An extension operator for Sobolev spaces with mixed weights
AAn extension operator for Sobolev spaces with mixed weights
Markus Hansen ∗† Cornelia Schneider ‡§ Flóra O. Szemenyei ¶ February 22, 2021
Abstract
We provide an extension operator for weighted Sobolev spaces on bounded poly-hedral cones K involving a mixture of weights, which measure the distance to thevertex and the edges of the cone, respectively. Our results are based on Stein’sextension operator [17] for Sobolev spaces and generalize [8]. In this article we construct an extension operator for weighted Sobolev spaces V l,pβ,δ ( K ) onpolyhedral cones K involving mixed weights, which measure the distance to the vertexand the edges of the cone as displayed by the parameters β and δ , respectively. Our con-struction is based on the original extension operator from Stein [17] for classical Sobolevspaces on sufficiently smooth domains, which was generalized in [8] by Hansen to thesetting of weighted Sobolev spaces involving one weight function only.Weighted Sobolev spaces are particularly important when it comes to regularity theoryfor solutions of partial differential equations (PDEs) on non-smooth domains: in this casesingularities of the solutions at the boundary may occur which diminish the Sobolev reg-ularity, cf. [9], but can be compensated with suitable weight functions. In particular,spaces with mixed weights are needed when studying stochastic PDEs as is demonstratedin [3, 2]. The weighted Sobolev spaces we are interested in can be seen as generalizations ∗ Philipps-University Marburg, FB12 Mathematics and Computer Science, Hans-Meerwein Straße,Lahnberge, 35032 Marburg, Germany. Email: [email protected] † The work of this author has been supported by Deutsche Forschungsgemeinschaft (DFG), Grant No.360/22-1. ‡ Friedrich-Alexander University Erlangen-Nuremberg, Applied Mathematics III, Cauerstr. 11, 91058Erlangen, Germany. Email: [email protected] § The work of this author has been supported by Deutsche Forschungsgemeinschaft (DFG), Grant No.SCHN 1509/1-2. ¶ Corresponding author . Friedrich-Alexander University Erlangen-Nuremberg, Applied MathematicsIII, Cauerstr. 11, 91058 Erlangen, Germany. Email: [email protected] Math Subject Classifications. Primary:
Secondary: Keywords and Phrases. weighted Sobolev spaces, mixed weights, extension operator, polyhedral cone. a r X i v : . [ m a t h . F A ] F e b f the so-called Kondratiev spaces which appeared in the 60s in [10, 11] and were studiedin detail in [6]. Later on more general spaces were considered by Kufner, Sändig [13],Babuska, Guo [1], Nistor, Mazzucato [15], and Costabel, Dauge, Nicaise [4], to mentionat least a few contributions in this context.The weighted Sobolev spaces V l,pβ,δ ( K ) with parameters l ∈ N , ≤ p < ∞ , β ∈ R , and δ = ( δ , . . . , δ n ) ∈ R n , were introduced and studied in detail by Maz’ya, Rossmann in [14]and contain all measurable functions u such that the norm (cid:107) u | V l,pβ,δ ( K ) (cid:107) := (cid:18) (cid:90) K (cid:88) | α |≤ l ρ ( x ) p ( β − l + | α | ) n (cid:89) k =1 (cid:18) r k ( x ) ρ ( x ) (cid:19) p ( δ k − l + | α | ) | ∂ α u ( x ) | p dx (cid:19) /p is finite. Here ρ denotes the regularized distance of a point x to the vertex of the cone K and r k ( x ) the (regularized) distances to the respective edges M k of the cone. It isprecisely this mixture of the different weights involved and their interplay, which allowsfor an even finer description when investigating the singularities of the solutions of PDEson polyhedral cones or even general domains of polyhedral type.Our construction of the extension operator in the weighted Sobolev spaces V l,pβ,δ will bedone in three steps: First we show that Stein’s definition of an extension operator forspecial Lipschitz domains remains bounded in the framework of weighted Sobolev spacesdenoted by V l,pδ where the weight involved (displayed by the parameter δ ∈ R ) measuresthe distance to the x -axis. In particular, the result holds for unbounded wedges (whichare special Lipschitz domains) where the weight in the respective Sobolev spaces nowmeasures the distance to the edge of the wedge. In the next step we establish an extensionoperator for a fixed layer K (cid:48) (bounded away from the vertex) of the cone K in the spaces V l,pβ,δ with mixed weights. The construction uses the first result, since the layer can becovered by finitely many diffeomorphic images of bounded wedges. Finally, in a thirdstep we decompose the cone into dyadic layers K j , where j ∈ Z , and construct a familyof extension operators for each layer, which is uniformly bounded in the norms of V l,pβ,δ with respect to j . Afterwards we glue together these operators properly and use thelocalization of the norm of V l,pβ,δ , which yields an extension operator for the whole cone.Our motivation for these kinds of studies is as follows: in a forthcoming paper we willuse regularity estimates for solutions of elliptic and parabolic PDEs on polyhedral conesin the spaces V l,pβ,δ ( K ) in order to study the smoothness α > of these solutions in thespecific scale B ατ,τ ( K ) , τ = αd + 1 p , (1)of Besov spaces. It is well known that the regularity α in these spaces determines theapproximation order that can be achieved by adaptive approximation schemes, see [5]for further details on this subject. In order to justify the use of adaptive algorithms(compared to non-adaptive ones) for the PDEs under investigation, one needs to knowthat the Besov regularity α of the solutions is sufficiently high. Our main tool here willbe an embedding for the weighted Sobolev spaces V l,pβ,δ into the scale of Besov spaces (1).In this context the extension operator we construct below will help to improve the upperbound of the smoothness α considerably, leading to a far better approximation rate ofthe adaptive algorithms compared to [7]. However, the results for the extension operator2stablished in this paper are of interest on their own, which is why we decided to publishthem separately.The paper is organized as follows. In Section 2 we introduce the weighted Sobolev spaceson polyhedral cones and study some of their relevant properties. Afterwards, in Section3, we construct an extension operator for these spaces in three steps as explained above. Notation
We start by collecting some general notation used throughout the paper.As usual, N stands for the set of all natural numbers, N = N ∪{ } , Z denotes the integers,and R d , d ∈ N , is the d -dimensional real Euclidean space with | x | , for x ∈ R d , denotingthe Euclidean norm of x . Let N d be the set of all multi-indices, α = ( α , . . . , α d ) with α j ∈ N and | α | := (cid:80) dj =1 α j . For partial derivatives ∂ α f = ∂ | α | f∂x α we will occasionally alsowrite f x α . Furthermore, B ( x, r ) is the open ball of radius r > centered at x , and for ameasurable set M ⊂ R d we denote by | M | its Lebesgue measure.We denote by c a generic positive constant which is independent of the main parameters,but its value may change from line to line. The expression A (cid:46) B means that A ≤ c B .If A (cid:46) B and B (cid:46) A , then we write A ∼ B .Throughout the paper ’domain’ always stands for an open and connected set. The testfunctions on a domain Ω are denoted by C ∞ (Ω) . Let L p (Ω) , ≤ p ≤ ∞ , be the Lebesguespaces on Ω as usual. We denote by C (Ω) the space of all bounded continuous func-tions f : Ω → R and C k (Ω) , k ∈ N , is the space of all functions f ∈ C (Ω) such that ∂ α f ∈ C (Ω) for all α ∈ N with | α | ≤ k , endowed with the norm (cid:80) | α |≤ k sup x ∈ Ω | ∂ α f ( x ) | .Let m ∈ N and ≤ p ≤ ∞ . Then W mp (Ω) denotes the standard L p -Sobolevspaces of order m on the domain Ω , equipped with the norm (cid:107) u | W mp (Ω) (cid:107) := (cid:16) (cid:80) | α |≤ m (cid:82) Ω | ∂ α u ( x ) | p dx (cid:17) /p . Polyhedral cone
We mainly consider function spaces defined on polyhedral cones inthe sequel. Let K := { x ∈ R : 0 < | x | < ∞ , x/ | x | ∈ Ω } be an infinite cone in R with vertex at the origin. Suppose that the boundary ∂K consistsof the vertex x = 0 , the edges (half lines) M , ..., M n , and smooth faces Γ , ..., Γ n . Hence, Ω ∩ S is a domain of polygonal type on the unit sphere S with sides Γ k ∩ S . Moreover,we consider the bounded polyhedral cone ˜ K obtained via truncation ˜ K := K ∩ B (0 , r ) . The singular points of the polyhedral cone K are those x ∈ ∂K for which for any ε > the set ∂K ∩ B ( x, ε ) is not smooth, i.e., the vertex and the edges M , ..., M n . When weconsider the bounded cone ˜ K we omit the non-smooth points induced by the truncationand also consider S = { }∪ M ∪ ... ∪ M n , which in this case is not the entire singularity set.3 M M M Ω Figure 1: Infinite polyhedral cone
K M M M M ρ ( x ) r ( x ) Ω x Figure 2: Bounded polyhedral cone ˜ K Definition 2.1.
