𝑋).

Other properties to be observed are the inequality 𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥)≤𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥),𝑥∈𝑋, (47) which holds, for any admissible 𝑓, and gives 𝑓_{ℒ𝑝,𝜙,𝜌}≤𝑓_{𝐿𝑝,𝜙,𝜌},𝐿_{𝑝,𝜙,𝜌}(𝑋)⊂ℒ_{𝑝,𝜙,𝜌}(𝑋), (48) and the continuous embeddings 𝐿_{𝑝1,𝜙,𝜌}(𝑋)⊂𝐿_{𝑝2,𝜙,𝜌}(𝑋),ℒ_{𝑝1,𝜙,𝜌}(𝑋)⊂ℒ_{𝑝2,𝜙,𝜌}(𝑋), (49) for 1≤𝑝₁<𝑝₂<∞, that follow from (18) and its version for 𝑀^#_{𝑝,𝜙,𝜌}.

When 𝜌₀≡0 and 𝑑 is a metric, for 𝑝=1 and 𝜙=𝟏, the space ℒ_1,𝟏,𝜌(𝑋) coincides with the local BMO space BMO_𝜌(𝑋)=BMO_𝜌(𝑋,𝑑,𝜇) defined and investigated in [4] in the setting of a metric measure space satisfying 𝜇(𝐵)>0, for every ball 𝐵.

Since ‖⋅‖_{ℒ𝑝,𝜙,𝜌} is merely a seminorm, a genuine norm is generated by considering the quotient space ℒ_{𝑝,𝜙,𝜌}(𝑋)/𝑊_0,𝜌, where the subspace 𝑊_0,𝜌 is 𝑊_0,𝜌=𝑓∈ℒ_{𝑝,𝜙,𝜌}(𝑋)∶𝑓_{ℒ𝑝,𝜙,𝜌}=0. (50) Unlikely to the case of 𝜌≡∞, 𝑊_0,𝜌 may be bigger than the space of constant functions. As it was explained in [4, p. 249], 𝑊_0,𝜌 coincides with the space of functions which are constant 𝜇-a.e. on each of 𝜌-components of 𝑋, where 𝜌-components are obtained by means of the equivalence relation ∼_𝜌 and 𝑥∼_𝜌𝑦 provided that there exist balls {𝐵₁,…,𝐵_𝑚}⊂ℬ_𝜌 such that 𝑥∈𝐵₁, 𝑦∈𝐵_𝑚, and 𝐵_𝑖∩𝐵_𝑖+1≠0/, 𝑖=1,…,𝑚−1.

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 𝑀_{𝑝,𝜙,𝜌} and 𝑀^#_{𝑝,𝜙,𝜌}, respectively. The resulting spaces are then denoted by 𝐿_{𝑝,𝜙,𝜌}(𝑋) and ℒ_{𝑝,𝜙,𝜌}(𝑋), respectively. Lemma 3 immediately leads to Corollary 6.

Corollary 6. Assume that (32) holds. Then, for 1≤𝑝<∞, we have 𝐿_{𝑝,𝜙,𝜌}(𝑋)=𝐿_{𝑝,𝜙,𝜌}(𝑋),ℒ_{𝑝,𝜙,𝜌}(𝑋)=ℒ_{𝑝,𝜙,𝜌}(𝑋), (51) 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, 𝜌≡∞. In the setting of ℝ^𝑛 equipped with the Euclidean distance and Lebesgue measure, the classical Morrey and Campanato spaces 𝐿_𝑝,𝜆 and ℒ_𝑝,𝜆 (in the notation from [16]) correspond to the choice of 𝜙=𝜙_𝑚,𝛼 (up to a multiplicative constant), where 1≤𝑝<∞, 𝛼=(𝜆/𝑛−1)/𝑝, and 0≤𝜆≤𝑛+𝑝, and are explicitely given by 𝐿_𝑝,𝜆=𝐿_𝑝,𝜆ℝ^𝑛=𝑓∶𝑓_(𝑝,𝜆)∶=sup_{𝑥∈ℝ𝑛,𝑟>0}1𝑟^𝜆_{𝐵(𝑥,𝑟)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝,ℒ<∞_𝑝,𝜆=ℒ_𝑝,𝜆ℝ^𝑛=𝑓∶𝑓_𝑝,𝜆∶=sup_{𝑥∈ℝ𝑛,𝑟>0}inf_𝑧∈ℂ1𝑟^𝜆_{𝐵(𝑥,𝑟)}𝑓𝑦−𝑧^𝑝𝑑𝑦^1/𝑝<∞. (52)
If 𝜆=0, then clearly ℒ_𝑝,0/𝒞≅𝐿_𝑝,0=𝐿^𝑝(ℝ^𝑛). It is also known (see [16] for references) that, for 0<𝜆<𝑛, ℒ_𝑝,𝜆/𝒞≅𝐿_𝑝,𝜆; for 𝜆=𝑛, ℒ_𝑝,𝑛=BMO(ℝ^𝑛) and 𝐿_𝑝,𝑛=𝐿^∞(ℝ^𝑛); and for 𝑛<𝜆≤𝑛+𝑝, ℒ_𝑝,𝜆=Lip_𝛼(ℝ^𝑛) with 𝛼=(𝜆−𝑛)/𝑝. Here, 𝒞=𝒞(ℝ^𝑛) denotes the space of all constant functions on ℝ^𝑛.
Recall that a quasimetric measure space (𝑋,𝑑,𝜇) is called a space of homogeneous type provided that 𝜇 is doubling; that is, it satisfies 𝜇(𝐵(𝑥,2𝑟))≲𝜇(𝐵(𝑥,𝑟)) (53) uniformly in 𝑥∈𝑋 and 𝑟>0; clearly, the doubling condition implies that 𝜌₀≡0.
In the framework of a space of homogeneous type (𝑋,𝑑,𝜇), 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 𝑋=ℝ^𝑛 and a Borel measure 𝜇 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 𝐿_{𝑝,𝜙,𝜌}(𝑋,𝑑,𝜇) is trivial in the sense that it contains only the null function. The triviality of 𝐿_{𝑝,𝜙,𝜌}(𝑋,𝑑,𝜇) is equivalent with the statement that, for every nonnull function 𝑓∈𝐿^𝑝_loc,𝜌(𝑋), there exists 𝑥₀∈𝑋 such that 𝑀_{𝑝,𝜙,𝜌}𝑓(𝑥₀)=∞. For instance, if 𝑋=ℝ^𝑛 with Lebesgue measure, 𝜌≡∞, 1≤𝑝<∞, and 𝜙=𝜙_𝑚,𝛼, with 𝛼<−1/𝑝, then 𝑀_𝑝,𝜙𝑓(𝑥₀)=∞, for every nonnull 𝑓∈𝐿^𝑝_loc(ℝ^𝑛) and every 𝑥₀∈ℝ^𝑛 (so that 𝐿_𝑝,𝜆(ℝ^𝑛)={0}, for every 𝜆<0). Similarly, it may happen that ℒ_{𝑝,𝜙,𝜌}(𝑋,𝑑,𝜇) is trivial in the sense that it consists of functions from 𝑊_0,𝜌 only. This time, the triviality of ℒ_{𝑝,𝜙,𝜌}(𝑋,𝑑,𝜇) is equivalent with the statement that, for every function 𝑓∈𝐿^𝑝_loc,𝜌(𝑋)⧵𝑊_0,𝜌, there exists 𝑥₀∈𝑋 such that 𝑀^#_{𝑝,𝜙,𝜌}𝑓(𝑥₀)=∞. For instance, if 𝑋=ℝ^𝑛 with Lebesgue measure, 𝜌≡∞, 1≤𝑝<∞, and 𝜙=𝜙_𝑚,𝛼, with 𝛼>−1, then 𝑀_𝑝,𝜙𝑓(𝑥₀)=∞, for every 𝑓∈𝐿^𝑝_loc(ℝ^𝑛)⧵𝒞(ℝ^𝑛) and every 𝑥₀∈ℝ^𝑛 (so that ℒ_𝑝,𝜆(ℝ^𝑛)=𝒞(ℝ^𝑛), for every 𝜆>𝑛+𝑝; in particular, (48) then implies that 𝐿_𝑝,𝜆(ℝ^𝑛)={0}, for 𝜆>𝑛+𝑝).
See also [18] for further remarks on triviality of 𝐿_𝑝,𝜙(ℝ^𝑛) (the global case; ℝ^𝑛 equipped with the Euclidean metric 𝑑⁽²⁾ and Lebesgue measure). In the same place, [18], the following interesting observation is made. Let 𝜙∶(0,∞)→(0,∞) be a function, 𝜙(𝐵)=𝜙(𝑟), for 𝐵=𝐵(𝑥,𝑟), 𝑥∈ℝ^𝑛, and let 1≤𝑝<∞ be given. If 𝜙(𝑟)=inf_{0<𝑡≤𝑟}𝜙(𝑡)>0, for every 𝑟>0, then 𝜙 is decreasing and 𝐿_𝑝,𝜙(ℝ^𝑛)=𝐿_𝑝,𝜙(ℝ^𝑛) with equivalency of norms. Similarly, if inf_{𝑟≤𝑡<∞}𝜙(𝑡)𝑡^𝑛/𝑝>0, for every 𝑟>0, then for 𝜙(𝑟)=𝑟^−𝑛/𝑝inf_{𝑟≤𝑡<∞}𝜙(𝑡)𝑡^𝑛/𝑝, 𝜙(𝑟)𝑟^𝑛/𝑝 is increasing and 𝐿_𝑝,𝜙(ℝ^𝑛)=𝐿_𝜙𝑝,(ℝ^𝑛) with equivalency of norms.

