Wavelet bases for function spaces on cellular domains

Transcription

1 Wavelet bases for function spaces on cellular domains Benjamin Scharf Friedrich Schiller University Jena January 14, 2011 Wavelet bases for function spaces on cellular domains Benjamin Scharf 1/28

2 Table of contents 1 Introduction and basic definitions Domains in R n The definition of function spaces on R n The definition of function spaces on domains Ω 2 Wavelets for function spaces on domains Ω Definition Wavelet results Traces on the boundary and wavelet-friendly extension operators Wavelets for cubes 3 The exceptional values of s Some known results Reinforced spaces Wavelet bases for function spaces on cellular domains Benjamin Scharf 2/28

3 Domains in R n (i) Let 2 n. A special Lipschitz (C -) domain in R n is the collection of all points x = (x, x n ) with x R n 1 and h(x ) < x n, where h is a Lipschitz (bounded C -) function. Wavelet bases for function spaces on cellular domains Benjamin Scharf 3/28

4 Domains in R n (i) Let 2 n. A special Lipschitz (C -) domain in R n is the collection of all points x = (x, x n ) with x R n 1 and h(x ) < x n, where h is a Lipschitz (bounded C -) function. (ii) Let 2 n. A bounded Lipschitz (C -) domain is a bounded domain Ω where the boundary Γ = Ω can be covered by finitely many open balls B j centred at Γ such that B j Ω = B j Ω j, where Ω j are rotations of suitable special Lipschitz (C -) domains. Wavelet bases for function spaces on cellular domains Benjamin Scharf 3/28

5 Domains in R n (i) Let 2 n. A special Lipschitz (C -) domain in R n is the collection of all points x = (x, x n ) with x R n 1 and h(x ) < x n, where h is a Lipschitz (bounded C -) function. (ii) Let 2 n. A bounded Lipschitz (C -) domain is a bounded domain Ω where the boundary Γ = Ω can be covered by finitely many open balls B j centred at Γ such that B j Ω = B j Ω j, where Ω j are rotations of suitable special Lipschitz (C -) domains. (iii) Let 2 n. A domain Ω in R n is called cellular if it is a bounded Lipschitz domain which can be represented as ( L ) Ω = Ω l with Ω l Ω l = if l l l=1 such that each Ω l is diffeomorphic to a polyhedron (cube). Wavelet bases for function spaces on cellular domains Benjamin Scharf 3/28

6 The definition of B s p,q(r n ) Let {ϕ j } j=0 be a resolution of unity. Let 0 < p, 0 < q and s R. For f S (R n ) we define f Bp,q(R s n ) ϕ := 2 jsq (ϕ jˆf )ˇ L p q (modified in case q = ) and j=0 B s,ϕ p,q (R n ) := { f S (R n ) : f B s p,q(r n ) ϕ < }. Then (Bp,q s,ϕ (R n ), Bp,q(R s n ) ϕ ) is a quasi-banach space. It does not depend on the choice of the resolution of unity {ϕ j } j=0 in the sense of equivalent norms. So we denote it shortly by Bp,q(R s n ). 1 q Wavelet bases for function spaces on cellular domains Benjamin Scharf 4/28

7 The definition of F s p,q(r n ) Let {ϕ j } j=0 be a resolution of unity. Let 0 < p <, 0 < q and s R. For f S (R n ) we define 1 q f Fp,q(R s n ) ϕ := 2 jsq (ϕ jˆf )ˇ E q Lp j=0 (modified in case q = ) and F s,ϕ p,q (R n ) := { f S (R n ) : f F s p,q(r n ) ϕ < }. Then (Fp,q s,ϕ (R n ), Fp,q(R s n ) ϕ ) is a quasi-banach space. It does not depend on the choice of the resolution of unity {ϕ j } j=0 in the sense of equivalent norms. So we denote it shortly by Fp,q(R s n ). Wavelet bases for function spaces on cellular domains Benjamin Scharf 5/28

8 The definition of A s p,q(ω), Ã s p,q(ω) and Ås p,q(ω) (i) Let Ω be an open set in R n. Denote by g Ω D (Ω) its restriction to Ω, hence (g Ω)(ϕ) = g(ϕ) for ϕ D(Ω). A s p,q(ω) := {f D (Ω) : f = g Ω for some g A s p,q(r n )}, f A s p,q(ω) = inf g A s p,q(r n ), where the infimum is taken over all g A s p,q(r n ) with g Ω = f. Wavelet bases for function spaces on cellular domains Benjamin Scharf 6/28