Let K be a (bounded or unbounded) polyhedral cone in R and S = { } ∪ M ∪ ... ∪ M n . Then the space V l,pβ,δ ( K, S ) is defined as the closure of the set C ∞∗ ( K, S ) := { u | K : u ∈ C ∞ ( R \ S ) } with respect to the norm (cid:107) u | V l,pβ,δ ( K, S ) (cid:107) := (cid:18) (cid:90) K (cid:88) | α |≤ l ρ ( x ) p ( β − l + | α | ) n (cid:89) k =1 (cid:18) r k ( x ) ρ ( x ) (cid:19) p ( δ k − l + | α | ) | ∂ α u ( x ) | p dx (cid:19) /p , (2) where β ∈ R , l ∈ N , δ = ( δ , ..., δ n ) ∈ R n , ≤ p < ∞ , ρ ( x ) = dist(0 , x ) denotes thedistance to the vertex, and r j ( x ) := dist( x, M j ) the distance to the edge M j . Remark 2.2.
We collect some remarks and properties concerning the weighted Sobolevspaces V l,pβ,δ ( K ) . • In the sequel if there is no confusion we omit S from the notation and write shortly V l,pβ,δ ( K ) . • The space V l,pβ,δ ( K ) is a Banach space for ≤ p < ∞ . The proof can be derived froma generalized result, see [12, p. 18, Theorem 3.6]. • V l,pβ,δ ( K ) ⊂ L p ( K ) for l ≥ β and l ≥ δ k , k = 1 , ..., n . • We have the following embeddings which one obtains easily from the definition ofthe spaces V l,pβ,δ ( K ) : V l,pβ,δ ( K ) ⊂ V l − ,pβ − ,δ − ( K ) ⊂ ... ⊂ V ,pβ − l,δ − l ( K ) . • There exist regularized versions of the distance functions ρ and r ,..., r n . We denotethem by (cid:101) ρ , (cid:101) r , ..., (cid:101) r n . According to [17, Theorem VI.2.2, p.171], these functionsdefined on K have nonnegative function values and are infinitely often differentiable.Moreover, there exist positive constants A , B , C α, such that A ρ ( x ) ≤ (cid:101) ρ ( x ) ≤ Bρ ( x ) , x ∈ K, nd for all α ∈ N n , | ∂ α (cid:101) ρ ( x ) | ≤ C α, ρ −| α | ( x ) , x ∈ K. One has similar results for (cid:101) r i , i = 1 , ..., n . • Replacing ρ , r , ..., r n by (cid:101) ρ , (cid:101) r , ..., (cid:101) r n in the norm of V l,pβ,δ ( K ) , one can prove that theoperator T β (cid:48) ,δ (cid:48) : V l,pβ,δ ( K ) −→ V l,pβ − β (cid:48) ,δ − δ (cid:48) ( K ) , u (cid:55)→ ρ β (cid:48) ( x ) n (cid:89) k =1 (cid:18) r k ( x ) ρ ( x ) (cid:19) δ (cid:48) k u, (3) is an isomorphism, where δ − δ (cid:48) = ( δ − δ (cid:48) , ..., δ n − δ (cid:48) n ) . The inverse operator is givenby ( T β (cid:48) ,δ (cid:48) ) − = T − β (cid:48) , − δ (cid:48) . • A function ϕ ∈ C l ( K ) is a pointwise multiplier in V l,pβ,δ ( K ) , i.e., for u ∈ V l,pβ,δ ( K ) wehave (cid:107) ϕu | V l,pβ,δ ( K ) (cid:107) ≤ c (cid:107) u | V l,pβ,δ ( K ) (cid:107) . (4)• The weighted Sobolev spaces V l,pβ,δ ( K ) are refinements of the Kondratiev spaces K ma,p ( K ) , which for m ∈ N , ≤ p < ∞ , and a ∈ R are defined as the collec-tion of all measurable functions such that (cid:107) u |K ma,p ( K ) (cid:107) := (cid:88) | α |≤ m (cid:90) K | ρ ( x ) | α |− a ∂ α u ( x ) | p dx /p < ∞ , with weight function ρ ( x ) := min(1 , dist( x, S )) for x ∈ K . In particular, the scalescoincide if m = l, δ = ( δ , ..., δ n ) = ( l − a, ..., l − a ) and β = l − a. To show this, one has to prove that the power of the weight function of K ma,p ( K ) is equivalent to the power of the weight function of V l,pβ,δ ( K ) with these parameterassumptions, i.e., ρ ( x ) β − l + | α | n (cid:89) k =1 (cid:18) r k ( x ) ρ ( x ) (cid:19) δ k − l + | α | ∼ ρ ( x ) | α |− a . This follows from [14, p. 90, Subsection 3.1.1, (3.1.2)]. • As an alternative approach to Definition 2.1 one could define spaces ˜ V l,pβ,δ ( K ) , whichcontain all measurable functions such that the norm (2) is finite, and investigate un-der which conditions the set C ∞∗ ( K, S ) is dense in ˜ V l,pβ,δ ( K ) , i.e., when these spacescoincide with the spaces V l,pβ,δ ( K ) . This interesting question is postponed to a forth-coming paper. V l,pβ,δ ( K ) . For this let ϕ j be infinitely differentiablefunctions depending only on ρ ( x ) = | x | such thatsupp ϕ j ⊂ { − j − < | x | < − j +1 } , | ∂ α ϕ j ( · ) | ≤ c α | α | j , ∞ (cid:88) j = j ϕ j = 1 , (5)where c α are constants independent of j . Remark 2.3. If K is a bounded polyhedral cone, then w.l.o.g. we assume j = 0 . If it isunbounded, then j = −∞ is required. The following lemma is a direct consequence of the localization result in [14, p. 91, Lemma3.1.2] and has a more convenient form for our setting in order to prove the existence ofthe extension operator.
Lemma 2.4.
Let K be a polyhedral cone, l ∈ N , β ∈ R , δ ∈ R n , and ≤ p < ∞ . Thenfor all u ∈ V l,pβ,δ ( K ) , (cid:107) u | V l,pβ,δ ( K ) (cid:107) p ∼ ∞ (cid:88) j = j (cid:107) ϕ j u | V l,pβ,δ ( K ) (cid:107) p , where ( ϕ j ) j ≥ j is resolution of unity satisfying (5) as above. Remark 2.5.
Note that Lemma 2.4 actually holds for every family ( ϕ j ) j ≥ j as above,which satisfies the weaker assumption ∞ (cid:88) j = j ϕ j ∼ . V l,pβ,δ ( K ) In this section we construct an extension operator for the space V l,pβ,δ ( K ) . Our main resultreads as follows: Theorem 3.1. (Extension operator)Let K ⊂ R be a polyhedral cone and let l ∈ N , β ∈ R , δ ∈ R n and ≤ p < ∞ . Thenthere exists a bounded linear extension operator E : V l,pβ,δ ( K, S ) → V l,pβ,δ ( R , S ) , where S is the singularity set of K . Remark 3.2.
The norm of the space V l,pβ,δ ( R , S ) is defined similarly as the norm of V l,pβ,δ ( K, S ) replacing the integral domain K by R . .1 Extension operator for special Lipschitz domains In order to prove Theorem 3.1 we need some preparations. We follow the proof of theextension theorem of Stein, who originally proved the theorem for classical Sobolev spaces.This was generalized by Hansen in [8] to the setting of Kondratiev spaces. Since we nowdeal with mixed weights in the context of the spaces V l,pβ,δ ( K ) , we need to make somecareful modifications. We start with an extension theorem for spaces defined on specialLipschitz domains, which is a counterpart of the corresponding theorem for Kondratievspaces [8, p. 582, Appendix]. For this we need to deal with the spaces V l,pδ ( D, R ∗ ) from[14, p. 24], where R ∗ := { x ∈ R : x = x = 0 } denotes the x -axis. Definition 3.3.
