Complex Interpolation of Lizorkin-Triebel-Morrey Spaces on Domains
aa r X i v : . [ m a t h . C A ] A ug Complex Interpolation of Lizorkin-Triebel-Morrey Spaces onDomains
Ciqiang Zhuo ∗ Marc Hovemann † Winfried Sickel ‡ August 4, 2020
Abstract
In this article the authors study complex interpolation of Sobolev-Morrey spacesand their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are consideredon bounded domains. Under certain conditions on the parameters the outcome belongsto the scale of the so-called diamond spaces.
Keywords: Morrey spaces, Lizorkin-Triebel-Morrey spaces, ± method of interpolation,Calder´on’s first and second complex interpolation method, diamond spaces, extensionoperators, Lipschitz domains.MSC subject class: 46B70, 46E35 One of the most popular formulas in interpolation theory is given by[ L p ( R d ) , L p ( R d )] Θ = L p ( R d ) , (1.1)where 1 ≤ p < p ≤ ∞ , 0 < Θ < p := − Θ p + Θ p . Here [ X , X ] Θ denotes Calderon’sfirst complex interpolation method or just the complex method. Morrey spaces M up ( R d )are generalizations of the Lebesgue spaces in view of M pp ( R d ) = L p ( R d ). Within the largerfamily of Morrey spaces the formula (1.1) is a singular point. Essentially as a result ofLemari´e-Rieusset [34], [35] it is known that[ M u p ( R d ) , M u p ( R d )] Θ = M up ( R d ) , (1.2)except the trivial cases given by either u = p , u = p , i.e., the Lebesgue case, or u = u , p = p . In [66] and [25] different explicit descriptions of the spaces [ M u p ( R d ) , M u p ( R d )] Θ can be found. The characterization given in [25] is the preferable one. When switchingfrom Lebesgue spaces to Morrey spaces we add two phenomena, one local and one global, ∗ Corresponding author; Key Laboratory of Computing and Stochastic Mathematics (Ministry of Educa-tion), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, People’sRepublic of China; E-mail: [email protected] † Friedrich-Schiller-University Jena, Ernst-Abbe-Platz 2, 07737 Jena, Germany; E-mail:[email protected] ‡ Friedrich-Schiller-University Jena, Ernst-Abbe-Platz 2, 07737 Jena, Germany; E-mail:[email protected] M u p ([0 , d ) , M u p ([0 , d )] Θ = ⋄ M up ([0 , d ) , (1.3)if 1 ≤ p < u < ∞ , < p < u < ∞ , p < p , < Θ < p u = p u , p := 1 − Θ p + Θ p , u := 1 − Θ u + Θ u . For a domain Ω ⊂ R d the space ⋄ M up (Ω) is defined as the closure of the smooth functionswith respect to the norm of the space M up (Ω). The aim of this paper will consist in anextension of (1.3) to smoothness spaces built on Morrey spaces, namely Lizorkin-Triebel-Morrey spaces E su,p,q (Ω), where Ω ⊂ R d is a bounded Lipschitz domain. For doing that wewill only investigate cases where the Lemari´e-Rieusset condition p u = p u is satisfied.Our main result reads as follows. Theorem 1.1.
Let Ω ⊂ R d be either a bounded Lipschitz domain if d ≥ or a boundedinterval if d = 1 . Under the following conditions on the parameters(a) ≤ p < p < ∞ , p ≤ u < ∞ , p ≤ u < ∞ ;(b) ≤ q , q ≤ ∞ , min( q , q ) < ∞ ;(c) p u = p u ;(d) s , s ≥ ; either s < s or < s = s and q ≤ q ;(e) < Θ < , p := − Θ p + Θ p , u := − Θ u + Θ u , q := − Θ q + Θ q and s := (1 − Θ) s + Θ s ;it holds [ E s u ,p ,q (Ω) , E s u ,p ,q (Ω)] Θ = ⋄ E su,p,q (Ω) . (1.4)Lizorkin-Triebel-Morrey spaces E su,p,q (Ω) are generalizations of Lizorkin-Triebel spaces F sp,q (Ω), more exactly, if u = p we have F sp,q (Ω) = E sp,p,q (Ω). Hence we get back thewell-known formula [ F s p ,q (Ω) , F s p ,q (Ω)] Θ = ⋄ F sp,q (Ω) = F sp,q (Ω) , (1.5)but under the extra condition (d). The Lemari´e-Rieusset condition (c) disappears in thiscase. There is a certain list of references for (1.5). Let us mention at least Triebel [57,Thm. 2.4.2.1] (Ω = R d or a bounded C ∞ domain), Frazier, Jawerth [16] (Ω = R d ), Kalton,Mayboroda, Mitrea [30] (Ω = R d ) and Triebel [59] (bounded Lipschitz domains). Thereis an interesting special case, given by the Sobolev-Morrey spaces, see Section 2.2 andLemma 2.9. 2 orollary 1.2. Let < Θ < , m ∈ N , m ∈ N , and either m < m or < m ≤ m .Let < p < p < ∞ , p < u < ∞ , p < u < ∞ and p u = p u . We define s := (1 − Θ) m + Θ m , p := 1 − Θ p + Θ p and u := 1 − Θ u + Θ u . Let Ω ⊂ R d be either a bounded Lipschitz domain if d ≥ or a bounded interval if d = 1 .Then we have (cid:2) W m M u p (Ω) , W m M u p (Ω) (cid:3) Θ = ⋄ E su,p, (Ω) . (1.6) In particular, if s = m ∈ N , then (cid:2) W m M u p (Ω) , W m M u p (Ω) (cid:3) Θ = ⋄ W m M up (Ω) (1.7) follows. There is another situation in which one can calculate [ E s u ,p ,q (Ω) , E s u ,p ,q (Ω)] Θ . Theorem 1.3.
Let Ω ⊂ R d be as in Theorem 1.1. Let the parameters satisfy the conditions(a), (b), (c) and (e). In addition we require(d’) s , s ∈ R and s − du > s − du . (1.8) Then (1.4) holds as well.
Clearly, in Theorem 1.3 we always have s > s . So there is no overlap with Theorem1.1. For convenience of the reader we add the consequences for the interpolation of Sobolev-Morrey spaces. Corollary 1.4.
Let < Θ < , < p < p < ∞ , p < u < ∞ , p < u < ∞ and p u = p u . Let m ∈ N , m ∈ N and m − du > m − du . We define s := (1 − Θ) m + Θ m , p := 1 − Θ p + Θ p and u := 1 − Θ u + Θ u . Let Ω ⊂ R d be either a bounded Lipschitz domain if d ≥ or a bounded interval if d = 1 .Then (1.6) holds. In particular, if s = m ∈ N , then also (1.7) is true. The formula (1.4) does not hold in general. There are many counterexamples.
Proposition 1.5.
Let Ω ⊂ R d be a domain. We assume that(a) ≤ p < p < ∞ , p < u < ∞ , p < u < ∞ ;(b) ≤ q , q ≤ ∞ ;(c) p u = p u .If < s < d/u and if s := s − d (cid:16) u − u (cid:17) > , (1.9) then with < Θ < , p := − Θ p + Θ p , u := − Θ u + Θ u , q := − Θ q + Θ q and s := (1 − Θ) s + Θ s it holds [ E s u ,p ,q (Ω) , E s u ,p ,q (Ω)] Θ ⋄ E su,p,q (Ω) . (1.10)3inally, we add a few comments concerning the situation on R d . • The conditions s , s ∈ R , 1 ≤ q , q ≤ ∞ together with (a),(c),(e) from Theorem1.1 guarantee the continuous embedding[ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ ֒ → E su,p,q ( R d ) . (1.11)We refer to Yang, Yuan, Zhuo [62]. • If 1 < p ≤ u < ∞ , 1 < q < q ≤ ∞ , s ∈ R , 0 < Θ < q := − Θ q + Θ q , then[ E su,p,q ( R d ) , E su,p,q ( R d )] Θ = E su,p,q ( R d ) (1.12)holds. We refer to Sawano and Tanaka [48].We supplement these assertions by one negative and one positive result. Proposition 1.6. (i)
Let s and s be positive real numbers. Let the conditions (a), (b),(c) and (e) from Theorem 1.1 be satisfied. Then ⋄ E su,p,q ( R d ) [ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ . (ii) Under the same restrictions as in Theorem 1.3 we have [ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ ֒ → ⋄ E su,p,q ( R d ) . This supplements the knowledge about Morrey spaces since it holds ⋄ M up ( R d ) [ M u p ( R d ) , M u p ( R d )] Θ ֒ → ⋄ M up ( R d ) , if 1 ≤ p < p < ∞ , p < u < ∞ , p < u < ∞ and p u = p u , see [66, Cor. 2.38]. Soall in all the general picture concerning complex interpolation of Lizorkin-Triebel-Morreyspaces seems to be more complicated than expected. Below we have tried to make thesituation on domains a bit more transparent. We shall plot an (1 /u, s ) diagram. Theinfluence of the parameters p , q , p , q is ignored. First we fix a point (1 /u , s ). Thenwe have indicated for which regions in the plane we may apply either Theorem 1.1 orTheorem 1.3 or Proposition 1.5. ts s = d/us /u
10 1 /u Theorem 1.1open Theorem 1.3The point t is given by the Sobolev-type embedding as t := s − d/u . In the open rectangle4 (1 /u , s ) : u < u , s > s } we can apply Theorem 1.1. In the open triangle withcorner points (0 , t ), (1 /u , s ), (0 , s ) we do not know [ E s u ,p ,q (Ω) , E s u ,p ,q (Ω)] Θ . Belowof the line connecting (0 , t ) and (1 /u , s ) we may apply Theorem 1.3. On this criticalline Proposition 1.5 applies.This article is organized as follows. In Section 2 we recall the definitions of Morrey spacesand Lizorkin-Triebel-Morrey spaces on the Euclidean space R d as well as on domains.In addition we introduce the diamond spaces. Moreover, a few basic properties of theseclasses are recalled as well.Section 3 is the most important one within this paper. We investigate the spaces ⋄ E su,p,q ( R d ) in detail. In the Lemmas 3.12 and 3.13 we characterize this space via differences,which is very important for us. We use this characterization to prove an embeddingproperty on the intersection of Lizorkin-Triebel-Morrey spaces in Lemma 3.17 below.Section 4 is devoted to the existence of an universal bounded linear extension operatorwhich maps E su,p,q (Ω) into E su,p,q ( R d ). Here we employ Rychkov’s method and construction.Interpolation will be the main topic in Section 5. Our treatment of the complexinterpolation of Lizorkin-Triebel-Morrey spaces will be reduced to the calculation of aclosure of some intersections by means of a formula due to Shestakov [49], [50].For convenience of the reader, in Section 6 we will give a short overview about inter-polation of smoothness Morrey spaces.Finally, in Section 7 a number of open problems is collected.But at first we want to fix some notation. Notation
For any x ∈ R d and r ∈ (0 , ∞ ) we use B ( x, r ) to denote the ball in R d centered at x with radius r , namely, B ( x, r ) := { y ∈ R d : | x − y | < r } . If α = ( α , . . . , α d ) ∈ N d and f : Ω → C , then we put D α f ( x ) = ∂ | α | f∂x α . . . x α d d ( x ) , x ∈ Ω . For a domain Ω ⊂ R d we define D (Ω) as the set of all functions f having derivatives up toany order and fulfill supp f ⊂ Ω. D ′ (Ω) is the dual space of D (Ω). The symbol L ( X → Y )denotes the set of all linear bounded operators from X to Y . By C ∞ ( R d ) we denotethe collection of all complex-valued infinitely differentiable functions on R d , by C ∞ ( R d )the subset consisting of those elements having compact support. Let S ( R d ) denote theSchwartz space of all complex-valued, rapidly decreasing and infinitely differentiable func-tions on R d . By S ′ ( R d ) we denote the collection of all tempered distributions on R d , i.e.,the topological dual of S ( R d ), equipped with the weak- ∗ topology. The symbol F refersto the Fourier transform, F − to its inverse transformation, both defined on S ′ ( R d ). Allfunction spaces which we consider in this paper are subspaces of S ′ ( R d ), i.e. spaces ofequivalence classes w.r.t. almost everywhere equality. However, if such an equivalenceclass contains a continuous representative, then usually we work with this representativeand call also the equivalence class a continuous function. The symbols C, C , c, c . . . de-note positive constants that depend only on the fixed parameters d, s, u, p, q and probably5n auxiliary functions. Unless otherwise stated their values may vary from line to line.Sometimes we also use the symbol . instead of ≤ . The meaning of A . B is given by:there exists a positive constant C such that A ≤ C B . Mainly in Section 4 we will use theabbreviation (with modification if q = ∞ ) k { f j } ∞ j =0 |M up ( ℓ sq ( R d )) k := (cid:13)(cid:13)(cid:13)(cid:16) ∞ X j =0 | js f j ( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) . In this section we recall the definitions of the function spaces under consideration.
