# Some Remarks on Spaces of Morrey Type

Authors: Loredana Caso; Roberta D'Ambrosio; Sara Monsurrò
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

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


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## Body

## 1. Introduction

LetΩ be an unbounded open subset of ℝn,n≥2. For p∈[1,+∞[ and λ∈[0,n[, we consider the space Mp,λ(Ω) of the functions g in Llocp(Ω̅) such that(1.1)∥g∥Mp,λ(Ω)p=supτ∈]0,1]x∈Ωτ-λ∫Ω∩B(x,τ)|g(y)|pdy<+∞,
where B(x,τ) is the open ball with center x and radius τ.This space of Morrey type, defined by Transirico et al. in [1], is a generalization of the classical Morrey space Lp,λ and strictly contains Lp,λ(ℝn) when Ω=ℝn. Its introduction is related to the solvability of certain elliptic problems with discontinuous coefficients in the case of unbounded domains (see e.g., [1–3]).In the first part of this work, we deepen the study of two subspaces ofMp,λ(Ω), denoted by M̃p,λ(Ω) and Mop,λ(Ω), that can be seen, respectively, as the closure of L∞(Ω) and Co∞(Ω) in Mp,λ(Ω).We start proving some characterization lemmas that allow us to construct suitable decompositions of functions in M̃p,λ(Ω) and Mop,λ(Ω). This is done in the spirit of the classical decomposition (L1,L∞), proved in [4] by Calderón and Zygmund for L1, where a given function in L1 is decomposed, for any t>0, in the sum of a part ft∈L∞ (whose norm can be controlled by ∥ft∥L∞(Ω)<c(n)·t) and a remaining one f-ft∈L1. Analogous decompositions can be found also for different functional spaces (see e.g., [5, 6] for decompositions (L1,L1,λ), (Lp,Sobolev), and (Lp,BMO)).The idea of our decomposition, both for ag in M̃p,λ(Ω) and Mop,λ(Ω), is the following: for any h∈ℝ+, the function g can be written as the sum of a “good” part gh, which is more regular, and of a “bad” part g-gh, whose norm can be controlled by means of a continuity modulus of the function g itself.Decompositions are useful in different contexts as the proof of interpolation results, norm inequalities and a priori estimates for solutions of boundary value problems.For instance, in the study of several elliptic problems with solutions in Sobolev spaces, it is sometimes necessary to establish regularity results and a priori estimates for a fixed operatorL. These results often rely on the boundedness and possibly on the compactness of the multiplication operator(1.2)u∈Wk,q(Ω)→gu∈Lq(Ω),
which entails the estimate(1.3)∥gu∥Lq(Ω)≤c·∥g∥V·∥u∥Wk,q(Ω),
where c∈ℝ+ depends on the regularity properties of Ω and on the summability exponents, and g is a given function in a normed space V satisfying suitable conditions. In some particular cases, this cannot be done for the operator L itself, but there is the need to introduce a suitable class of operators Lh, whose coefficients, more regular, approximate the ones of L. This “deviation” of the coefficients of Lh from the ones of L needs to be done controlling the norms of the approximating coefficients with the norms of the given ones. Hence, it is necessary to obtain estimates where the dependence on the coefficients is expressed just in terms of their norms. Decomposition results play an important role in this approximation process, providing estimates where the constants involved depend just on the norm of the given coefficients and on their moduli of continuity and do not depend on the considered decomposition.In the framework of Morrey type spaces, in [1], the authors studied, for k=1, the operator defined in (1.2), generalizing a well-known result proved by Fefferman in [7] (cf. also [8]). They established conditions for the boundedness and compactness of this operator. In [2], the boundedness result and the straightforward estimates have been extended to any k∈ℕ.In view of the above considerations, the second part of this work is devoted to a further analysis of the multiplication operator defined in (1.2), for functions g in Mp,λ(Ω). By means of our decomposition results, we are allowed to deduce a compactness result for the operator given in (1.2). The obtained estimates can be used in the study of elliptic problems to prove that the considered operators have closed range or are semi-Fredholm.The deeper examination of the structure ofMp,λ(Ω) and of its subspaces leads us to the definition of a new functional space, that is a weighted Morrey type space, denoted by Mρp,λ(Ω).In literature, several authors have considered different kinds of weighted spaces of Morrey type and their applications to the study of elliptic equations, both in the degenerate case and in the nondegenerate one (see e.g., [9–11]).In this paper, given a weightρ in a class of measurable functions G(Ω) (see § 6 for its definition), we prove that the corresponding weighted space Mρp,λ(Ω) is a space settled between Mop,λ(Ω) and M̃p,λ(Ω). In particular, we provide some conditions on ρ that entail Mop,λ(Ω)=Mρp,λ(Ω).Taking into account the results of this paper, we are now in position to approach the study of some classes of elliptic problems with discontinuous coefficients belonging to the weighted Morrey type spaceMρp,λ(Ω).