Let l ∈ N , δ ∈ R , and ≤ p < ∞ . Furthermore, let D ⊂ R be a specialLipschitz domain, i.e., D = { x ∈ R : x = ( x (cid:48) , x ) , x (cid:48) ∈ R , x > ω ( x (cid:48) ) } , (6) for some Lipschitz-continuous function ω : R → R . Assume R ∗ ⊂ ∂D and let r ( x ) denote the distance of x to R ∗ . Then the space V l,pδ ( D, R ∗ ) is defined as the closure of C ∞∗ ( D, R ∗ ) = { u (cid:12)(cid:12) D : u ∈ C ∞ ( R \ R ∗ ) } with respect to the norm (cid:107) u | V l,pδ ( D, R ∗ ) (cid:107) = (cid:90) D (cid:88) | α |≤ l r p ( δ − l + | α | ) | ∂ α u ( x ) | p dx /p . Remark 3.4.
More general, let D be a domain of polyhedral type where for the edge M we have M ⊂ ∂D . Then the space V l,pδ ( D, M ) can be defined in a similar manner. Inthis case the function r denotes the distance to the edge M . Furthermore, the norm ofthe space V l,pδ ( R , R ∗ ) is the counterpart of the norm of V l,pδ ( D, R ∗ ) replacing the integraldomain D by R . Now we can state and proof an extension theorem for the above spaces defined on specialLipschitz domain.
Theorem 3.5.
Let l ∈ N , δ ∈ R , and ≤ p < ∞ . Moreover, let D ⊂ R be a specialLipschitz domain and assume R ∗ ⊂ ∂D . Then there exists a universal linear and boundedextension operator E : V l,pδ ( D, R ∗ ) → V l,pδ ( R , R ∗ ) . Remark 3.6. n particular, note that Theorem 3.5 holdsfor wedges. In this case we choose ω as theLipschitz function ω ( x , x ) = | x | , and describe the wedge as a special Lipschitzdomain according to (6) , i.e., x x x SW = { x ∈ R : x = ( x , x , x ) , x > ω ( x , x ) } . Thus, the weight in the V l,pδ ( W, R ∗ ) -norm measures the distance of a point to the x -axis,which represents the edge (and therefore the singularity set) of the wedge.Proof. For the proof we rely on calculations of Pieperbeck [16] and Hansen [8]. Since bydefinition the set C ∞∗ ( D, R ∗ ) is dense in V l,pδ ( D, R ∗ ) , it suffices to prove the theorem forthis dense subset. Step 1 (Preparations):
We set δ ( ξ ) = dist( ξ, ∂D ) as the distance of the point ξ ∈ D c to the boundary of D . Its regularized version is denoted by ∆( ξ ) , cf. Stein [17, p. 171,Theorem 2]. In particular, it holds C δ ( ξ ) ≤ ∆( ξ ) ≤ C δ ( ξ ) , and (cid:12)(cid:12) ∂ αξ ∆( ξ ) (cid:12)(cid:12) ≤ B α ( δ ( ξ )) −| α | , ξ ∈ D c , (7)for constants C , C and B α independent of D . We consider the point ξ = ( x , ω ( x )) ∈ ∂D and denote by Γ ξ = { ξ = ( x (cid:48) , x ) : x < ω ( x ) , | x − ω ( x ) | > M | x (cid:48) − x |} the lower cone with vertex at the point ξ , where M is the Lipschitz constant of ω and x (cid:48) ∈ R , x ∈ R . Then we clearly have Γ ξ ∩ D = { ξ } . Moreover, the point p = ( x , y ) , y < ω ( x ) , lies on the axis of the cone. We make some elementary geometric calculationsusing the notation from the figure below. One can check easily that δ ( p ) ≥ h , and we seethat q = (cid:16) ω ( x ) − y + M | x | M , y (cid:17) ∈ ∂ Γ ξ according to the definition of Γ ξ .Furthermore, we have a = ω ( x ) − y,b = ω ( x ) − yM ,c = (cid:32)(cid:18) ω ( x ) − yM (cid:19) + ( ω ( x ) − y ) (cid:33) / , and hence, h = abc = ω ( x ) − y ( M +1) / . x (cid:48) x D Γ ξ ξ = ( x , ω ( x )) qcbpa hx y From these observations we obtain δ ( x (cid:48) , x ) ≥ ω ( x (cid:48) ) − x (1+ M ) / , for arbitrary ( x (cid:48) , x ) ∈ D c , whichimplies ω ( x (cid:48) ) − x ≤ C ∆( ξ ) . δ ∗ ( ξ ) = 2 C ∆( ξ ) and obtain the estimate δ ∗ ( ξ ) ≥ ω ( x (cid:48) ) − x ) . From thedefinition of δ and since D is a special Lipschitz domain we obtain δ ( x (cid:48) , x ) ≤ d (( x (cid:48) , x ) , ( x (cid:48) , ω ( x (cid:48) ))) = | x − ω ( x (cid:48) ) | = ω ( x (cid:48) ) − x for all points ( x (cid:48) , x ) ∈ D c . It further follows for all λ > that x + λδ ∗ ( x (cid:48) , x ) ≥ x + δ ∗ ( x (cid:48) , x ) ≥ x + 2( ω ( x (cid:48) ) − x ) = 2 ω ( x (cid:48) ) − x . (8)Finally, we also have δ ∗ ( x (cid:48) , x ) = 2 C ∆( x (cid:48) , x ) ≤ C C δ ( x (cid:48) , x ) ≤ C C ( ω ( x (cid:48) ) − x ) . (9) Step 2 (Stein’s extension operator):
Stein defined the operator E for a special Lipschitzdomain D and a function u = g | D where g ∈ C ∞ ( R d ) by E u ( x (cid:48) , x ) = (cid:90) ∞ u ( x (cid:48) , x + λδ ∗ ( x (cid:48) , x )) ψ ( λ ) dλ, x (cid:48) ∈ R , x ∈ R , x < ω ( x (cid:48) ) , (10)where ψ : [1 , ∞ ) → R is a rapidly decaying smooth function with (cid:90) ∞ ψ ( λ ) dλ = 1 and (cid:90) ∞ λ k ψ ( λ ) dλ = 0 for all k ∈ N . Such a function ψ exists indeed according to [17, p.182, Chapter VI, Lemma 1]. Now we fix a point ( x , ω ( x )) ∈ ∂D. The properties of thefunction ψ imply that | ψ ( λ ) | ≤ Aλ − for some constant A . Using this and the previousestimates for δ ∗ we obtain for x < ω ( x ) , | E u ( x , x ) | ≤ A (cid:90) ∞ | u ( x , x + λδ ∗ ( x , x )) | dλλ = A (cid:90) ∞ x + δ ∗ ( x ,x ) | u ( x , s ) | δ ∗ ( x , x ) − dsδ ∗ ( x , x ) − ( s − x ) ≤ Aδ ∗ ( x , x ) (cid:90) ∞ x + δ ∗ ( x ,x ) | u ( x , s ) | ( s − x ) − ds ≤ Aδ ∗ ( x , x ) (cid:90) ∞ ω ( x ) − x | u ( x , s ) | ( s − x ) − ds (cid:46) ( ω ( x ) − x ) (cid:90) ∞ ω ( x ) − x | u ( x , s ) | ( s − x ) − ds, (11)where we used the integral substitution s := x + λδ ∗ ( x , x ) with ds = δ ∗ dλ as well asformulas (8) and (9) in the last two lines.This pointwise estimate is now the basis for proving the relevant estimates in weightedSobolev spaces. Step 3 (Hardy’s inequality and an estimate for α = 0 ): The weight function of the space V l,pδ ( D, R ∗ ) is given by ρ ( x ) := r ( x ) δ − l + | α | , (12)9here r ( x ) denotes the distance of a point x to R ∗ (i.e. the x -axis), thus r ( x ) = x + x .In this step we focus on the case