9 The definition of A s p,q(ω), à s p,q(ω) and Ås p,q(ω) (i) Let Ω be an open set in R n. Denote by g Ω D (Ω) its restriction to Ω, hence (g Ω)(ϕ) = g(ϕ) for ϕ D(Ω). A s p,q(ω) := {f D (Ω) : f = g Ω for some g A s p,q(r n )}, f A s p,q(ω) = inf g A s p,q(r n ), where the infimum is taken over all g A s p,q(r n ) with g Ω = f. Moreover, let à s p,q( Ω) := {f A s p,q(r n ) : supp f Ω} with the quasi-norm from A s p,q(r n ). Then à s p,q(ω) := {f D (Ω) : f = g Ω for some g à s p,q( Ω)}, f à s p,q(ω) = inf g à s p,q( Ω) where the infimum is taken over all g à s p,q( Ω) with g Ω = f. Wavelet bases for function spaces on cellular domains Benjamin Scharf 6/28

10 The definition of A s p,q(ω), à s p,q(ω) and Ås p,q(ω) (ii) It holds A s p,q(ω) = A s p,q(r n )/à s p,q(ω C ), à s p,q(ω) = à s p,q( Ω)/à s p,q( Ω) in the sense of quotient spaces and norms. Hence A s p,q(ω) and à s p,q(ω) are quasi-banach spaces. Wavelet bases for function spaces on cellular domains Benjamin Scharf 7/28

11 The definition of A s p,q(ω), à s p,q(ω) and Ås p,q(ω) (ii) It holds A s p,q(ω) = A s p,q(r n )/à s p,q(ω C ), à s p,q(ω) = à s p,q( Ω)/à s p,q( Ω) in the sense of quotient spaces and norms. Hence A s p,q(ω) and à s p,q(ω) are quasi-banach spaces. Let Ω be a cellular domain and let ( ( ) 1 0 < p, 0 < q, max n p 1, 1 ) p 1 < s (p < for the F -spaces). Then by T08, Prop à s p,q(ω) = Ãs p,q( Ω) in the sense of Ãs p,q( Ω) = {0}. Wavelet bases for function spaces on cellular domains Benjamin Scharf 7/28

12 The definition of A s p,q(ω), Ã s p,q(ω) and Ås p,q(ω) (iii) Furthermore, let Å s p,q(ω) be the completion of D(Ω) with respect to A s p,q(ω). Theorem (The starting stripe - T08, Prop ) Let Ω be a cellular domain and let Then 0 < p <, 0 < q <, max ( n A s p,q(ω) = Å s p,q(ω) = Ã s p,q(ω). ( ) 1 p 1, 1 ) p 1 < s < 1 p Wavelet bases for function spaces on cellular domains Benjamin Scharf 8/28

13 The definition of A s p,q(ω), Ã s p,q(ω) and Ås p,q(ω) (iii) Furthermore, let Å s p,q(ω) be the completion of D(Ω) with respect to A s p,q(ω). Theorem (The starting stripe - T08, Prop ) Let Ω be a cellular domain and let Then 0 < p <, 0 < q <, max ( n A s p,q(ω) = Å s p,q(ω) = Ã s p,q(ω). ( ) 1 p 1, 1 ) p 1 < s < 1 p Proof: Use that χ Ω is a pointwise multiplier in these spaces. Wavelet bases for function spaces on cellular domains Benjamin Scharf 8/28

14 The starting stripe s 0 1 n+1 n 1 p -1 Wavelet bases for function spaces on cellular domains Benjamin Scharf 9/28

15 Wavelets for function spaces on domains Ω (i) From now on let Ω be a cellular domain and Γ = Ω. Let } Z Ω = {x j l Ω : j N 0, l = 1,..., N j, (typically N j 2 jn ), such that for some c 1 > 0 x j l x j l c 1 2 j, j N 0, l l. Wavelet bases for function spaces on cellular domains Benjamin Scharf 10/28

16 Wavelets for function spaces on domains Ω (i) From now on let Ω be a cellular domain and Γ = Ω. Let } Z Ω = {x j l Ω : j N 0, l = 1,..., N j, (typically N j 2 jn ), such that for some c 1 > 0 x j l x j l c 1 2 j, j N 0, l l. We can introduce the usual sequence spaces bp,q(z s Ω ) and fp,q(z s Ω ) adapted to Z Ω and abbreviated by ap,q(z s Ω ). For example, 1 λ f s p,q(z Ω ) := N j 2 jsq λ j l χ j,l q j=0 l=1 q L p (Ω), where χ j,l is the characteristic function of B(x j l, c 32 j ) or B(x j l, c 32 j ) Ω with another constant c 3. Wavelet bases for function spaces on cellular domains Benjamin Scharf 10/28

