Morrey Spaces on Domains: Different Approaches and Growth Envelopes

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Introduction
In this paper, we study Morrey spaces on bounded domains ⊂ R n defined by different approaches. Originally, these spaces were introduced by Morrey in [21], when studying solutions of second-order quasi-linear elliptic equations in the framework of Lebesgue spaces. They can be understood as a complement (generalization) of the Lebesgue spaces L p ( ). In particular, the Morrey space M u, p ( ), 0 < p ≤ u < ∞, is defined as the collection of all complex-valued Lebesgue measurable functions on such that cf. [14]. Obviously, M p, p ( ) = L p ( ), since we may assume for bounded domains that diam( ) ≤ 1. As can be seen from the definition, Morrey spaces investigate the local behaviour of the L p norm, which makes them useful when describing the local behaviour of solutions of non-linear partial differential equations, cf. [13,[15][16][17][18][19][20]29]. Furthermore, applications in harmonic analysis and potential analysis can be found in the papers [2][3][4][5]. Interpolation results of these and related spaces are established in [28]. For more information we refer to the books [1] and [27]. Our aim here is to compare the spaces M u, p ( ) with two other approaches for Morrey spaces on domains as can be found in the literature and characterize the unboundedness of functions belonging to the Morrey spaces M u, p ( ) in some further detail.
To be more precise, we consider Morrey spaces M u, p ( ), defined and studied in [23,24], where in contrast to (1.1) the supremum is now taken over balls B(x, 2 − j ) fully contained in . We remark that in the original definition cubes were considered but the change to balls is immaterial here. On the other hand, we deal with the spaces M u, p ( ) introduced in [32], which differ from (1.1) by the fact that the supremum is now only taken over balls B(x, 2 − j ) having distance at least 2 − j to the boundary of . Clearly, by their definitions, we have embeddings M u, p ( ) → M u, p ( ) → M u, p ( ). Our main result in Theorem 2.7 now clarifies the connections and diversities explicitly. In particular, it turns out that Surprisingly, we can see that the behaviour of the spaces changes with respect to the interplay of the parameters n, u, p. Furthermore, in Theorem 2.3 it is established that for so-called type A domains, cf. Remark 2.2, the spaces M u, p ( ) can be characterized by spaces M u, p (R n ) via restriction to the domain.
Apart from these considerations, we would like to understand the 'quality' of unboundedness, which is admitted in the spaces M u, p ( ) and M u, p ( ). This contributes to the problem of optimal embeddings. We have which leads to the question whether the L p ( ) spaces on the right-hand side are indeed the best possible Lebesgue-type spaces in which the Morrey spaces can be embedded. These kind of questions can be investigated with the help of the growth its growth envelope function, and u X G ∈ (0, ∞] is some additional index providing a finer description. Here f * denotes the non-increasing rearrangement of f . These concepts were introduced in [31] and [8], where the latter book also contains a recent survey of the present state-of-the-art (concerning extensions and more general approaches) as well as applications and further references. Therefore, our second main result can be formulated as cf. Theorem 2.13. In contrast to this we obtain for the spaces M u, p ( ) in Theorem 2.15 that Again, from the envelope results above it can also be seen that the interplay between the parameters in terms of n u < 1 p and n u > 1 p plays a decisive role in the behaviour of the Morrey spaces.
The paper is organized as follows. First we present three different approaches for Morrey spaces on domains and discuss these concepts in terms of their connections and diversities. Then we turn to the concept of growth envelopes and present and prove our main results, finally obtaining some sharp embedding results and Hardytype inequalities.
We are very grateful to Professor Hans Triebel who introduced us in personal communications to some of his ideas contained in the unpublished notes [32]. He granted us permission to use some of his arguments and, moreover, present part of his results in the context of this paper.

Different Approaches: Connection and Diversity
Preliminaries We shall adopt the following general notation: N denotes the set of all natural numbers, N 0 = N ∪ {0}, R n , n ∈ N, denotes the n-dimensional real Euclidean space. Furthermore, μ = | · | stands for the Lebesgue measure. For a real number a, let a + := max(a, 0) and let a denote its integer part. For p ∈ (0, ∞], the number p is defined by 1/ p := (1 − 1/ p) + with the convention that 1/∞ = 0. By c, c 1 , c 2 , etc. we denote positive constants independent of appropriate quantities. For two non-negative expressions (i.e. functions or functionals) A, B, the symbol A B (or A B) means that A ≤ c B (or c A ≥ B). If A B and A B, we write A ∼ B and say that A and B are equivalent. Given two quasi-Banach spaces X and Y , we write X → Y if X ⊂ Y and the natural embedding is bounded.