## 2. Notation and Preliminary Results

LetG be a Lebesgue measurable subset of ℝn and Σ(G) be the σ-algebra of all Lebesgue measurable subsets of G. Given F∈Σ(G), we denote by |F| its Lebesgue measure and by χF its characteristic function. For every x∈F and every t∈ℝ+, we set F(x,t)=F∩B(x,t), where B(x,t) is the open ball with center x and radius t, and in particular, we put F(x)=F(x,1).The class of restrictions toF of functions ζ∈Co∞(ℝn) with F¯∩suppζ⊆F is denoted by 𝔇(F) and, for p∈[1,+∞[, Llocp(F) is the class of all functions g:F→ℝ such that ζg∈Lp(F) for any ζ∈𝔇(F).Let us recall the definition of the classical Morrey spaceLp,λ(ℝn).Forn≥2, λ∈[0,n[ and p∈[1,+∞[, Lp,λ(ℝn) is the set of the functions g∈Llocp(ℝn) such that(2.1)∥g∥Lp,λ(ℝn)=supτ>0x∈ℝnτ-λ/p∥g∥Lp(B(x,τ))<+∞,
equipped with the norm defined by (2.1).IfΩ is an unbounded open subset of ℝn and t is fixed in ℝ+, we can consider the space Mp,λ(Ω,t), which is larger than Lp,λ(ℝn) when Ω=ℝn. More precisely, Mp,λ(Ω,t) is the set of all functions g in Llocp(Ω¯) such that (2.2)∥g∥Mp,λ(Ω,t)=supτ∈]0,t]x∈Ωτ-λ/p∥g∥Lp(Ω(x,τ))<+∞,
endowed with the norm defined in (2.2).We explicitly observe that a diadic decomposition gives for everyt1,t2∈ℝ+ the existence of c1,c2∈ℝ+, depending only on t1,t2, and n, such that(2.3)c1∥g∥Mp,λ(Ω,t1)≤∥g∥Mp,λ(Ω,t2)≤c2∥g∥Mp,λ(Ω,t1),∀g∈Mp,λ(Ω,t1).
All the norms being equivalent, from now on, we consider the space(2.4)Mp,λ(Ω)=Mp,λ(Ω,1).For the reader's convenience, we briefly recall some properties of functions inLp,λ(ℝn) and Mp,λ(Ω) needed in the sequel.The first lemma is a particular case of a more general result proved in [12, Proposition 3].Lemma 2.1.
Let(Jh)h∈ℕ be a sequence of mollifiers in ℝn. If g∈Lp,λ(ℝn) and
(2.5)limy→0∥g(x-y)-g(x)∥Lp,λ(ℝn)=0,
then
(2.6)limh→+∞∥g-Jh*g∥Lp,λ(ℝn)=0.The second results concerns the zero extensions of functions inMp,λ(Ω) (see also  [1, Remark 2.4]).Remark 2.2.
Letg∈Mp,λ(Ω). If we denote by g0 the zero extension of g outside Ω, then g0∈Mp,λ(ℝn) and for every τin]0,1](2.7)∥g0∥Mp,λ(ℝn,τ)≤c1∥g∥Mp,λ(Ω,τ),
where c1∈ℝ+ is a constant independent of g, Ω and τ.
Furthermore, ifdiam(Ω)<+∞, then g0∈Lp,λ(ℝn) and
(2.8)∥g0∥Lp,λ(ℝn)≤c2∥g∥Mp,λ(Ω),
where c2∈ℝ+ is a constant independent of g and Ω.For a general survey on Morrey and Morrey type spaces, we refer to [1, 2, 13, 14].