Morrey spaces can be understood as a replacement (or a generalization) of the Lebesguespaces L p ( R d ). This is immediate in view of their definition. Definition 2.1.
Let ≤ p ≤ u < ∞ . Then the Morrey space M up ( R d ) is defined as thecollection of all locally Lebesgue-integrable functions f on R d such that k f |M up ( R d ) k := sup B | B | u − p (cid:20)Z B | f ( x ) | p dx (cid:21) p < ∞ , (2.1) where the supremum is taken over all balls B in R d . Clearly, there is a big difference between the cases | B | > | B | ≤
1. We have astrong local condition combined with a weak global condition. Later we shall need someknowledge about certain subspaces of Morrey spaces. Therefore we give the followingdefinition.
Definition 2.2.
Let X be a Banach space of distributions or functions. (i) By ⋄ X we denote the closure in X of the set of all infinitely often differentiablefunctions f that fulfill D α f ∈ X for all α ∈ N d . (ii) Let C ∞ ( R d ) ֒ → X . Then by ˚ X we denote the closure of C ∞ ( R d ) in X . The next lemma gives explicit descriptions of ˚ M up ( R d ) and ⋄ M up ( R d ), very much in thespirit of the original definition of Morrey spaces, see [66, Lemma 2.33]. Lemma 2.3.
Let ≤ p < u < ∞ . (i) ˚ M up ( R d ) is equal to the collection of all f ∈ M up ( R d ) having the following properties: lim r ↓ | B ( y, r ) | u − p "Z B ( y,r ) | f ( x ) | p dx p = 0 , (2.2)lim r →∞ | B ( y, r ) | u − p "Z B ( y,r ) | f ( x ) | p dx p = 0 , (2.3) both uniformly in y ∈ R d , and lim | y |→∞ | B ( y, r ) | u − p "Z B ( y,r ) | f ( x ) | p dx p = 0 (2.4)6 niformly in r ∈ (0 , ∞ ) . (ii) ⋄ M up ( R d ) is equal to the collection of all f ∈ M up ( R d ) such that (2.2) holds true uni-formly in y ∈ R d . R d In what follows we will define the Lizorkin-Triebel-Morrey spaces E su,p,q ( R d ). For thatpurpose we need some additional notation. Let ϕ ∈ C ∞ ( R d ) be a non-negative functionsuch that ϕ ( x ) = 1 if | x | ≤ ϕ ( x ) = 0 if | x | ≥ /
2. For k ∈ N we define ϕ k ( x ) = ϕ (2 − k x ) − ϕ (2 − k +1 x ) , x ∈ R d . Because of ∞ X k =0 ϕ k ( x ) = 1 , x ∈ R d , and supp ϕ k ⊂ (cid:8) x ∈ R d : 2 k − ≤ | x | ≤ · k − (cid:9) , k ∈ N , we call the system ( ϕ k ) k ∈ N a smooth dyadic decomposition of unity on R d . Clearly, bythe Paley-Wiener-Schwarz theorem, F − [ ϕ k F f ] is a smooth function for all f ∈ S ′ ( R d ). Definition 2.4.
Let ≤ p ≤ u < ∞ , ≤ q ≤ ∞ and s ∈ R . Let ( ϕ k ) k ∈ N be the abovesystem. Then the Lizorkin-Triebel-Morrey space E su,p,q ( R d ) is the collection of all tempereddistributions f ∈ S ′ ( R d ) such that k f |E su,p,q ( R d ) k ϕ := (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ∞ X k =0 ksq |F − [ ϕ k F f ]( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) < ∞ (with usual modification if q = ∞ ). Remark 2.5.
The spaces E su,p,q ( R d ) are Banach spaces. They do not depend on the chosengenerator ϕ of the smooth dyadic decomposition of unity in the sense of equivalent norms.We refer, e.g., to [64] or [61]. For this reason we will drop the dependence on ϕ in notationsand simply write k · |E su,p,q ( R d ) k .Now we want to collect some basic properties of the spaces E su,p,q ( R d ). Most of themwill be used later. At first we recall a characterization of E su,p,q ( R d ), due to Tang and Xu[56], in terms of lower order derivatives. Lemma 2.6.
Let m ∈ N , s ∈ R , ≤ p ≤ u < ∞ and ≤ q ≤ ∞ . Then we have f ∈ E su,p,q ( R d ) if, and only if, the tempered distribution f and its distributional derivatives ∂ m f∂x mj , j = 1 , . . . , d , belong to E s − mu,p,q ( R d ) . Furthermore, the norms k f |E su,p,q ( R d ) k and k f |E s − mu,p,q ( R d ) k + d X j =1 (cid:13)(cid:13)(cid:13) ∂ m f∂x mj (cid:12)(cid:12)(cid:12) E s − mu,p,q ( R d ) (cid:13)(cid:13)(cid:13) are equivalent. The classical forerunner of Lemma 2.6 can be found in [58, Thm. 2.3.8].7 emma 2.7.
Let s ∈ R , ≤ p ≤ u < ∞ and ≤ q ≤ ∞ . If f ∈ E su,p,q ( R d ) , then D α f ∈ E s −| α | u,p,q ( R d ) for all α ∈ N d . Furthermore, there exists a constant c α such that k D α f |E s −| α | u,p,q ( R d ) k ≤ c α k f |E su,p,q ( R d ) k holds for all f ∈ E su,p,q ( R d ) . In case u = p this can be found in [58, Thm. 2.3.8]. The generalization to p = u can bedone in the same way as the proof of Lemma 2.6.Lizorkin-Triebel-Morrey spaces are generalizations of Sobolev-Morrey spaces. Definition 2.8.
Let m ∈ N and ≤ p ≤ u < ∞ . Then the Sobolev-Morrey space W m M up ( R d ) is the collection of all functions f ∈ M up ( R d ) such that all distributionalderivatives D α f of order | α | ≤ m belong to M up ( R d ) . We put k f | W m M up ( R d ) k := X | α |≤ m k D α f |M up ( R d ) k . It will be convenient to use W M up ( R d ) := M up ( R d ). Lemma 2.9.
Let < p ≤ u < ∞ and m ∈ N . Then E mu,p, ( R d ) = W m M up ( R d ) in thesense of equivalent norms.Proof. Mazzucato has proved the Littlewood-Paley characterization of Morrey spaces in[39], i.e., she proved that E u,p, ( R d ) = M up ( R d ), 1 < p ≤ u < ∞ , holds in the sense ofequivalent norms. Combined with Lemma 2.6 this proves Lemma 2.9. Remark 2.10.
The spaces E su,p, ( R d ) with 1 < p < u < ∞ and s ∈ R are investigated inAdams [1], see also Adams, Xiao [2] and Triebel [61, Rem. 3.68].For the next result we refer to [64, Prop. 2.6]. Lemma 2.11.
Let s ∈ R , ≤ p ≤ u < ∞ and ≤ q ≤ ∞ . Then E su,p,q ( R d ) ֒ → B s − d/u ∞ , ∞ ( R d ) . (2.5) Remark 2.12. (i) It is well-known that B s − d/u ∞ , ∞ ( R d ) ֒ → L ∞ ( R d ) holds if s > d/u .(ii) Also in case of the Sobolev-Morrey spaces W m M up ( R d ) one knows that W m M u ( R d ) ֒ → L ∞ ( R d ) if m > d/u , see [15]. An important inequality
Later on we shall need the following lemma, see [47, Thm. 2.4]. For ν ∈ R let H ν ( R d )denote the Bessel-potential space, defined as the collection of all f ∈ S ′ ( R d ) with k f | H ν ( R d ) k = k (1 + | · | ) ν ( F f )( · ) | L ( R d ) k < ∞ . Lemma 2.13.
Let ≤ q ≤ ∞ , ≤ p ≤ u < ∞ and ν > d . Let ( R j ) ∞ j =0 ⊂ [1 , ∞ ) .Suppose ( h j ) ∞ j =0 ⊂ H ν ( R d ) and ( f j ) ∞ j =0 ⊂ M up ( R d ) such that supp F f j ⊂ B (0 , R j ) . Thenthere is a constant c > , independent of ( R j ) ∞ j =0 , ( h j ) ∞ j =0 and ( f j ) ∞ j =0 , such that (cid:13)(cid:13)(cid:13)(cid:16) ∞ X j =0 |F − [ h j F f j ]( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c (cid:16) sup j ∈ N k h j ( R j · ) | H ν ( R d ) k (cid:17) (cid:13)(cid:13)(cid:13)(cid:16) ∞ X j =0 | f j ( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) olds. Remark 2.14.
Those vector-valued Fourier multiplier assertions are standard tools inthe theory of function spaces, see, e.g, [58, 1.6.3] for the classical case p = u . In our article spaces on domains are defined by restrictions. For us this is the mostconvenient way. Here, for all domains Ω ⊂ R d and g ∈ S ′ ( R d ) by g | Ω we denote therestriction of g to Ω. Definition 2.15.
Let X ( R d ) be a normed space of tempered distributions such that X ( R d ) ֒ → S ′ ( R d ) . Let Ω denote an open, nontrivial subset of R d . Then X (Ω) is de-fined as the collection of all f ∈ D ′ (Ω) such that there exists a distribution g ∈ X ( R d ) satisfying f ( ϕ ) = g ( ϕ ) for all ϕ ∈ D (Ω) . Here ϕ ∈ D (Ω) is extended by zero on R d \ Ω . We put k f | X (Ω) k := inf n k g | X ( R d ) k : g | Ω = f o . Clearly, in the case of Morrey spaces this means the following.Let 1 ≤ p ≤ u < ∞ and Ω ⊂ R d be bounded. Then the Morrey space M up (Ω) is thecollection of all f ∈ L ℓocp (Ω) such that k f |M up (Ω) k := sup x ∈ Ω sup r ∈ (0 , ∞ ) | B ( x, r ) ∩ Ω | u − p "Z B ( x,r ) ∩ Ω | f ( y ) | p dy p < ∞ . In this paper we will concentrate on Lipschitz domains Ω ⊂ R d . We follow Stein, see[55, VI.3.2]. Definition 2.16.
By a Lipschitz domain, we mean either a special or a bounded Lipschitzdomain. (i) A special Lipschitz domain is an open set Ω ⊂ R d lying above the graph of a Lipschitzfunction ω : R d − → R , namely, Ω := { ( x ′ , x d ) ∈ R d : x d > ω ( x ′ ) } , where ω satisfies that, for all x ′ , y ′ ∈ R d − , | ω ( x ′ ) − ω ( y ′ ) | ≤ A | x ′ − y ′ | with a positive constant A independent of x ′ and y ′ . (ii) A bounded Lipschitz domain is a bounded domain Ω ⊂ R d whose boundary ∂ Ω can becovered by a finite number of open balls B k such that, for each k , after a suitable rotation, ∂ Ω ∩ B k is a part of the graph of a Lipschitz function. For notational simplicity we shall use the convention, that a bounded Lipschitz domainin R is just a bounded interval. 9 The diamond space associated to E su,p,q ( R d ) In this section we will investigate the properties of the spaces ⋄ E su,p,q ( R d ), see Definition2.2. This is very important in order to prove our main results. First, we recall two resultsfrom [66], see Lemmas 2.25. and 2.26. Lemma 3.1.
Let s ∈ R , ≤ p ≤ u < ∞ and ≤ q ≤ ∞ . Then ˚ E su,p,q ( R d ) = ⋄ E su,p,q ( R d ) ifand only if u = p . Even more important is the following.
Lemma 3.2.
Let s ∈ R , ≤ p ≤ u < ∞ and ≤ q ≤ ∞ . Then ⋄ E su,p,q ( R d ) = E su,p,q ( R d ) if and only if u = p and q ∈ [1 , ∞ ) . Remark 3.3.