Different Approaches
In this section we discuss three different approaches for Morrey spaces on domains. They provide intrinsic and extrinsic characterizations and we show below that under some restrictions on the parameters involved, the introduced spaces may coincide or differ.
We assume throughout this paper that the domain ⊂ R n is bounded.
If is a Lipschitz domain, then Let M( ) be the collection of all equivalence classes of complex-valued Lebesgue measurable functions on . There are several equivalent definitions of Morrey spaces. One can take averages over balls or cubes, or dyadic cubes. Below we give the definition of the spaces using balls B(x, 2 − j ) centred at x ∈ and of radius 2 − j , j ∈ N 0 , but in some proofs we use also the equivalent norm that uses dyadic cubes. Definition 2.1 Let ⊂ R n be a bounded domain and 0 < p ≤ u < ∞.
(i) The Morrey space M u, p ( ) is defined to be the set of all functions f ∈ M( ) such that where for x ∈ , by j x we denote the smallest number such that The definition of the spaces M u, p ( ) was already considered in [23], and the last approach for M u, p ( ) was considered in [32], where also growth envelopes for these spaces were studied. Our approach differs from the above ones in the sense that we consider parameters 0 < p ≤ u < ∞, which is more convenient for us, whereas the above references deal with 0 < p < ∞ and parameters λ := −np 1 u − 1 p or σ = − n u , resulting in the conditions 0 ≤ λ ≤ n or − n p ≤ σ ≤ 0, respectively. (ii) Clearly we have the embeddings which follows directly from the definitions of the spaces. Obviously, M u, p (R n ) = M u, p (R n ). (iii) In order to be able to compare the Morrey spaces M u, p ( ) as defined in (i) with the other two Morrey spaces on domains, we shall restrict ourselves to so-called domains of type A meaning that there exists a constant A > 0 such that for every x ∈ and all j ≥ j 0 we have This approach already appears in [33,Ch. 1] for the definition of Morrey spaces (when p = 2). In this case (2. 2) reduces to For example a square in the plane is a set of type A with A = 1 2 , whereas the domain = {(x, y) ∈ R 2 : 0 < x < 1, 0 < y < x 2 } is not of type A for any A > 0 (since the origin is a cuspidal point of the boundary of ). The situations are illustrated below.
Furthermore, our definitions in (i), (ii) differ from the ones used in [14,23] in the sense that we take balls with radii 2 − j , j ≥ j 0 instead of r ∈ (0, δ). Furthermore, we take the supremum over all j ∈ N 0 instead of j ≥ j 0 only, since for functions f ∈ L p ( ) we clearly have that the term with j = 0 is finite and for 0 ≤ j ≤ j 0 we have which differs from the j = 0 term only by some constant depending on j 0 .
We proceed by demonstrating that the spaces M u, p ( ) can be characterized by spaces via restriction to the domain.
where the infimum is taken over Clearly f |M u, p ( ) ≤ f |M u, p (R n ) so we are left to prove the converse. By definition First we argue why it is always sufficient to consider x ∈ instead of x ∈ R n . Let x / ∈ .
It remains to show that the supremum is attained for some j ∈ N 0 . Since is bounded, w.l.o.g. we can assume that it can be covered by some ball with radius 1. Then for big radii corresponding to which corresponds to some term which can be expressed by level j = 0. Therefore, we have shown that where we have finally used the assumption on to be a domain of type A. This completes the proof.
Next we briefly report on a result of Piccinini in [23], see also [24], for spaces M u, p ( ). We adapt the formulation to our setting and extend it to the quasi-Banach case which causes no difficulties looking at the proof. Let Q ⊂ R n be some cube, and 0 < p i ≤ u i < ∞, i = 1, 2. Then if, and only if, The result (2.8) was extended to R n by Rosenthal in [26, Satz 1.6], if, and only if, Remark 2.4 Note that in [24] also Morrey spaces of type M u, p ( ) for domains of type A are studied, whereas in [23] the setting is restricted to cubes only which simplifies the situation. Furthermore, in [24] one can find further generalizations of this approach, as well as related interpolation results.
whereas for u < p the corresponding norm becomes but there is no longer additional local information as in (2.4). The following theorem collects some embedding assertions obtained by Triebel [32].
Proof (i) The embedding follows from the definition of the spaces and Hölder's inequality.
To be more precise, p 1 ≥ p 2 implies that (ii) The proof can be found in [32,Th. 2.15] and uses arguments from interpolation theory. We sketch the main ideas. Let and S J as in (2.1) with |S J | ∼ 2 −J . Then where L 1 ,∞ ( ) denotes a Lorentz space. Recall that This is well-known, a short detailed proof of this assertion can also be found in [7,Lem. 2.12] and is based on Hölder's inequality and real interpolation of Lebesgue and Lorentz spaces. Then it follows from (2.13) that Using again the fact that is a bounded Lipschitz domain one obtains which implies r = u n in (2.14). If n u > 1 p , then = n u − 1 p > 0 as requested in (2.12). Combining (2.15) and (2.16) we have Let L u, p ( ) be a space quasi-normed by the right-hand side of the last inequality. Thus (2.17) means that L u, p ( ) → L u n ,∞ ( ) . (2.18) We take now (2.18) as a starting point for real interpolation. The interpolation of spaces L u, p ( ) can be described in the same way as the interpolation of weighted sequence spaces so we recall it briefly. Let A be a quasi-Banach space, 0 < q ≤ ∞ and δ ∈ R. Then δ q (A) is the quasi-Banach space consisting of all Then in [22], cf. also [ Adopting the proof of Theorem 5.6.1 in [6] to our situation we get Using the well-known interpolation properties of Lorentz spaces we finally obtain the desired result They are obviously contained in M u, p ( ), since the ∞ -norm in (2.4) is replaced now by its p counterpart in (2.19). Part (i) of Theorem 2.5 is literally the same for the spaces M * u, p ( ), whereas part (ii), that is, (2.11), has to be replaced by where is a bounded Lipschitz domain, 0 < p ≤ u < ∞. In particular,