## 3. The SpacesM̃p,λ(Ω) and Mop,λ(Ω)

This section is devoted to the study of two subspaces ofMp,λ(Ω), denoted by M̃p,λ(Ω) and Mop,λ(Ω). Here, we point out the peculiar characteristics of functions belonging to these sets by means of two characterization lemmas.Let us put, forh∈ℝ+ and g∈Mp,λ(Ω),(3.1)F[g](h)=supE∈Σ(Ω)supx∈Ω|E(x)|≤1/h∥gχE∥Mp,λ(Ω).Lemma 3.1.
Letλ∈[0,n[, p∈[1,+∞[, and g∈Mp,λ(Ω). The following properties are equivalent:
(3.2)gis  in  the  closure  ofL∞(Ω)inMp,λ(Ω),(3.3)limh→+∞F[g](h)=0,(3.4)limh→+∞(supE∈Σ(Ω)supx∈Ωτ∈]0,1]τ-λ|E(x,τ)|≤1/h∥gχE∥Mp,λ(Ω))=0.We denote byM̃p,λ(Ω) the subspace of Mp,λ(Ω) made up of functions verifying one of the above properties.Proof of Lemma3.1.
The equivalence between (3.2) and (3.3) is proved in of [1, Lemma 1.3]. Let us show that (3.2) entails (3.4) and vice versa.
Fixg in the closure of L∞(Ω) in Mp,λ(Ω), then for each ε>0, there exists a function gε∈L∞(Ω) such that(3.5)∥g-gε∥Mp,λ(Ω)<ε2.
Fixed E∈Σ(Ω), from (3.5), it easily follows that
(3.6)∥gχE∥Mp,λ(Ω)≤∥(g-gε)χE∥Mp,λ(Ω)+∥gεχE∥Mp,λ(Ω)<ε2+∥gεχE∥Mp,λ(Ω).
On the other hand
(3.7)∥gεχE∥Mp,λ(Ω)=supτ∈]0,1]x∈Ωτ-λ/p∥gεχE∥Lp(Ω(x,τ))≤∥gε∥L∞(Ω)supτ∈]0,1]x∈Ω(τ-λ|E(x,τ)|)1/p.
Therefore, if we set
(3.8)1hε=(ε2∥gε∥L∞(Ω))p,
from(3.7), we deduce that, if supτ∈]0,1],x∈Ωτ-λ|E(x,τ)|≤1/hε, then
(3.9)∥gεχE∥Mp,λ(Ω)≤ε2.
Putting together (3.6) and (3.9), we get (3.4).
Conversely, if we take a functiong∈Mp,λ(Ω) satisfying (3.4), for any ε>0, there exists hε∈ℝ+ such that if E∈Σ(Ω) with supτ∈]0,1],x∈Ωτ-λ|E(x,τ)|≤1/hε, then ∥gχE∥Mp,λ(Ω)<ε.
For eachk∈ℝ+, we set
(3.10)Ek={x∈Ω∣|g(x)|≥k}.
Observe that
(3.11)∥g∥Mp,λ(Ω)≥supτ∈]0,1]x∈Ωτ-λ/p∥g∥Lp(Ek(x,τ))≥ksupτ∈]0,1]x∈Ω(τ-λ|Ek(x,τ)|)1/p.
Therefore, if we put
(3.12)kε=∥g∥Mp,λ(Ω)hε1/p,
from (3.11), it follows that
(3.13)supτ∈]0,1]x∈Ωτ-λ|Ekε(x,τ)|≤1hε,
and then
(3.14)∥gχEkε∥Mp,λ(Ω)<ε.
To end the proof, we define the function gε=g-gχEkε. Indeed, by construction  gε∈L∞(Ω) and by (3.14), one gets that ∥g-gε∥Mp,λ(Ω)<ε.Remark 3.2.
It is easily seen (see also [1]) that if g∈M̃p,λ(Ω), then
(3.15)limt→0∥g∥Mp,λ(Ω,t)=0.Now, we introduce two classes of applications needed in the sequel.Forh∈ℝ+, we denote by ζh a function of class Co∞(ℝn) such that(3.16)0≤ζh≤1,ζh∣B(0,h)¯=1,suppζh⊂B(0,2h).
To define the second class, we first fix f in 𝔇(ℝ+̅) satisfying (3.17)0≤f≤1,f(t)=1ift≤12,f(t)=0ift≥1,
and α∈C∞(Ω)∩C0,1(Ω¯) equivalent to dist(·,∂Ω) (for more details on the existence of such an α, see for instance [15]). Hence, for h∈ℝ+, we put(3.18)ψh:x∈Ω¯→(1-f(hα(x)))f(|x|2h).
It is easy to prove that ψh belongs to Co∞(Ω), for any h∈ℝ+. Moreover,(3.19)0≤ψh≤1,ψh|Ω¯h=1,suppψh⊂Ω¯2h,
where(3.20)Ωh={x∈Ω∣|x|<h,α(x)>1h}.Lemma 3.3.
Letλ∈[0,n[, p∈[1,+∞[, and g∈Mp,λ(Ω). The following properties are equivalent:
(3.21)gis in the closure ofCo∞(Ω)inMp,λ(Ω),(3.22)limh→+∞(∥(1-ζh)g∥Mp,λ(Ω)+F[g](h))=0,(3.23)limh→+∞(∥(1-ψh)g∥Mp,λ(Ω)+F[g](h))=0,(3.24)limt→0∥g∥Mp,λ(Ω,t)+lim|x|→+∞(supτ∈]0,1]τ-λ/p∥g∥Lp(Ω(x,τ)))=0,(3.25)g∈M̃p,λ(Ω),lim|x|→+∞(supτ∈]0,1]τ-λ/p∥g∥Lp(Ω(x,τ)))=0.The subspace ofMp,λ(Ω) of the functions satisfying one of the above properties will be denoted by Mop,λ(Ω).Proof of Lemma3.3.
The equivalence between (3.21) and (3.22) is a consequence of (3.3) and of [1, Lemmas 2.1 and 2.5]. The one between (3.21) and (3.24) follows from of [1, Remark 2.2]. Always in [1], see Lemma 2.1 and Remark 2.2, it is proved that (3.21) entails (3.25) and vice versa. Let us show that (3.21) and (3.23) are equivalent too.
Let us firstly assume thatg belongs to the closure of Co∞(Ω) in Mp,λ(Ω).
Clearly, this entails thatg is in the closure of L∞(Ω) in Mp,λ(Ω), thus by Lemma 3.1, one has that(3.26)limh→+∞F[g](h)=0.
It remains to show that
(3.27)limh→+∞∥(1-ψh)g∥Mp,λ(Ω)=0.
To this aim, observe that fixed ε>0, there exists gε∈Co∞(Ω) such that
(3.28)∥g-gε∥Mp,λ(Ω)<ε.
On the other hand, if we consider the sets Ωh defined in (3.20), one has
(3.29)Ω/Ωh={x∈Ω∣|x|≥h}∪{x∈Ω∣α(x)≤1h}.
Therefore, since gε has a compact support, there exists hε∈ℝ+(3.30)(Ω/Ωh)∩suppgε=∅,∀h≥hε.
Then, since ψh|Ωh=1, one has that supp(1-ψh)⊂Ω∖Ωh, hence (1-ψh)gε=0 for allh≥hε.
The above considerations together with (3.28) give, for any h≥hε,(3.31)∥(1-ψh)g∥Mp,λ(Ω)=∥(1-ψh)(g-gε)∥Mp,λ(Ω)≤∥g-gε∥Mp,λ(Ω)<ε,
that is, (3.27).
Conversely, assume thatg∈Mp,λ(Ω) and that (3.23) holds.
First of all, we observe that denoted bygo the zero extension of g to ℝn, by (2.7) of Remark 2.2, there exists a positive constant c1, independent of g, ψh and of Ω, such that(3.32)∥(1-ψh)go∥Mp,λ(ℝn)≤c1∥(1-ψh)g∥Mp,λ(Ω).
Furthermore, by (3.23), we get that fixed ε>0, there exists hε such that
(3.33)∥(1-ψhε)g∥Mp,λ(Ω)<ε2c1.
Therefore,
(3.34)∥(1-ψhε)go∥Mp,λ(ℝn)<ε2.
Set
(3.35)Φε=ψhεgo,
by construction
(3.36)suppΦε⊂suppψhε⊂Ω¯2hε.
Hence, taking into account (2.8) of Remark 2.2, one has that
(3.37)Φε∈Lp,λ(ℝn).
On the other hand, (3.23) together with Lemma 3.1 give that g∈M̃p,λ(Ω), then from Remark 3.2, we get
(3.38)limt→0∥g∥Mp,λ(Ω,t)=0.
So, using (2.7) of Remark 2.2, we have that Φε∈Mp,λ(ℝn) and
(3.39)limt→0∥Φε∥Mp,λ(ℝn,t)=0.
Arguing as in [16, Lemma 1.2], from (3.36)–(3.39), we conclude that
(3.40)limy→0∥Φε(x-y)-Φε(x)∥Lp,λ(ℝn)=0.
We are now in the hypotheses of Lemma 2.1. Hence, denoted by (Jk)k∈ℕ a sequence of mollifiers in ℝn, we can find a positive integer kε>hε such that
(3.41)∥Φε-Jkε*Φε∥Lp,λ(ℝn)<ε2.
Set gε=Jkε*Φε, one has gε∈Co∞(Ω). Furthermore, using (3.34) and (3.41), we get
(3.42)∥g-gε∥Mp,λ(Ω)≤∥go-Jkε*Φε∥Mp,λ(ℝn)≤∥go-Φε∥Mp,λ(ℝn)+∥Φε-Jkε*Φε∥Mp,λ(ℝn)≤∥go-ψhεgo∥Mp,λ(ℝn)+∥Φε-Jkε*Φε∥Lp,λ(ℝn)≤∥(1-ψhε)go∥Mp,λ(ℝn)+ε2<ε,
this concludes the proof.