α = 0 , and also additionally assume δ ≥ l .Now we multiply both sides of (11) with ρ ( x , x ) , take the p -th power on both sides andafterwards integrate w.r.t. x < ω ( x ) . This leads to (cid:90) ω ( x ) −∞ ρ ( x , x ) p | E u ( x , x ) | p dx (cid:46) (cid:90) ω ( x ) −∞ ( ω ( x ) − x ) p ρ ( x , x ) p (cid:18)(cid:90) ∞ ω ( x ) − x | u ( x , s ) | ( s − x ) − ds (cid:19) p dx . We make an integral substitution (cid:101) x = ω ( x ) − x with d (cid:101) x = − dx and (cid:101) s = s − ω ( x ) with d (cid:101) s = ds . Then we obtain (cid:90) ω ( x ) −∞ ( ω ( x ) − x ) p ρ ( x , x ) p (cid:18)(cid:90) ∞ ω ( x ) − x | u ( x , s ) | ( s − x ) − ds (cid:19) p dx = (cid:90) ∞ (cid:101) x p ρ ( x , ω ( x ) − (cid:101) x ) p (cid:18)(cid:90) ∞ ω ( x )+ (cid:102) x | u ( x , s ) | ( s − ω ( x ) + (cid:101) x ) − ds (cid:19) p d (cid:101) x ≤ (cid:90) ∞ (cid:101) x p ρ ( x , ω ( x ) − (cid:101) x ) p (cid:18)(cid:90) ∞ (cid:102) x | u ( x , (cid:101) s + ω ( x )) | (cid:101) s − d (cid:101) s (cid:19) p d (cid:101) x . (13)We now intend to apply the following version of Hardy’s inequality from [16, p. 21]: Let v, w : R → R be measurable functions, f ∈ L + p ( R ) = { g ∈ L p ( R ) : g is non-negative } , ≤ p < ∞ and p + q = 1 . Then there exists some finite constant C ( v, w ) such that (cid:18)(cid:90) ∞ (cid:12)(cid:12)(cid:12)(cid:12) v ( x ) (cid:90) ∞ x f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:19) /p ≤ C ( v, w ) (cid:18)(cid:90) ∞ | w ( x ) f ( x ) | p dx (cid:19) /p , (14)if it holds that B ( v, w ) := sup a> (cid:18)(cid:90) a | v ( x ) | p dx (cid:19) /p (cid:18)(cid:90) ∞ a | w ( x ) | − q dx (cid:19) /q < ∞ . (15)We thus need to check the condition (15), where we choose the functions v and w as v ( x ) = xr ( x , ω ( x ) − x ) γ ,w ( x ) = x r ( x , x + ω ( x )) γ , for some parameter γ > specified below. The assumption R ∗ ⊂ ∂D entails ω ( x ,
0) = 0 for all x ∈ R , hence we find | ω ( x ) | = | ω ( x , x ) − ω ( x , | ≤ M | ( x , x ) − ( x , | = M | x | , which implies r ( x , ω ( x ) ± x ) ∼ = | x | + | ω ( x ) ± x | ≤ | x | + ( M | x | + x ) (cid:46) max( | x | , x ) . (16)10urthermore, it holds | x | ≤ r ( x , ω ( x ) ± x ) . (17)Using (16) we estimate the integral (cid:90) a | v ( x ) | p dx = (cid:90) a x p r ( x , ω ( x ) − x ) γp dx (cid:46) max( | x | , a ) γp (cid:90) a x p dx (cid:46) max( | x | , a ) γp a p +1 . (18)Now we treat the second integral in (15). We consider two cases in order to obtain theneeded estimate. Case 1:
For a < (2 M + 1) | x | we conclude with (17), that (cid:90) ∞ a | w ( x ) | − q dx = (cid:90) ∞ a x − q r ( x , x + ω ( x )) − γq dx (cid:46) | x | − γq a − q +1 . Case 2:
For a ≥ (2 M + 1) | x | we have x ≥ a ≥ M | x | ≥ | ω ( x ) | , from which weconclude r ( x , ω ( x ) + x ) ≥ | ω ( x ) + x | ≥ | x − | ω ( x ) || ≥ x , and hence, (cid:90) ∞ a | w ( x ) | − q dx ≤ (cid:90) ∞ a x − q (cid:16) x (cid:17) − γq dx (cid:46) (cid:90) ∞ a x − (2+ γ ) q dx (cid:46) a − (2+ γ ) q +1 . From the two cases above we now see that (cid:90) ∞ a | w ( x ) | − q dx (cid:46) a − q +1 max( a, | x | ) − γq . (19)Now we estimate B ( v, w ) using (18), (19) and the fact that p + q = 1 . This gives B ( v, w ) (cid:46) sup a> (cid:16) max( | x | , a ) γ a p +1 p max( | x | , a ) − γ a − q +1 q (cid:17) = sup a> < ∞ , independent of x , and therefore, condition (15) is satisfied for our choice of functions v and w . Applying now Hardy’s inequality (14) with f ( x ) = x − | u ( x , ω ( x ) + x ) | andweights v and w for γ = δ − l > to (13) we obtain (cid:90) ∞ (cid:101) x p ρ ( x , ω ( x ) − (cid:101) x ) p (cid:18)(cid:90) ∞ (cid:102) x | u ( x , (cid:101) s + ω ( x )) | (cid:101) s − d (cid:101) s (cid:19) p d (cid:101) x (cid:46) (cid:90) ∞ ρ ( x , (cid:101) x + ω ( x )) p | u ( x , (cid:101) x + ω ( x )) | p d (cid:101) x = (cid:90) ∞ ω ( x ) ρ ( x , x ) p | u ( x , x ) | p dx .
11n conclusion, we obtain the inequality (cid:90) ω ( x ) −∞ ρ ( x , x ) p | E u ( x , x ) | p dx (cid:46) (cid:90) ∞ ω ( x ) ρ ( x , x ) p | u ( x , x ) | p dx . (20)Since E u = u for x > ω ( x ) we trivially have (cid:90) ∞ ω ( x ) ρ ( x , x ) p | E u ( x , x ) | p dx = (cid:90) ∞ ω ( x ) ρ ( x , x ) p | u ( x , x ) | p dx . (21)Summing (20) and (21) and integrating w.r.t. x ∈ R we obtain (cid:90) R (cid:90) R ρ ( x , x ) | E u ( x , x ) | p dx dx (cid:46) (cid:90) R (cid:90) ∞ ω ( x ) ρ ( x , x ) | u ( x , x ) | p dx dx . (22)Taking the /p -th power on both sides and applying Fubini’s theorem yields (cid:107) ρ E u | L p ( R ) (cid:107) (cid:46) (cid:107) ρu | L p ( R ) (cid:107) , which takes care of the term with α = 0 in the norm-estimate for E u . Step 4 (The case α (cid:54) = 0 ): Next we need to investigate the partial derivatives of E u , i.e.we consider now ∂ α ( E u ) , where < | α | ≤ l : Since u is a test function of C ∞ ( R \ R ∗ ) and, according to its definition, ψ is a rapidly decaying function (i.e. decays faster thanevery polynomial), applying the Dominated Convergence Theorem we obtain that all ofthe derivatives of u ( · ) ψ ( · ) are bounded, and hence we can change the order of integrationand derivation as follows: ∂ α ( E u )( x , x ) = ∂ α (cid:18)(cid:90) ∞ u ( x , x + λδ ∗ ( x , x )) ψ ( λ ) dλ (cid:19) = (cid:90) ∞ ψ ( λ ) ∂ α (cid:2) u ( x , x + λδ ∗ ( x , x )) (cid:3) dλ. We consider the first-order derivatives of E u using the chain rule. This yields ∂ ( E u ) ∂x i ( x , x )= (cid:90) ∞ ψ ( λ ) ∂ x i (cid:2) u ( x , x + λδ ∗ ( x , x )) (cid:3) dλ = (cid:90) ∞ ψ ( λ ) (cid:16) λδ ∗ x i ( x , x ) u x ( x , x + λδ ∗ ( x , x )) + u x i ( x , x + λδ ∗ ( x , x )) (cid:17) dλ (23)for ≤ i ≤ and ∂ ( E u ) ∂x ( x , x ) = (cid:90) ∞ ψ ( λ ) ∂ x (cid:2) u ( x , x + λδ ∗ ( x , x )) (cid:3) dλ = (cid:90) ∞ ψ ( λ ) (cid:0) λδ ∗ x ( x , x ) + 1 (cid:1) u x ( x , x + λδ ∗ ( x , x )) dλ. (24)12learly higher-order derivative can be calculated similar to (23) and (24). Exemplary, wefind, once again using the chain rule, ∂ ( E u ) ∂x ( x , x ) = ∂∂x (cid:18) ∂ ( E u ) ∂x ( x , x ) (cid:19) = ∂∂x (cid:18)(cid:90) ∞ ψ ( λ )(( λδ ∗ x ( x , x ) + 1) u x ( x , x + λδ ∗ ( x , x ))) dλ (cid:19) = (cid:90) ∞ ψ ( λ ) ∂∂x (cid:18) ( λδ ∗ x ( x , x ) + 1) u x ( x , x + λδ ∗ ( x , x )) (cid:19) dλ = (cid:90) ∞ ψ ( λ )(( λδ ∗ x ( x , x ) + 1) u x ( x , x + λδ ∗ ( x , x ))+ λδ ∗ x ( x , x ) u x ( x , x + λδ ∗ ( x , x ))) dλ = (cid:90) ∞ ψ ( λ ) λ δ ∗ x ( x , x ) u x ( x , x + λδ ∗ ( x , x )) dλ (cid:124) (cid:123)(cid:122) (cid:125) =: I + 2 (cid:90) ∞ ψ ( λ ) λδ ∗ x ( x , x ) u x ( x , x + λδ ∗ ( x , x )) dλ (cid:124) (cid:123)(cid:122) (cid:125) =: II + (cid:90) ∞ ψ ( λ ) u x ( x , x + λδ ∗ ( x , x )) dλ (cid:124) (cid:123)(cid:122) (cid:125) =: III + (cid:90) ∞ ψ ( λ ) λδ ∗ x ( x , x ) u x ( x , x + λδ ∗ ( x , x )) dλ (cid:124) (cid:123)(cid:122) (cid:125) =: IV . (25)Derivatives of higher order just result in a higher number of terms of similar form. Toderive norm-estimates, every term will be treated separately, using the moment conditionsfor ψ together with suitable Taylor expansions. This will be illustrated again by discussingthe case ∂ ( E u ) ∂x in detail. Thus we need to derive estimates for the integrals I–IV in (25).We first recall that according to (7) we have ∂ α δ ∗ ≤ c α ( δ ∗ ) −| α | (26)and | ψ ( λ ) | ≤ A k λ − k for any k ∈ N . We consider I for x < ω ( x ) and obtain with asimilar argument as in the case α = 0 [cf. (11)] | I | ≤ (cid:90) ∞ A λ − λ c x | u x ( x , x + λδ ∗ ( x , x )) | dλ (cid:46) A c x ( ω ( x ) − x ) (cid:90) ∞ ω ( x ) − x | u x ( x , s ) | ( s − x ) − ds. II we similarly find | II | ≤ A (cid:90) ∞ λ − λc x | u x ( x , x + λδ ∗ ( x , x )) | dλ (cid:46) A c x ( ω ( x ) − x ) (cid:90) ∞ ω ( x ) − x | u x ( x , s ) | ( s − x ) − ds. Moreover, concerning
III we get [cf. (11)], | III | ≤ A (cid:90) ∞ λ − | u x ( x , x + λδ ∗ ( x , x )) | dλ (cid:46) A ( ω ( x ) − x ) (cid:90) ∞ ω ( x ) − x | u x ( x , s ) | ( s − x ) − ds. The last term IV needs a bit more care, because it contains the term δ ∗ x and for that theestimate (26) alone is not enough. Instead we use a Taylor expansion to rewrite u x as u x ( x , x + λδ ∗ ( x , x )) = u x ( x , x + δ ∗ ( x , x )) + (cid:90) x + λδ ∗ ( x ,x ) x + δ ∗ ( x ,x ) u x ( x , t ) dt. Due to the properties of ψ , in particular (cid:90) ∞ λψ ( λ ) dλ = 0 , we calculate for the term IV , (cid:90) ∞ ψ ( λ ) λδ ∗ x ( x , x ) u x ( x , x + λδ ∗ ( x , x )) dλ = (cid:90) ∞ ψ ( λ ) λδ ∗ x ( x , x ) u x ( x , x + δ ∗ ( x , x )) dλ + (cid:90) ∞ ψ ( λ ) λδ ∗ x ( x , x ) (cid:90) x + λδ ∗ ( x ,x ) x + δ ∗ ( x ,x ) u x ( x , t ) dt dλ = δ ∗ x ( x , x ) u x ( x , x + δ ∗ ( x , x )) (cid:90) ∞ ψ ( λ ) λ dλ + (cid:90) ∞ ψ ( λ ) λδ ∗ x ( x , x ) (cid:90) x + λδ ∗ ( x ,x ) x + δ ∗ ( x ,x ) u x ( x , t ) dt dλ = (cid:90) ∞ ψ ( λ ) λδ ∗ x ( x , x ) (cid:90) x + λδ ∗ ( x ,x ) x + δ ∗ ( x ,x ) u x ( x , t ) dt dλ. This leads to the following estimate | IV | (cid:46) ( δ ∗ ( x , x )) − A (cid:90) ∞ (cid:32)(cid:90) x + λδ ∗ ( x ,x ) x + δ ∗ ( x ,x ) | u x ( x , t ) | dt (cid:33) λ − dλ = ( δ ∗ ( x , x )) − A (cid:90) ∞ x + δ ∗ ( x ,x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32)(cid:90) ∞ t − x δ ∗ ( x ,x λ − dλ (cid:33) u x ( x , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ∼ ( δ ∗ ( x , x )) − (cid:90) ∞ x + δ ∗ ( x ,x ) ( δ ∗ ( x , x )) | u x ( x , t ) | dt ( t − x ) (cid:46) ( ω ( x ) − x ) (cid:90) ∞ ω ( x ) − x | u x ( x , s ) | ( s − x ) − ds . I - IV imply (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( E u ) ∂x ( x , x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ( ω ( x ) − x ) (cid:90) ∞ ω ( x ) − x | u x ( x , s ) | ( s − x ) − ds, which is the counterpart of the pointwise estimate (11). From there on, the estimate (cid:13)(cid:13)(cid:13) ρ ∂ ( E u ) ∂x (cid:12)(cid:12)(cid:12) L p ( R ) (cid:13)(cid:13)(cid:13) (cid:46) (cid:13)(cid:13)(cid:13) ρ ∂ u∂x (cid:12)(cid:12)(cid:12) L p ( R ) (cid:13)(cid:13)(cid:13) follows mutatis mutandis as in Step 3, simply by replacing u by u x and with ρ ( x ) = r ( x ) δ − l +2 . Since the higher order partial derivatives of ( E u ) can be treated similarly asexplained above, the theorem is proved for δ ≥ l . Step 5 (The case δ < ): We modify the estimate (11) in order to allow negative δ asfollows: According to the conditions of the function ψ for Stein’s extension operator wecan replace | ψ ( λ ) | ≤ Aλ − by | ψ ( λ ) | ≤ A κ λ − κ , where κ ∈ N can be chosen sufficientlylarge. This implies an analogous estimate for E u ( x , x ) as in (11), in particular, as asubstitute we obtain | E u ( x , x ) | (cid:46) ( ω ( x ) − x ) κ − (cid:90) ∞ ω ( x ) − x | u ( x , s ) | ( s − x ) − κ ds. (27)Then, when applying Hardy’s inequality (14), we now choose the functions v ( x ) = x κ − r ( x , ω ( x ) − x ) γ and w ( x ) = x κ r ( x , x + ω ( x )) γ , where now γ = δ − l < . Similar calculations as in Step 3 above yield that the condition(15) of Hardy’s inequality holds if ( κ + γ ) q > . Therefore, we can take any negative γ < if κ is sufficiently large, which results in the estimate (22) with ρ ( x ) = r ( x ) δ − l and δ < and proves the case α = 0 . Concerning α (cid:54) = 0 , the arguments have to be adaptedaccording to Step 4. Remark 3.7.