Connection and Diversity
Now we take a closer look at the connections and diversities of these spaces refining the embedding result (2.6). Surprisingly, it turns out that depending on the parameters n, u, and p the three approaches might coincide altogether or differ completely. The precise results can be found below.

Theorem 2.7
Let ⊂ R n be a bounded Lipschitz domain, and 0 < p ≤ u < ∞. Proof Note that our assumption of to be a bounded Lipschitz domain implies that is also a type A domain, cf. [33, Ch. 1, p.32].
Step 1. We first show (i). By (2.6) it suffices to show that for any f ∈ M u, p ( ) we have Having a closer look at the norms of the two spaces we need to show that balls B(x, 2 − j ) ⊂ , which can be arbitrarily close to the boundary ∂ and are considered in the supremum of M u, p ( ), can be 'compensated' somehow with the help of balls B(x, 2 − j ) with j ≥ j x as allowed in the supremum of M u, p ( ). This can be seen as follows. Consider the sets S J from (2.1). We cover balls B(x, 2 − j ) ⊂ on their intersection with S J by balls B(x, 2 −J ) withx ∈ S J , J ≥ Jx ≥ j, and control the number of balls of radius 2 −J we need. Since the domain is bounded and Lipschitz the volume of the intersection is at most 2 −J 2 − j (n−1) . So the intersection can be covered by C2 J (n−1) 2 − j (n−1) balls of radius 2 −J , where the constant C is independent of j and J .
This leads to the estimate (2.21) where in the second but last step we used that the exponent of our geometric series is negative since n u < 1 p . Bringing the weight factor in (2.21) to the left-hand side we obtain the desired result, To show the coincidence M u, p ( ) = M u, p ( ) in (i) we may stress the same arguments as above, the only difference being (in the picture) that now we cover a ball centred at x ∈ ∂ with balls B(x, 2 −J ), where J ≥ J x . The calculations remain the same.
Step 2. As for (ii) it will be enough to show that we can find a function f ∈ .
where in the second step we used as estimate the largest value of d(y) in the ball On the other hand we have f / ∈ M u, p ( ) which can be seen as follows. Consider the disjoint sets where we assume that Then we calculate since n u ≥ 1 p implies np u ≥ 1 and therefore, the sum in the last line above diverges.

Growth Envelopes for Morrey Spaces M u, p ( )
We now turn our attention towards the Morrey spaces M u, p ( ), 0 < p ≤ u < ∞.
One can easily see that In particular, the embedding on the right-hand side follows immediately from the definition. Our aim now is to tackle the question whether L p ( ) is indeed the best Lebesgue-type space in which the Morrey spaces can be embedded. We will study embeddings into the scale of Lorentz spaces (which can be considered as refined L p spaces) and try to obtain some optimal (sharp) results. This problem can be rephrased in terms of growth envelopes as defined by Haroske and Triebel (see [8,31], where more details and references on the subject can be found). Therefore, we shall briefly recall the concept before we present our results. As an application of the computed growth envelopes we will obtain some answers regarding sharp embeddings and Hardy-type inequalities for Morrey spaces.
Let for some measurable f ∈ M( ) its decreasing rearrangement f * be defined as usual,

Definition 2.9
Let X ⊂ M( ) be some quasi-Banach function space on .
The growth envelope function of X is the class [E X G ] of functions g : (0, ε] → [0, ∞), for some ε > 0, such that g(·) ∼ E X G (·) in (0, ε]. For convenience, we do not distinguish between representative and equivalence class. Therefore, any representative function of the class will be called as well growth envelope function and sometimes we also denote a particular representative by E X G . (ii) Assume X → L ∞ ( ). Let E X G (or an equivalent function) be continuously differentiable. Then the number u X (with the usual modification if v = ∞) holds for some c > 0 and all f ∈ X . The couple is called (local) growth envelope for the function space X .
We recall some useful properties of growth envelopes.