## 4. Decompositions of Functions inM̃p,λ(Ω) and Mop,λ(Ω)

The characterizations of the spacesM̃p,λ(Ω) and Mop,λ(Ω) naturally lead us to the introduction of the following moduli of continuity.Letg be a function in M̃p,λ(Ω). A modulus of continuity of g in M̃p,λ(Ω) is a map σ̃p,λ[g]:ℝ+→ℝ+ such that(4.1)F[g](h)≤σ̃p,λ[g](h),limh→+∞σ̃p,λ[g](h)=0.
If g belongs to Mop,λ(Ω), a modulus of continuity of g in Mop,λ(Ω) is an application σop,λ[g]:ℝ+→ℝ+ such that(4.2)∥(1-ζh)g∥Mp,λ(Ω)+F[g](h)≤σop,λ[g](h),limh→+∞σop,λ[g](h)=0.
Let us show now the decomposition results.Lemma 4.1.
Letλ∈[0,n[, p∈[1,+∞[, and g∈M̃p,λ(Ω). For any h∈ℝ+, one has
(4.3)g=gh′+gh′′,
with gh′′∈L∞(Ω) and
(4.4)∥gh′∥Mp,λ(Ω)≤σ̃p,λ[g](h),∥gh′′∥L∞(Ω)≤h1/p∥g∥Mp,λ(Ω).Proof.
Giveng∈M̃p,λ(Ω) and h∈ℝ+, we introduce the set
(4.5)Eh={x∈Ω∣|g(x)|≥h1/p∥g∥Mp,λ(Ω)}.
Observe that
(4.6)|Eh(x)|≤∫Ω(x)∩Eh|g(y)|p∥g∥Mp,λ(Ω)phdy≤1∥g∥Mp,λ(Ω)ph∫Ω(x)|g(y)|pdy≤1∥g∥Mp,λ(Ω)phsupτ∈]0,1]x∈Ωτ-λ∥g∥Lp(Ω(x,τ))p=1h.
Set
(4.7)gh′=gχEh={gifx∈Eh,0ifx∈Ω/Eh,gh′′=g-gχEh={0ifx∈Eh,gifx∈Ω/Eh.
In view of (4.6),
(4.8)∥gh′∥Mp,λ(Ω)=∥gχEh∥Mp,λ(Ω)≤F[g](h)≤σ̃p,λ[g](h),
this gives the first inequality in (4.4), the second one easily follows from (4.5).Lemma 4.2.
Letλ∈[0,n[, p∈[1,+∞[ and g∈Mop,λ(Ω). For any h∈ℝ+, one has
(4.9)g=ϕh′+ϕh′′,
with
(4.10)∥ϕh′∥Mp,λ(Ω)≤σop,λ[g](h),|ϕh′′|≤ζhh1/p∥g∥Mp,λ(Ω).Proof.
To prove this second decomposition result, we exploit again the definition of the setEh introduced in (4.5) and inequality (4.6).
In this case, for anyh∈ℝ+, we define (4.11)ϕh′=g(1-ζh)+ζhgχEh={gifx∈Eh,g(1-ζh)ifx∈Ω/Eh,ϕh′′=ζh(g-gχEh)={0ifx∈Eh,gζhifx∈Ω/Eh.
To obtain the first inequality in (4.10), we observe that (4.6) gives
(4.12)∥ϕh′∥Mp,λ(Ω)≤∥g(1-ζh)∥Mp,λ(Ω)+∥ζhgχEh∥Mp,λ(Ω)≤∥g(1-ζh)∥Mp,λ(Ω)+∥gχEh∥Mp,λ(Ω)≤∥g(1-ζh)∥Mp,λ(Ω)+F[g](h)≤σop,λ[g](h).
The second one is a consequence of (4.5).