17 Wavelets for function spaces on domains Ω (ii) Let u N 0 (now incorporating Haar-Wavelets). Then } Φ = {Φ j l : j N 0, l = 1,..., N j C u (Ω) is called a u-wavelet system in Ω if it fulfils support conditions: For some c 3 > 0 let supp Φ j l B(x j l, c 32 j ) Ω, j N 0, l = 1,..., N j, Wavelet bases for function spaces on cellular domains Benjamin Scharf 11/28

18 Wavelets for function spaces on domains Ω (ii) Let u N 0 (now incorporating Haar-Wavelets). Then } Φ = {Φ j l : j N 0, l = 1,..., N j C u (Ω) is called a u-wavelet system in Ω if it fulfils support conditions: For some c 3 > 0 let supp Φ j l B(x j l, c 32 j ) Ω, j N 0, l = 1,..., N j, derivative conditions: For some c 4 > 0 and all α N n 0 with 0 α u let D α Φ j l (x) c4 2 j n 2 +j α, j N 0, l = 1,..., N j, x Ω. Wavelet bases for function spaces on cellular domains Benjamin Scharf 11/28

19 Wavelets for function spaces on domains Ω (iii) Additionally, the u-wavelet system is called oscillating if it fulfils (substitute) moment conditions: Let c 5 and c 6 < c 7 be constants such that dist(b(xl 0, c 3), Γ) c 6, for l = 1,..., N 0 and ψ(x)φ j l (x) dx c 5 2 j n 2 ju ψ C u (Ω) for all ψ C u (Ω) Ω for all Φ j l with j N and c 62 j dist(b(x 0 l, c 3), Γ) c 7 2 j. Wavelet bases for function spaces on cellular domains Benjamin Scharf 12/28

20 Wavelets for function spaces on domains Ω (iii) Additionally, the u-wavelet system is called oscillating if it fulfils (substitute) moment conditions: Let c 5 and c 6 < c 7 be constants such that dist(b(xl 0, c 3), Γ) c 6, for l = 1,..., N 0 and ψ(x)φ j l (x) dx c 5 2 j n 2 ju ψ C u (Ω) for all ψ C u (Ω) Ω for all Φ j l with j N and c 62 j dist(b(x 0 l, c 3), Γ) c 7 2 j. An oscillating u-wavelet system is called interior if it fulfils (further) interior support conditions, namely dist(b(x j l, c 32 j ), Γ) c 6 2 j, j N 0, l = 1,..., N j. Wavelet bases for function spaces on cellular domains Benjamin Scharf 12/28

21 Wavelet bases in L 2 (Ω) - the starting point Theorem (T08 - Theorem 2.33) Let Ω be an arbitrary domain in R n. For any u N 0 there is an orthonormal basis in L 2 (Ω) which is simultaneously an interior u-wavelet system. It even fulfils true moment conditions inside the domain. For u = 0 one can take the Haar Wavelet suitably restricted. Wavelet bases for function spaces on cellular domains Benjamin Scharf 13/28

22 Wavelet bases in L 2 (Ω) - the starting point Theorem (T08 - Theorem 2.33) Let Ω be an arbitrary domain in R n. For any u N 0 there is an orthonormal basis in L 2 (Ω) which is simultaneously an interior u-wavelet system. It even fulfils true moment conditions inside the domain. For u = 0 one can take the Haar Wavelet suitably restricted. Proof (main idea): First start with wavelets with small enough support. Take a wavelet basis in L 2 (R n ), restrict them suitably to Ω adapted to a Whitney decomposition. Orthonormalize at the boundary (losing moment conditions there). In the special case of the Haar Wavelet no further orthonormalisation is necessary because of no overlapping. Wavelet bases for function spaces on cellular domains Benjamin Scharf 13/28

23 Wavelet bases in à s p,q(ω) and L p Theorem (T08 - Theorem 2.36, Prop. 3.10) Let Ω be a Lipschitz (E-thick) domain in R n. Let u > s > σ p resp. > σ p,q. Then the interior u-wavelet system orthonormal in L 2 (Ω) is a Riesz basis for L p (Ω) = F 0 p,2 ( Ω), 1 < p <, for B s p,q(ω) and F s p,q(ω). This means f L loc 1 (resp. D (Ω)) is an element of à s p,q(ω) if and only if it can be represented as f = N j λ j r 2 jn 2 Φ j r, λ ap,q(z s Ω ). j=0 r=1 Moreover, the representation is unique with λ j r (f ) = 2 jn 2 (f, Φ j r ) and f à s p,q(ω) λ a s p,q(z Ω ). Wavelet bases for function spaces on cellular domains Benjamin Scharf 14/28