In the Euclidean setting of ℝ^𝑛 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 𝑑⁽²⁾ or 𝑑^(∞). In the general setting, we consider two equivalent quasimetrics on 𝑋 and possibly different 𝜌 and 𝜙 functions.

The result that follows compares generalized local Morrey and Campanato spaces for the given system (𝑋,𝑑,𝜇,𝜌,𝜙) with these of (𝑋,𝑑^′,𝜇,𝜌^′,𝜙^′) under convenient and, in some sense, natural assumptions.

Proposition 9. Let 1≤𝑝<∞ and the system (𝑋,𝑑,𝜇,𝜌,𝜙) be given, and suppose that the triple (𝑑^′,𝜌^′,𝜙^′) is different from (𝑑,𝜌,𝜙). Assume also that there exists 𝑛₀∈ℕ such that, for any ball 𝐵∈ℬ_𝜌,𝑑, there exists a covering {𝐵^′₁,…,𝐵^′_𝑛0} of 𝐵 consisting of balls from ℬ_{𝜌′,𝑑′} such that 𝜙^′𝐵^′_𝑗𝜇𝐵^′_𝑗^1/𝑝≲𝜙(𝐵)𝜇(𝐵)^1/𝑝,𝑗=1,…,𝑛₀. (54) Then, 𝑓_{𝐿𝑝,𝜙,𝜌,𝑑}≲𝑓_{𝐿′′′𝑝,𝜙,𝜌,𝑑},𝑓∈𝐿_{𝑝,𝜙′,𝜌′,𝑑′}(𝑋), (55) and consequently 𝐿_{𝑝,𝜙′,𝜌′,𝑑′}(𝑋)⊂𝐿_{𝑝,𝜙,𝜌,𝑑}(𝑋). Similarly, if, for any ball 𝐵∈ℬ_𝜌,𝑑, there exists a ball 𝐵^′∈ℬ_{𝜌′,𝑑′} such that 𝐵⊂𝐵^′ and 𝜙^′𝐵^′𝜇𝐵^′1/𝑝≲𝜙(𝐵)𝜇(𝐵)^1/𝑝, (56) then 𝑓_{ℒ𝑝,𝜙,𝜌,𝑑}≲𝑓_{ℒ′′′𝑝,𝜙,𝜌,𝑑},𝑓∈ℒ_{𝑝,𝜙′,𝜌′,𝑑′}(𝑋), (57) and, hence, ℒ_{𝑝,𝜙′,𝜌′,𝑑′}(𝑋)⊂ℒ_{𝑝,𝜙,𝜌,𝑑}(𝑋).

Proof. To prove the first claim, take 𝑓∈𝐿_{𝑝,𝜙′,𝜌′,𝑑′}(𝑋) and 𝐵∈ℬ_𝜌,𝑑, and consider a covering of 𝐵, 𝐵⊂∪^𝑛0_𝑗=1𝐵^′_𝑗, consisting of balls from ℬ_{𝜌′,𝑑′} and satisfying (54). We have 1𝜙(𝐵)1𝜇(𝐵)_𝐵𝑓^𝑝𝑑𝜇^1/𝑝≤^𝑛0_𝑗=11𝜙(𝐵)𝜇(𝐵)^{1/𝑝𝐵′}_𝑗𝑓^𝑝𝑑𝜇^1/𝑝≲_𝑗1𝜙^′𝐵^′_𝑗𝜇𝐵^′_𝑗^{1/𝑝𝐵′𝑗}𝑓^𝑝𝑑𝜇^1/𝑝≤𝑛₀𝑓_{𝐿′′′𝑝,𝜙,𝜌,𝑑}, (58) where in the second sum summation goes only over these 𝑗's for which 𝜇(𝐵^′_𝑗)>0. Taking the supremum over the relevant balls 𝐵 on the left hand side shows the required estimate and, hence, the inclusion. To prove the second claim, take 𝑓∈ℒ_{𝑝,𝜙′,𝜌′,𝑑′}(𝑋) and 𝐵∈ℬ_𝜌,𝑑, and consider 𝐵^′, 𝐵⊂𝐵^′, satisfying (56). Then inf_𝑧∈ℂ1𝜙(𝐵)1𝜇(𝐵)_𝐵𝑓−𝑧^𝑝𝑑𝜇^1/𝑝≲inf_𝑧∈ℂ1𝜙^′𝐵^′𝜇𝐵^{′1/𝑝𝐵′}𝑓−𝑧^𝑝𝑑𝜇^1/𝑝≤𝑓_{ℒ′′′𝑝,𝜙,𝜌,𝑑}. (59) Taking again the supremum over the relevant balls 𝐵 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 (𝑑,𝜌,𝜙) and (𝑑^′,𝜌^′,𝜙^′) switched, we have 𝐿_{𝑝,𝜙,𝜌,𝑑}(𝑋)=𝐿_{𝑝,𝜙′,𝜌′,𝑑′}(𝑋),ℒ_{𝑝,𝜙,𝜌,𝑑}(𝑋)=ℒ_{𝑝,𝜙′,𝜌′,𝑑′}(𝑋), (60) with equivalency of the corresponding norms.

Remark 11. In the case when, in the system (𝑋,𝑑,𝜇,𝜌,𝜙), only 𝜙 is replaced by 𝜙^′, it may happen that 𝜙^′(𝐵)≲𝜙(𝐵) uniformly in 𝐵∈ℬ_𝜌. 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 𝑛₀=1 and the covering of 𝐵 consisting of {𝐵}.

