Embedding of vector-valued Morrey spaces and separable differential operators

The paper is the first part of a program devoted to the study of the behavior of operator-valued multipliers in Morrey spaces. Embedding theorems and uniform separability properties involving E-valued Morrey spaces are proved. As a consequence, maximal regularity for solutions of infinite systems of anisitropic elliptic partial differential equations are established.


Introduction
The aim of this note is to study the behavior of some differential operators in Morrey spaces. Useful tools to achieve this goal are embedding properties of these spaces studied in [33][34][35]. It is worth to mention that weighted spaces are used, in order to introduce weighted variational and quasi-variational inequalities and kinetic equations (see [4][5][6] 1 The interest of such a general setting raises from the following considerations. Fourier multipliers, in vector-valued function spaces, has been well studied (see e.g. [29,45]) as well as operator-valued Fourier multipliers [7,15,22,25,46]. On the other hand, the study of Morrey spaces has received considerable attention in the last thirty years in different research areas (see e.g. [8][9][10]16,17,[19][20][21]23,28,31,36,43]). A further motivation comes from the fact that, to our knowledge, nothing is known concerning Morrey estimates for such operator-valued Fourier multipliers and embedding properties of abstract Sobolev-Morrey spaces. Lebesgue multipliers of the Fourier transformation are, in a clear way and in detail, treated in [45], §2.2.1- §2.2.4. We also mention the papers [24,37,47] where boundary value problems (BVPs) for differentialoperator equations (DOEs) have been studied.
Our main results are operator-valued multiplier theorems in E-valued Morrey spaces L p,λ ( ; E) . To develop this study, the authors consider the E-valued Sobolev-Morrey type function space W l, p,λ ( ; E 0 , E) = W l, p,λ ( ; E) ∩ L p,λ ( ; E 0 ), where is a domain in R n , E 0 and E are two Banach spaces and E 0 is continuously and densely embedded into E.
Let us introduce the set E A θ as the space D A θ equipped with the following norm Let E 1 and E 2 be two Banach spaces and θ and p such that 0 < θ < 1 and 1 ≤ p ≤ ∞. Let us denote by (E 1 , E 2 ) θ, p the interpolation space obtained from {E 1 , E 2 } by the K -method ( [45] §1.3.1), for the above values of p and θ .
In Theorems 4.2 and 4.6 the authors prove that the most regular class of interpolation space E α , between E 0 and E, is the one such that the mixed differential operators D α are bounded from W l, p,λ ( ; E 0 , E) to L p,λ ( ; E α ), where α = (α 1 , α 2 , . . . , α n ) and l = (l 1 , l 2 , . . . , l n ) are n-tuples of nonnegative integer numbers such that |α : l| = n k=1 α k l k ≤ 1, and are compact from W l, p,λ ( ; E 0 , E) to L p,λ ( ; E α ) if the last inequality is strict, that is, if |α : l| = n k=1 α k l k < 1. We point out that these results are sharp because, among the spaces E α such that the following embedding holds the space (E (A) , E) k, p is the most smooth, i.e. (E (A) , E) k, p ⊂ E α for all kind of spaces E α such that the above embedding is valid. The undertaken study has the purpose to refine and improve the outcomes contained in [3] §9, [42] §1.7 for scalar Sobolev spaces, the upshot contained in [26] for one dimensional vector function spaces, and the achievements obtained in [39][40][41] for Hilbert-space valued class.
Throughout the paper we refer to the following parameter-dependent differentialoperator equation where ν is a positive parameter, a α are complex numbers, A and A α (x) are linear operators in a Banach space E. We notice that, for l 1 = l 2 = · · · = l n = 2m, Eq.
(1.1) can be written as the following elliptic DOE We establish that Eq. (1.1) is L p,λ (R n ; E)-separable, namely, we show that, for all f ∈ L p,λ (R n ; E), there exists a unique solution u ∈ W l, p,λ (R n ; E (A) , E) satisfying (1.1) almost everywhere on R n and there exists a positive constant C independent of f , such that the following coercive estimate: is true. This enables us to state that if f ∈ L p,λ (R n ; E) and u is the solution of (1.1), then all the terms of Eq. (1.1) belong to L p,λ (R n ; E) or, equivalently, that all the terms are separable in L p,λ (R n ; E).
Moreover, we point out that the above estimate implies that the inverse of the differential operator generated by ( The paper is organized as follows. In Sect. 2 we mention the necessary tools from Banach space theory and some background materials. Section 3 is devoted to the proof of multiplier theorems. In Sect. 4 we study continuity and compactness of embedding operators in E-valued Sobolev-Morrey spaces. In Sect. 5 we obtain separability properties and, finally, in Sect. 6 maximal regularity properties of infinite systems of anisotropic