In the sequel we will (in slight abuse) frequently use the notation u ∈ V l,pβ,δ ( K (cid:48) , S (cid:48) ) , where K (cid:48) denotes a fixed layer of the cone and S (cid:48) := K (cid:48) ∩ S the correspondingpart of the singularity set, when dealing with functions u ∈ V l,pβ,δ ( K, S ) , where we onlyconsider their values on K (cid:48) . We remark that in view of Lemma 3.9 below this is in goodagreement with the definition of the spaces V l,pβ,δ ( K, S ) , since the expression (cid:107) u | V l,pβ,δ ( K (cid:48) , S (cid:48) ) (cid:107) = (cid:18) (cid:90) K (cid:48) (cid:88) | α |≤ l ρ ( x ) p ( β − l + | α | ) n (cid:89) k =1 (cid:18) r (cid:48) k ( x ) ρ ( x ) (cid:19) p ( δ k − l + | α | ) | ∂ α u ( x ) | p dx (cid:19) /p < ∞ , (29) where r (cid:48) k denotes the distance to the edge M k ∩ K (cid:48) is equivalent with (cid:107) u | V l,pβ,δ ( K (cid:48) , S ) (cid:107) .Moreover, if u has support in K (cid:48) we also see that the expressions (cid:107) u | V l,pβ,δ ( K (cid:48) , S (cid:48) ) (cid:107) and (cid:107) u | V l,pβ,δ ( K, S ) (cid:107) are equivalent.
15e now consider a fixed layer K (cid:48) ⊂ K of the cone, i.e., K (cid:48) = { x ∈ K : C < | x | < C } (28)for some constants < C < C < ∞ and give an exten-sion for functions belonging to V l,pβ,δ from the layer K (cid:48) to R . In the proof we use the extension operator for specialLipschitz domains constructed in Theorem 3.5. S (cid:48) K (cid:48) C C We now construct an extension operator from the layer K (cid:48) to R in our weighted Sobolevspaces as follows. Lemma 3.8.
Let K ⊂ R be a polyhedral cone and let K (cid:48) ⊂ K be a fixed layer as in (28) .Furthermore, let l ∈ N , δ ∈ R n , β ∈ R , and ≤ p < ∞ . Then there exists a boundedlinear extension operator E : V l,pβ,δ ( K (cid:48) , S (cid:48) ) −→ V l,pβ,δ ( R , S ) , where S = M ∪ ... ∪ M n and S (cid:48) := K (cid:48) ∩ S .Proof. Step 1: One can see easily that the layer K (cid:48) can be covered by finitely manydiffeomorphic images of bounded wedges W , ..., W n . Therefore, we first give an extensionfor the single wedges W k w.r.t. the spaces V l,pδ k , where δ k ∈ R . Then, using a localizationargument we provide an extension operator for the layer K (cid:48) in the space V l,pβ,δ with β ∈ R , δ = ( δ , ..., δ n ) based on the operators for the single wedges.Let Ω be a bounded wedge representing the neighbourhoodof the edge M k on K (cid:48) , which in cylinder coordinates isgiven by Ω = { x = ( r, ϕ, x ) : 0 < x < , < r < , < ϕ < ϕ } for some < ϕ < π . 10 ϕ Ω V l,pδ k , ρ ( x ) := r k ( x ) δ k − l + | α | ∼ r ( x ) δ k − l + | α | , (30)is equivalent to the power of the distance of a point x to the edge of the bounded wedge Ω (represented by the x -axis in cylinder coordinates).Via multiplication with a cut-off function, we can reducethe problem to considering the half-unbounded wedge Ω + = { x = ( r, ϕ, x ) : x > , r > , < ϕ < ϕ } . Ω + ϕ η be a smooth function on R such that η ( x ) = 1 for x ≥ / and η ( x ) = 0 for x ≤ / , and put η = 1 − η . Then we decompose u = η u + η u . Clearly,we can extend η u to x < and η u to x > , respectively, by zero — both situationsthen obviously being equivalent to discussing functions on Ω + .We proceed as follows. First we give an extension fromthe half-unbounded wedge to the unbounded wedge, Ω = { x = ( r, ϕ, x ) : x ∈ R , r > , < ϕ < ϕ } , and subsequently refer to Theorem 3.5 for anotherextension to R for the special Lipschitz domain Ω (seeRemark 3.6), where the x -axis is rotated to the edge ofthe wedge. ϕ Ω We summarize the problem in a diagram: Ω cut-off −−−−−→ function Ω + Steps −−−−−−−−−→ Ω Theorem 3.5 −−−−−−−→ R Thus, we only need to show the second arrow and use Stein’s extension operator in orderto give this extension.
Step 2:
We need some more notation and preparatory remarks: Let G be a specialLipschitz domain as in Theorem 3.5, G = { x ∈ R : x = ( x (cid:48) , x ) , x (cid:48) ∈ R , x > ω ( x (cid:48) ) } , where ω : R −→ R is a Lipschitz function. Now we consider the sets G − x (cid:48) = { ( x (cid:48) , x ) : x < ω ( x (cid:48) ) } and G + x (cid:48) = { ( x (cid:48) , x ) : x > ω ( x (cid:48) ) } for fixed x (cid:48) ∈ R .We see that Stein’s extension operator from (10)in points x ∈ G − x (cid:48) uses only function values of u in G + x (cid:48) . This allows us to apply the same definitionalso for functions given on the restricted specialLipschitz domain G = { x ∈ R : x = ( x (cid:48) , x ) , x (cid:48) ∈ G , x > ω ( x (cid:48) ) } , where G ⊂ R is a sufficiently smooth domain,which we can choose according to our needs and ω is a Lipschitz function on G . R R x (cid:48) x G G + x (cid:48) G − x (cid:48) ω ( x (cid:48) ) Figure 3: Special Lipschitz domain G above graph of ω ( x (cid:48) ) This particularly applies to half-unbounded wedges by choosing G = { ( r, ϕ ) : r > , <ϕ < ϕ } , where ( r, ϕ ) represents x (cid:48) ∈ R in polar coordinates and the function ω can bedefined as ω ( x (cid:48) ) ≡ for all x (cid:48) ∈ G . Ω + is a restricted special Lipschitz domain in view of G := Ω + = G × R + = { ( x (cid:48) , x ) : x (cid:48) ∈ G , x > ω ( x (cid:48) ) = 0 } . Step 3:
Now we can apply Theorem 3.5 on the domain Ω + = G × R + with some carefulmodifications: For the norm estimates in the spaces V l,pδ k we use Theorem 3.5, in particular,the inequalities (20) and (21) with ω ( x (cid:48) ) = 0 and weight ρ ( x (cid:48) , x ) ≡ (note that the Hardyinequality (14) also works with v = x and w = x when ρ ≡ ). This gives (cid:90) −∞ | E u ( x (cid:48) , x ) | p dx (cid:46) (cid:90) ∞ | u ( x (cid:48) , x ) | p dx . (31)and (cid:90) ∞ | E u ( x (cid:48) , x ) | p dx = (cid:90) ∞ | u ( x (cid:48) , x ) | p dx . (32)Then we sum up (31) and (32), multiply with the weight ρ ( x (cid:48) , x ) = | x (cid:48) | δ k (measuring thedistance to the x -Axis) and integrate w.r.t. x ∈ G . Since the weight is independent of x this leads to (cid:90) G (cid:90) R ρ ( x (cid:48) , x ) | E u ( x (cid:48) , x ) | p dx dx (cid:48) (cid:46) (cid:90) G (cid:90) ∞ ρ ( x (cid:48) , x ) | u ( x (cid:48) , x ) | p dx dx (cid:48) . (33)Hence, we obtain (cid:107) E u | V ,pδ k (Ω , R ∗ ) (cid:107) (cid:46) (cid:107) u | V ,pδ k (Ω + , R ∗ ) (cid:107) . For l > we can use a similar argument as in Theorem 3.5. Summarizing, Stein’sextension operator can be modified to give an extension from Ω + to Ω together withnorm estimates in associated weighted Sobolev spaces V l,pδ k , δ k ∈ R . Finally, on Ω wecan apply the results of Theorem 3.5 to extend functions from Ω to R .We denote corresponding extension operators for the wedges W , ..., W n by E , ..., E n . Step 4:
The results for the single wedges can now be transferred to the layer K (cid:48) , andspaces V l,pβ,δ , δ ∈ R n , where the weight function contains r (cid:48) , ..., r (cid:48) n , i.e., the distances to theedges of all wedges along the layer. In order to glue these operators properly together weneed some preparations: Let K (cid:48) ⊂ n (cid:91) j =1 U j , where W j = K (cid:48) ∩ U j is a wedgeas in Step 1 and U j does not intersect the edges M i for i (cid:54) = j . Since K (cid:48) is a boundedLipschitz domain, the sets U j can be chosen such that there exists an ε > with B ( x, ε ) ⊂ U j for some j for every x ∈ ∂K (cid:48) . Then we define U εj = { x ∈ U j : B ( x, ε ) ⊂ U j } . Moreover,let ϕ , ..., ϕ n be non-negative smooth functions with ϕ j ( x ) = 1 on U ε/ j and supp ϕ j ⊂ U j . B ( x, ε ) U j ∂K (cid:48) W j Figure 4: The set U j ϕ j ∂K (cid:48) U εj U ε/ j U j Figure 5: The function ϕ j Furthermore, let Ψ be a smooth function withsupp Ψ ⊂ K (cid:48) and Ψ( x ) = 1 for x ∈ K (cid:48) with dist( x, ∂K (cid:48) ) > ε/ , and let Φ be another smooth function with Φ = 1 − Ψ on K (cid:48) and supp Φ ⊂ { x ∈ R : dist( x, ∂K (cid:48) ) < ε/ } ⊂ (cid:91) j U ε/ j .ε/ ε/ K (cid:48) Ψ = 1 supp Ψ supp ΦΦ = 1 − Ψ Figure 6: The functions Ψ and Φ from above point of viewWe glue together the operators for the wedges and define E u ( x ) = Ψ( x ) u ( x ) + Φ( x ) (cid:80) nj =1 ϕ j ( x ) E j ( ϕ j u )( x ) (cid:80) nj =1 ϕ j ( x ) . The second term is well-defined, since for every x ∈ supp Φ we have ϕ j ( x ) = 1 for at leastone j . Since E u ( x ) = u ( x ) for x ∈ K (cid:48) (according to the definition of the functions Ψ and Φ ) and E u = 0 on { x ∈ R \ K (cid:48) : dist( x, ∂K (cid:48) ) > ε/ } , (34)we see that E indeed defines an extension from the layer K (cid:48) to R , which is continuousbecause every term of E is continuous. 19t remains to verify the corresponding norm estimate in the spaces V l,pβ,δ with δ = ( δ , ..., δ n ) ,i.e., (cid:107) E u | V l,pβ,δ ( R , S ) (cid:107) (cid:46) (cid:107) u | V l,pβ,δ ( K (cid:48) , S (cid:48) ) (cid:107) . (35)The functions Ψ , Φ , ϕ j are smooth with compact supports, and are therefore multipliersin the spaces V l,pβ,δ , cf. (4). An easy calculation gives (cid:107) E u | V l,pβ,δ ( R , S ) (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ψ u + Φ (cid:80) nj =1 ϕ j E j ( ϕ j u ) (cid:80) nj =1 ϕ j (cid:12)(cid:12)(cid:12) V l,pβ,δ ( R , S ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:46) (cid:107) Ψ u | V l,pβ,δ ( R , S ) (cid:107) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:88) j =1 ϕ j E j ( ϕ j u ) | V l,pβ,δ ( R , S ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:46) (cid:107) u | V l,pβ,δ ( K (cid:48) , S ) (cid:107) + n (cid:88) j =1 (cid:107) ϕ j E j ( ϕ j u ) | V l,pβ,δ ( R , S ) (cid:107) =: I + II.
Since for the first term I we have (cid:107) u | V l,pβ,δ ( K (cid:48) , S ) (cid:107) ∼ (cid:107) u | V l,pβ,δ ( K (cid:48) , S (cid:48) ) (cid:107) it suffices to considerthe second term II . In the estimation of II we will use the following norm estimates ofthe operators for single wedges, (cid:107) E k u | V l,pδ k ( R , M k ) (cid:107) (cid:46) (cid:107) u | V l,pδ k ( W k , M k ) (cid:107) ∼ (cid:107) u | V l,pδ k ( W k , M k ∩ W k ) (cid:107) ∀ k ∈ { , ..., n } . Moreover, by (34) we have supp( E u ) ⊂ (cid:83) nj =1 U j , i.e., the support is contained in a smallneighbourhood of K (cid:48) . Since on U j (neighborhood of the wedge W j ) we have for thedistance functions ρ , r , ..., r j − , r j +1 , ..., r n ∼ , it follows that on U j the norms of V l,pδ j and V l,pβ,δ are equivalent w.r.t. S . A similar statement holds for ρ (cid:48) , r (cid:48) , ..., r (cid:48) n replacing S by S (cid:48) . This ultimately yields n (cid:88) j =1 (cid:107) ϕ j E j ( ϕ j u ) | V l,pβ,δ ( R , S ) (cid:107)∼ n (cid:88) j =1 (cid:107) ϕ j E j ( ϕ j u ) | V l,pδ j ( R , M j ) (cid:107) (cid:46) n (cid:88) j =1 (cid:107) E j ( ϕ j u ) | V l,pδ j ( R , M j ) (cid:107) (cid:46) n (cid:88) j =1 (cid:107) ϕ j u | V l,pδ j ( W j , W j ∩ M j ) (cid:107) (cid:46) n (cid:88) j =1 (cid:107) ϕ j u | V l,pβ,δ ( K (cid:48) , S (cid:48) ) (cid:107) (cid:46) (cid:107) u | V l,pβ,δ ( K (cid:48) , S (cid:48) ) (cid:107) . Therefore, (35) holds, which finally completes the proof.
Having established an extension operator for our weighted Sobolev spaces on the fixedlayer K (cid:48) ⊂ K of the cone in Lemma 3.8 we are now going to construct an extensionoperator on the whole cone K as follows. We first decompose the cone K into severallayers K j and use extension operators on each layer (which are uniformly bounded w.r.t.20 ). Afterwards we glue these extension operators together in a suitable way to obtain anextension for the whole cone. For this let K j = { x ∈ K : 2 − j − < | x | < − j +1 } (36)be a (dyadic) layer of the cone. The following lemma shows that on these layers thedistance of a point x ∈ K j to the restricted set S j := K j ∩ S is equivalent to its distanceto the whole set S . Lemma 3.9.
For the layer E j := { x ∈ R : 2 − j − < | x | < − j +1 } and the set S j definedabove consider the respective distance function on E j w.r.t. the edge M k , r k,j ( x ) := dist( x, S j ∩ M k ) . Then we have r k,j ( x ) ∼ r k ( x ) for all x ∈ E j , i.e., the distance of a point x ∈ E j to the set S j ∩ M k is equivalent to the distance of x tothe whole edge M k .Proof. We decompose x as x = x (cid:107) + x ⊥ with x (cid:107) being the component parallel to M k andconsider the difference ∆ = 2 − j − − | x (cid:107) | . Case 1:
Let x ∈ E j and ∆ > , i.e. | x (cid:107) | < − j − . It is clear that r k ( x ) ≤ r k,j ( x ) . (37)By geometric considerations we see that for x ∈ E j it holds r k ( x ) + | x (cid:107) | = | x | ≥ (2 − j − ) = (∆ + | x (cid:107) | ) , hence r k ( x ) ≥ ∆ + 2∆ | x (cid:107) | ≥ ∆ . (38) − j +1 − j − M k ∩ E j E j ∩ K ∆ r k,j ( x ) x ⊥ = r k ( x ) x (cid:107) x (cid:107) x ⊥ | x | x ∈ E j Since r k,j ( x ) = r k ( x ) + ∆ , using (38) we obtain r k,j ( x ) ≤ r k ( x ) , i.e., r k,j ( x ) (cid:46) r k ( x ) . (39)Therefore (37) and (39) imply that r k,j ( x ) ∼ r k ( x ) for x ∈ E j , ∆ > . ase 2: Let x ∈ E j and ∆ ≤ , then − j − ≤ | x (cid:107) | ≤ − j +1 . Obviously in this case we have r k ( x ) = r k,j ( x ) . − j +1 − j − M k ∩ E j E j ∩ K x (cid:107) | x | x ⊥ = r k ( x ) = r k,j ( x ) x ∈ E j Together Cases 1 and 2 yield r k,j ∼ r k on E j . Remark 3.10. (i) Below we use the notation V l,pβ,δ ( K j , S j ) . This is to be understoodin the same way as already explained in Remark 3.7. In particular, in view of thedefinition of the spaces V l,pβ,δ ( K, S ) , we replace the weight functions r k by r k,j (whichaccording to Lemma 3.9 are equivalent on the layers K j ), and the integral domain K by K j in the norm, respectively.(ii) Let θ be the angle of the edge M k and of theline −→ x . Then sin θ = r k ( x ) ρ ( x ) = r k (2 j x ) ρ (2 j x ) and we see that the distance functions r k (2 j x ) = 2 j | x | sin θ = 2 j r k ( x ) and ρ (2 j x ) = | j x | = 2 j | x | = 2 j ρ ( x ) are homogeneous w.r.t. isotropic dilations. x x x M k x j x r k ( x ) r k (2 j x ) θ After these preparations we can prove Theorem 3.1.