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

Example 12. Let (𝑋,𝑑,𝜇) be a space of homogeneous type. Assume that 𝑑^′ is a quasimetric equivalent with 𝑑 and 𝜌=𝜌^′≡∞. Given 𝛼∈ℝ, let 𝜙=𝜙^(𝑑)_𝑚,𝛼 and 𝜙^′=𝜙^(𝑑′)_𝑚,𝛼. Then, for 1≤𝑝<∞ and 𝛼≥−1/𝑝, we have 𝑀_{𝑝,𝜙,𝑑}𝑓(𝑥)≃𝑀_{𝑝,𝜙′,𝑑′}𝑓(𝑥),𝑀^#_{𝑝,𝜙,𝑑}𝑓(𝑥)≃𝑀^#_{𝑝,𝜙′,𝑑′}𝑓(𝑥), (61) uniformly in 𝑓 and 𝑥∈𝑋, and consequently 𝐿_{𝑝,𝜙,𝑑}(𝑋)=𝐿_{𝑝,𝜙′,𝑑′}(𝑋),ℒ_{𝑝,𝜙,𝑑}(𝑋)=ℒ_{𝑝,𝜙′,𝑑′}(𝑋), (62) with equivalency of the corresponding norms. Indeed, assuming that 𝑐⁻¹𝑑^′≤𝑑≤𝑐𝑑^′, for a 𝑐>1, we have (in what follows 𝐵^′ means a ball related to 𝑑^′) 𝐵(𝑥,𝑟)⊂𝐵^′(𝑥,𝑐𝑟)⊂𝐵𝑥,𝑐²𝑟 (63) and, hence, we take {𝐵^′(𝑥,𝑐𝑟)} as a covering of 𝐵(𝑥,𝑟)∈ℬ_𝑑. The doubling property of 𝜇 then implies 𝜇𝐵^′(𝑥,𝑐𝑟)≤𝜇𝐵𝑥,𝑐²𝑟≤𝐶𝜇(𝐵(𝑥,𝑟)) (64) and, therefore, (54) follows with 𝜙 and 𝜙^′ declared as above. The “dual” estimate follows analogously.

Remark 13. Sawano and Tanaka [3] defined and investigated Morrey spaces in the setting of (ℝ^𝑛,𝑑^(∞),𝜇), where 𝜇 is a Borel measure on ℝ^𝑛 finite on bounded sets (recall that every such measure is automatically a Radon measure) which may be nondoubling.
For a parameter 𝑘>1 and 1≤𝑝≤𝑞<∞, the Morrey space ℳ^𝑞_𝑝(𝑘,𝜇) (in the notation of [3] but with the roles of 𝑝 and 𝑞 switched) is the space of functions on ℝ^𝑛 satisfying sup_𝜇(𝑄)>0𝜇(𝑘𝑄)^{1/𝑞−1/𝑝}_𝑄𝑓^𝑝𝑑𝜇^1/𝑝<∞, (65) where the supremum is taken over all (closed) cubes with the property 𝜇(𝑄)>0. The space ℳ^𝑞_𝑝(𝑘,𝜇) coincides with our space 𝐿_𝑝,𝜙(ℝ^𝑛,𝑑^(∞),𝜇) (i.e., 𝜌≡∞), where 𝜙(𝑄)=𝜇(𝑄)^−1/𝑝𝜇(𝑘𝑄)^{1/𝑝−1/𝑞}. (66) It was proved in [3, Proposition 1.1] (the growth condition (23) did not intervene there) that ℳ^𝑞_𝑝(𝑘,𝜇) does not depend on the choice of 𝑘>1. This corresponds to the situation of 𝑑=𝑑^′=𝑑^(∞), 𝜌=𝜌^′≡∞, 𝜙 as above and 𝜙^′(𝑄)=𝜇(𝑄)^−1/𝑝𝜇(𝑘^′𝑄)^{1/𝑝−1/𝑞}, 𝑘^′>1, in Corollary 10 since, as it can be easily observed, for 1<𝑘<𝑘^′<∞ say, we have 𝜙≤𝜙^′ and, on the other hand, the assumption of Proposition 9 is satisfied due to simple geometrical properties of cubes in ℝ^𝑛 (see [3, p. 1536] for details).

Remark 14. Sawano [17] defined and investigated the so-called generalized Morrey spaces in the same setting of (ℝ^𝑛,𝑑^(∞),𝜇) (with closed cubes). For a parameter 𝑘>1 and a nondecreasing function 𝜓∶(0,∞)→(0,∞), the space ℒ^𝑝,𝜓(𝑘,𝜇) was defined as the space of functions on ℝ^𝑛 satisfying sup_𝜇(𝑄)>01𝜓𝜇(𝑘𝑄)_𝑄𝑓^𝑝𝑑𝜇^1/𝑝<∞. (67) The space ℒ^𝑝,𝜓(𝑘,𝜇) coincides with our space 𝐿_𝑝,𝜙(ℝ^𝑛,𝑑^(∞),𝜇), where 𝜙(𝑄)=𝜓𝜇(𝑘𝑄)𝜇(𝑄)^1/𝑝. (68) (Note that, for 𝜓(𝑡)=𝑡^{1−𝑝/𝑞}, 1≤𝑝≤𝑞<∞, we also have ℒ^𝑝,𝜓(𝑘,𝜇)=ℳ^𝑞_𝑝(𝑘,𝜇)). It was proved in [1, Proposition 1.2] (again the growth assumption did not intervene there) that ℒ^𝑝,𝜓(𝑘,𝜇) is independent of 𝑘>1 with equivalency of norms. This result may also be seen as a consequence of Corollary 10. Indeed, by taking 𝜙^′(𝑄)=(𝜓(𝜇(𝑘^′𝑄))/𝜇(𝑄))^1/𝑝 in this corollary, for 1<𝑘<𝑘^′<∞, we have 𝜙≤𝜙^′. On the other hand, the assumption of Proposition 9 is satisfied by the argument already mentioned in Remark 13 (geometrical properties of cubes in ℝ^𝑛).