Notation and background
Let us introduce the main tools and briefly discuss some consequence of them. Given ⊂ R n a measurable set, E a Banach space and, for x = (x 1 , x 2 , . . . , x n ), γ = γ (x) a positive measurable function on , we set L p,γ ( ; E) for the Banach space of strongly measurable E-valued functions defined in , endowed with the norm We note L p = L p ( ; E), the space L p,γ ( ; E) when γ (x) ≡ 1. Let us consider 1 < p < ∞ and 0 ≤ λ < n. We use the notation L p,λ (R n ; E), for the E-valued Morrey Space of those functions f ∈ L 1 loc (R n ; E) for which the following quantity is finite It is worth emphasize that a Banach space E is a ζ -convex space if there exists a symmetric real-valued function ζ (u, v), defined in E × E, that is convex with respect to each variable and that satisfies the following properties We mention that a ζ -convex Banach space E is usually called a UMD space, see for instance [11]. We also recall that E is a UMD space if and only if the Hilbert operator Note that L p and p spaces, as well as Lorentz spaces L pq , p, q ∈ (1, ∞), belong to the class of UMD spaces. We refer the reader to [11] for further information on the above definitions and comments.
In what follows we need the following definitions.
We assume that a Banach space E has the h p,γ property if the Hilbert operator is bounded in L p,γ (R n ; E) , for all p ∈ (1, ∞). Let C be the set of complex numbers and 0 ≤ ϕ < π. We set A linear operator A is said to be positive in a Banach space E and has bound M > 0, if its domain D (A) is dense in E and where I is the identity operator in E and B (E) is the space of bounded linear operators on E. The constant M is dependent only on ϕ but, since we consider ϕ a fixed angle, we do not need uniformly estimate with respect to ϕ. Without ambiguity we only write A + ξ instead of A + ξ I and denote it by A ξ . It is useful to recall ([45] §1.15.1) that there exist fractional powers A θ of the positive operator A, −∞ < θ < ∞.
We need to introduce the following definition, that hereafter plays an important role.
Denoting by F the Fourier transformation, a function ∈ L ∞ (R n ; L (E 1 , E 2 )) is called a multiplier from L p,λ (R n ; E 1 ) to L q,λ (R n ; E 2 ), provided there exists a positive constant C such that In the sequel let us consider H a generic set, h a parameter in H and [15,22,46]), if there exists a positive constant C such that for all T 1 , T 2 , . . . , T m ∈ K and u 1, where r j is a sequence of independent symmetric [−1, 1]-valued random variables on [0, 1]. The smallest constant C is called the R-bound of K and is denoted by where the constant C is independent of the parameter h, that is In a similar way we can introduce the multipliers in the weighted spaces L p,γ (R n ; E) and define M p,γ p,γ (E) as the collection of multipliers in L p,γ (R n ; E).
In view of the next definition we set Definition 2.2 A Banach space E satisfies a multiplier condition, with respect to p ∈ (1, ∞) and a weight function γ, if for every ∈ C n (R n \ {0}; B (E)) such that
For instance, the space L p ( ), 1 ≤ p < ∞, verify Property (α). A Banach space E is said to have local unconditional structure (in short l.u.st.) (see [32]) if there exists a positive constant C with the following property: given any finite dimensional subspace F ⊂ E, there exists a space U , with an unconditional basis {u n }, and operators A from F to U and B from U to E such that B A is the identity on F and A · B · χ {u n } ≤ C.
Let us recall that a function γ is a Muckenhoupt A p weight (see [30] for all balls Q ⊂ R n . The next remark shows a useful property that correlates the above definitions ( [38], Theorem 3.7).