## 5. A Compactness Result

In this section, as application, we use the previous results to prove the compactness of a multiplication operator on Sobolev spaces.To this aim, let us recall an imbedding theorem proved in [2, Theorem 3.2].Let us specify the assumptions:(h1)
Ω is an open subset of ℝn having the cone property with cone C, the parameters k,r,p,q,λ satisfy one of the following conditions:(h2)
k∈ℕ,1≤p≤q≤r<+∞,0≤λ<n,γ=1/q-1/p+k/n>0, with r>q when p=n/k>1 and λ=0, and with λ>n(1-rγ) when rγ<1,(h3)
k=1,1<p=q<r≤n,λ=n-r.Theorem 5.1.
Under hypothesis(h1) and if (h2) or (h3) holds, for any u∈Wk,p(Ω) and for any g∈Mr,λ(Ω), one has gu∈Lq(Ω). Moreover, there exists a constant c∈ℝ+, depending on n,k,p,q,r,λ, and C, such that
(5.1)∥gu∥Lq(Ω)≤c∥g∥Mr,λ(Ω)∥u∥Wk,p(Ω).Putting together Lemma4.1 and Theorem 5.1, we easily have the following result.Corollary 5.2.
Under hypothesis(h1) and if (h2) or (h3) holds, for any g∈M̃r,λ(Ω) and for any h∈ℝ+, one has
(5.2)∥gu∥Lq(Ω)≤c·σ̃r,λ[g](h)·∥u∥Wk,p(Ω)+h1/r·∥g∥Mr,λ(Ω)·∥u∥Lq(Ω),
for each u∈Wk,p(Ω), where c∈ℝ+ is the constant of (5.1).Ifg is in Mor,λ(Ω), the previous estimate can be improved as showed in the corollary below.Corollary 5.3.
Under hypothesis(h1) and if (h2) or (h3) holds, for any g∈Mor,λ(Ω) and for any h∈ℝ+, there exists an open set Ah⊂⊂Ω with the cone property, such that
(5.3)∥gu∥Lq(Ω)≤c·σor,λ[g](h)·∥u∥Wk,p(Ω)+h1/r·∥g∥Mr,λ(Ω)·∥u∥Lq(Ah),
for each u∈Wk,p(Ω), where c∈ℝ+ is the constant of (5.1).Proof.
Fixg∈Mor,λ(Ω) and h∈ℝ+. In view of Lemma 4.2 and Theorem 5.1, for any u∈Wk,p(Ω), we have
(5.4)∥gu∥Lq(Ω)≤∥ϕh′u∥Lq(Ω)+∥ϕh′′u∥Lq(Ω)≤c∥ϕh′∥Mr,λ(Ω)·∥u∥Wk,p(Ω)+∥ϕh′′u∥Lq(Ω)≤c·σor,λ[g](h)·∥u∥Wk,p(Ω)+∥ϕh′′u∥Lq(Ω).
Using again Lemma 4.2, we obtain
(5.5)∥ϕh′′u∥Lq(Ω)≤∥g∥Mr,λ(Ω)h1/r(∫Ω|ζhu|qdx)1/q≤∥g∥Mr,λ(Ω)h1/r(∫suppζh|u|qdx)1/q.
Putting together (5.4) and (5.5), we get (5.3), with Ah obtained as follows: fixed dh∈]0,dist(suppζh,∂Ω)/2[ and θ∈]0,π/2[, the set Ah is union of the open cones 𝒞⊂⊂Ω with opening θ, height dh and such that 𝒞∩suppζh≠∅.We are now in position to prove the compactness result.Corollary 5.4.
Suppose that condition(h1) is satisfied, that (h2) or (h3) holds, and fix g∈Mor,λ(Ω). Then, the operator
(5.6)u∈Wk,p(Ω)→gu∈Lq(Ω)
is compact.Proof.
Observe that ifΩ'⊂⊂Ω is a bounded open set with the cone property, the operator(5.7)u∈Wk,p(Ω)→u∈Lq(Ω′)
is compact.
Indeed, ifΩ'⊂⊂Ω is a bounded open set, the operator(5.8)u∈Wk,p(Ω)→u|Ω'∈Wk,p(Ω′)
is linear and bounded. Moreover, since Ω' has the cone property, the Rellich-Kondrachov Theorem (see e.g., [17]) applies and gives that the operator
(5.9)w∈Wk,p(Ω′)→w∈Lq(Ω′)
is compact.
Let us consider now a sequence(un)n∈ℕ bounded in Wk,p(Ω), and let M∈ℝ+ be such that ∥un∥Wk,p(Ω)≤Mforalln∈ℕ. According to the above considerations, fixed ε>0, there exist a subsequence (unm)m∈ℕ and ν∈ℕ such that(5.10)∥unm-unl∥Lq(Ω′)≤ε,∀m,l>ν.
On the other hand, given g∈Mor,λ(Ω) and h∈ℝ+, in view of Corollary 5.3, there exists a constant c∈ℝ+ and an open set Ah⊂⊂Ω with the cone property, independent of un, such that
(5.11)∥gun∥Lq(Ω)≤c·σor,λ[g](h)·∥un∥Wk,p(Ω)+h1/r·∥g∥Mr,λ(Ω)·∥un∥Lq(Ah).
From (5.11) and (5.10) written for ε=(c·σor,λ[g](h))/(h1/r·∥g∥Mr,λ(Ω)) and Ω'=Ah, for m,l>ν, one has
(5.12)∥gunm-gunl∥Lq(Ω)≤c·σor,λ[g](h)·(2M+1).
By (5.12) and (4.2), we conclude that (gunm)m∈ℕ is a Cauchy sequence in Lq(Ω), which gives the compactness of the operator defined in (5.6).