Remark 15. Recently, Liu et al. [6] defined and investigated the local Morrey spaces in the setting of a locally doubling metric measure space (𝑋,𝑑,𝜇). The latter means that the measure 𝜇 possesses the doubling and the reverse doubling properties only on a class of admissible balls. This class, ℬ_𝑎, is defined with an aid of an admissible function 𝑚∶𝑋→(0,∞) and a parameter 𝑎∈(0,∞) and agrees with our class ℬ_𝜌 for the locality function 𝜌(𝑥)=𝑎𝑚(𝑥) (in [6], an assumption of geometrical nature is imposed on 𝑚). Then, the Morrey-type space ℳ_ℬ^𝑝,𝑞_𝑎(𝑋), 1≤𝑞≤𝑝<∞, was defined as the space of functions on 𝑋 satisfying sup_{𝐵∈ℬ𝑎}𝜇(𝐵)^1/𝑝1𝜇(𝐵)_𝐵𝑓^𝑝𝑑𝜇^1/𝑝<∞. (69) The investigations in the general setting were next specified in [6] to the important example of the Gauss measure space (ℝ^𝑛,𝑑⁽²⁾,𝛾_𝑛), where 𝛾_𝑛 denotes the Gauss measure 𝑑𝛾_𝑛(𝑥)=𝜋^−𝑛/2exp(−‖𝑥‖²₂)𝑑𝑥. The importance of this example lies in the fact that the measure space (ℝ^𝑛,𝛾_𝑛) is the natural environment for the Ornstein-Uhlenbeck operator −(1/2)Δ+𝑥⋅∇. In the context of (ℝ^𝑛,𝑑⁽²⁾,𝛾_𝑛), the Campanato-type space ℰ_ℬ^𝑝,𝑞_𝑎(𝛾_𝑛) was also defined as the space of functions on 𝑋 satisfying 𝑓_{𝐿1(𝛾𝑛)}+sup_{𝐵∈ℬ𝑎}𝜇(𝐵)^1/𝑝1𝜇(𝐵)_𝐵𝑓−𝑓_𝐵^𝑝𝑑𝜇^1/𝑝<∞ (70) (the additional summand ‖𝑓‖_{𝐿1(𝛾𝑛)} 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 𝜇 in ℝ² was provided (𝜇 being absolutely continuous with respect to Lebesgue measure) such that BMO⁽²⁾(ℝ²,𝜇) (=ℒ_1,𝟏(ℝ²,𝑑⁽²⁾,𝜇)) and BMO^(∞)(ℝ²,𝜇) (=ℒ_1,𝟏(ℝ²,𝑑^(∞),𝜇)) 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 𝑋=ℕ, 𝑑 to be the 0−1 metric on ℕ, and 𝜇 to be the measure on ℕ such that 𝜇({𝑛})=𝑎_𝑛, where 𝑎_𝑛>0 and ∑𝑎_𝑛=1 (so that 𝜇(ℕ)=1). Note that 𝜇 is nondoubling; it is not even locally doubling and if 𝐵 is a ball, then either 𝐵={𝑛}, for some 𝑛∈ℕ, or 𝐵=ℕ, and, hence, 0<𝜇(𝐵)<∞, for every ball 𝐵. Then 𝐿^𝑝_loc(ℕ)=ℓ^𝑝(ℕ,𝜇), for 1≤𝑝<∞, and 𝜌₀≡0. For simplicity, we now treat the case 𝜙=𝟏 only.
Consider first 𝜌≡∞. Then, for any 𝑓=(𝑓(𝑗))_𝑗∈ℕ∈ℓ^𝑝(ℕ,𝜇), 𝑀^#_𝑝,𝟏𝑓 and 𝑀^#,𝑐_𝑝,𝟏𝑓 are constant functions: 𝑀^#_𝑝,𝟏𝑓(𝑛)=𝑀^#,𝑐_𝑝,𝟏𝑓(𝑛)=_𝑗𝑓𝑗−𝑓_ℕ^𝑝𝑎_𝑗^1/𝑝,𝑛∈ℕ, (71) where ⟨𝑓⟩_ℕ=∑_𝑗𝑓(𝑗)𝑎_𝑗. Hence, ℒ_𝑝,𝟏(ℕ)=ℓ^𝑝(ℕ,𝜇), ‖𝑓‖_ℒ𝑝,𝟏≃‖𝑓−⟨𝑓⟩_ℕ‖_{ℓ𝑝(ℕ,𝜇)}, 𝑊₀=𝒞, and ℒ_𝑝,𝟏(ℕ)/𝒞 is identified with ℓ^𝑝(ℕ,𝜇), where 𝒞=𝒞(ℕ) denotes the space of constant sequences. Similarly, for any 𝑓∈ℓ^𝑝(ℕ,𝜇), 𝑀_𝑝,𝟏𝑓(𝑛)=𝑀^𝑐_𝑝,𝟏𝑓(𝑛)=max𝑓(𝑛),_𝑗𝑓𝑗^𝑝𝑎_𝑗^1/𝑝,𝑛∈ℕ, (72) and, hence, 𝐿_𝑝,𝟏(ℕ)=ℓ^∞(ℕ)∩ℓ^𝑝(ℕ,𝜇) with identity of norms.
Consider now the case of 𝜌≡1. Then ℬ_𝜌 consists of balls of the form 𝐵={𝑛}, 𝑛∈ℕ, 𝐿^𝑝_loc,𝜌(ℕ)=𝑠(ℕ), where 𝑠(ℕ) denotes the space of all sequences on ℕ, and 𝑀^#_{𝑝,𝟏,𝜌}𝑓(𝑛)=0, for any 𝑓∈𝑠(ℕ) and 𝑛∈ℕ, and, hence, ℒ_{𝑝,𝟏,𝜌}(ℕ)=𝑠(ℕ) and ‖𝑓‖_{ℒ𝑝,𝟏,𝜌}=0, for 𝑓∈𝑠(ℕ). In addition, every 𝜌-component is of the form {𝑛}, 𝑛∈ℕ, and, hence, 𝑊_0,𝜌=𝑠(ℕ). Similarly, for any 𝑓∈𝑠(ℕ), we have 𝑀_{𝑝,𝟏,𝜌}𝑓(𝑛)=|𝑓(𝑛)|, 𝑛∈ℕ, and, hence, 𝐿_{𝑝,𝟏,𝜌}(ℕ)=ℓ^∞(ℕ) and ‖⋅‖_{L𝑝,𝟏,𝜌}=‖⋅‖_ℓ^∞.

3.1. Morrey and Campanato Spaces on Open Proper Subsets of ℝ^𝑛

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

As it was mentioned in [4, p. 259], indecisions accompany choosing a suitable definition of BMO(Ω) for a general open proper subset Ω of ℝ^𝑛 equipped with Lebesgue measure. The “right” way seems to be the following: BMO(Ω) consists of those functions 𝑓∈𝐿¹_loc(Ω) such that 𝑓_BMO(Ω)∶=sup_𝐵⊂Ω1𝐵^𝐵𝑓−𝑓_𝐵<∞, (73) where the supremum is taken over all closed balls (or closed cubes, if one prefers; then the character 𝐵 should be replaced by 𝑄), entirely contained in Ω; see [20]. Throughout this section |𝐴| stands for the Lebesgue measure of 𝐴, a measurable subset of Ω. Note that such a definition has a local flavor: the locality function entering the scene is 𝜌_Ω(𝑥)=𝑑(𝑥,𝜕Ω),𝑥∈Ω, (74) where the distance from 𝑥∈Ω to 𝜕Ω is given by 𝑑(𝑥,𝜕Ω)∶=inf𝑑𝑥,𝑦∶𝑦∈𝜕Ω, (75) and 𝑑=𝑑⁽²⁾ or 𝑑=𝑑^(∞).

Similar indecisions accompany the process of choosing a suitable definition of Morrey and Campanato spaces for a general open proper subset Ω⊂ℝ^𝑛. The spaces 𝐿^𝑝,𝜆(Ω), 1≤𝑝<∞, and 0≤𝜆≤𝑛, determined by 𝑓_{𝐿𝑝,𝜆(Ω)}∶=sup_𝑥∈Ωsup_0<𝑟<𝑟01𝑟^𝜆_{𝐵(𝑥,𝑟)∩Ω}𝑓𝑦^𝑝𝑑𝑦^1/𝑝<∞, (76) where 𝑟₀=diamΩ (see also [21]), were originally introduced by Morrey [22] (with a restriction to open and bounded subsets). For a definition of ℒ^𝑝,𝜆(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^𝑛 is by using our general approach with the locality function 𝜌_Ω given above. To fix the attention let us assume, for a moment, that 𝑑=𝑑^(∞). Thus, for a given function 𝜙∶ℬ_𝜌(Ω)→(0,∞), we define 𝐿_𝑝,𝜙(Ω)∶=𝐿_{𝑝,𝜙,𝜌Ω}(Ω) and ℒ_𝑝,𝜙(Ω)∶=ℒ_{𝑝,𝜙,𝜌Ω}(Ω). Explicitely, this means that, for 𝜌∶=𝜌_Ω, 𝐿_𝑝,𝜙(Ω)=𝑓∶𝑓_{𝐿𝑝,𝜙(Ω)}∶=sup_{𝐵∈ℬ𝜌(Ω)}1𝜙𝐵1𝐵^𝐵𝑓𝑦^𝑝𝑑𝑦^1/𝑝,ℒ<∞_𝑝,𝜙(Ω)=𝑓∶𝑓_{ℒ𝑝,𝜙(Ω)}∶=sup_{𝐵∈ℬ𝜌(Ω)}inf_𝑧∈ℂ1𝜙𝐵1𝐵^𝐵𝑓𝑦−𝑧^𝑝𝑑𝑦^1/𝑝<∞; (77) by the definition of 𝜌, ℬ_𝜌(Ω) is the family of all closed balls entirely contained in Ω.