Remark 2.4
If E is a UMD space having Property (α), it satisfies the multiplier condition with respect to γ ∈ A p , for p ∈ (1, ∞).
It is well known (see [25,27]) that any Hilbert space satisfies the multiplier condition. There are, however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition, for example UMD spaces (see [15,22,46]).

Definition 2.5 We say that a positive operator
In a Hilbert space, every norm bounded set is R-bounded. As a consequence, in a Hilbert space all positive operators are R-positive.
Let us now consider a domain in R n and l = (l 1 , l 2 , . . . , l n ). We define and equipped with the norm given by: Let us recall the definition of a Hardy-Littlewood Maximal function, a notion which is very important in various areas of analysis including harmonic analysis, PDE's and function theory (see e.g. [18]).
Let us consider the following anisotropic partial differential equation (PDE) |α:l|≤1 where a α are complex numbers. It is anisotropic elliptic if, for all ξ ∈ R n , there exists a positive constant C such that |α:l|=1 The term anisotropic means that the principal part could contain generally, different differentiation with respect to different variables.

Multiplier theorems
Our aim in this section is to prove a sufficient condition to have multipliers in Evalued Morrey spaces L p,λ (R n ; E). In order to obtain this result we make use of the concepts of Hardy-Littlewood maximal function, Muckenhoupt weights A p and Fourier multipliers theorems in E-valued in L p spaces. We refer the reader to [2,12,48] for related results.
Theorem 3.1 Assume that the following conditions are verified: Moreover, if the quantity If n = 1 then, the result remains true for all the UMD spaces E and E 1 .
Proof The theorem is proved, in a similar way as in [1].

Remark 3.2
It is easily verifiable that Theorem 3.1 is true if multiplier functions are not dependent on a parameter.

Theorem 3.3
Let us suppose that all conditions of Theorem 3.1 are true. Then, { h } h∈H is a uniform collection of multipliers in L p,λ (R n ; E), for every 1 < p < ∞ and 0 < λ < n.
Proof We recall that a function ∈ L ∞ (R n ; L (E)) is a multiplier in the space L p,γ (R n ; E) if there exists a positive constant C such that the operator u → F −1 (ξ ) Fu is bounded in L p,γ (R n ; E). This is equivalent to say that the convolution operator u → K u = F −1 (ξ ) * u is bounded in L p,γ (R n ; E) i.e.
for all u ∈ L p,γ (R n ; E).
We get the required result if we prove that estimate (3.1) implies K u L p,λ (R n ;E) ≤ C u L p,λ (R n ;E) .

Embedding theorems in abstract Morrey spaces
In this section, continuity and compactness of embedding operators in E-valued Sobolev-Morrey spaces are derived. Specifically, boundedness and compactness of mixed differential operators in the framework of abstract interpolation of Banach spaces are shown. Theorem 4.1 Let 1 < p < ∞, γ ∈ A p , 0 < λ < n, l = (l 1 , l 2 , . . . , l n ) and E be a Banach space. Let us also assume that ⊂ R n is a region such that there exists a bounded linear extension operator from W l p,γ ( ; E) to W l p,γ (R n ; E). Then, there exists a bounded linear extension operator from W l, p,λ ( ; E) to W l, p,λ (R n ; E).
Proof From the assumptions we know that there exists a bounded extension operator P acting from W l p,γ ( , E) to W l p,γ (R n , E), i.e.
for all u ∈ W l p,γ ( , E) . Let us fix anyγ ∈]λ/n; 1[; we know that [(Mχ B r (x 0 ) )γ ](x) ∈ A 1 ⊆ A p , for every ball B r = B r (x 0 ) having center x 0 ∈ R n and radius r > 0. Then, we have Repeating the same arguments for the generalized derivatives D l k k Pu we obtain the requested inequality for all u ∈ W l, p,λ ( , E).
Proof We distinguish two cases. First case: = R n . We have that Additionally, for every u ∈ W l, p,λ (R n ; E (A), E), we see that u W l, p,λ (R n ;E(A),E) = u L p,λ (R n ;E(A)) + n k=1 D l k k u L p,λ (R n ;E) .