## 6. The SpaceMρp,λ(Ω)

In this section, we introduce some weighted spaces of Morrey type settled betweenMop,λ(Ω) and M̃p,λ(Ω). To this aim, given d∈ℝ+, we consider the set G(Ω,d) defined in [18] as the class of measurable weight functions ρ:Ω→ℝ+ such that(6.1)supx,y∈Ω|x-y|<dρ(x)ρ(y)<+∞.
It is easy to show that ρ∈G(Ω,d) if and only if there exists γ∈ℝ+, independent on x and y, such that(6.2)γ-1ρ(y)≤ρ(x)≤γρ(y),∀y∈Ω,∀x∈Ω(y,d).
Furthermore,(6.3)ρ,ρ-1∈Lloc∞(Ω¯).
We put(6.4)G(Ω)=⋃d>0G(Ω,d).
For p∈[1,+∞[,s∈ℝ, and ρ∈G(Ω), we denote by Lsp(Ω) the Banach space made up of measurable functions g:Ω→ℝ such that ρsg∈Lp(Ω) equipped with the norm(6.5)∥g∥Lsp(Ω)=∥ρsg∥Lp(Ω).
It can be proved that the space Co∞(Ω) is dense in Lsp(Ω) (see e.g., [18, 19]).From now on, we considerρ∈G(Ω)∩L∞(Ω), and we denote by d the positive real number such that ρ∈G(Ω,d).Lemma 6.1.
Letλ∈[0,n[, p∈[1,+∞[ and g∈Mp,λ(Ω). The following properties are equivalent:
(6.6)gis  in  the  closure ofL-1/p∞(Ω)inMp,λ(Ω),(6.7)limh→+∞(supE∈Σ(Ω)supx∈Ωτ∈]0,d]τ-λρ(x)|E(x,τ)|≤1/h∥gχE∥Mp,λ(Ω))=0,(6.8)limh→+∞(supE∈Σ(Ω)supx∈Ωρ(x)|E(x,d)|≤1/h∥gχE∥Mp,λ(Ω))=0.We denote byMρp,λ(Ω) the set of functions satisfying one of the above properties.Proof of Lemma6.1.
We start proving the equivalence between (6.6) and (6.7). This proof is in the spirit of the one of Lemma 3.1. For the reader's convenience, we write down just few lines pointing out the main differences.
If (6.6) holds, fixed ε>0, there exists a function gε∈L-1/p∞(Ω) such that(6.9)∥g-gε∥Mp,λ(Ω)<ε2.
From (6.9), we get that for any E∈Σ(Ω),
(6.10)∥gχE∥Mp,λ(Ω)<ε2+∥gεχE∥Mp,λ(Ω).
Furthermore, in view of the equivalence of the spaces Mp,λ(Ω,d) and Mp,λ(Ω) given by (2.3) and taking into account (6.2),
(6.11)∥gεχE∥Mp,λ(Ω)≤c1∥gεχE∥Mp,λ(Ω,d)=c1supτ∈]0,d]x∈Ωτ-λ/p∥gεχE∥Lp(Ω(x,τ))≤c1γ1/p∥gε∥L-1/p∞(Ω)supτ∈]0,d]x∈Ω(τ-λρ(x)|E(x,τ)|)1/p,
where c1∈ℝ+ depends only on n and d. Hence, set
(6.12)1hε=(ε2c1γ1/p∥gε∥L-1/p∞(Ω))p,
from (6.11) we deduce that if supτ∈]0,d],x∈Ωτ-λρ(x)|E(x,τ)|≤1/hε, then
(6.13)∥gεχE∥Mp,λ(Ω)≤ε2.
Putting together (6.10) and (6.13), we obtain (6.7).
Now, assume thatg is a function in Mp,λ(Ω) and that (6.7) holds. Then, for any ε>0, there exists hε∈ℝ+ such that if E∈Σ(Ω) with supτ∈]0,d],x∈Ωτ-λρ(x)|E(x,τ)|≤1/hε, then(6.14)∥gχE∥Mp,λ(Ω)<ε.
For each k∈ℝ+, we define the set
(6.15)Gk={x∈Ω∣ρ-1/p(x)|g(x)|≥k}.
Using again (2.3), there exists c2∈ℝ+ depending on the same parameters as c1 such that
(6.16)∥g∥Mp,λ(Ω)≥c2∥g∥Mp,λ(Ω,d)≥c2supτ∈]0,d]x∈Ωτ-λ/p∥g∥Lp(Gk(x,τ))≥c2γ-1/pksupτ∈]0,d]x∈Ω(τ-λρ(x)|Gk(x,τ)|)1/p.
Therefore, if we put
(6.17)kε=γ1/phε1/p∥g∥Mp,λ(Ω)c2,
from (6.16), we obtain
(6.18)supτ∈]0,d]x∈Ωτ-λρ(x)|Gkε(x,τ)|≤1hε,
and then
(6.19)∥gχGkε∥Mp,λ(Ω)<ε.
We conclude setting gε=g-gχGkε. Indeed, by (6.15), gε∈L-1/p∞(Ω) and (6.19) gives that ∥g-gε∥Mp,λ(Ω)<ε.
Arguing similarly, we prove also that (6.6) entails (6.8) and vice versa. Indeed, if g∈Mp,λ(Ω) and (6.6) holds, we can obtain as before (6.10) and (6.11).