In particular this implies ⋄ F sp,q ( R d ) = ˚ F sp,q ( R d ) = F sp,q ( R d ) if 1 ≤ p, q < ∞ .Now we turn to some further descriptions of the diamond spaces. The diamond spacesare defined as a closure. So it is most natural to look for characterizations in form ofapproximations. A first characterization is using the Littlewood-Paley decomposition. Let ( ϕ j ) ∞ j =0 be a smooth dyadic decomposition of unity. Then we put S N f ( x ) := N X j =0 F − [ ϕ j F f ]( x ) , N ∈ N . Of course, by the Paley-Wiener-Schwarz Theorem, S N f are smooth functions. Lemma 3.4.
Let ≤ p ≤ u < ∞ , ≤ q ≤ ∞ and s ∈ R . Let f ∈ E su,p,q ( R d ) . Then thesequence ( S N f ) ∞ N =0 has the following properties:(i) S N f ∈ E σu,p,q ( R d ) for all σ ∈ R .(ii) For all α ∈ N d we have D α ( S N f ) ∈ E su,p,q ( R d ) .(iii) For all α ∈ N d we have D α ( S N f ) ∈ L ∞ ( R d ) .(iv) The following identity holds S N f ( x ) = F − [ ϕ (2 − N ξ ) F f ( ξ )]( x ) , x ∈ R d , N ∈ N . (v) There exists a constant c , independent on f , such that sup N ∈ N k S N f |E su,p,q ( R d ) k ≤ c k f |E su,p,q ( R d ) k . (3.1) Proof.
Part (i) is a consequence of the estimate k S N f |E σu,p,q ( R d ) k ≤ c (cid:13)(cid:13)(cid:13)(cid:16) N +1 X j =0 jσq |F − [ ϕ j F f ]( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) c independent of f and N ∈ N , see Lemma 2.13. From (i) we derive that S N f ∈ E s + mu,p,q ( R d ) with m ∈ N . Next we use Lemma 2.6 obtaining D α ( S N f ) ∈ E su,p,q ( R d )for | α | = m . To show (iii) it is enough to apply Lemma 2.11. The next part (iv) isan elementary conclusion of the definition of the functions ϕ j with j ∈ { , , . . . , N } .Finally, (v) follows from the generalized Minkowski inequality combined with a standardconvolution inequality: k S N f |E su,p,q ( R d ) k = k F − [ ϕ (2 − N ξ ) F f ( ξ )]( · ) |E su,p,q ( R d ) k≤ k F − ϕ | L ( R d ) k k f |E su,p,q ( R d ) k . The proof is complete.Associated to the definition of ⋄ E su,p,q ( R d ) we need a further abbreviation. Definition 3.5.
Let ≤ p < u < ∞ , ≤ q ≤ ∞ and s ≥ . The set E su,p,q ( R d ) is thecollection of all functions f ∈ E su,p,q ( R d ) such that D α f ∈ E su,p,q ( R d ) for all α ∈ N d . As an immediate consequence we get E su,p,q ( R d ) k · |E su,p,q ( R d ) k = ⋄ E su,p,q ( R d ) . Moreover, by Lemma 3.4, for any f ∈ E su,p,q ( R d ) we have S N f ∈ E su,p,q ( R d ). This will beof some use later on. Proposition 3.6.
Let ≤ p ≤ u < ∞ , ≤ q ≤ ∞ and s ∈ R . Then ⋄ E su,p,q ( R d ) is thecollection of all f ∈ E su,p,q ( R d ) such that lim N →∞ k f − S N f |E su,p,q ( R d ) k = 0 . (3.2) Proof.
Clearly, if (3.2) holds, then f ∈ ⋄ E su,p,q ( R d ) follows.Now, let us suppose that f ∈ ⋄ E su,p,q ( R d ). By ( f ℓ ) ℓ we denote a sequence in E su,p,q ( R d ) suchthat lim ℓ →∞ k f − f ℓ |E su,p,q ( R d ) k = 0 . Without loss of generality we may assume k f − f ℓ |E su,p,q ( R d ) k < ℓ , ℓ ∈ N . Let σ ∈ R with σ > s . We use a standard Fourier multiplier assertion from Lemma 2.13.Then we obtain k f ℓ − S N f ℓ |E su,p,q ( R d ) k = (cid:13)(cid:13)(cid:13) ∞ X j = N +1 F − [ ϕ j F f ℓ ]( · ) (cid:12)(cid:12)(cid:12) E su,p,q ( R d ) (cid:13)(cid:13)(cid:13) ≤ c (cid:13)(cid:13)(cid:13) (cid:16) ∞ X j = N jsq |F − [ ϕ j F f ℓ ]( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c N ( s − σ ) (cid:13)(cid:13)(cid:13) (cid:16) ∞ X j = N jσq |F − [ ϕ j F f ℓ ]( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c N ( s − σ ) k f ℓ |E σu,p,q ( R d ) k . N →∞ k f ℓ − S N f ℓ |E su,p,q ( R d ) k = 0 for any ℓ ∈ N . Hence, for ℓ ∈ N there exists some N ( ℓ ) such that k f ℓ − S N ( ℓ ) f ℓ |E su,p,q ( R d ) k < ℓ . This yields k f − S N ( ℓ ) f |E su,p,q ( R d ) k ≤ k f − f ℓ |E su,p,q ( R d ) k + k f ℓ − S N ( ℓ ) f ℓ |E su,p,q ( R d ) k + k S N ( ℓ ) f ℓ − S N ( ℓ ) f |E su,p,q ( R d ) k≤ ℓ + k S N ( ℓ ) ( f ℓ − f ) |E su,p,q ( R d ) k≤ cℓ , where c is the constant from (3.1). Hence, we have the convergence of an appropriatesubsequence ( S N ( ℓ ) f ) ∞ ℓ =1 . It remains to switch from a subsequence to the whole sequence.Therefore we assume that our sequence ( N ( ℓ )) ℓ satisfies N ( ℓ + 1) − N ( ℓ ) > ℓ. Furthermore we will use the following identity (cid:13)(cid:13)(cid:13) N X j = M F − [ ϕ j F f ] (cid:12)(cid:12)(cid:12) E su,p,q ( R d ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) N − X m = M +1 msq |F − [ ϕ m F f ]( · ) | q + 2 Msq |F − [ ϕ M ( ϕ M + ϕ M +1 ) F f ]( · ) | q + 2 ( M − sq |F − [ ϕ M − ϕ M F f ]( · ) | q + 2 Nsq |F − [ ϕ N ( ϕ N − + ϕ N ) F f ]( · ) | q + 2 ( N +1) sq |F − [ ϕ N +1 ϕ N F f ]( · ) | q (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) , valid for all natural numbers M and N such that 2 ≤ M + 1 < N −
1. This follows from ϕ m · (cid:16) N X j = M ϕ j (cid:17) = ϕ m if M < m < N ; ϕ M − ϕ M if m = M − ϕ M ( ϕ M + ϕ M +1 ) if m = M ; ϕ N ( ϕ N − + ϕ N ) if m = N ; ϕ N +1 ϕ N if m = N + 1;0 otherwise . A standard convolution inequality combined with the generalized Minkowski inequalityyields k F − [ ϕ j ϕ ℓ F f ]( · ) |M up ( R d ) k≤ k F − ϕ j | L ( R d ) k k F − [ ϕ ℓ F f ]( · ) |M up ( R d ) k . Applying a homogeneity argument we find k F − ϕ j | L ( R d ) k = k F − ϕ | L ( R d ) k , j ∈ N . (cid:13)(cid:13)(cid:13)(cid:16) N − X m = M +1 msq |F − [ ϕ m F f ]( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) N X j = M F − [ ϕ j F f ] (cid:12)(cid:12)(cid:12) E su,p,q ( R d ) (cid:13)(cid:13)(cid:13) ≤ c (cid:13)(cid:13)(cid:13)(cid:16) N +1 X m = M − msq |F − [ ϕ m F f ]( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) (3.3)with some constant c independent on M, N and f . Let1 ≤ N ( ℓ ) ≤ M − < N − < N + 2 ≤ N ( ℓ + 1) . Then (3.3) implies k S N f − S M − f |E su,p,q ( R d ) k = (cid:13)(cid:13)(cid:13) N X j = M F − [ ϕ j F f ] (cid:12)(cid:12)(cid:12) E su,p,q ( R d ) (cid:13)(cid:13)(cid:13) ≤ c (cid:13)(cid:13)(cid:13)(cid:16) N +1 X m = M − msq |F − [ ϕ m F f ] | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c (cid:13)(cid:13)(cid:13) N ( ℓ +1) X j = N ( ℓ )+1 F − [ ϕ j F f ] (cid:12)(cid:12)(cid:12) E su,p,q ( R d ) (cid:13)(cid:13)(cid:13) = c k S N ( ℓ +1) f − S N ( ℓ ) f |E su,p,q ( R d ) k . Repeating the argument we conclude that k S N f − S M − f |E su,p,q ( R d ) k ≤ c k S N ( ℓ +2) f − S N ( ℓ − f |E su,p,q ( R d ) k for all M, N such that N ( ℓ ) ≤ M ≤ N ≤ N ( ℓ + 1) with c independent of ℓ . Consequently( S N f ) ∞ N =0 is a Cauchy sequence in E su,p,q ( R d ). This proves the claim. Remark 3.7.
Proposition 3.6 is not new, we refer to Hakim, Nogayama and Sawano [26,Thm. 1.1]. However, our proof is slightly different and covers the cases p = 1 ≤ u < ∞ .In fact, it extends without any change to 0 < p ≤ u < ∞ .Later on we shall need the following consequence of Proposition 3.6. Proposition 3.8.
Let ≤ p ≤ u < ∞ , ≤ q , q ≤ ∞ and s , s ∈ R with s < s . Thenwe have the continuous embedding E s u,p,q ( R d ) ֒ → ⋄ E s u,p,q ( R d ) . Proof.
Let f ∈ E s u,p,q ( R d ). Lemma 2.13 yields k f − S N f |E s u,p,q ( R d ) k ≤ c (cid:13)(cid:13)(cid:13)(cid:16) ∞ X j = N js q |F − [ ϕ j F f ] | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c − ( s − s ) N (cid:13)(cid:13)(cid:13) sup j ≥ N js |F − [ ϕ j F f ] | (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) with constants c , c independent of f and N . Because of E s u,p,q ( R d ) ֒ → E s u,p, ∞ ( R d ), for N → ∞ this implies (3.2) and therefore f ∈ ⋄ E s u,p,q ( R d ).13 .2 A characterization using mollifiers It is possible to describe the spaces ⋄ E su,p,q ( R d ) by using mollifiers. In this section we willbriefly collect the main ideas concerning that topic. For that purpose we need some morenotation. Therefore let ̺ ∈ D ( R d ) be a function satisfying Z R d ̺ ( x ) dx = 1 and supp ̺ ⊂ B (0 , . We put ρ j ( x ) := 2 jd ̺ (2 j x ) with x ∈ R d and j ∈ N . For a Banach space X that iscontinuously embedded into S ′ ( R d ) we define X ℓoc as the collection of all f ∈ S ′ ( R d ) suchthat the pointwise product fulfills ψ · f ∈ X for all ψ ∈ D ( R d ). Convergence of a sequence { f j } ∞ j =1 with limit f in X ℓoc is defined aslim j →∞ k f ψ − f j ψ | X k = 0 for all ψ ∈ D ( R d ) . Lemma 3.9.
Let ≤ p ≤ u < ∞ , ≤ q ≤ ∞ and s > . Let f ∈ E su,p,q ( R d ) . Then thesequence { f ∗ ρ j } ∞ j =1 has the following properties:(i) For all α ∈ N d and all j ∈ N we have D α ( f ∗ ρ j ) ∈ E su,p,q ( R d ) , i.e., ( f ∗ ρ j ) ∈ E su,p,q ( R d ) .(ii) For all α ∈ N d and all j ∈ N we have D α ( f ∗ ρ j ) ∈ L ∞ ( R d ) .(iii) For all j ∈ N we have ( f ∗ ρ j ) ∈ C ∞ ( R d ) .(iv) For all j ∈ N we have ( f ∗ ρ j ) ∈ E σu,p,q ( R d ) for all σ ∈ R .(v) There exists a constant c , independent on f , such that sup j ∈ N k f ∗ ρ j |E su,p,q ( R d ) k ≤ c k f |E su,p,q ( R d ) k . (3.4)Essentially all of Lemma 3.9 is known. So we skip the proof. There is a counterpart ofProposition 3.6 that reads as follows. Proposition 3.10.