Then, prove (4.2) is equivalent to show
for a suitable positive constant C μ . We obtain inequality (4.3), at once, if we prove that Q 0h = ξ α Q h (ξ ) and Q kh = ξ l k k Q h (ξ ) are uniform collections of multipliers in This fact is proved in a similar way as in [1], Theorem A 2 . Really, to achieve this, we prove that the sets uniformly in h. Using the R-positivity assumption of the operator A and from the above estimate we obtain that the following sets are R-bounded, uniformly respect to h. Furthermore, for u 1, u 2 , . . . , u m ∈ E, m ∈ N and ξ j = ξ 1 j , ξ 2 j , . . . , ξ nj ∈ R n \ {0}, we get m j=1 r j (y) h ξ j u j where r j is a sequence of independent From (4.4), combining the above estimate and product properties of the collection of R-bounded operators (see e.g. [15], Proposition 3.4), we get that the set Second case: is a generic set in R n . Let us set B a bounded linear extension operator from W l, p,λ ( ; E(A), E) to W l, p,λ (R n ; E(A), E), and let B be the restriction operator from R n to . Then, for any u ∈ W l, p,γ ( ; E(A), E), we have from which follows estimate (4.2).
Let us now prove the next compactness result, using the s-horn condition (see definition in [3], §7).

Theorem 4.5 Let E and E 0 be two Banach spaces such that the embedding E 0 ⊂ E is compact. Let also
⊂ R n be a bounded region satisfying the s-horn condition, 1 l k < 1 and η ≤ n 1 − q 1 q . Then, the embedding Proof Using Rellich's Theorem, we have that W l, p ( ; E 0 , E) is compactly embedded in L q ( ; E) , for every q ∈ [1, p * [. We also have that According to the fact that W l, p,λ ( ; E 0 , E) ⊂ W l, p ( ; E 0 , E) , the compactness is established.

Theorem 4.6
Suppose that E is a Banach space, ⊂ R n is a bounded region satisfying the s-horn condition and A −1 is a compact operator in E. Let us also assume 0 < λ < n, 1 < p < n : . We obtain, for 0 ≤ μ ≤ 1 − κ, the following multiplicative inequality Assuming, in Theorem 4.5, q 1 = p and λ = η, we get that the following embedding W l, p,λ ( ; E(A), E) ⊂ L p,λ ( ; E) is compact.
Then, for any bounded sequence {u k } k∈N ⊂ W l, p,λ ( ; E (A), E) there exists a subsequence u k j k j ∈N which converges in L p,λ ( ; E) to an element u. Furthermore, the boundedness of the set {u k } k∈N in W l, p,λ ( ; E (A), E) and the estimate (4.2) imply the boundedness of the set {D α u k } k∈N in L p,λ ( ; E), for κ ≤ 1, i.e. this set is weakly compact in L p,λ ( ; E). hence, generalized derivatives D α u of the limit function u exist and verify D α u ∈ L p,λ ( ; E). Moreover, due to closedness of A we get Au ∈ L p,λ ( ; E), i.e. u ∈ W l, p,λ ( ; E(A), E). Then, from (4.5), for In a similar way we obtain the following result.