Proof of Theorem 3.1.
We decompose K into the dyadic layers K j from (36), K = (cid:91) j ≥ j K j , where j ∈ Z is fixed and depends on the cone. Step 1:
Let E denote the extension operator from Lemma 3.8 with regard to the layer K . Then we define an extension operator on K j via E j = T − j ◦ E ◦ T j , where the dilation operator T j is defined as T j u ( x ) = u (2 j x ) . The inverse operator isgiven by T − j = T − j . In order to prove uniform boundedness of the family of extension22perators ( E j ) j we first need to calculate the operator norms of the family of dilationoperators ( T j ) j . In particular, for u ∈ V l,pβ,δ ( K , S ) using the homogeneity of r k and ρ ,we compute (cid:107) T j u | V l,pβ,δ ( K j , S j ) (cid:107) p ∼ (cid:88) | α |≤ l (cid:90) K j ρ ( x ) ( β − l + | α | ) p n (cid:89) k =1 (cid:18) r k ( x ) ρ ( x ) (cid:19) ( δ k − l + | α | ) p j | α | p | ( ∂ α u )(2 j x ) | p dx = (cid:88) | α |≤ l (cid:90) K j − j ( β − l + | α | ) p ρ (2 j x ) ( β − l + | α | ) p n (cid:89) k =1 (cid:18) r k (2 j x ) ρ (2 j x ) (cid:19) ( δ k − l + | α | ) p j | α | p | ( ∂ α u )(2 j x ) | p dx = 2 − j ( β − l ) p − j (cid:107) u | V l,pβ,δ ( K , S ) (cid:107) p , with the integral substitution y := 2 j x and dy = 2 j dx . Hence, (cid:107) T j (cid:107) ∼ − j ( β − l + p ) ∀ j ∈ Z . (40)A similar estimate holds for T − j . We conclude (cid:107) E j : V l,pβ,δ ( K j , S j ) → V l,pβ,δ ( R , S ) (cid:107) (cid:46) (cid:107) T − j || · (cid:107) E : V l,pβ,δ ( K , S ) → V l,pβ,δ ( R , S ) (cid:107) · (cid:107) T j (cid:107) = (cid:107) E : V l,pβ,δ ( K , S ) → V l,pβ,δ ( R , S ) (cid:107) , independent of j ∈ Z . Since E is a bounded operator according to Lemma 3.8, ( E j ) j ∈ Z is a family of uniformly bounded operators and (cid:107) E j ϕ | V l,pβ,δ ( R , S ) (cid:107) (cid:46) (cid:107) ϕ | V l,pβ,δ ( K j , S j ) (cid:107) ∀ j ∈ Z . Step 2:
We define the extension operator E via a suitable combination of the operators E j . Substep 2.1:
Let ϕ : R −→ [0 , be a radially symmetric smooth function, such thatsupp ϕ ⊂ (cid:26) < | x | < (cid:27) =: E , and ϕ ≡ on (cid:26)
12 + ε < | x | < − ε (cid:27) , for some ε > sufficiently small. Moreover, we put ϕ j ( x ) := ϕ (2 j x ) , for which we havesupp ϕ j ⊂ { − j − < | x | < − j +1 } =: E j , and ϕ j ≡ on (cid:26) − j (cid:18)
12 + ε (cid:19) < | x | < − j (2 − ε ) (cid:27) . (cid:88) j ∈ Z ϕ j ( x ) ∼ for all x ∈ R . (41)Furthermore, since ϕ is smooth and has compact support, we see that | ∂ α ϕ | ≤ c α and | ∂ α ϕ j ( x ) | = | ∂ α ( ϕ (2 j x )) | ≤ c α j | α | , (42)for some constant c α . By (41) and (42) the family ( ϕ j ) j ∈ Z forms a suitable localizationfor the cone K , where supp ϕ j | K ⊂ K j , and we can apply Lemma 2.4. Substep 2.2:
Now we put Φ( x ) = (cid:88) j ∈ Z ϕ j ( x ) , x (cid:54) = 0 , and define the extension operator via E u ( x ) := 1Φ( x ) (cid:88) j ∈ Z ϕ j ( x ) E j ( ϕ j | K u )( x ) . The mapping E is well-defined because from the construction we deduce Φ( x ) ∼ for all x (cid:54) = 0 . Moreover, it is immediately verified that E u defines an extension of u to R . Step 3:
It remains to show that (cid:107) E u | V l,pβ,δ ( R , S ) (cid:107) (cid:46) (cid:107) u | V l,pβ,δ ( K, S ) (cid:107) . (43)For the calculations to come we use the fact that the localization of Lemma 2.4 also worksfor the spaces V l,pβ,δ ( R , S ) and that the functions ϕ j , j ∈ Z , are multipliers in the spaces V l,pβ,δ . With this we estimate (cid:107) E u | V l,pβ,δ ( R , S ) (cid:107) p ∼ (cid:88) j ≥ j (cid:107) ϕ j E u | V l,pβ,δ ( R , S ) (cid:107) p ∼ (cid:88) j ≥ j − j ( β − l ) p − j (cid:107) ( ϕ j E u )(2 − j · ) | V l,pβ,δ ( R , S ) (cid:107) p = (cid:88) j ≥ j − j ( β − l ) p − j (cid:107) ϕ ( · ) E u (2 − j · ) | V l,pβ,δ ( R , S ) (cid:107) p = (cid:88) j ≥ j − j ( β − l ) p − j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ϕ ( x ) (cid:124) (cid:123)(cid:122) (cid:125) supp ·⊂ E − j x ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ (cid:88) m ∈ Z ϕ m (2 − j x ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:54) =0 for | m − j |≤ E m ( ϕ m | K u )(2 − j x ) (cid:12)(cid:12)(cid:12)(cid:12) V l,pβ,δ ( R , S ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p (cid:46) (cid:88) j ≥ j − j ( β − l ) p − j (cid:107) E j ( ϕ j | K u )(2 − j · ) | V l,pβ,δ ( R , S ) (cid:107) p ∼ (cid:88) j ≥ j (cid:107) E j ( ϕ j | K u ) | V l,pβ,δ ( R , S ) (cid:107) p (cid:46) (cid:88) j ≥ j (cid:107) ϕ j | K u | V l,pβ,δ ( K j , S j ) (cid:107) p ∼ (cid:88) j ≥ j (cid:107) ϕ j u | V l,pβ,δ ( K, S ) (cid:107) p ∼ (cid:107) u | V l,pβ,δ ( K, S ) (cid:107) p , which completes the proof. 24 eferences [1] Babuska, I. and Guo, B. (1997). Regularity of the solutions for elliptic problemson nonsmooth domains in R , Part I: countably normed spaces on polyhedraldomains. Proc. Roy. Soc. Edinburgh Sect. A , 77–126.[2] Cioica-Licht, P. A. (2020). An L p -theory for the stochastic heat equationon angular domains in R with mixed weights . Preprint, arXiv:2003.03782v2[math.PR].[3] Cioica-Licht, P. A., Kim, K.-H., and Lee, K. (2019). On the regularity of thestochastic heat equation on polygonal domains in R . J. Differential Equations , , 6447–6479.[4] Costabel, M., Dauge, M., and Nicaise, S. (2010). Mellin analysis of weightedSobolev spaces with nonhomogeneous norms on cones. Around the research ofVladimir Maz’ya I, Int. Math. Ser. (N. Y.) , Springer, New York, 105–136.[5] Dahlke, S. and DeVore, R. A. (1997). Besov regularity for elliptic boundaryvalue problems. Comm. Partial Differential Equations , No. 1-2, 1–16.[6] Dahlke, S., Hansen, M., Schneider, C., Sickel, W. (2020). Properties of Kon-dratiev spaces.
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