Let ≤ p ≤ u < ∞ , ≤ q ≤ ∞ and s > . Let f ∈ E su,p,q ( R d ) . Thenthe following assertions are equivalent.(i) f ∈ ⋄ E su,p,q ( R d ) ;(ii) lim j →∞ k f ∗ ̺ j − f |E su,p,q ( R d ) k = 0 . We will not use Proposition 3.10 in what follows. Therefore we will drop the proof.
It is possible to describe the spaces E su,p,q ( R d ) and also the spaces ⋄ E su,p,q ( R d ) in terms ofdifferences. For that purpose we need some additional notation. Let f : R d → C be afunction. Then for x, h ∈ R d we define the difference of the first order by ∆ h f ( x ) := f ( x + h ) − f ( x ). Let N ∈ N with N >
1. Then we define the difference of the order N by ∆ Nh f ( x ) := (∆ h (∆ N − h f ))( x ). Now at first we recall the following characterization of E su,p,q ( R d ). We refer to [28] and [64, 4.3.1]. 14 roposition 3.11. Let ≤ p < u < ∞ , ≤ q ≤ ∞ and s > . Let N ∈ N such that s < N . Then E su,p,q ( R d ) is the collection of all f ∈ M up ( R d ) such that k f |M up ( R d ) k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) is finite (with equivalent norms). In the case q = ∞ the usual modifications have to bemade. Next we turn to the spaces ⋄ E su,p,q ( R d ). Very much in the spirit of Lemma 2.3 is thefollowing observation. Lemma 3.12.
Let ≤ p < u < ∞ , ≤ q ≤ ∞ and s > . Let N ∈ N such that s < N .Then ⋄ E su,p,q ( R d ) is contained in the set of all f ∈ E su,p,q ( R d ) such that lim r ↓ | B ( y, r ) | u − p (cid:16) Z B ( y,r ) | f ( x ) | p dx (cid:17) p = 0 (3.5) and lim r ↓ | B ( y, r ) | u − p (cid:20) Z B ( y,r ) (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p = 0 , (3.6) both uniformly in y ∈ R d .Proof. Step 1. In a first step we deal with functions f belonging to E su,p,q ( R d ). Clearly,those functions are uniformly Lipschitz continuous on R d , see the proof of Lemma 3.4. Tosee (3.5) in this situation we argue as follows. Obviously we have (cid:16) Z B ( y,r ) | f ( x ) | p dx (cid:17) p ≤ k f | L ∞ ( R d ) k | B ( y, r ) | p . Multiplying this inequality by | B ( y, r ) | /u − /p it follows for u < ∞ that the right-handside tends to 0 (uniformly in y ) if r ↓
0. The argument for deriving (3.6) is quite similar.Recall that for a smooth function we have with N ∈ N | ∆ Nh f ( x ) | ≤ c (cid:16) max | α |≤ N sup y ∈ R d | D α f ( y ) | (cid:17) | h | N , x, h ∈ R d , with a constant c independent of f, x and h . Hence (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) q ≤ c (cid:16) Z t − sq t Nq dtt (cid:17) q ≤ c < ∞ for some c independent of x . This implies | B ( y, r ) | u − p (cid:20) Z B ( y,r ) (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p ≤ c | B ( y, r ) | u and therefore the claim follows. Step 2.
Now we turn to the general case. Let f ∈ ⋄ E su,p,q ( R d ) and let ε > M ∈ N it follows | B ( y, r ) | u − p (cid:20) Z B ( y,r ) (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p ≤ k f − S M f |E su,p,q ( R d ) k + | B ( y, r ) | u − p (cid:20) Z B ( y,r ) (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh ( S M f )( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p . The second term on the righ-hand side becomes smaller than ε > r ≤ r ( ε ) since S M f ∈ E su,p,q ( R d ) and therefore we may use Step 1. The first term on the right-hand sidewill be smaller than ε > M ≥ M ( ε ) thanks to Proposition 3.6. Both statements holduniformly in y . This proves (3.6). The convergence in (3.5) can be proved in a similarway.Now we turn to the converse. Lemma 3.13.
Let ≤ p < u < ∞ , ≤ q < ∞ and s > . Let N ∈ N such that s < N . Let f ∈ E su,p,q ( R d ) be a function with compact support and such that (3.5) , (3.6) hold uniformly in y ∈ R d . Then f ∈ ⋄ E su,p,q ( R d ) .Proof. Because of Proposition 3.6 it is enough to provelim M →∞ k f − S M f |E su,p,q ( R d ) k = 0 . Using Proposition 3.11 this can be reduced to showlim M →∞ k f − S M f |M up ( R d ) k = 0 (3.7)and lim M →∞ (cid:13)(cid:13)(cid:13)(cid:16) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh ( f − S M f )( x ) | dh (cid:17) q dtt (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) = 0 . (3.8) Step 1.
We shall show (3.7). Let f ∈ E su,p,q ( R d ). Let M ∈ N and 0 < σ < s . Then we find k f − S M f |M up ( R d ) k ≤ (cid:13)(cid:13)(cid:13) ∞ X j = M +1 |F − [ ϕ j F f ]( · ) | (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c − Mσ (cid:13)(cid:13)(cid:13) ∞ X j = M +1 jσ |F − [ ϕ j F f ]( · ) | (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c − Mσ k f |E σu,p, ( R d ) k , ≤ c − Mσ k f |E su,p,q ( R d ) k . We used Definition 2.4. Here c is independent of f and M ∈ N . So because of f ∈E su,p,q ( R d ) if M tends to infinity (3.7) follows. Step 2.
Next we prove (3.8). Let B stand for every ball in R d . Since f satisfies (3.6), forevery ε >
0, we find some δ > | B | <δ | B | u − p (cid:20) Z B (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p ≤ ε. (cid:20) Z B (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh S M f ( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p ≤ c (cid:20) Z B (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p with c independent of B and f . Consequently we get (cid:13)(cid:13)(cid:13)(cid:16) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh ( f − S M f )( x ) | dh (cid:17) q dtt (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) (3.9) ≤ c ε + sup | B |≥ δ | B | u − p (cid:20) Z B (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh ( f − S M f )( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p . Since f ∈ E su,p,q ( R d ) the supremum on the right-hand side is finite. By the definition of thesupremum there exists a sequence of balls B j := B ( y j , r j ) with j ∈ N and | B ( y j , r j ) | ≥ δ such thatsup | B |≥ δ | B | u − p (cid:20) Z B (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh ( f − S M f )( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p ≤ j + | B j | u − p (cid:20) Z B j (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh ( f − S M f )( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p ≤ j + | B j | u − p (cid:20) Z R d (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh ( f − S M f )( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p ≤ j + c δ u − p k f − S M f | F sp,q ( R d ) k . (3.10)Here in the last step we used Proposition 3.11 for the original Lizorkin-Triebel spaces F sp,q ( R d ), i.e., in case p = u , see also [58, 2.5.11]. Substep 2.1 . We claim that a function f ∈ E su,p,q ( R d ) with compact support belongs to F sp,q ( R d ) as well. We may assume supp f ⊂ B (0 , R ) for some R >
1. Based on Proposition3.11 we observe that k f | L p ( R d ) k + (cid:13)(cid:13)(cid:13)(cid:16) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:17) q (cid:12)(cid:12)(cid:12) L p ( R d ) (cid:13)(cid:13)(cid:13) = k f | L p ( B (0 , R )) k + (cid:13)(cid:13)(cid:13)(cid:16) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:17) q (cid:12)(cid:12)(cid:12) L p ( B (0 , R + N )) (cid:13)(cid:13)(cid:13) ≤ | B (0 , R ) | − u + p k f |M up ( R d ) k + | B (0 , R + N ) | − u + p (cid:13)(cid:13)(cid:13)(cid:16) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c ( R + N ) d ( p − u ) k f |E su,p,q ( R d ) k . (3.11)Hence f ∈ F sp,q ( R d ). Substep 2.2.
Next we shall use Lemma 3.2 and Proposition 3.6. Because of f ∈ F sp,q ( R d ) = E sp,p,q ( R d ) = ⋄ E sp,p,q ( R d ) and 1 ≤ p, q < ∞ we getlim M →∞ k f − S M f | F sp,q ( R d ) k = 0 . (3.12)17inally, we collect (3.9)-(3.12) together and find for fixed ε and associated δ (cid:13)(cid:13)(cid:13)(cid:16) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh ( f − S M f )( x ) | dh (cid:17) q dtt (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) ≤ c ε + 1 j + c δ u − p k f − S M f | F sp,q ( R d ) k ≤ c ε + 1 j if M is chosen large enough. So if j tends to infinity this proves (3.8). The proof iscomplete.To continue we have to deal with the following subspaces of E su,p,q ( R d ). Definition 3.14.
Let ≤ p < u < ∞ , ≤ q < ∞ and s > . Let B be a ball in R d . Then E su,p,q ( R d ; B ) is the collection of all f ∈ E su,p,q ( R d ) satisfying supp f ⊂ B . Putting together Lemma 3.12 and 3.13 we obtain the following theorem which will beour main tool for what follows.
Theorem 3.15.
Let ≤ p < u < ∞ , ≤ q < ∞ , s > and let B be a ball in R d .Then f ∈ E su,p,q ( R d ; B ) belongs to ⋄ E su,p,q ( R d ) if and only if (3.5) and (3.6) hold uniformlyin y ∈ R d . Intersections of Lizorkin-Triebel-Morrey spaces will play a role in the description of theinterpolation spaces, see Section 5. In particular we are interested in properties of E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ). Lemma 3.16.
Let Θ ∈ (0 , , s i ∈ R , p i ∈ [1 , ∞ ) , q i ∈ [1 , ∞ ] and u i ∈ [ p i , ∞ ) with i ∈ { , } such that s = (1 − Θ) s + Θ s , p = 1 − Θ p + Θ p , q = 1 − Θ q + Θ q and u = 1 − Θ u + Θ u . Then we have E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ) ֒ → E su,p,q ( R d ) . Proof.
Because of our assumptions and H¨older’s inequality we have k (2 js a j ) ∞ j =0 | ℓ q k ≤ k (2 js a j ) ∞ j =0 | ℓ q k − Θ k (2 js a j ) ∞ j =0 | ℓ q k Θ . This will be applied with a j := F − [ ϕ j F f ] and j ∈ N . We continue by a furtherapplication of H¨older’s inequality and find (cid:13)(cid:13)(cid:13) k (2 js a j ) ∞ j =0 | ℓ q k (cid:12)(cid:12)(cid:12) L p ( B ( y, r )) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) k (2 js a j ) ∞ j =0 | ℓ q k − Θ k (2 js a j ) ∞ j =0 | ℓ q k Θ (cid:12)(cid:12)(cid:12) L p ( B ( y, r )) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) k (2 js a j ) ∞ j =0 | ℓ q k (cid:12)(cid:12)(cid:12) L p ( B ( y, r )) (cid:13)(cid:13)(cid:13) − Θ (cid:13)(cid:13)(cid:13) k (2 js a j ) ∞ j =0 | ℓ q k (cid:12)(cid:12)(cid:12) L p ( B ( y, r )) (cid:13)(cid:13)(cid:13) Θ , which proves the claim.We need to improve Lemma 3.16. Therefore we have to accept stronger restrictions.18 emma 3.17. Let Θ ∈ (0 , , ≤ s ≤ s , ≤ p < p < ∞ , ≤ q , q ≤ ∞ , min( q , q ) < ∞ , p < u , p < u and u < u , such that s = (1 − Θ) s + Θ s , p = 1 − Θ p + Θ p , q = 1 − Θ q + Θ q and u = 1 − Θ u + Θ u . In addition we assume either s < s or < s = s and q ≤ q . Let B be a ball in R d .Then we have E s u ,p ,q ( R d ; B ) ∩ E s u ,p ,q ( R d ; B ) ֒ → ⋄ E su,p,q ( R d ) . Proof.