Theorem 4.7
Suppose that all assumptions of Theorem 4.6 are satisfied.
Then, for 0 < μ ≤ 1 − κ, the embedding We highlight that for the isotropic case and n = 1. From Theorem 4.7, we obtain the following result.
We point out that l 0 p = p . Let us also set A an infinite matrix defined in the space q such that D (A) = s q , A = δ i j 2 si , where δ i j = 0 for i = j and δ i j = 1 if i = j, being i, j = 1, 2, . . . , ∞. Since the operator A is R-positive in q , from Theorems 4.2 and 4.7, we have: α k l k ≤ 1, 1 < p < n and 0 < λ ≤ n 1 − p q the embedding D α W l, p,λ ; s q , q ⊂ L p,λ ; s(1−κ) q is continuous and there exists a positive constant C μ such that for all u ∈ W l, p,λ ; s q , q and h > 0; (2) for κ < 1, 0 < λ < n and 0 ≤ μ ≤ 1 − κ the following embedding D α W l, p,λ ; s q , q ⊂ L p,λ ; It should be noted that the above embedding has not been obtained by the authors using a classical method concerning the integral representation of differentiable functions.

Separable differential in Morrey spaces
Let us consider the parameter-dependent principal equation where a α are complex numbers, ν is a complex parameter and A is a linear operator defined in a Banach space E. We want to highlight the fact that A could be an unbounded operator. By reasoning as in [1] Theorem A 4 we have the following result.
Theorem 5.1 Let us assume that the following assumptions are true: (1) E is a Banach space satisfying the multiplier condition with respect to p ∈ (1, ∞) and 0 < λ < n; Then, for every f ∈ L p,λ (R n ; E) there is a unique solution u of equation (5.1) that belongs to the space W l, p,λ (R n ; E (A) , E) and the following coercive uniform estimate holds true: Proof Applying Fourier transform to Eq. (5.1) it follows where Since K (ξ ) ∈ S (ϕ), for every ξ ∈ R n , we derive that the operator A + K (ξ ) is invertible in E. Then, the solution of (5.3) can be expressed as Thanks to this expression of u, we have Then, it is enough to prove that are multipliers in L p,λ (R n ; E). Therefore, we must show that the following collections are R-bounded in E, uniformly in ν ∈ S (ϕ 1 ). Thanks to the R-positivity of A, the set is R-bounded, uniformly with respect to parameter ν. Similarly to the proof of Theorem 4.2 and having in mind hypothesis (2), we have that the set is R-bounded. Furthermore, making use of Kahane's contraction principle, product properties of the collection of R-bounded operators (see e.g. [15], Lemma 3.5, Proposition 3.4) and R-positivity of operator A, we have Estimates (5.5) imply that the functions σ 1 (ξ ) and σ 2 (ξ ) are L p,λ (E) multipliers. The proof is achieved.
Let us denote by L 0 the differential operator in L p,λ (R n ; E) generated by (5.1) that is Proof The left part of the chain comes from Theorem 5.1. The right side is obtained from Theorem 4.2. Indeed, according to the last mentioned result, for all u ∈ W l, p,λ (R n ; E (A) , E), we have L 0 u L p,λ (R n ;E) ≤ |α:l|=1 |a α | D α u L p,λ (R n ;E) + Au L p,λ (R n ;E) ≤ max α |a α | |α:l|=1 D α u L p,λ (R n ;E) + Au L p,λ (R n ;E) ≤ M 2 u W l, p,λ (R n ;E(A),E) .

Corollary 5.3
Let us suppose that all assumptions of Theorem 5.1 are satisfied.
As a natural consequence of Corollary 5.3 we have the following result.

Corollary 5.4
Let us suppose that all conditions of Theorem 5.1 are true. Then, the operator L 0 is positive in L p,λ (R n ; E).
Let us call L the differential operator in L p,λ (R n ; E) generated by (1.1). Namely, and its domain D (L) is the set W l, p,λ (R n , E(A), E).

holds.
Proof Let E = l q , A and A α (x) be infinite matrices, such that The operator A is obviously positive in q . Thus, thanks to Theorem 5.5, the assertion is immediate.

Remark 6.2
As an application of the above results, considering concrete Banach spaces instead of E, and concrete R-positive differential, pseudo differential operators, or finite, infinite matrices instead of operator A, on the differential-operator equation (1.1), by virtue of Theorem 5.5 we catch different classes of maximal regular partial differential equations or systems of equations.