Given a parameter 0<𝑘≤1, we now define the locality function 𝜌_𝑘=𝜌_𝑘,Ω as 𝜌_𝑘(𝑥)=𝑘⋅𝑑(𝑥,𝜕Ω),𝑥∈Ω, (78) so that 𝜌₁=𝜌_Ω. Then, for a function 𝜙 as above, we define 𝐿_{𝑝,𝜙,𝑘}(Ω)∶=𝐿_{𝑝,𝜙,𝜌𝑘}(Ω) and ℒ_{𝑝,𝜙,𝑘}(Ω)∶=ℒ_{𝑝,𝜙,𝜌𝑘}(Ω). The structure of the above definition of 𝐿_{𝑝,𝜙,𝑘}(Ω) and ℒ_{𝑝,𝜙,𝑘}(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, 𝑓∈𝐿_{𝑝,𝜙,𝑘}(Ω), Ω=⋃_𝑗∈𝐽Ω_𝑗 (𝐽 is finite or countable), where each Ω_𝑗 is a connected component of Ω, and 𝑓_𝑗 denotes the restriction of 𝑓 to Ω_𝑗, then 𝑓_{𝐿𝑝,𝜙,𝑘(Ω)}=sup_𝑥∈Ωsup_{𝑥∈𝐵∈ℬ𝑘(Ω)}1𝜙𝐵1𝐵^𝐵𝑓^𝑝1/𝑝=sup_𝑗∈𝐽sup_{𝑥∈Ω𝑗}sup_{𝑥∈𝐵∈ℬ𝑘(Ω𝑗)}1𝜙𝐵1𝐵^𝐵𝑓^𝑝1/𝑝=sup_𝑗∈𝐽𝑓_{𝑗𝐿𝑗𝑝,𝜙,𝑘(Ω𝑗)}. (79) 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 𝑑=𝑑⁽²⁾. To distinguish between the two cases corresponding to the choice of 𝑑⁽²⁾ or 𝑑^(∞), when necessary, we shall write 𝜌_𝑘⁽²⁾ and 𝜌_𝑘^(∞), 𝐿^ℬ_{𝑝,𝜙,𝑘}(Ω) and 𝐿^𝒬_{𝑝,𝜙,𝑘}(Ω), and so forth. Also, the family of balls related to 𝜌_𝑘⁽²⁾ will be denoted by ℬ^Ω_𝑘, while the family of cubes related to 𝜌_𝑘^(∞) will be denoted by 𝒬^Ω_𝑘.

In what follows, rather than considering a general 𝜙, we limit ourselves to the specific case of 𝜙=𝜙_𝑚,𝛼. Clearly, 𝜙_𝑚,𝛼 satisfies (32) and, hence, distinguishing between open or closed balls (or open or closed cubes) is not necessary. We write 𝐿^𝒬_{𝑝,𝛼,𝑘}(Ω) in place of 𝐿^𝒬_{𝑝,𝜙𝑚,𝛼,𝑘}(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≤𝑝<∞ and 𝛼≥−1/𝑝 be given. The spaces 𝐿^𝒬_{𝑝,𝛼,𝑘}(Ω) are independent of the choice of the scale parameter 𝑘∈(0,1) with equivalence of the corresponding norms. The analogous statement is valid for the spaces ℒ^𝒬_{𝑝,𝛼,𝑘}(Ω).

Proof. Let 0<𝑘<𝑚<1. We shall prove the inequalities 𝑓_{𝐿𝒬𝑝,𝛼,𝑚(Ω)}≤𝐶𝑓_{𝐿𝒬𝑝,𝛼,𝑘(Ω)},𝑓∈𝐿^𝒬_{𝑝,𝛼,𝑘}(Ω),𝑓_{ℒ𝒬𝑝,𝛼,𝑚(Ω)}≤𝐶𝑓_{ℒ𝒬𝑝,𝛼,𝑘(Ω)},𝑓∈ℒ^𝒬_{𝑝,𝛼,𝑘}(Ω), (80)(81) which give the inclusions 𝐿^𝒬_{𝑝,𝛼,𝑘}(Ω)⊂𝐿^𝒬_{𝑝,𝛼,𝑚}(Ω) and ℒ^𝒬_{𝑝,𝛼,𝑘}(Ω)⊂ℒ^𝒬_{𝑝,𝛼,𝑚}(Ω). The inequalities opposite to (80) and (81) (with 𝐶=1) are obvious, and thus the opposite inclusions follow.
Consider first the case of (80). There exists 𝑁=𝑁(𝑚,𝑘,𝑛) such that bisecting any cube 𝑄∈𝒬^Ω_𝑚  𝑁 times results in obtaining a family {𝑄_𝑗} of 2^𝑛𝑁 congruent subcubes of 𝑄 each of them in 𝒬^Ω_𝑘. Thus 1|𝑄|^𝛼1|𝑄|_𝑄𝑓^𝑝1/𝑝≤1|𝑄|^𝛼_𝑗1|𝑄|_𝑄𝑗𝑓^𝑝1/𝑝≤_𝑗1𝑄_𝑗^𝛼1𝑄_𝑗^𝑄_𝑗𝑓^𝑝1/𝑝≤2^𝑛𝑁𝑓_{𝐿𝒬𝑝,𝛼,𝑘(Ω)} (82) and the result follows.
Considering (81), we shall apply the procedure similar to that used in the proof of [24, Theorem 3.5]. Take 𝑓∈ℒ^𝒬_{𝑝,𝛼,𝑘}(Ω) and 𝑄∈𝒬^Ω_𝑚⧵𝒬^Ω_𝑘. Then inf_𝑧∈ℂ1|𝑄|^𝛼1|𝑄|_𝑄𝑓−𝑧^𝑝1/𝑝=1|𝑄|^𝛼+1/𝑝𝑓_{𝐿𝑝(𝑄)/ℂ}, (83) where 𝐿^𝑝(𝑄)/ℂ is the quotient space and ‖⋅‖_𝐿^𝑝_(𝑄)/ℂ is the quotient norm. Since the dual to 𝐿^𝑝(𝑄)/ℂ is identified with 𝐿^𝑝′₀(𝑄), where 𝑝^′ is the exponent conjugate to 𝑝, 1/𝑝+1/𝑝^′=1, and 𝐿^𝑝′₀(𝑄) denotes the subspace of 𝐿^𝑝(𝑄) consisting of functions 𝑓 with ∫_𝑄𝑓=0, therefore 𝑓_{𝐿𝑝(𝑄)/ℂ}=sup_𝑄𝑓𝜓∶𝜓∈𝐿^𝑝′₀(𝑄),𝜓_𝑝′≤1. (84) According to [24, Lemma 3.1], there exist constants 𝐶=𝐶(𝑘,𝑚,𝑝^′,𝑛) and 𝑁=𝑁(𝑘,𝑚,𝑛) such that, for every 𝑄∈𝒬^Ω_𝑚 and every function 𝜓∈𝐿^𝑝′₀(𝑄), there exist subcubes {𝑄_𝑗∈𝒬^Ω_𝑘∶1≤𝑗≤𝑁} of 𝑄 and functions {𝜓_𝑗∈𝐿^𝑝′₀(𝑄)∶1≤𝑗≤𝑁} such that supp(𝜓_𝑗)⊂𝑄_𝑗 and ‖𝜓_𝑗‖_𝑝′≤𝐶‖𝜓‖_𝑝′, for 𝑗=1,…,𝑁, and 𝜓=∑^𝑁_𝑗=1𝜓_𝑗. Take 𝜓∈𝐿^𝑝′₀(𝑄) satisfying ‖𝜓‖_𝑝′≤1, where 𝑄 is the cube chosen earlier, and select subcubes {𝑄_𝑗}^𝑁₁ and functions {𝜓_𝑗}^𝑁₁ with properties as above. Then 1|𝑄|^{𝛼+1/𝑝𝑄}=1𝑓𝜓|𝑄|^{𝛼+1/𝑝𝑄}𝑓^𝑁_𝑗=1𝜓_𝑗≤1|𝑄|^{𝑁𝛼+1/𝑝}_𝑗=1^𝑄_𝑗𝑓𝜓_𝑗=1|𝑄|^{𝑁𝛼+1/𝑝}_𝑗=1^𝑄_𝑗𝑓−𝑓_𝑄𝑗𝜓_𝑗≤𝐶^𝑁_𝑗=11𝑄_𝑗^{𝛼+1/𝑝𝑄𝑗}𝑓−𝑓_𝑄𝑗^𝑝1/𝑝≤𝐶𝑁𝑓_{ℒ𝒬𝑝,𝛼,𝑘(Ω)}. (85) Hence, 1|𝑄|^𝛼+1/𝑝𝑓_{𝐿1(𝑄)/ℂ}≤𝐶𝑁𝑓_{ℒ𝒬𝑝,𝛼,𝑘(Ω)}, (86) and, consequently, ‖𝑓‖_ℒ^𝒬_{𝑝,𝛼,𝑚(Ω)}≤𝐶𝑁‖𝑓‖_ℒ^𝒬_{𝑝,𝛼,𝑘(Ω)}.