On the other hand, there exists a constantc3=c3(n) such that(6.20)supτ∈]0,d]x∈Ω(τ-λ·ρ(x)·|E(x,τ)|)1/p≤∥ρ∥L∞(Ω)λ/npsupτ∈]0,d]x∈Ωτ-λ/p·ρ(n-λ)/np(x)·|E(x,τ)|λ/np·|E(x,τ)|(n-λ)/np≤c3·∥ρ∥L∞(Ω)λ/npsupτ∈]0,d]x∈Ω(ρ(x)·|E(x,τ)|)(n-λ)/np.
Putting together (6.11) and (6.20), we obtain
(6.21)∥gεχE∥Mp,λ(Ω)≤c4γ1/p∥gε∥L-1/p∞(Ω)∥ρ∥L∞(Ω)λ/npsupτ∈]0,d]x∈Ω(ρ(x)|E(x,τ)|)(n-λ)/np,
where c4=c1·c3. Now, set
(6.22)1hε=(ε2c4γ1/p∥gε∥L-1/p∞(Ω)∥ρ∥L∞(Ω)λ/np)np/(n-λ),
from (6.21), we deduce that if supτ∈]0,d],x∈Ωρ(x)|E(x,τ)|≤1/hε, then
(6.23)∥gεχE∥Mp,λ(Ω)≤ε2.
From (6.10) and (6.23), we obtain (6.8).
Conversely, assume that (6.8) holds. We consider again the sets Gk introduced in (6.15). From (6.16), we get(6.24)∥g∥Mp,λ(Ω)≥c2∥g∥Mp,λ(Ω,d)≥c2d-λ/pγ-1/pksupx∈Ω(ρ(x)|Gk(x,d)|)1/p.
Therefore, if we put
(6.25)kε=dλ/pγ1/phε1/p∥g∥Mp,λ(Ω)c2,
from (6.24), we obtain
(6.26)supx∈Ωρ(x)|Gkε(x,d)|≤1hε,
and then, (6.8) being verified,
(6.27)∥gχGkε∥Mp,λ(Ω)<ε.
We conclude the proof setting gε=g-gχGkε. Indeed, clearly gε∈L-1/p∞(Ω) and (6.27) gives ∥g-gε∥Mp,λ(Ω)<ε.Arguing in the spirit of Section4, we want to obtain a decomposition result also for functions in Mρp,λ(Ω). To this aim, we put for h∈ℝ+ and g∈Mp,λ(Ω)(6.28)D[g](h)=supE∈Σ(Ω)supx∈Ωρ(x)|E(x,d)|≤1/h∥gχE∥Mp,λ(Ω)
In view of the previous lemma, we can define a modulus of continuity of a function g in Mρp,λ(Ω) as a map σρp,λ[g]:ℝ+→ℝ+ such that(6.29)D[g](h)≤σρp,λ[g](h),limh→+∞σρp,λ[g](h)=0.Lemma 6.2.
Letλ∈[0,n[, p∈[1,+∞[ and g∈Mρp,λ(Ω). For any h∈ℝ+, one has
(6.30)g=φh′+φh′′,
with φh′′∈L-1/p∞(Ω) and
(6.31)∥φh′∥Mp,λ(Ω)≤σρp,λ[g](h),∥φh′′∥L-1/p∞(Ω)≤cγ1/ph1/p∥g∥Mp,λ(Ω),
where c is a positive constant only depending on n,d,p, and λ and where γ is that of (6.2).Proof.
Fixg∈Mρp,λ(Ω), for any h∈ℝ+, we set
(6.32)φh′=gχGh={gifx∈Gh,0ifx∈Ω/Gh,φh′′=g-gχGh={0ifx∈Gh,gifx∈Ω/Gh,
where
(6.33)Gh={x∈Ω∣ρ-1/p(x)|g(x)|≥dλ/pγ1/ph1/p∥g∥Mp,λ(Ω,d)}.
The thesis followed by (6.2) and (2.3) arguing as in the proof of Lemma 4.1.Let us show the following inclusion.Lemma 6.3.
Letλ∈[0,n[ and p∈[1,+∞[. Then, L-α∞(Ω)∩Mp,λ(Ω)⊂Mρp,λ(Ω),forallα∈ℝ+.Proof.
Forα≥1/p, clearly L-α∞(Ω)⊂L-1/p∞(Ω) and then (6.6) holds. On the other hand, for α<1/p, we can show that if g∈L-α∞(Ω)∩Mp,λ(Ω), then (6.7) holds. Indeed, observe that by (2.3), there exists a constant c1=c1(n,d) such that for any E∈Σ(Ω)(6.34)∥gχE∥Mp,λ(Ω)≤c1∥gχE∥Mp,λ(Ω,d)=c1supτ∈]0,d]x∈Ωτ-λ/p∥gχE∥Lp(Ω(x,τ))≤c1γα∥g∥L-α∞(Ω)supτ∈]0,d]x∈Ωτ-λ/pρα(x)|E(x,τ)|1/p.
Moreover, there exists a constant c2=c2(n) such that
(6.35)supτ∈]0,d]x∈Ωτ-λ/pρα(x)|E(x,τ)|1/p=supτ∈]0,d]x∈Ω(τ-λρ(x)|E(x,τ)|)α(τ-λ|E(x,τ)|)1/p-α≤c2d(n-λ)(1/p-α)supτ∈]0,d]x∈Ω(τ-λρ(x)|E(x,τ)|)α.
Hence, fixed ε>0 and set
(6.36)1hε=(εc1·c2γα∥g∥L-α∞(Ω)d(n-λ)(1/p-α))1/α,
we deduce that, if supτ∈]0,d],x∈Ωτ-λρ(x)|E(x,τ)|≤1/hε, then ∥gχE∥Mp,λ(Ω)≤ε.Now, we can prove a further characterization ofMρp,λ(Ω).Lemma 6.4.
Letλ∈[0,n[,p∈[1,+∞[.Then, Mρp,λ(Ω) is the closure of ⋃α∈ℝ+L-α∞(Ω)∩Mp,λ(Ω)in Mp,λ(Ω).Proof.