By Lemma 3.16 we already know that E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ) ֒ → E su,p,q ( R d ) . Now we want to employ Theorem 3.15. This is possible because we have s > q < ∞ .Let f ∈ E s u ,p ,q ( R d ; B ) ∩ E s u ,p ,q ( R d ; B ). Using p < p < p and H¨older’s inequality wefind | B ( y, r ) | u − p "Z B ( y,r ) | f ( x ) | p dx p ≤ | B ( y, r ) | u − p "Z B ( y,r ) | f ( x ) | p dx p = | B ( y, r ) | u − u | B ( y, r ) | u − p "Z B ( y,r ) | f ( x ) | p dx p ≤ | B ( y, r ) | u − u k f |M u p ( R d ) k , (3.13)which tends to zero if r → u < u < u . Now we proceed similarly with the term I ( f, y, r, s, u, p, q ) given by I ( f, y, r, s, u, p, q ) := | B ( y, r ) | u − p (cid:20) Z B ( y,r ) (cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) pq dx (cid:21) p with N > s . We observe I ( f, y, r, s, u, p, q ) ≤ | B ( y, r ) | u − p (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) L p ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ | B ( y, r ) | u − u (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f ( x ) | dh (cid:17) q dtt (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) M u p ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ c | B ( y, r ) | u − u k f |E su ,p ,q ( R d ) k≤ c | B ( y, r ) | u − u k f |E s u ,p ,q ( R d ) k , (3.14)where we used Proposition 3.11 and the elementary embedding E s u ,p ,q ( R d ) ֒ →E su ,p ,q ( R d ), see Proposition 2.1 in [64]. As in (3.13) it is obvious that the right-handside tends to zero for r → y . Hence, by Theorem 3.15, (3.13) and (3.14)we finally proved f ∈ ⋄ E su,p,q ( R d ). 19 .5 Some test functions In this subsection we shall investigate some families of test functions. There are two reasonsfor doing that. On the one hand it allows to get a feeling for the spaces under consideration.On the other hand these families will be used in the proofs of the Propositions 1.5 and 1.6.It is well-known that the function f ( x ) := | x | − d/u , x ∈ R d \ { } , is an extremal function for M up ( R d ). Here we shall deal with a few modifications of this extremal function. Withinthis subsection we shall work with a smooth cut-off function ψ supported around theorigin. More exactly, ψ ∈ C ∞ ( R d ), radial-symmetric, real-valued, 0 ≤ ψ ( x ) ≤ x , ψ ( x ) = 1 if | x | ≤ ψ ( x ) = 0 if | x | ≥ / Lemma 3.18.
Let ≤ p < u < ∞ and ≤ q ≤ ∞ . (i) Then the function h u ( x ) := (1 − ψ ( x )) | x | − du , x ∈ R d , (3.15) belongs to ⋄ W m M up ( R d ) for all m ∈ N . (ii) For all s ∈ R we have h u ∈ ⋄ E su,p,q ( R d ) .Proof. We will concentrate on (ii). Temporarily we assume s >
0. Clearly, h u is a C ∞ ( R d )function. Let α ∈ N d be a multi-index. Then we claim that D α h u belongs to E su,p,q ( R d ) forall α , i.e., we claim that h u ∈ E su,p,q ( R d ) ⊂ ⋄ E su,p,q ( R d ), see Definition 3.5. We shall workwith Proposition 3.11. Therefore we have to deal with k D α h u |M up ( R d ) k (3.16)and (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh D α h u ( x ) | dh (cid:17) q dtt (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) (3.17)with N > s . Let us start with (3.16). Since h u is smooth, estimates with respect to smallballs are no problem. By means of the radial symmetry and h u ∈ C ∞ ( R d ) elementarycalculations show that it will be sufficient to estimate I = sup r> | B (0 , r ) | u − p (cid:16) Z < | x | 1) = ∅ again some elementary calculations show that it will be sufficient to estimate I = sup r> N r du − dp (cid:16) Z N< | x | 1. Using thisand s < N we obtain I ≤ c sup r> N r du − dp (cid:16) Z N< | x | N < | x | < r we define z ′ := x ( | x |− N ) | x | . Then because of | z ′ | = | x | − N and | z ′ − x | = N we obtainsup | x − z |≤ N | z | − dpu − pN − p | α | = (cid:12)(cid:12)(cid:12) x ( | x | − N ) | x | (cid:12)(cid:12)(cid:12) − dpu − pN − p | α | = ( | x | − N ) − dpu − pN − p | α | . Now we insert this in our estimate and use | α | = M ∈ N to find I ≤ c sup r> N r du − dp (cid:16) Z N< | x | 0. In the case s ≤ ⋄ E s u,p,q ( R d ) ֒ → ⋄ E s u,p,q ( R d ) with s < s .Observe that (i) follows from (3.16).Next we consider the function f α ( x ) := ψ ( x ) | x | − α , x ∈ R d \ { } , α > . (3.20)Already in [66], page 1849, one can find that in case 1 ≤ p < u < ∞ we have f α ∈ M up ( R d )if and only if α ≤ du . Moreover we have f d/u / ∈ ⋄ M up ( R d ).As a consequence of Lemma 2.3 one obtains a characterization of ⋄ W m M up ( R d ). Lemma 3.19. Let ≤ p < u < ∞ and m ∈ N . Then ⋄ W m M up ( R d ) is equal to thecollection of all f ∈ W m M up ( R d ) such that, for any β ∈ N d with | β | ≤ m , lim r ↓ | B ( y, r ) | u − p "Z B ( y,r ) | D β f ( x ) | p dx p = 0 (3.21) uniformly in y ∈ R d . f α with respect to the scale ⋄ W m M up ( R d ). Lemma 3.20. Let ≤ p < u < ∞ , m ∈ N and m < du . Then (i) f α ∈ W m M up ( R d ) if and only if m + α ≤ du ; (ii) f α / ∈ ⋄ W m M up ( R d ) if m + α = du .Proof. Step 1. Proof of (i). Let β ∈ N d with | β | ≤ m . It follows from the Leibniz rule,(3.18) and the smoothness of ψ that | D β ( f α )( x ) | ≤ C α,β | x | − ( α + | β | ) , | x | < , with an appropriate constant C α,β . Hence with m + α ≤ du we find D β f α ∈ M up ( R d ) andtherefore f α ∈ W m M up ( R d ).Conversely, let f α ∈ W m M up ( R d ). We fix β := ( m, , · · · , m even and m odd. If m = 2 m ′ , then D β ( f α )( x ) = D β ( | x | − α ) = m ′ X i =0 c i x i | x | α +2 m ′ +2 i , | x | < , where { c i } m ′ i =0 are appropriate constants independent of x . If m = 2 m ′ + 1, then D β ( f α )( x ) = D β ( | x | − α ) = m ′ X i =0 d i x i +11 | x | α +2 m ′ +1+2 i , where { d i } m ′ i =0 are appropriate constants independent of x . Observe that the terms x j | x | α +2 m ′ + j are ordered, i.e., | x | j +2 | x | α +2 m ′ + j +2 ≤ | x | j | x | α +2 m ′ + j . Now we choose a subset A of R d and a constant c > A := n x ∈ R d : | x | < , | x | ≥ max( | x | , · · · , | x d | ) c o . Let E denote the minimum of those constants c , . . . , c m ′ , d , . . . , d m ′ , which are positive.Then c ≥ (cid:12)(cid:12)(cid:12) m ′ X i =0 c i x i | x | α +2 m ′ +2 i (cid:12)(cid:12)(cid:12) ≥ E | x m ′ || x | α +4 m ′ , x ∈ A , if m = 2 m ′ and (cid:12)(cid:12)(cid:12) m ′ X i =0 d i | x | i +1 | x | α +2 m ′ +2 i +1 (cid:12)(cid:12)(cid:12) ≥ E | x | m ′ +1 | x | α +4 m ′ +1 , x ∈ A , if m = 2 m ′ + 1. Then for r ∈ (0 , 1) and β as above we have k f α | W m M up ( R d ) k ≥ | B (0 , r ) | u − p (cid:16) Z B (0 ,r ) ∩ A (cid:12)(cid:12)(cid:12) D β ( | x | − α ) (cid:12)(cid:12)(cid:12) p dx (cid:17) p ≥ E | B (0 , r ) | u − p (cid:16) Z B (0 ,r ) ∩ A | x | − ( α + m ) p dx (cid:17) p ≥ E r du − ( α + m ) (3.22)22or appropriate positive constants E , E independent of r . On the one hand this yieldsnecessity of α + m ≤ du in (i), on the other hand we get f du − m ⋄ W m M up ( R d ), see Lemma3.19.Now we turn to the case of fractional smoothness. This will be a little bit moretechnical than the previous proof. Lemma 3.21. Let s > , ≤ p < u < ∞ and ≤ q ≤ ∞ . Then we have(i) f α ∈ E su,p,q ( R d ) if and only if α + s ≤ d/u .(ii) f α ⋄ E su,p,q ( R d ) if α + s = d/u .Proof. Step 1. Proof of (i). We will use Proposition 3.11. Substep 1.1. Sufficiency. By means of the elementary embedding E su,p,q ( R d ) ֒ → E su,p, ∞ ( R d )we may restrict us to the case q < ∞ . The membership of f α in Morrey spaces is alreadyinvestigated. It remains to deal with (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) Z t − sq (cid:16) t − d Z B (0 ,t ) | ∆ Nh f α ( x ) | dh (cid:17) q dtt (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) , (3.23)where we assume α + s ≤ d/u and N > s . Because of the compact support it will beenough to deal with small balls. Furthermore, because of the radial symmetry, it will besufficient to study the balls B (0 , r ) with 0 < r < 1, i.e., we are interested insup 1. Here we observe Z t − sq (cid:16) t − d Z | x |≤| h | It remains to deal with | x | / (2 N ) ≤ | h | < | x | . Temporarily we assume 2 | x | < Z t − sq (cid:16) t − d Z | h | 24n the other hand we observe that Z | x | / (2 N ) t − sq (cid:16) t − d Z | h | Necessity. Let α + s > d/u . By means of the elementary embedding25 su,p,q ( R d ) ֒ → E su,p, ∞ ( R d ) it will be enough to consider the case q = ∞ . We claim thatsup 1, it follows the existence of a positive constant C such that | Ω( t ) | ≥ C t d . (3.30)Let 2 − j − ≤ | x | ≤ − j for some j ∈ N and min k x k ≥ 0. Moreover we assume 2 − i ≤ t < − i +1 for some i ∈ N with 1 ≤ i < j − L ′ , where L ′ ∈ N will be chosen later. Now let h ∈ Ω(2 − i ). Then for k ∈ { , , . . . , d } , because of j − i > L ′ , we observe2 x k h k ≥ x k − i √ d ≥ x k j − i √ d ≥ x k L ′ √ d . We choose L ∈ N such that 2 L ′ √ d ≥ L . Hence( x k + h k ) = x k + 2 x k h k + h k ≥ x k h k ≥ L x k and therefore | x + h | α ≥ αL/ | x | α . The restrictions x k , h k ≥ k ∈ { , , . . . , d } also imply | x + ℓh | α ≥ | x + h | α ≥ αL/ | x | α , ℓ ∈ { , . . . , N } , which results in | x | − α ≥ αL/ | x + ℓh | − α , ℓ ∈ { , . . . , N } . (3.31)Now we are able to find an appropriate estimate of | ∆ Nh f α | . Under the constraints collectedabove we obtain | ∆ Nh f α ( x ) | ≥ | x | − α − (cid:16) N − X ℓ =0 (cid:18) Nℓ (cid:19) | x + ( N − ℓ ) h | − α (cid:17) ≥ | x | − α − N | x + h | − α ≥ | x | − α − N − αL/ | x | − α = | x | − α (1 − N − αL/ ) . Now we choose L ∈ N as small as possible such that 1 − N − αL/ ≥ / L only depends on N and α and we get | ∆ Nh f α ( x ) | ≥ | x | − α . Choosing L ′ ∈ N as the smallest number that fulfills 2 L ′ √ d ≥ L we derive with2 − j − ≤ | x | ≤ − j < − − L ′ , min k x k ≥ , 0, we concludesup 0. By means of the elementary embedding ⋄ E su,p,q ( R d ) ֒ → ⋄ E su,p, ∞ ( R d ) it will be enough to concentrate on q = ∞ . Here we can applyStep 1.2, in particular (3.33). It followssup Let ≤ p ≤ u < ∞ , ≤ q ≤ ∞ and s ∈ R . Let λ ∈ S ( R d ) be a functionsuch that Z R d λ ( x ) dx = 0 , (4.1) L λ ≥ [ s ] , where λ ( · ) := λ ( · ) − − d λ ( · / . (4.2) Then the Lizorkin-Triebel-Morrey space E su,p,q ( R d ) is the collection of all tempered distri-butions f ∈ S ′ ( R d ) such that k f |E su,p,q ( R d ) k λ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ∞ X k =0 ksq |F − [ λ k F f ]( · ) | q (cid:17) q (cid:12)(cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) < ∞ in the sense of equivalent norms (with usual modification if q = ∞ ).Proof. In principle the proof follows the same lines as in case p = u , see [8], [9] and [45,Prop. 1.2]. So we skip the details.For a special Lipschitz domain Ω, one can find a narrow vertically directed cone K with vertex at origin that its shifts x + K are in Ω for every x ∈ Ω. For example, we maytake K := { ( x ′ , x d ) ∈ R d : | x ′ | < A − x d } , where A denotes the Lipschitz constant of ω . Let − K := {− x : x ∈ K } be the ”reflected”cone. Then for every test function γ ∈ D ( − K ) and f ∈ D ′ (Ω), the convolution γ ∗ f ( x ) = h f, γ ( x − · ) i is well defined in Ω since supp γ ( x − · ) ⊂ Ω for x ∈ Ω. Proposition 4.2. Let Ω ⊂ R d be a special Lipschitz domain and let K be one associatedcone as above. Let ϕ ∈ D ( − K ) have nonzero integral and let ϕ ( · ) := ϕ ( · ) − − d ϕ ( · / .Then for any given L ∈ N there exist functions ψ , ψ ∈ D ( − K ) such that L ψ ≥ L and f = ∞ X j =0 ψ j ∗ ϕ j ∗ f (4.3) for all f ∈ D ′ (Ω) . f : Ω → C ,denote by f Ω its extension from Ω to all of R d by zero. In addition, if g : R d → C , then g | Ω denotes the restriction of g to Ω. This notation will be also used for distributions. Theorem 4.3. Let Ω ⊂ R d be a special Lipschitz domain and K its associated cone. Let s ∈ R , q ∈ [1 , ∞ ] and ≤ p ≤ u < ∞ . Let ϕ ∈ D ( − K ) satisfy (4.1) and (4.2) . Let ψ , ψ ∈ D ( − K ) be given by Proposition 4.2 such that L ψ > d/ min( p, q ) . Then the map E defined by Ef := ∞ X j =0 ψ j ∗ ( ϕ j ∗ f ) Ω , f ∈ D ′ (Ω) , (4.4) induces a linear and bounded extension operator from E su,p,q (Ω) into the space E su,p,q ( R d ) .Moreover, for any f ∈ D ′ (Ω) we have E ( f ) | Ω = f .Proof. We follow Rychkov [45]. Only a few modifications have to be made. The parameters s, p, u, q, d are considered to be fixed in what follows. Step 1. Let X be the space of all sequences { g j } j ∈ N of locally integrable functions on R d such that k { g j } ∞ j =0 | X k := (cid:13)(cid:13)(cid:13)(cid:16) ∞ X j =0 | js G ( g j ) | q (cid:17) q (cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13) < ∞ , where G ( g j ) denotes the Peetre maximal function of g j , namely, G ( g j )( x ) := sup y ∈ R d | g j ( y ) | (1 + 2 j | x − y | ) N , x ∈ R d . The natural number N will be chosen such that d min( p, q ) < N ≤ L ψ . (4.5)We claim that, for any { g j } ∞ j =0 ∈ X , the series P ∞ j =0 ψ j ∗ g j converges in S ′ ( R d ) and (cid:13)(cid:13)(cid:13) ∞ X j =0 ψ j ∗ g j (cid:12)(cid:12)(cid:12) E su,p,q ( R d ) (cid:13)(cid:13)(cid:13) . k { g l } ∞ l =0 | X k . (4.6)By [45, (2.14)] we know that, if L ϕ ≥ [ s ] and L ψ ≥ N , then there exists some σ ∈ (0 , ∞ ),such that 2 ls | ϕ l ∗ ψ j ∗ g j ( x ) | . −| l − j | σ js G ( g j )( x ) (4.7)with hidden constant independent of x ∈ R d , l, j ∈ N and { g j } ∞ j =0 . By Lemma 4.1 wemay assume that E su,p,q ( R d ) is equipped with the norm generated by ϕ . Thus, for any j ∈ N , we have k ψ j ∗ g j |E s − σu,p,q ( R d ) k . (cid:13)(cid:13)(cid:13)(cid:13) ∞ X l =0 − (2 l + | l − j | ) σq [2 js G ( g j )] q ! q (cid:12)(cid:12)(cid:12)(cid:12) M up ( R d ) (cid:13)(cid:13)(cid:13)(cid:13) . From this we conclude that, for any j ∈ N , k ψ j ∗ g j |E s − σu,p,q ( R d ) k . − jσ k js G ( g j ) |M up ( R d ) k . − jσ k { G ( g l ) } ∞ l =0 |M up ( ℓ sq ( R d )) k . k , k ∈ N , k < k , we find (cid:13)(cid:13)(cid:13) k X j = k ψ j ∗ g j (cid:12)(cid:12)(cid:12) E s − σu,p,q ( R d ) (cid:13)(cid:13)(cid:13) . k X j = k − jσ k { G ( g l ) } ∞ l =0 |M up ( ℓ sq ( R d )) k . − k σ . Hence, P ∞ j =0 ψ j ∗ g j converges in E s − σu,p,q ( R d ) and therefore in S ′ ( R d ), since E s − σu,p,q ( R d ) ֒ →S ′ ( R d ). Now we turn to the norm estimate. By (4.7) we also have for any l ∈ N and any x ∈ R d , 2 ls (cid:12)(cid:12)(cid:12)(cid:12) ϕ l ∗ (cid:16) ∞ X j =0 ψ j ∗ g j (cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . ∞ X j =0 −| l − j | σ js G ( g j )( x ) . Taking the M up ( ℓ q )-norm on both sides it is easy to see that (4.6) holds true. Step 2. Now we aim to prove that f ∈ E su,p,q (Ω) implies E ( f ) ∈ E su,p,q ( R d ) and k E ( f ) |E su,p,q ( R d ) k . k f |E su,p,q (Ω) k . (4.8)By definition, for any ε ∈ (0 , ∞ ), there exists g ∈ E su,p,q ( R d ) such that g | Ω = f in D ′ (Ω)and k g |E su,p,q ( R d ) k ≤ k f |E su,p,q (Ω) k + ε. (4.9)Let g j := ( ϕ j ∗ f ) Ω with j ∈ N . We will show that k { ( ϕ j ∗ f ) Ω } ∞ j =0 | X k . k g |E su,p,q ( R d ) k . (4.10)Again we apply an inequality due to Rychkov [45, p. 248]. We havesup y ∈ R d | ( ϕ j ∗ f ) Ω ( y ) | (1 + 2 j | x − y | ) N . sup y ∈ Ω | ϕ j ∗ f ( y ) | (1+2 j | x − y | ) N if x ∈ Ω;sup y ∈ Ω | ϕ j ∗ f ( y ) | (1+2 j | e x − y | ) N if x / ∈ Ω . Here e x := ( x ′ , w ( x ′ ) − x d ) ∈ Ω is the point symmetric to x / ∈ Ω with respect to ∂ Ω. Sincethe convolution of ϕ j with f in Ω is only using values in Ω we obtain ϕ j ∗ f ( x ) = ϕ j ∗ g ( x ) for any x ∈ Ω . Hence sup y ∈ R d | ( ϕ j ∗ f ) Ω ( y ) | (1 + 2 j | x − y | ) N . ( G ( ϕ j ∗ g )( x ) if x ∈ Ω; G ( ϕ j ∗ g )( e x ) if x / ∈ Ω . Obviously, for any ball B ( z, r ) ⊂ R d , we know that | B ( z, r ) | u − p (cid:13)(cid:13)(cid:13) (cid:16) ∞ X j =0 (cid:12)(cid:12)(cid:12) js sup y ∈ R d ( ϕ j ∗ f ) Ω (1 + 2 j | · − y | ) N (cid:12)(cid:12)(cid:12) q (cid:17) q (cid:12)(cid:12)(cid:12) L p ( B ( z, r )) (cid:13)(cid:13)(cid:13) . | B ( z, r ) | u − p (cid:13)(cid:13)(cid:13) (cid:16) ∞ X j =0 (cid:12)(cid:12)(cid:12) js G ( ϕ j ∗ g )( x ) (cid:12)(cid:12)(cid:12) q (cid:17) q (cid:12)(cid:12)(cid:12) L p ( B ( z, r ) ∩ Ω) (cid:13)(cid:13)(cid:13) + | B ( z, r ) | u − p (cid:13)(cid:13)(cid:13) (cid:16) ∞ X j =0 (cid:12)(cid:12)(cid:12) js G ( ϕ j ∗ g )( e x ) (cid:12)(cid:12)(cid:12) q (cid:17) q (cid:12)(cid:12)(cid:12) L p ( B ( z, r ) ∩ Ω ∁ ) (cid:13)(cid:13)(cid:13) =: I + II . . k { G ( ϕ j ∗ g ) } ∞ j =0 |M up ( ℓ sq ( R d )) k . Concerning II we argue as follows. Let x ∈ B ( z, r ) ∩ Ω ∁ . Independent on the situation( z ∈ Ω or z Ω) we associate to z the vector e z := ( z ′ , ω ( z ′ ) − z d ). Here ω refers to thefunction occuring in the definition of a special Lipschitz domain, see Definition 2.16. Itfollows that | e z − e x | ≤ | z ′ − x ′ | + (2 A | z ′ − x ′ | + | z d − x d | ) < max(2 A, r , i.e., e x ∈ B ( e z, max(2 A, r ). By Rademacher’s Theorem ω is differentiable almost every-where in R d − . Using this we observe that the transformation T ( x ) = e x with x ∈ R d hasJacobi determinant | det J T ( x ) | = 1 almost everywhere. Thus, it follows from a change ofvariable formula, see, e.g., [19], [6], that Z B ( z,r ) ∩ Ω ∁ (cid:16) ∞ X j =0 (cid:12)(cid:12)(cid:12) js G ( ϕ j ∗ g )( T ( x )) (cid:12)(cid:12)(cid:12) q (cid:17) pq dx . Z B ( e z, max(2 A, r ) (cid:16) ∞ X j =0 (cid:12)(cid:12)(cid:12) js G ( ϕ j ∗ g )( e x ) (cid:12)(cid:12)(cid:12) q (cid:17) pq d e x . Applying this inequality we deriveII . | B ( z, r ) | u − p (cid:18) Z B ( e z, max(2 A, r ) (cid:16) ∞ X j =0 (cid:12)(cid:12)(cid:12) js G ( ϕ j ∗ g )( e x ) (cid:12)(cid:12)(cid:12) q (cid:17) pq d e x (cid:19) p . | B ( e z, max(2 A, r ) | u − p × (cid:18) Z B ( e z, max(2 A, r ) (cid:16) ∞ X j =0 (cid:12)(cid:12)(cid:12) js G ( ϕ j ∗ g )( e x ) (cid:12)(cid:12)(cid:12) q (cid:17) pq d e x (cid:19) p . k { G ( ϕ j ∗ g ) } ∞ j =0 |M up ( ℓ sq ( R d )) k . From this, combined with the characterization of E su,p,q ( R d ) via the Peetre maximal func-tion with N > d/ min( p, q ) (see, for example, [36, Subsection 11.2]), we further deducethat II . k g |E su,p,q ( R d ) k . Thus, (4.10) is proved. By Step 1, (4.8), and (4.10) we concludethat k E ( f ) |E su,p,q ( R d ) k . k{ ( ϕ j ∗ f ) Ω } ∞ j =0 |M up ( ℓ sq ( R d )) k . k f |E su,p,q (Ω) k + ε. When ε tends to zero, we find that E is a bounded linear operator from E su,p,q (Ω) into E su,p,q ( R d ). Step 3. Let ρ ∈ D (Ω). Then supp Z R d ψ j ( x − · ) ρ ( x ) dx ⊂ Ω , where we used that the supports of ψ and ψ are lying in − K . Hence Z R d (cid:16) Z R d ψ j ( x − y ) ρ ( x ) dx (cid:17) ( ϕ j ∗ f ) Ω ( y ) dy = Z R d (cid:16) Z R d ψ j ( x − y ) ρ ( x ) dx (cid:17) ( ϕ j ∗ f )( y ) dy . E ( f ) | Ω = ∞ X j =0 ψ j ∗ ϕ j ∗ f = f in D ′ (Ω) . This finishes the proof of Theorem 4.3.We remark that the extension operator E in Theorem 4.3 depends on p, q and s . Moreprecisely, we need to have[ s ] ≤ L ϕ and min( p, q ) > dL ψ . (4.11)However, Rychkov [45] has shown how to overcome these restrictions. He constructed anuniversal extension operator, i.e., an extension operator, which works for all admissibleparameter constellations simultaneously. In view of (4.11), one is tempted to take L ϕ = L ψ = ∞ , which is certainly impossible for compactly supported functions, but can beachieved with ϕ, ψ rapidly decreasing at infinity.Let Ω and K be as above. By S ′ (Ω) we denote the subset of D ′ (Ω) consisting of alldistributions having finite order and at most polynomial growth at infinity. More precisely, f ∈ S ′ (Ω) if and only if the estimate |h f, γ i| ≤ c sup x ∈ Ω , | α |≤ M | D α γ ( x ) | (1 + | x | ) M , for all γ ∈ D (Ω) , (4.12)is true with some constants c and M ∈ N depending on f . Remark 4.4. By [45, p. 250], we find that f ∈ S ′ (Ω) if and only if there exists a g ∈ S ′ ( R d )such that g | Ω = f . In particular, E su,p,q (Ω) is a subset of S ′ (Ω).The following lemma is just [45, Theorem 