Proposition 19. Let 1≤𝑝<∞ and 𝛼≥−1/𝑝 be given. The spaces 𝐿^ℬ_{𝑝,𝛼,𝑘}(Ω) are independent of the choice of the scale parameter 𝑘∈(0,1) with equivalence of the corresponding norms. The analogous statement is valid for the spaces ℒ^ℬ_{𝑝,𝛼,𝑘}(Ω).

Proof. The present proof mimics the one of Proposition 18, since essentially it suffices to replace the character 𝒬 by ℬ and, assuming 0<𝑘<𝑚<1 are given, to use the following geometrical properties of Euclidean balls. The first one says that there exists 𝑁=𝑁(𝑘,𝑚,𝑛) such that every ball 𝐵∈ℬ^Ω_𝑚 may be covered by a family {𝐵_𝑗}^𝑁₁ of balls each of them in ℬ^Ω_𝑘 and with radii smaller than that of 𝐵. The second one (more sophisticated) is contained in [24, Lemma 3.8] and says that there exist constants 𝐶=𝐶(𝑘,𝑚,𝑝^′,𝑛) and 𝑁=𝑁(𝑘,𝑚,𝑛) such that, for every 𝐵∈ℬ^Ω_𝑚 and every function 𝜓∈𝐿^𝑝′₀(𝐵), there exist balls {𝐵_𝑗∈ℬ^Ω_𝑘∶1≤𝑗≤𝑁} with radii smaller than that of 𝐵 and functions {𝜓_𝑗∈𝐿^𝑝′₀(𝐵)∶1≤𝑗≤𝑁} such that supp(𝜓_𝑗)⊂𝐵_𝑗 and ‖𝜓_𝑗‖_𝑝′≤𝐶‖𝜓‖_𝑝′, for 𝑗=1,…,𝑁, and 𝜓=∑^𝑁_𝑗=1𝜓_𝑗. (The fact that radii of 𝐵_𝑗 are smaller than the radius of 𝐵 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 0≤𝑅^′<𝑅 and 𝜀>1−𝑅^′/𝑅, there exists 𝑁^′=𝑁^′(𝑅^′/𝑅,𝜀,𝑛) such that, for any sphere 𝑆(𝑥₀,𝑅)={𝑥∈ℝ^𝑛∶‖𝑥−𝑥₀‖₂=𝑅}, one can find points 𝑥₁,…,𝑥_𝑁′ on that sphere such that 𝐵𝑥₀,2𝑅−𝑅^′⧵𝐵𝑥₀,𝑅^′⊂∪^𝑁′_𝑗=1𝐵𝑥_𝑗,𝜀𝑅 (87) (if 𝑅^′=0, then we set 𝐵(𝑥₀,0)=0/).
Now, take any 𝐵=𝐵(𝑥₀,𝑟)∈ℬ^Ω_𝑚. In fact, we shall prove the aforementioned property for the “maximal” ball 𝐵(𝑥₀,𝑟₀) with 𝑟₀∶=𝑚𝑑⁽²⁾(𝑥₀,𝜕Ω). Let 𝑅_𝑗=(𝑗/𝑠)𝑟₀, 𝑗∈ℕ, where 𝑠∈ℕ is large enough (to be determined in the last step of the argument). Using the above geometrical fact, on each sphere 𝑆_𝑗=𝑆(𝑥₀,𝑅_𝑗), 𝑗=1,2,…,𝑠−1, we choose finite number of points such that the balls centered at these points and with radii equal (2/𝑠)𝑟₀ covering the annulus 𝐵(𝑥₀,𝑅_𝑗+1)⧵𝐵(𝑥₀,𝑅_𝑗−1). More precisely, given 𝑗=1,2,…,𝑠−1, we apply the geometrical fact with 𝑅=𝑅_𝑗=(𝑗/𝑠)𝑟₀ and 𝑅^′=𝑅_𝑗−1=((𝑗−1)/𝑠)𝑟₀ (so that 𝑅^′/𝑅=((𝑗−1)/𝑗)) and 𝜀=2/𝑗 (so that 𝜀>1−𝑅^′/𝑅). It is clear that the union of all chosen balls covers 𝐵(𝑥₀,𝑟₀) and there is 𝑁=∑^𝑠−1_𝑗=1𝑁^′((𝑗−1)/𝑗,2/𝑗,𝑛) of them. To verify that each of these balls is in ℬ^Ω_𝑘, take 𝐵(𝑥^∗₀,𝑟^∗₀) with center lying on the sphere 𝑆(𝑥₀,𝑅_𝑠−1) (this is the worst case). Since, for 𝜔∈Ω^𝑐, we have 𝑑⁽²⁾(𝜔,𝑥₀)≤𝑑⁽²⁾(𝜔,𝑥^∗₀)+𝑑⁽²⁾(𝑥^∗₀,𝑥₀) and 𝑑⁽²⁾(𝑥^∗₀,𝑥₀)=((𝑠−1)/𝑠)𝑟₀=((𝑠−1)/𝑠)𝑚𝑑⁽²⁾(𝑥₀,𝜕Ω), it is clear that 𝑑⁽²⁾𝑥^∗₀,𝜕Ω≥𝑑⁽²⁾𝑥₀,𝜕Ω1−𝑚𝑠−1𝑠. (88) Hence, if 𝑠 is chosen to be the least positive integer with the property (2/𝑠)(1/(1−𝑚))<𝑘, then 𝑟^∗₀=2𝑠𝑟₀=2𝑠𝑚𝑑⁽²⁾𝑥₀<2,𝜕Ω𝑠1𝑑1−𝑚⁽²⁾𝑥^∗₀,𝜕Ω≤𝑘𝑑⁽²⁾𝑥^∗₀,𝜕Ω (89) and the required property follows (note that 𝑠 depends on 𝑘 and 𝑚, and, hence, 𝑁 depends on 𝑘, 𝑚, and 𝑛, as claimed).

The results of Propositions 18 and 19 allow us to define 𝐿^𝒬_𝑝,𝛼(Ω)∶=𝐿^𝒬_{𝑝,𝛼,1/2}(Ω) and ‖⋅‖_𝐿^𝒬_{𝑝,𝛼(Ω)}∶=‖⋅‖_𝐿^𝒬_{𝑝,𝛼,1/2(Ω)} (the choice of 𝑘=1/2 being “random”) and similarly for 𝐿^ℬ_𝑝,𝛼(Ω), ℒ^𝒬_𝑝,𝛼(Ω), ℒ^ℬ_𝑝,𝛼(Ω), and the corresponding norms. The following theorem partially contains [24, Theorem 4.2] as a special case.

