Some Remarks on Spaces of Morrey Type

and Applied Analysis 3 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 weight ρ in a class of measurable functions G Ω see § 6 for its definition , we prove that the corresponding weighted space M ρ Ω is a space settled between M o Ω and M̃ Ω . In particular, we provide some conditions on ρ that entail M p,λ o Ω M p,λ ρ 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 spaceM ρ Ω . 2. Notation and Preliminary Results Let G be a Lebesgue measurable subset of R 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 to F of functions ζ ∈ C∞ o R with F∩supp ζ ⊆ F is denoted by D F and, for p ∈ 1, ∞ , Lploc F is the class of all functions g : F → R such that ζ g ∈ L F for any ζ ∈ D F . Let us recall the definition of the classical Morrey space L R . For n ≥ 2, λ ∈ 0, n and p ∈ 1, ∞ , L R is the set of the functions g ∈ Lploc R such that ∥ ∥g ∥ ∥ Lp,λ Rn sup τ>0 x∈Rn τ−λ/p ∥ ∥g ∥ ∥ Lp B x,τ < ∞, 2.1 equipped with the norm defined by 2.1 . IfΩ is an unbounded open subset of R and t is fixed in R , we can consider the space M Ω, t , which is larger than L R whenΩ R. More precisely,M Ω, t is the set of all functions g in Lploc Ω such that ∥ ∥g ∥ ∥ Mp,λ Ω,t sup τ∈ 0,t x∈Ω τ−λ/p ∥ ∥g ∥ ∥ Lp Ω x,τ < ∞, 2.2 endowed with the norm defined in 2.2 . We explicitly observe that a diadic decomposition gives for every t1, t2 ∈ R the existence of c1, c2 ∈ R , depending only on t1, t2, and n, such that c1 ∥ ∥g ∥ ∥ Mp,λ Ω,t1 ≤ ∥∥g∥∥Mp,λ Ω,t2 ≤ c2 ∥ ∥g ∥ ∥ Mp,λ Ω,t1 , ∀g ∈Mp,λ Ω, t1 . 2.3 All the norms being equivalent, from now on, we consider the space M Ω M Ω, 1 . 2.4 4 Abstract and Applied Analysis For the reader’s convenience, we briefly recall some properties of functions in L R andM 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 R . If g ∈ L R and lim y→ 0 ∥ ∥g ( x − y) − g x ∥∥Lp,λ Rn 0, 2.5


Introduction
Let Ω be an unbounded open subset of R n , n ≥ 2. For p ∈ 1, ∞ and λ ∈ 0, n , we consider the space M p,λ Ω of the functions g in L p loc Ω such that This space of Morrey type, defined by Transirico et al. in 1 , is a generalization of the classical Morrey space L p,λ and strictly contains L p,λ R n when Ω R n .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 of M p,λ Ω , denoted by M p,λ Ω and M p,λ o Ω , that can be seen, respectively, as the closure of L ∞ Ω and C ∞ o Ω in M p,λ Ω .We start proving some characterization lemmas that allow us to construct suitable decompositions of functions in M p,λ Ω and M p,λ o Ω .This is done in the spirit of the classical decomposition L 1 , L ∞ , proved in 4 by Calder ón and Zygmund for L 1 , where a given function in L 1 is decomposed, for any t > 0, in the sum of a part f t ∈ L ∞ whose norm can be controlled by f t L ∞ Ω < c n • t and a remaining one f − f t ∈ L 1 .Analogous decompositions can be found also for different functional spaces see e.g., 5, 6 for decompositions L 1 , L 1,λ , L p , Sobolev , and L p , BMO .
The idea of our decomposition, both for a g in M p,λ Ω and M p,λ o Ω , is the following: for any h ∈ R , the function g can be written as the sum of a "good" part g h , which is more regular, and of a "bad" part g − g h , 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 operator L. These results often rely on the boundedness and possibly on the compactness of the multiplication operator u ∈ W k,q Ω −→ gu ∈ L q Ω , 1.2 which entails the estimate 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 L h , whose coefficients, more regular, approximate the ones of L. This "deviation" of the coefficients of L h 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 M p,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 of M p,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 M p,λ ρ 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 weight ρ in a class of measurable functions G Ω see § 6 for its definition , we prove that the corresponding weighted space M p,λ ρ Ω is a space settled between M p,λ o Ω and M p,λ Ω .In particular, we provide some conditions on ρ that entail ρ 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 space M p,λ ρ Ω .