4.1]. Lemma 4.5. Let Ω ⊂ R d be a special Lipschitz domain and K its associated cone. Thereexist four functions ϕ , ϕ, ψ , ψ ∈ S ( R d ) supported in − K such that L ϕ = L ψ = ∞ and (4.3) holds in D ′ (Ω) for any f ∈ S ′ (Ω) . Theorem 4.6. Let ϕ , ϕ, ψ , ψ ∈ S ( R d ) be as in Lemma 4.5. Then the map E definedby Ef := ∞ X j =0 ψ j ∗ ( ϕ j ∗ f ) Ω , f ∈ S ′ (Ω) , (4.13) yields a linear and bounded extension operator from E su,p,q (Ω) into E su,p,q ( R d ) for all admis-sible values of p, q, u and s .Proof. The proof is based on that of Theorem 4.3 and similar to that of [45, Theorem4.1(b)]. Let f ∈ E su,p,q (Ω). Then f ∈ S ′ (Ω) follows, see Remark 4.4. By Lemma 4.5 wehave ∞ X j =0 ψ j ∗ ϕ j ∗ f = f in D ′ (Ω) . Moreover, since the supports of ψ and ψ lie in − K , it follows that E ( f ) | Ω = ∞ X j =0 ψ j ∗ ϕ j ∗ f = f. 32t remains to prove that the series in (4.13) converges in S ′ ( R d ) and k E ( f ) |E su,p,q ( R d ) k . k f |E su,p,q (Ω) k . (4.14)Observe that for any l, j ∈ N and x ∈ R d we have | ϕ l ∗ ψ j ∗ ( ϕ j ∗ f ) Ω ( x ) | ≤ Z R d | ϕ l ∗ ψ j ( z ) || ( ϕ j ∗ f ) Ω ( x − z ) | dz ≤ G (( ϕ j ∗ f ) Ω )( x ) Z R d | ϕ l ∗ ψ j ( z ) | (1 + 2 j | z | ) N dz , where N is chosen as in (4.5). By [8, Lemma 2.1], see also [45, (4.8)], we know that forany M ∈ N and any l, j ∈ N , Z R d | ϕ l ∗ ψ j ( z ) | (1 + 2 j | z | ) N dz . −| l − j | M . Thus, there is a σ > ls | ϕ l ∗ ψ j ∗ ( ϕ j ∗ f ) Ω ( x ) | . −| l − j | σ js G (( ϕ j ∗ f ) Ω )( x ) , x ∈ R d . Now by an argument similar to that used in the proof of Theorem 4.3 above, we concludethat the series in (4.13) converges in S ′ ( R d ) and that (4.14) holds.Since multiplication by smooth functions in D (Ω) preserves E su,p,q ( R d ) (see [64, The-orem 6.1]), a standard procedure (see [55] or [45, p. 244]) allows to reduce the case ofa bounded Lipschitz domain to a special Lipschitz domain. Now we are in position toformulate the final result of this subsection. Corollary 4.7. Let Ω ⊂ R d be either a bounded Lipschitz domain if d ≥ or a boundedinterval if d = 1 . Then there exists a linear and bounded extension operator E Ω such that E Ω ∈ L ( E su,p,q (Ω) → E su,p,q ( R d )) simultaneously for all admissible values of p, q, u and s . In addition, for any f ∈ S ′ (Ω) we have E Ω ( f ) | Ω = f in D ′ (Ω) . Remark 4.8. A different extension operator for Morrey smoothness spaces has beeninvestigated in Moura, Neves, Schneider [41], but restricted to a smaller class of domains.Let Ω ⊂ R d be a bounded domain. We shall call Ω an extension domain for E su,p,q ( R d )if there exists a linear and continuous extension operator E ∈ L ( E su,p,q (Ω) → E su,p,q ( R d )). In this section we will prove our main results. For that purpose we have to deal withcomplex interpolation. Let ( X , X ) be an interpolation couple of Banach spaces. By[ X , X ] Θ we denote the result of the complex interpolation of these spaces. We refer toCalder´on [13], Bergh, L¨ofstr¨om [4], Kre˘ın, Petunin, Semenov [33], Lunardi [38] and Triebel[57] for the basics. All our investigations will be based on the following essentially knownformulas. 33 roposition 5.1. Let Θ ∈ (0 , , s i ∈ R , p i ∈ [1 , ∞ ) , q i ∈ [1 , ∞ ] and u i ∈ [ p i , ∞ ) with i ∈ { , } . Let p u = p u and s = (1 − Θ) s + Θ s as well as p = 1 − Θ p + Θ p , q = 1 − Θ q + Θ q and u = 1 − Θ u + Θ u . (i) Then we have (cid:2) E s u ,p ,q ( R d ) , E s u ,p ,q ( R d ) (cid:3) Θ = E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ) k · |E su,p,q ( R d ) k . (5.1)(ii) Let Ω ⊂ R d be either a bounded Lipschitz domain if d ≥ or a bounded interval if d = 1 . Then (cid:2) E s u ,p ,q (Ω) , E s u ,p ,q (Ω) (cid:3) Θ = E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) k · |E su,p,q (Ω) k holds.Proof. Essentially (5.1) is proved in [66]. However, the last step, i.e., writing down theexplicit formula, has not been done there. For convenience of the reader we will sketch aproof. Step 1. Proof of (i). We need to switch to the associated sequence spaces e s u ,p ,q ( R d )based on appropriate wavelet isomorphisms. We refer to Rosenthal [42], Sawano [46] andTriebel [61] for more details and proofs. The advantage of these sequence spaces e su,p,q ( R d )compared with the function spaces E su,p,q ( R d ) is that they are Banach lattices. Calder´onproducts X − Θ0 X Θ1 are well-defined for Banach lattices, see Calder´on [13]. Shestakov[49, 50] has proved the following useful identity. Let ( X , X ) be an interpolation coupleof Banach lattices and Θ ∈ (0 , X , X ] Θ = X ∩ X k · | X − Θ0 X Θ1 k . Because of e s u ,p ,q ( R d ) − Θ e s u ,p ,q ( R d ) Θ = e su,p,q ( R d ) , see Yang, Yuan, Zhuo [62], we find under the same restrictions as in Proposition 5.1[ e s u ,p ,q ( R d ) , e s u ,p ,q ( R d )] Θ = e s u ,p ,q ( R d ) ∩ e s u ,p ,q ( R d ) k · | e su,p,q ( R d ) k . Complex interpolation spaces are invariant under isomorphisms. Again based on appro-priate wavelet isomorphisms we can turn back to the spaces E su,p,q ( R d ). This proves (i). Step 2. Proof of (ii). We employ a standard method, see e.g., [4, Thm. 6.4.2], [57,Thm. 1.2.4] or [59]. Suppose that E is our universal extension operator with respect to Ωthat was constructed in Corollary 4.7. Then we have E ∈ L ( E s u ,p ,q (Ω) → E s u ,p ,q ( R d ))and E ∈ L ( E s u ,p ,q (Ω) → E s u ,p ,q ( R d )) as well as E ∈ L ( E su,p,q (Ω) → E su,p,q ( R d )). It fol-lows that E is a coretraction to the restriction R with respect to Ω. It is R ◦ E = I . Here I denotes the identity on the space defined on the domain. At the same time E is a linearand continuous extension operator in L ( X → Y ) where X := E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) k · |E su,p,q (Ω) k and Y := E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ) k · |E su,p,q ( R d ) k . Furthermore, the restriction R applied to Y leads to X . Hence, Theorem 1.2.4 in [57]together with Step 1 yield (ii). 34 emark 5.2. (i) The formula (5.1) itself is explicitely stated in Hakim, Nogayama,Sawano [26, Thm. 1.5], but under slightly more restrictive conditions. Whereas Hakim etal [26] reduced (5.1) to results on the second complex interpolation method of Calder´onand an abstract result of Bergh [3], we employed Calder´on products and an abstract resultof Shestakov [49, 50].(ii) The interesting formula (5.1) has several forerunners. It has been used before in Lu,Yang, Yuan [37] (restricted to Morrey spaces), in Sickel, Skrzypczak, Vyb´ıral [53, 4.3](restricted to the classical situation p = u ) and in Yuan, Sickel, Yang [66, 2.4.3] (generalcase). To prove Theorem 1.1 we need the counterpart of Definition 3.5 for domains. Let s ≥ ≤ p < u < ∞ and 1 ≤ q ≤ ∞ . We put E su,p,q (Ω) := n f ∈ D ′ (Ω) : ∃ g ∈ E su,p,q ( R d ) such that f = g on Ω o . It is not difficult to see that we also can write E su,p,q (Ω) = n f ∈ E su,p,q (Ω) : D α f ∈ E su,p,q (Ω) for all α ∈ N d o . But we know even more. There is the following counterpart of Proposition 4.21 in [60]with almost identical proof. Lemma 5.3. Let Ω ⊂ R d be either a bounded Lipschitz domain if d ≥ or a boundedinterval if d = 1 . Let s ≥ , ≤ p < u < ∞ and ≤ q ≤ ∞ . Then the set E su,p,q (Ω) isindependent of the parameters s, u, p and q . Indeed, it holds E su,p,q (Ω) = n f ∈ C ∞ (Ω) : D α f ∈ L ∞ (Ω) for all α ∈ N d o . Let us continue with the proof of the main Theorem 1.1. Proof. Step 1. Based on Proposition 5.1 we have to calculate E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) k · |E su,p,q (Ω) k . Lemma 5.3 yields E su,p,q (Ω) = E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) . Therefore just by the definition of the space ⋄ E su,p,q (Ω), Definition 3.5 and the trivial em-beddings E s i u i ,p i ,q i (Ω) ֒ → E s i u i ,p i ,q i (Ω) with i ∈ { , } we find ⋄ E su,p,q (Ω) = E su,p,q (Ω) k · |E su,p,q (Ω) k ֒ → E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) k · |E su,p,q (Ω) k . (5.2) Step 2. Recall, we assume that either 0 ≤ s < s or 0 < s = s and q ≤ q , i.e., theconditions of Lemma 3.17 are satisfied. We claim that E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) ֒ → ⋄ E su,p,q (Ω) . (5.3)Let E denote the common extension operator. Let f ∈ E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω). Then Ef ∈ E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ). Let ψ be a function in D ( R d ) such that ψ ( x ) = 1 on Ω.35hen the operator h ψ · h belongs to L ( E σx,y,z ( R d ) → E σx,y,z ( R d )) for all admissible tuples( σ, x, y, z ). Hence g := ψ · Ef ∈ E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ). Obviously ( ψ · Ef ) | Ω = f in D ′ (Ω). Let B be a ball such that Ω ⊂ supp ψ ⊂ B . Hence g ∈ E s u ,p ,q ( R d ; B ) ∩ E s u ,p ,q ( R d ; B ) ֒ → ⋄ E su,p,q ( R d ) , see Lemma 3.17. Obviously this means f ∈ ⋄ E su,p,q (Ω) and this proves (5.3). Step 1 andStep 2 combined with Theorem 4.6 prove Theorem 1.1. Proof of Corollary 1.2 . The Corollary is a direct consequence of Theorem 1.1 and Lemma2.9. First we recall some well-known embedding relations. The new restriction (d’) guaranteesthe continuous embedding E s u ,p ,q ( R d ) ֒ → E tu ,p ,q ( R d ) ֒ → E s u ,p ,q ( R d ) , t := s − d (cid:16) u − u (cid:17) , we refer to [64, Cor. 2.2] and [27]. In addition we get E s u ,p ,q ( R d ) ֒ → E t Θ u,p,q ( R d ) ֒ → E su,p,q ( R d ) , t Θ := s − d (cid:16) u − u (cid:17) , since p < p < p , u < u < u , u p = p u and s − du < s − du = (1 − Θ) (cid:16) s − du (cid:17) + Θ (cid:16) s − du (cid:17) < s − du . Because of t Θ > s we may apply Proposition 3.8 and obtain E s u ,p ,q ( R d ) ֒ → E t Θ u,p,q ( R d ) ֒ → ⋄ E su,p,q ( R d ) . Consequently we have (cid:16) E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ) (cid:17) = E s u ,p ,q ( R d ) ֒ → ⋄ E su,p,q ( R d ) . Hence E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ) k · |E su,p,q ( R d ) k ֒ → ⋄ E su,p,q ( R d ) . (5.4)Employing the universal extension operator E from Corollary 4.7 we conclude E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) k · |E su,p,q (Ω) k ֒ → ⋄ E su,p,q (Ω) . To prove the reverse embedding we argue as before. By Lemma 5.3 we have E su,p,q (Ω) = (cid:16) E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) (cid:17) ⊂ (cid:16) E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) (cid:17) , which yields ⋄ E su,p,q (Ω) ֒ → [ E s u ,p ,q (Ω) , E s u ,p ,q (Ω)] Θ . Proof of Corollary 1.4 . The Corollary is a direct consequence of Theorem 1.3 andLemma 2.9. 