Clearly, ifg∈Mρp,λ(Ω) by (6.6), one has also that g is in the closure of ⋃α∈ℝ+L-α∞(Ω)∩Mp,λ(Ω) in Mp,λ(Ω).
Conversely, let us prove that ifg belongs to the closure of ⋃α∈ℝ+L-α∞(Ω)∩Mp,λ(Ω) in Mp,λ(Ω), then (6.8) holds. Indeed, given ε>0, there exists a function gε∈L-α∞(Ω)∩Mp,λ(Ω), for an α∈ℝ+, such that(6.37)∥g-gε∥Mp,λ(Ω)<ε2.
Hence, given E∈Σ(Ω)(6.38)∥gχE∥Mp,λ(Ω)≤∥(g-gε)χE∥Mp,λ(Ω)+∥gεχE∥Mp,λ(Ω)<ε2+∥gεχE∥Mp,λ(Ω).
Now, observe that since gε∈L-α∞(Ω)∩Mp,λ(Ω) by Lemma 6.3, we get gε∈Mρp,λ(Ω), and therefore, using (6.8) of Lemma 6.1, we obtain that if supx∈Ωρ(x)|E(x,d)|≤1/h, then
(6.39)∥gεχE∥Mp,λ(Ω)≤ε2.
This, together with (6.38), ends the proof.A straightforward consequence of the definitions (3.21) of Lemma 3.3, (6.6) of Lemma 6.1, and (3.2) of Lemma 3.1 is given by the following result.Lemma 6.5.
Letλ∈[0,n[ and p∈[1,+∞[. Then, Mop,λ(Ω)⊂Mρp,λ(Ω)⊂M̃p,λ(Ω).Let us show that ifρ vanishes at infinity, the first inclusion stated in the lemma above becomes an identity.Lemma 6.6.
Letλ∈[0,n[ and p∈[1,+∞[. If ρ is such that
(6.40)lim|x|→+∞ρ(x)=0,
then Mop,λ(Ω)=Mρp,λ(Ω).Proof.
We show the inclusionMρp,λ(Ω)⊂Mop,λ(Ω), the converse being stated in Lemma 6.5. In view of Lemma 6.4, it is enough to verify that if (6.40) holds, then L-α∞(Ω)∩Mp,λ(Ω)⊂Mop,λ(Ω), for any α∈ℝ+.
To this aim, givenα∈ℝ+, we fix g∈L-α∞(Ω)∩Mp,λ(Ω), and we prove that (3.25) is satisfied. Observe that by Lemmas 6.3 and 6.5   g∈M̃p,λ(Ω). Moreover, for any x∈Ω and if 1≤d there exists a constant c=c(n) such that(6.41)supτ∈]0,1]τ-λ/p∥g∥Lp(Ω(x,τ))≤γα∥g∥L-α∞(Ω)supτ∈]0,1]τ-λ/pρα(x)|Ω(x,τ)|1/p≤cγα∥g∥L-α∞(Ω)supτ∈]0,1]τ(n-λ)/pρα(x)=cγα∥g∥L-α∞(Ω)ρα(x).
On the other hand, if d<1, clearly one has
(6.42)supτ∈]0,1]τ-λ/p∥g∥Lp(Ω(x,τ))=max{supτ∈]0,d]τ-λ/p∥g∥Lp(Ω(x,τ)),supτ∈]d,1]τ-λ/p∥g∥Lp(Ω(x,τ))}.
We can treat the first term on the right-hand side of this last equality as done in (6.41) obtaining
(6.43)supτ∈]0,d]τ-λ/p∥g∥Lp(Ω(x,τ))≤d(n-λ)/pcγα∥g∥L-α∞(Ω)ρα(x),
the constant c=c(n) being the one of (6.41).
Concerning the second one, observe that for anyx∈Ω and τ∈]d,1], we have the inclusion Ω(x,τ)⊂Q(x,τ), where Q(x,τ) denotes an n-dimensional cube of center x and edge 2τ. Now, there exists a positive integer k such that we can decompose the cube Q(x,1) in k cubes of edge less than d/2 and center xi, with xi∈Ω for i=1,…,k. Therefore, Q(x,1)⊂⋃i=1kB(xi,d/2). Hence, for any x∈Ω and τ∈]d,1], we have, arguing as before with opportune modifications,(6.44)τ-λ/p∥g∥Lp(Ω(x,τ))≤d-λ/p∑i=1k∥g∥Lp(Ω(xi,d/2))≤kd(n-λ)/pcγα∥g∥L-α∞(Ω)ρα(x),
the constant c=c(n) being the same of (6.41).
The thesis follows then from (6.41), (6.42), (6.43), and (6.44) passing to the limit as |x|→+∞, as a consequence of hypothesis (6.40).From the latter result, we easily obtain the following lemma.Lemma 6.7.
Letλ∈[0,n[ and p∈[1,+∞[. If ρ,σ∈G(Ω)∩L∞(Ω) and
(6.45)lim|x|→+∞ρ(x)=lim|x|→+∞σ(x)=0,
then Mρp,λ(Ω)=Mσp,λ(Ω).



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