Notation and Preliminary Results
Let G be a Lebesgue measurable subset of R n 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 χ F 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 Let us recall the definition of the classical Morrey space equipped with the norm defined by 2.1 .
If Ω is an unbounded open subset of R n and t is fixed in R , we can consider the space M p,λ Ω, t , which is larger than L p,λ R n when Ω R n .More precisely, M p,λ Ω, t is the set of all functions g in L p loc Ω such that endowed with the norm defined in 2.2 .We explicitly observe that a diadic decomposition gives for every t 1 , t 2 ∈ R the existence of c 1 , c 2 ∈ R , depending only on t 1 , t 2 , and n, such that

2.3
All the norms being equivalent, from now on, we consider the space M p,λ Ω M p,λ Ω, 1 .

2.4
For the reader's convenience, we briefly recall some properties of functions in L p,λ R n and M p,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 J h h∈N be a sequence of mollifiers in R n .If g ∈ L p,λ R n and The second results concerns the zero extensions of functions in M p,λ Ω see also 1, Remark 2.4 .
Remark 2.2.Let g ∈ M p,λ Ω .If we denote by g 0 the zero extension of g outside Ω, then g 0 ∈ M p,λ R n and for every τ in 0, 1 where c 1 ∈ R is a constant independent of g, Ω and τ.Furthermore, if diam Ω < ∞, then g 0 ∈ L p,λ R n and where c 2 ∈ 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 .

Ω
This section is devoted to the study of two subspaces of M p,λ Ω , denoted by M p,λ Ω and M p,λ o Ω .Here, we point out the peculiar characteristics of functions belonging to these sets by means of two characterization lemmas.
Let us put, for h ∈ R and g ∈ M p,λ Ω , ∞ , and g ∈ M p,λ Ω .The following properties are equivalent: 3.4 We denote by M p,λ Ω the subspace of M p,λ Ω made up of functions verifying one of the above properties.
Proof of Lemma 3.1.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.
Fix g in the closure of L ∞ Ω in M p,λ Ω , then for each ε > 0, there exists a function Fixed E ∈ Σ Ω , from 3.5 , it easily follows that On the other hand Therefore, if we set 3.9 Putting together 3.6 and 3.9 , we get 3.4 .
Conversely, if we take a function g ∈ M p,λ Ω satisfying 3.4 , for any ε > 0, there exists For each k ∈ R , we set Observe that Therefore, if we put and then gχ E kε M p,λ Ω < ε.

3.14
To end the proof, we define the function g ε g − gχ E kε .Indeed, by construction g ε ∈ L ∞ Ω and by 3.14 , one gets that g − g ε M p,λ Ω < ε.
Now, we introduce two classes of applications needed in the sequel.
To define the second class, we first fix for more details on the existence of such an α, see for instance 15 .Hence, for h ∈ R , we put It is easy to prove that ψ h belongs to where ∞ , and g ∈ M p,λ Ω .The following properties are equivalent:

3.25
The subspace of M p,λ Ω of the functions satisfying one of the above properties will be denoted by M p,λ o Ω .
Proof of Lemma 3.3.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 that g belongs to the closure of

8 Abstract and Applied Analysis
To this aim, observe that fixed ε > 0, there exists On the other hand, if we consider the sets Ω h defined in 3.20 , one has Therefore, since g ε has a compact support, there exists The above considerations together with 3.28 give, for any Conversely, assume that g ∈ M p,λ Ω and that 3.23 holds.
First of all, we observe that denoted by g o the zero extension of g to R n , by 2.7 of Remark 2.2, there exists a positive constant c 1 , independent of g, ψ h and of Ω, such that 3.32 Furthermore, by 3.23 , we get that fixed ε > 0, there exists h ε such that

3.40
We are now in the hypotheses of Lemma 2.1.Hence, denoted by J k k∈N a sequence of mollifiers in R n , we can find a positive integer k ε > h ε such that Furthermore, using 3.34 and 3.41 , we get this concludes the proof.Let g be a function in M p,λ Ω .A modulus of continuity of g in M p,λ Ω is a map σ p,λ g : R → R such that

Decompositions of Functions in
Let us show now the decomposition results.

4.7
In view of 4.6 ,

4.10
Proof.To prove this second decomposition result, we exploit again the definition of the set E h introduced in 4.5 and inequality 4.6 .
In this case, for any h ∈ R , we define

4.11
To obtain the first inequality in 4.10 , we observe that 4.6 gives

4.12
The second one is a consequence of 4.5 .

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: is an open subset of R n having the cone property with cone C, the parameters k, r, p, q, λ satisfy one of the following conditions: > 0, with r > q when p n/k > 1 and λ 0, and with λ > n 1 − rγ when rγ < 1, Theorem 5.1.Under hypothesis h 1 and if h 2 or h 3 holds, for any u ∈ W k,p Ω and for any g ∈ M r,λ Ω , one has gu ∈ L q Ω .Moreover, there exists a constant c ∈ R , depending on n, k, p, q, r, λ, and C, such that Putting together Lemma 4.1 and Theorem 5.1, we easily have the following result.
Corollary 5.2.Under hypothesis h 1 and if h 2 or h 3 holds, for any g ∈ M r,λ Ω and for any h ∈ R , one has If g is in M r,λ o Ω , the previous estimate can be improved as showed in the corollary below.