36 .3 Proof of Proposition 1.5 For the complex method it is well-known that X ∩ X is a dense subset of [ X , X ] Θ ,see, e.g., [4, Thm. 4.2.2] or [57, Thm. 1.9.3]. Let the restrictions of Proposition 1.5 withrespect to p , p , u , u , q , q , s , s and Θ be satisfied. The parameters p, u, q and s arethen fixed as well. Without loss of generality we may assume that Ω contains the ball B (0 , α := du − s . By assumption α > α = du − s = du − s . Thus Lemma 3.21 implies f α ∈ E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) and f α ⋄ E su,p,q (Ω) . This proves the claim. Step 1. Proof of (i). It will be enough to show that there exists a function h ∈ ⋄ E su,p,q ( R d )such that h [ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ . Therefore we will work with the family of testfunctions h u we investigated in Lemma 3.18. Let σ > 0, 1 ≤ y ≤ x < ∞ and 1 ≤ z ≤ ∞ .Then there is the embedding E σx,y,z ( R d ) ֒ → M xy ( R d ). Hence we find (cid:16) E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ) (cid:17) ⊂ (cid:16) M u p ( R d ) ∩ M u p ( R d ) (cid:17) . Moreover we observe E s u ,p ,q ( R d ) ∩ E s u ,p ,q ( R d ) k · |E su,p,q ( R d ) k ⊂ M u p ( R d ) ∩ M u p ( R d ) k · |E su,p,q ( R d ) k ⊂ M u p ( R d ) ∩ M u p ( R d ) k · |M up ( R d ) k . Lemma 3.18 yields h u ∈ ⋄ E su,p,q ( R d ). But we have h u 6∈ M u p ( R d ) ∩ M u p ( R d ) k · |M up ( R d ) k . This has been proved in [66], see the proof of Corollary 2.38, Step 3, page 1891. Step 2. Proof of (ii). This has been proved in (5.4). In this section we will collect some more material concerning interpolation of Morreyspaces, smoothness Morrey spaces and their relatives. Let us start with two papers ofLemari´e-Rieusset [34, 35]. Based on earlier work, see Ruiz, Vega [43] and Blasco, Ruiz,Vega [5], he was able to show the importance of the restriction u p = u p . Under therestrictions 1 < p ≤ u < ∞ , 1 < p ≤ u < ∞ , 0 < Θ < p := − Θ p + Θ p and u := − Θ u + Θ u he proved that there exists an interpolation functor F of exponent Θ suchthat F ( M u p ( R d ) , M u p ( R d )) = M up ( R d ) if and only if u p = u p . In the meanwhile twointerpolation functors are known which have this property, namely the ± method of Gus-tavsson and Peetre and the second complex interpolation method introduced by Calder´on.37e refer to Lu, Yang, Yuan [37] and Lemari´e-Rieusset [35], respectively. Concerning the ± method, denoted by h · , · , Θ i , Yuan, Sickel, Yang, [66] have shown that D E s u ,p ,q ( R d ) , E s u ,p ,q ( R d ) , Θ E = E su,p,q ( R d )holds subject to the restrictions(a) 0 < p < p < ∞ , p ≤ u < ∞ , p ≤ u < ∞ ;(b) 0 < q , q ≤ ∞ ;(c) p u = p u ;(d) s , s ∈ R ;(e) 0 < Θ < p := − Θ p + Θ p , u := − Θ u + Θ u , q := − Θ q + Θ q , s := (1 − Θ) s + Θ s .Concerning the second complex interpolation method, denoted by [ · , · ] Θ , Hakim,Nagoyama and Sawano [26] proved[ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ = E su,p,q ( R d ) , provided that (a)-(e) are satisfied and in addition p , p , q , q ∈ (1 , ∞ ).Let us come back to the first complex interpolation method. Together with the real inter-polation method of Lions-Peetre it is the most important interpolation method. Thereforeit is of interest for its own to understand the spaces [ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ . In caseof the Morrey spaces different characterizations of [ M u p ( R d ) , M u p ( R d )] Θ can be found inYuan, Sickel, Yang [66] and Hakim, Nakamura, Sawano [25]. There is a certain number ofpublications dealing with the interpolation of subspaces of either Morrey or of Lizorkin-Triebel-Morrey spaces. In particular (but not only), • D ˚ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d ) , Θ E , D ˚ E s u ,p ,q ( R d ) , ˚ E s u ,p ,q ( R d ) , Θ E ; • [˚ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ , [˚ E s u ,p ,q ( R d ) , ˚ E s u ,p ,q ( R d )] Θ ; • [˚ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ , [˚ E s u ,p ,q ( R d ) , ˚ E s u ,p ,q ( R d )] Θ ; • [ ⋄ E s u ,p ,q ( R d ) , ⋄ E s u ,p ,q ( R d )] Θ ; • [ ⋄ E s u ,p ,q ( R d ) , ⋄ E s u ,p ,q ( R d )] Θ and similarly for Morrey spaces, we refer to [37], [62], [66], [21], [22], [25], [20], [26], [23]and [24].Probably it is of certain interest to notice that the diamond spaces on domains form ascale under complex interpolation, i.e.,[ ⋄ E s u ,p ,q (Ω) , ⋄ E s u ,p ,q (Ω)] Θ = ⋄ E su,p,q (Ω) , (6.1)at least under the restrictions in Theorem 1.1 or in Theorem 1.3. This follows from E su,p,q (Ω) = E s u ,p ,q (Ω) ∩ E s u ,p ,q (Ω) ⊂ [ E s u ,p ,q (Ω) , E s u ,p ,q (Ω)] Θ = ⋄ E su,p,q (Ω) , ⋄ E su,p,q (Ω) = E su,p,q (Ω) k · |E su,p,q (Ω) k and ⋄ E su,p,q (Ω) k · |E su,p,q (Ω) k = ⋄ E su,p,q (Ω) . Let us add a few references to the real method as well. First results on real interpolationof Besov-Morrey spaces can be found in Kozono, Yamazaki [32]. Mazzucato [40] was thefirst who had dealt with the real interpolation of Sobolev-Morrey spaces W m M up ( R d ) andtheir generalizations to the classes E su,p, ( R d ) with 1 < p < u < ∞ . Her result is containedin N su,p,q ( R d ) = ( E s u,p,q ( R d ) , E s u,p,q ( R d )) θ,q if s , s ∈ R , s < s , 0 < p < u < ∞ , 0 < q , q , q ≤ ∞ and 0 < Θ < 1, see [52].Recently Burenkov, Ghorbanalizadeh, Sawano [12] described the K -functional for the pair( M up ( a, b ) , ˙ W m M up ( a, b )). Here ˙ W m M up ( a, b ) refers to the homogeneous Sobolev space. In[10], [11] Burenkov et al studied the real interpolation of slightly modified spaces, so-calledlocal Morrey spaces. They behave much better under real interpolation than the originalMorrey spaces.Finally, we mention that the interpolation property has been investigated, e.g., inAdams, Xiao [2], Adams [1] and Yuan, Sickel, Yang, [66], where also further referencescan be found. At the end of our paper we would like to address a few open problems which could be ofcertain interest.1. A general question is about the role of the Lemari´e-Rieusset condition u p = p u .How do the interpolation spaces look like if this condition is violated? There arespecial cases which one should investigate first like the following. Let p = p and u < u . How do the interpolation spaces[ W m M u p (Ω) , W m M u p (Ω)] Θ look like in the case m < m ?2. What happens if s − d ( u − u ) < s < s and u p = p u ? These cases arenot treated in the Theorems 1.1 and 1.3. We refer to the picture at the end of theIntroduction.3. Find a characterization of [ E s u ,p ,q ( R d ) , E s u ,p ,q ( R d )] Θ for all admissible constella-tions of the parameters. The answer could become technical.4. We always had to exclude the case q = q = ∞ . Under necessary additionalrestrictions it is known that[ E s p ,p , ∞ ( R d ) , E s p ,p , ∞ ( R d )] Θ = [ F s p , ∞ ( R d ) , F s p , ∞ ( R d )] Θ = ⋄ F s p , ∞ ( R d ) , see [53] and [66]. So the question is about the characterization of[ E s u ,p , ∞ (Ω) , E s u ,p , ∞ (Ω)] Θ . 39. In contrast to the classical case there are two Besov counterparts of the Lizorkin-Triebel-Morrey spaces, namely B s,τp,q (Ω) and N su,p,q (Ω), respectively. In case of theso-called Besov-Morrey spaces N su,p,q (Ω) one knows the counterpart of Theorem 1.1,see Theorem 2.45 and Corollary 2.65 in [66]. Let Ω ⊂ R d be a bounded interval if d = 1 or a bounded Lipschitz domain if d ≥ 2. Assume that 0 < p i ≤ u i < ∞ , s , s ∈ R and q i ∈ (0 , ∞ ), i ∈ { , } . Let s := (1 − Θ) s + Θ s , p := − Θ p + Θ p and q := − Θ q + Θ q . If u p = u p , then[ N s u ,p ,q (Ω) , N s u ,p ,q (Ω)] Θ = ⋄ N su,p,q (Ω)holds true for all Θ ∈ (0 , s and s . The main reason for this more simple behavior can be found in ⋄ N su,p,q ( R d ) = N su,p,q ( R d ) if and only if q ∈ (0 , ∞ ) . The behavior of the Besov-type spaces B s,τp,q (Ω) under complex interpolation seemsto be widely open.6. Let us turn to Corollary 1.2. Obviously the case p = 1 has been left out. Whathappens if p = 1 ?7. Probably even more difficult is the question around the use of the extension propertyof our function spaces on domains. Is there a wider class of domains than boundedLipschitz domains allowing the validity of Theorem 1.1 ?8. We concentrated on Banach spaces in our paper. There is a well developed theoryof the function spaces also for values u, p, q ∈ (0 , Acknowledgements Ciqiang Zhuo is supported by the Construct Program of the Key Discipline in Hu-nan Province, the National Natural Science Foundation of China (Grant Nos. 11701174,11831007, 11871100 and China Scholarship Council (Grant No. 201906725036).Marc Hovemann is funded by a Landesgraduiertenstipendium which is a scholarshipfrom the Friedrich-Schiller university and the Free State of Thuringia. References [1] D. R. Adams, Morrey spaces, Birkh¨auser, Cham, 2015.[2] D. R. Adams, J. Xiao, Morrey spaces in harmonic analysis, Ark. Mat. 50 (2012),201-230.[3] J. Bergh, Relation between the two complex methods of interpolation, Indiana Univ.Math. J. 28(5) (1979), 775-778.[4] J. Bergh, J. L¨ofstr¨om, Interpolation Spaces. An Introduction. Springer, New York,1976. 405] O. Blasco, A. Ruiz, L. Vega, Non-interpolation in Morrey-Campanato and blockspaces, Ann. Scuola Norm Sup. Pisa Cl. 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