24 Wavelet bases in A s p,q(ω) Theorem (T08 - Prop. 3.13) Let Ω be a Lipschitz (E-thick) domain in R n. Let s < 0 and u > σ p s resp. u > σ p,q s. Then the interior u-wavelet system orthonormal in L 2 (Ω) is a Riesz basis for B s p,q(ω) and F s p,q(ω). This means f D (Ω)) is an element of A s p,q(ω) if and only if it can be represented as f = N j λ j r 2 jn 2 Φ j r, λ ap,q(z s Ω ). j=0 r=1 Moreover, the representation is unique with λ j r (f ) = 2 jn 2 (f, Φ j r ) and f A s p,q(ω) λ a s p,q(z Ω ). Wavelet bases for function spaces on cellular domains Benjamin Scharf 15/28

25 Wavelet bases in A s p,q(ω), Ã s p,q(ω) and L p Proof (main idea) of the theorems: Wavelets serve as atoms and local means. To generate moment conditions for atoms (s < 0) or local means (s > 0) one has to extend the boundary wavelets. Then f needs support in Ω for s > 0. Wavelet bases for function spaces on cellular domains Benjamin Scharf 16/28

26 Wavelet bases in A s p,q(ω), à s p,q(ω) and L p Proof (main idea) of the theorems: Wavelets serve as atoms and local means. To generate moment conditions for atoms (s < 0) or local means (s > 0) one has to extend the boundary wavelets. Then f needs support in Ω for s > 0. Corollary (Homogeneity property - T08, Theorem 2.11.) Let U λ be either the balls or cubes with radius resp. diameter λ. Then for s > σ p resp. s > σ p,q For s < 0 it holds f (λ ) à s p,q(u 1 ) λ s n p f à s p,q(u λ ). f (λ ) A s p,q(u 1 ) λ s n p f A s p,q(u λ ), where the equivalence constants in both cases are independent of 0 < λ 1 and of f. Wavelet bases for function spaces on cellular domains Benjamin Scharf 16/28

27 Wavelet bases in A s p,q(ω), à s p,q(ω) and L p Proof (main idea) of the theorems: Wavelets serve as atoms and local means. To generate moment conditions for atoms (s < 0) or local means (s > 0) one has to extend the boundary wavelets. Then f needs support in Ω for s > 0. Corollary (Homogeneity property - T08, Theorem 2.11.) Let U λ be either the balls or cubes with radius resp. diameter λ. Then for s > σ p resp. s > σ p,q For s < 0 it holds f (λ ) à s p,q(u 1 ) λ s n p f à s p,q(u λ ). f (λ ) A s p,q(u 1 ) λ s n p f A s p,q(u λ ), where the equivalence constants in both cases are independent of 0 < λ 1 and of f. Proof idea: The wavelet representations of U 2 j and U 1 can be transformed into each other by dilation with 2 j. Wavelet bases for function spaces on cellular domains Benjamin Scharf 16/28

28 Wavelet bases in the starting stripe Corollary (T09 - Theorem 2.13., T08 - Prop ) Let Ω be a cellular domain and let 1 p <, 0 < q < resp. 1 q <. Then the interior u-wavelet system (and the Haar Wavelet system) are Riesz bases for A s p,q(ω) in the starting stripe 1 p 1 < s < 1 p. Wavelet bases for function spaces on cellular domains Benjamin Scharf 17/28

29 Wavelet bases in the starting stripe Corollary (T09 - Theorem 2.13., T08 - Prop ) Let Ω be a cellular domain and let 1 p <, 0 < q < resp. 1 q <. Then the interior u-wavelet system (and the Haar Wavelet system) are Riesz bases for A s p,q(ω) in the starting stripe 1 p 1 < s < 1 p. Proof idea: By A s p,q(ω) = Ãs p,q(ω) the cases with s 0 follow from the theorems before. The rest is a matter of interpolation. For s > 1 p there are no interior u-wavelet systems for As p,q(ω) because of boundary values. So we have to find wavelet systems having values at the boundary and omit interior. Wavelet bases for function spaces on cellular domains Benjamin Scharf 17/28