Theorem 20. Let 1≤𝑝<∞ and 𝛼≥−1/𝑝 be given. Then, we have 𝐿^𝒬_𝑝,𝛼(Ω)=𝐿^ℬ_𝑝,𝛼(Ω),ℒ^𝒬_𝑝,𝛼(Ω)=ℒ^ℬ_𝑝,𝛼(Ω), (90) 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 𝑄 or a ball 𝐵, by 𝐵_𝑄 or 𝑄_𝐵, we will denote the ball circumscribed on 𝑄 or the cube circumscribed on 𝐵, respectively. By the inequality ‖⋅‖_∞≤‖⋅‖₂≤√𝑛‖⋅‖_∞, it is clear that, for 0<𝑘≤1/√𝑛, 𝐵_𝑄∈ℬ^Ω_√𝑛𝑘, if 𝑄∈𝒬^Ω_𝑘, and 𝑄_𝐵∈𝒬^Ω_√𝑛𝑘, if 𝐵∈ℬ^Ω_𝑘. Moreover, |𝐵_𝑄|=𝑐₁|𝑄| and |𝑄_𝐵|=𝑐₂|𝐵|, where 𝑐₁ and 𝑐₂, depend on the dimension 𝑛 only.
Fix 0<𝑘<1/√𝑛 and take 𝑓∈ℒ^𝒬_{√𝑝,𝛼,𝑛𝑘}(Ω). For any 𝐵 and 𝑄_𝐵 defined above, inf_𝑧∈ℂ1|𝐵|^𝛼1|𝐵|_𝐵𝑓−𝑧^𝑝1/𝑝≤𝑐^′₂inf_𝑧∈ℂ1𝑄_𝐵^𝛼1𝑄_𝐵𝑄𝐵𝑓−𝑧^𝑝1/𝑝. (91) Consequently, ‖𝑓‖_ℒ^ℬ_{𝑝,𝛼,𝑘(Ω)}≤𝑐^′₂‖𝑓‖_ℒ^𝒬_{√𝑝,𝛼,𝑑𝑘(Ω)} which also shows that ℒ^𝒬_{√𝑝,𝛼,𝑛𝑘}(Ω)⊂ℒ^ℬ_{𝑝,𝛼,𝑘}(Ω). The results of Propositions 18 and 19 now give ℒ^𝒬_𝑝,𝛼(Ω)⊂ℒ^ℬ_𝑝,𝛼(Ω) and ‖⋅‖_ℒ^ℬ_{𝑝,𝛼(Ω)}≤𝐶‖⋅‖_ℒ^𝒬_{𝑝,𝛼(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^𝑛 may be generalized to open proper subsets of a general quasimetric space 𝑋. 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^𝑛 with the locality function Ω∋𝑥↦dist(𝑥,ℝ^𝑛⧵Ω).

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 (𝑋,𝑑,𝜇,𝜌) 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 𝜏_𝐾=𝐾(2𝐾+1), where 𝐾 is the constant from the quasitriangle inequality (if 𝑑 is a metric, then 𝐾=1 and 𝜏_𝐾=3). Observe that the assumptions we impose in Proposition 21 are satisfied, for instance, when 𝜇(𝐵)≃𝑟(𝐵)^𝑛, for some 𝑛>0, uniformly in 𝐵∈ℬ_𝜌, 𝜑(𝑟)=𝑟^𝛼, −𝑛/𝑝≤𝛼≤0, and 𝜇 satisfies the 𝜌-local 𝜏_𝐾-condition. Recall also that when it comes to the boundedness of 𝑀_1,𝟏,𝜌 on 𝐿^𝑝(𝑋,𝜇), we have the conclusion of Proposition 4 to our disposal.

Proposition 21. Let 1≤𝑝<∞ and 𝜑∶(0,∞)→(0,∞) be a nonincreasing function such that 𝜙(𝐵)=𝜑(𝑟(𝐵)) satisfies 𝜙𝜏_𝐾𝐵𝜇𝜏_𝐾𝐵^1/𝑝≲𝜙(𝐵)𝜇(𝐵)^1/𝑝, (92) uniformly in 𝐵∈ℬ_𝜌. If 𝑀_1,𝟏,𝜌 is bounded on 𝐿^𝑝(𝑋,𝜇), then, it is also bounded from 𝐿_{𝑝,𝜙,𝜏𝐾𝜌}(𝑋) to 𝐿_{𝑝,𝜙,𝜌}(𝑋).

Proof. For the notational convention, let 𝑀_𝜌∶=𝑀_1,𝟏,𝜌; that is, 𝑀_𝜌𝑓(𝑥)=sup_{𝑥∈𝐵∈ℬ𝜌}1𝜇(𝐵)_𝐵𝑓𝑑𝜇. (93)
Take 𝑓∈𝐿_{𝑝,𝜙,𝜏𝐾𝜌}(𝑋), fix a ball 𝐵∈ℬ_𝜌, and consider the splitting 𝑓=𝑓₁+𝑓₂ adjusted to 𝐵 in the sense that 𝑓₁=𝑓𝜒_𝜏𝐾𝐵. Then, for any 𝑥∈𝐵, 𝑀_𝜌𝑓₂(𝑥)≤sup_𝐵′1𝜇𝐵^′𝐵′𝑓𝑑𝜇, (94) where the supremum is taken over all balls 𝐵^′∈ℬ_𝜌 such that 𝑥∈𝐵^′ and 𝑟(𝐵^′)>𝑟(𝐵). If 𝑥∈𝐵 and 𝐵^′ is one of such balls, then the fact that 𝜑 is nonincreasing gives 1𝜙(𝐵)1𝜇𝐵^′𝐵′𝑓1𝑑𝜇≤𝜑(𝑟(𝐵1))𝜇𝐵^′𝐵′𝑓^𝑝𝑑𝜇^1/𝑝≤1𝜙𝐵^′1𝜇𝐵^′𝐵′𝑓^𝑝𝑑𝜇^1/𝑝≤𝑓_{𝐿𝑝,𝜙,𝜌}. (95) Consequently, 1𝜙(𝐵)𝑀_𝜌𝑓₂(𝑥)≤𝑓_{𝐿𝑝,𝜙,𝜌},𝑥∈𝐵. (96) This estimate, subadditivity of 𝑀_𝜌, and the assumption that 𝑀_𝜌 is bounded on 𝐿^𝑝(𝑋,𝜇) give 1𝜙(𝐵)1𝜇(𝐵)_𝐵𝑀_𝜌𝑓^𝑝𝑑𝜇^1/𝑝≤_𝑖=1,21𝜙(𝐵)1𝜇(𝐵)_𝐵𝑀_𝜌𝑓_𝑖^𝑝𝑑𝜇^1/𝑝≤𝑐_𝑝1𝜙(𝐵)𝜇(𝐵)^1/𝑝𝜏_𝐾^𝐵𝑓^𝑝𝑑𝜇^1/𝑝+𝑓_{𝐿𝑝,𝜙,𝜌}≤𝑐^′_𝑝1𝜙𝜏_𝐾𝐵𝜇𝜏_𝐾𝐵^1/𝑝𝜏_𝐾^𝐵𝑓^𝑝𝑑𝜇^1/𝑝+𝑓_{𝐿𝐾𝜌𝑝,𝜙,𝜏}≤𝑐_𝑝^′′𝑓_{𝐿𝐾𝜌𝑝,𝜙,𝜏}. (97) This shows the required estimate ‖𝑀_𝜌𝑓‖_{𝐿𝑝,𝜙,𝜌}≲‖𝑓‖_{𝐿𝐾𝜌𝑝,𝜙,𝜏}.