Corollary 5.3. Under hypothesis h 1 and if h 2 or h 3 holds, for any g ∈ M r,λ
o Ω and for any h ∈ R , there exists an open set A h ⊂⊂ Ω with the cone property, such that for each u ∈ W k,p Ω , where c ∈ R is the constant of 5.1 .
Proof.Fix g ∈ M r,λ o Ω and h ∈ R .In view of Lemma 4.2 and Theorem 5.1, for any u ∈ W k,p Ω , we have

5.4
Using again Lemma 4.2, we obtain We are now in position to prove the compactness result.
Corollary 5.4.Suppose that condition h 1 is satisfied, that h 2 or h 3 holds, and fix g ∈ M r,λ o Ω .Then, the operator Proof.Observe that if Ω ⊂⊂ Ω is a bounded open set with the cone property, the operator 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 w ∈ W k,p Ω −→ w ∈ L q Ω 5.9 is compact.
Let us consider now a sequence u n n∈N bounded in W k,p Ω , and let M ∈ R be such that u n W k,p Ω ≤ M for all n ∈ N. According to the above considerations, fixed ε > 0, there exist a subsequence u n m m∈N and ν ∈ N such that 5.10 On the other hand, given g ∈ M r,λ o Ω and h ∈ R , in view of Corollary 5.3, there exists a constant c ∈ R and an open set A h ⊂⊂ Ω with the cone property, independent of u n , such that

5.11
From 5.11 and 5.10 written for ε c 12 By 5.12 and 4.2 , we conclude that gu n m m∈N is a Cauchy sequence in L q Ω , which gives the compactness of the operator defined in 5.6 .

Ω
In this section, we introduce some weighted spaces of Morrey type settled between M p,λ o Ω and M p,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 : 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 We put For p ∈ 1, ∞ , s ∈ R, and ρ ∈ G Ω , we denote by L p s Ω the Banach space made up of measurable functions g : Ω → R such that ρ s g ∈ L p Ω equipped with the norm It can be proved that the space C ∞ o Ω is dense in L p s Ω see e.g., 18, 19 .From now on, we consider ρ ∈ G Ω ∩ L ∞ Ω , and we denote by d the positive real number such that ρ ∈ G Ω, d .Lemma 6.1.Let λ ∈ 0, n , p ∈ 1, ∞ and g ∈ M p,λ Ω .The following properties are equivalent: We denote by M p,λ ρ Ω the set of functions satisfying one of the above properties.
Proof of Lemma 6.1.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 6.9 From 6.9 , we get that for any E ∈ Σ Ω , Furthermore, in view of the equivalence of the spaces M p,λ Ω, d and M p,λ Ω given by 2.3 and taking into account 6.2 , where c 1 ∈ R depends only on n and d.Hence, set 6.12 from 6.11 we deduce that if sup τ∈ 0,d Putting together 6.10 and 6.13 , we obtain 6.7 .Now, assume that g is a function in M p,λ Ω and that 6.7 holds.Then, for any ε > 0, there exists 6.14 For each k ∈ R , we define the set Using again 2.3 , there exists c 2 ∈ R depending on the same parameters as c 1 such that and then gχ G kε M p,λ Ω < ε.
Arguing similarly, we prove also that 6.6 entails 6.8 and vice versa.Indeed, if g ∈ M p,λ Ω and 6.6 holds, we can obtain as before 6.10 and 6.11 .
On the other hand, there exists a constant c 3 c 3 n such that sup

6.20
Putting together 6.11 and 6.20 , we obtain where Therefore, if we put

6.32
where 6.33 The thesis followed by 6.2 and 2.3 arguing as in the proof of Lemma 4.1.
Let us show the following inclusion.
p Ω and then 6.6 holds.On the other hand, for α < 1/p, we can show that if g ∈ L ∞ −α Ω ∩ M p,λ Ω , then 6.7 holds.Indeed, observe that by 2.3 , there exists a constant c 1 c 1 n, d such that for any E ∈ Σ Ω

6.34
Moreover, there exists a constant c 2 c 2 n such that
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.τ −λ/p g L p Ω x,τ .

6.42
We can treat the first term on the right-hand side of this last equality as done in 6.41 obtaining sup τ∈ 0,d τ −λ/p g L p Ω x,τ ≤ d n−λ /p cγ α g L ∞ −α Ω ρ α x , 6.43 the constant c c n being the one of 6.41 .
Concerning the second one, observe that for any x ∈ 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 x i , with x i ∈ Ω for i 1, . . ., k.Therefore, Q x, 1 ⊂ k i 1 B x i , d/2 .Hence, for any x ∈ Ω and τ ∈ d, 1 , we have, arguing as before with opportune modifications, τ −λ/p g L p Ω x,τ ≤ d −λ/p k i 1 g L p Ω x i ,d/2 ≤ kd n−λ /p cγ α g L ∞ −α Ω ρ α x , 6.44 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.

where
B x, τ is the open ball with center x and radius τ.

Ω
M p,λ Ω and M p,λ o The characterizations of the spaces M p,λ Ω and M p,λ o Ω naturally lead us to the introduction of the following moduli of continuity.10 Abstract and Applied Analysis