Proposition 2.11
(i) Let X i → L ∞ , i = 1, 2, be some function spaces on . Then X 1 → X 2 implies that there is some positive constant c such that for all t > 0, for some ε > 0. Then we get for the corresponding indices u X i G , i = 1, 2, that This result coincides with [8,Props. 3.4,4.5].
Example 2.12 If X = L p,q ( ), 0 < p < ∞, 0 < q ≤ ∞, are the usual Lorentz spaces, then it is shown in [8,Thm. 4.7,Cor. 10.14] that Recall that left-hand side of (2.23) with v = q is an equivalent quasi-norm in L p,q ( ), We now study growth envelopes of the Morrey spaces M u, p ( ). The problem is delicate. On R n the results from [9, Th. 3.7] establish the non-existence of growth envelopes, since it is shown there that whenever 0 < p < u < ∞, then However, the situation for bounded domains is completely different. In this case we have the embeddings L u ( ) → M u, p ( ) → L p ( ), which immediately give upper and lower bounds for the growth envelope function. The ideas for the theorem to come are taken from [32].

Theorem 2.13 Let ⊂ R n be a bounded domain of type A, and let
(2.28)

Proof
Step 1. We assume in the proof that p < u. The case p = u is known since as desired. In order to compute the lower estimate we assume for simplicity that the domain contains the unit cube Q 0,0 = [0, 1] n , otherwise one can rescale the argument. Let Q j,k , j ∈ N 0 and k ∈ Z n , denote the dyadic cube by 2 − j k + [0, 2 − j ] n . We adopt the method used in the proof of Theorem 3.1 in [11], cf. also the proof of Theorem 3.2 in [10]. For 0 < ν we put where x = max{l ∈ Z : l ≤ x}. Then 1 ≤ k ν < 2 nν and there exists c p,u > 0 such that For convenience let us assume that c p,u = 1 (otherwise the argument below has to be modified in an obvious way). For any j > 0 we define a finite sequence λ j,m , where m ∈ {k : Q j,k ⊂ Q 0,0 }. The sequence takes only two values 0 and 1. Moreover the value 1 is taken k j times. It was proved in [11], cf. also [10], that the sequence can be chosen in such a way that for any 0 < ν < j and any cube Q ν,k ⊂ Q 0,0 , the subsequence {λ j,m : Q j,m ⊂ Q ν,k } contains at most k j−ν elements that equal 1. We consider the functions The function f j belongs to M u, p ( ), which can be seen as follows. Since an equivalent norm in M u, p ( ) can be defined by taking the supremum over dyadic cubes we see that Now, the desired estimate from below follows from (2.30) and (2.31).
We define the function f supported in the cube Q 0,0 via the formula We postpone the definition of the sequence (γ j ) j for a moment. The function f takes the value 0 on ν cubes of size 2 −nν , the value γ 1 on k ν ν cubes of size 2 −2nν and, by induction, the value γ 1 + · · · + γ on k ν ν cubes of size 2 −( +1)νn . We choose λ > 0 such that λp > 1 and put γ 1 = 2 ν n u , γ 2 = 2 2ν n u 2 −λ − γ 1 = 2 ν n u 2 ν n u 2 −λ − 1 , If Q = Q ν,m and λ ν,m = 1, then analogously  . So in this case the corresponding integral is smaller. If Q is any dyadic cube in Q 0,0 , then one can find ∈ N such that Q ν,m ⊂ Q ⊂ Q ( −1)ν,k for some m, k ∈ Z n . So it follows from (2.33) that (2.34) For the rearrangement f * let first t = k j ν 2 −( j+1)nν ν = 2 −nν j p u ν 2 −νn . Then by the construction hence, Let v < p and choose λ such that vλ = 1 < λp. Then one has  Proof This is an immediate consequence of the growth envelope results in Theorem 2.15.

Remark 2.20
Plainly one can also formulate counterparts of the above corollaries for spaces M * u, p ( ), n u > 1 p , and M u, p ( ), n u < 1 p , in view of (2.37) and (2.38), respectively. This is left to the reader.