Remark 22. Consider the global case, 𝜌≡∞. 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 𝐿_𝑝,𝜆(ℝ^𝑛), 1<𝑝<∞ and 0<𝜆<𝑛 (see Remark 7), take 𝜑(𝑟)=𝑟^{(𝜆−𝑛)/𝑝} which is decreasing. The assumption (92), with 𝜇 being Lebesgue measure and the metric being 𝑑^(∞) is obviously satisfied (clearly the usual Hardy-Littlewood operator is also bounded on 𝐿^𝑝(ℝ^𝑛)).
Similarly, if (𝑋,𝑑,𝜇) is a space of homogeneous type and 𝜑(𝑟)=𝑟^𝛼, 𝛼≤0, so that 𝜙=𝜙_𝑟,𝛼, then condition (92) is satisfied and, hence, 𝑀_𝜇=𝑀_1,𝟏, the Hardy-Littlewood operator associated to 𝜇, maps boundedly 𝐿_{𝑝,𝜙𝑟,𝛼} 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 (𝑋,𝑑,𝜇) with 𝜇 satisfying the upper growth condition (23) with 𝜏=𝑛. For any appropriate function 𝑓 and 0<𝛽<𝑛, we define the fractional integral operator 𝐼_𝛽 by letting 𝐼_𝛽𝑓(𝑥)=_𝑋𝑓𝑦𝑑𝑥,𝑦^𝑛−𝛽𝑑𝜇𝑦,𝑥∈𝑋. (98) For functions 𝜑,𝜓∶(0,∞)→(0,∞), we shall consider the following conditions (compare them with the assumptions imposed in [32]): 𝜑(𝑟)𝑟isnonincreasing;_∞𝑟𝑡^𝛽−1𝜑(𝑡)𝑑𝑡≲𝜓(𝑟)𝑟>0. (99)

Proposition 23. Let 1≤𝑝,𝑞<∞ and 𝜑,𝜓∶(0,∞)→(0,∞) be functions satisfying (99). In addition, assume that 𝜙(𝐵)=𝜑(𝑟(𝐵)) and Ψ(𝐵)=𝜓(𝑟(𝐵)) satisfy 𝜙𝜏_𝐾𝐵𝜇𝜏_𝐾𝐵^1/𝑝≲Ψ(𝐵)𝜇(𝐵)^1/𝑞, (100) uniformly in 𝐵∈ℬ. If 𝐼_𝛽 is bounded from 𝐿^𝑝(𝑋,𝜇) to 𝐿^𝑞(𝑋,𝜇), then, for any 𝜌, it is also bounded from 𝐿_𝑝,𝜙(𝑋) to 𝐿_𝑞,Ψ,𝜌(𝑋).

Proof. Since ‖⋅‖_{𝐿𝑞,Ψ,𝜌}≤‖⋅‖_𝐿𝑞,Ψ, it is sufficient to consider the case 𝜌≡∞. The estimate to be proved is 1Ψ(𝐵)1𝜇(𝐵)_𝐵𝐼_𝛽𝑓^𝑞𝑑𝜇^1/𝑞≲𝑓_{𝐿𝑝,𝜙}, (101) uniformly in 𝐵∈ℬ and 𝑓∈𝐿_𝑝,𝜙(𝑋). Take 𝐵=𝐵(𝑎,𝑟) and 𝑓∈𝐿_𝑝,𝜙(𝑋) and consider the decomposition 𝑓=𝑓₁+𝑓₂, 𝑓₁=𝑓𝜒_𝜏𝐾𝐵, and 𝜏_𝐾∶=2𝐾. It suffices to verify (101) with 𝑓 replaced by 𝑓_𝑖, 𝑖=1,2, on the left hand side of this estimate. For 𝑓₁, using the assumption on the 𝐿^𝑝−𝐿^𝑞 boundedness of 𝐼_𝛽 and (100), we write 1Ψ(𝐵)1𝜇(𝐵)_𝐵𝐼_𝛽𝑓₁^𝑞𝑑𝜇^1/𝑞≤1Ψ(𝐵)𝜇(𝐵)^1/𝑞𝐼_𝛽𝑓_{1𝐿𝑞(𝜇)}≲1Ψ(𝐵)𝜇(𝐵)^1/𝑞𝑓_{1𝐿𝑝(𝜇)}≲1𝜙𝜏_𝐾𝐵1𝜇𝜏_𝐾𝐵_𝜏𝐾𝐵𝑓^𝑝𝑑𝜇^1/𝑝≲𝑓_{𝐿𝑝,𝜙}. (102) For 𝑓₂, note that, for any 𝑥∈𝐵, we have 𝐵(𝑎,𝜏_𝐾𝑟)^𝑐⊂𝐵(𝑥,𝑟)^𝑐 (𝐴^𝑐 denotes the completion of 𝐴 in 𝑋), and, therefore, 𝐼_𝛽𝑓₂(𝑥)≤_{(𝜏𝐾𝐵)𝑐}𝑓𝑦𝑑𝑥,𝑦^𝑛−𝛽𝑦≤𝑑𝜇_{𝐵(𝑥,𝑟)𝑐}𝑓𝑦𝑑𝑥,𝑦^𝑛−𝛽𝑦=𝑑𝜇_{∞𝑗=0{𝑦∶2𝑗𝑟≤𝑑(𝑥,𝑦)<2𝑗+1𝑟}}𝑓𝑦𝑑𝑥,𝑦^𝑛−𝛽𝑦≤𝑑𝜇_∞𝑗=02^𝑗𝑟^𝛽−𝑛_{{𝑦∶2𝑗𝑟≤𝑑(𝑥,𝑦)<2𝑗+1𝑟}}𝑓𝑦𝑦≲𝑑𝜇_∞𝑗=02^𝑗𝑟^𝛽1𝜇𝐵𝑥,2^𝑗+1𝑟_𝐵(𝑥,2^𝑗+1_𝑟)𝑓𝑦𝑦≲𝑑𝜇_∞𝑗=02^𝑗𝑟^𝛽𝜙𝐵𝑥,2^𝑗+1𝑟𝑓_{𝐿𝑝,𝜙}≲^∞_𝑟𝑡^𝛽−1𝜑(𝑡)𝑑𝑡𝑓_{𝐿𝑝,𝜙}≲𝜓(𝑟)𝑓_{𝐿𝑝,𝜙}. (103) With this pointwise estimate, it follows that 1Ψ(𝐵(1𝑎,𝑟))𝜇(𝐵(𝑎,𝑟))_{𝐵(𝑎,𝑟)}𝐼_𝛽𝑓₂^𝑞𝑑𝜇^1/𝑞≲𝑓_{𝐿𝑝,𝜙}. (104) The proof is complete.

Remark 24. García-Cuerva and Gatto proved that [33, Corollary 3.3], for a metric measure space (𝑋,𝑑,𝜇), satisfying (23), 𝐼_𝛽 is bounded from 𝐿^𝑝(𝜇) to 𝐿^𝑞(𝜇) provided that 1<𝑝<𝑛/𝛽 and 1/𝑞=1/𝑝−𝛽/𝑛. The assumption that 𝑑 is a metric may be relaxed; see [1], and in fact we can assume (𝑋,𝑑,𝜇) to be a quasi-metric measure space satisfying (23). Let 𝛼<−𝛽. Then, 𝜑(𝑟)∶=𝑟^𝛼 and 𝜓(𝑟)∶=𝑟^𝛼+𝛽 satisfy (99). In addition, if we assume that 𝜇 satisfies 𝜇(𝐵(𝑥,𝑟))≃𝑟^𝑛, uniformly in 𝑥∈𝑋 and 𝑟>0, then (100) holds with constraints on 𝑝,𝑞,𝛽 as above. Therefore, with all these assumptions, 𝐼_𝛽 is bounded from 𝐿_𝑝,𝜙(𝑋) to 𝐿_𝑞,Ψ(𝑋).

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.