30 Traces on the boundary of cubes (i) Let Q = {x R n : x = (x 1,..., x n ), 0 < x m < 1, m = 1,..., n). The boundary Γ = Q of Q can be represented as Γ = n 1 l=0 Γ l with Γ l Γ l = for l l, where Γ l = N l j=0 Q l,j consists of all l-dimensional faces Q l,j of Q, which are disjoint cubes of dimension l. Wavelet bases for function spaces on cellular domains Benjamin Scharf 18/28

31 Traces on the boundary of cubes (i) Let Q = {x R n : x = (x 1,..., x n ), 0 < x m < 1, m = 1,..., n). The boundary Γ = Q of Q can be represented as Γ = n 1 l=0 Γ l with Γ l Γ l = for l l, where Γ l = N l j=0 Q l,j consists of all l-dimensional faces Q l,j of Q, which are disjoint cubes of dimension l. Let tr l,j be the restriction of f A s p,q(r n ) to Q l,j and tr r l : f TRr l (f ) := { trl,j D α γ f : α r, j = 0,..., N l }, where only derivatives perpendicular to Q l,j are admitted. Then we consider the composite mapping for r = ( r l 0,... r n 1), l 0 n 1 and r l = s n l p. n 1 tr r Γ : f TR rl l (f ). l=l 0 Wavelet bases for function spaces on cellular domains Benjamin Scharf 18/28

32 Traces on the boundary of cubes (ii) Theorem Let 1 p <, 0 < q <, s > 1 p and s k p / N 0 for k = 1,..., n. Let l 0 = 0 if s > n p. Otherwise l 0 N is chosen such that Then tr r Γ :Bs p,q(q) tr r Γ :F s p,q(q) 0 < s n l 0 p < 1 p. n 1 N l s n l p B α l=l 0 j=0 α r l p,q (Q l,j ), n 1 N l s n l p B α l=l 0 j=0 α r l p,p (Q l,j ). Wavelet bases for function spaces on cellular domains Benjamin Scharf 19/28

33 Traces on the boundary of cubes (iii) Theorem Let 1 p <, 0 < q <, s > 1 p and s k p / N 0 for k = 1,..., n. Let l 0 = 0 if s > n p. Otherwise l 0 N is chosen such that Then 0 < s n l 0 p < 1 p. B s p,q(q) = { f B s p,q(q) : tr r Γ = 0}, F s p,q(q) = { f F s p,q(q) : tr r Γ = 0}. Wavelet bases for function spaces on cellular domains Benjamin Scharf 20/28

34 Extension operators for the boundary of cubes (i) Theorem Let p, q, s, l 0 as in the theorem before and additionally s < u. Then there is a wavelet-friendly extension operator n 1 Ext r,u Γ : n 1 Ext r,u Γ : N l l=l 0 j=0 α r l N l l=l 0 j=0 α r l B B s n l p α p,q s n l p α p,p (Q l,j ) B s p,q(q), (Q l,j ) F s p,q(q). It holds tr r Γ ext r,u Γ = id. Wavelet bases for function spaces on cellular domains Benjamin Scharf 21/28

35 Extension operators for the boundary of cubes (ii) Theorem Furthermore, B s p,q(q) = B s p,q(q) Ext r,u Γ F s p,q(q) = F s p,q(q) Ext r,u Γ n 1 N l l=l 0 j=0 α r l n 1 N l l=l 0 j=0 α r l s n l α p B p,q (Q l,j ), s n l α p B p,p (Q l,j ). Wavelet bases for function spaces on cellular domains Benjamin Scharf 22/28

36 Extension operators for the boundary of cubes (ii) Theorem Furthermore, B s p,q(q) = B s p,q(q) Ext r,u Γ F s p,q(q) = F s p,q(q) Ext r,u Γ n 1 N l l=l 0 j=0 α r l n 1 N l l=l 0 j=0 α r l s n l α p B p,q (Q l,j ), s n l α p B p,p (Q l,j ). Proof (sketch): Take first the traces to the faces of dimension l 0, then extend it back to Bp,q(Q). s These maps are projections. The complementary space is { f Bp,q(Q) s : tr f = 0 }. Now take the trace of this space to the faces of dimension l = l of this n l s p space. By boundary considerations this is B p,q (Q l,j ). And so on. Wavelet bases for function spaces on cellular domains Benjamin Scharf 22/28

37 Wavelets for cubes Theorem (T08 - Theorem 6.30) Let A s p,q(q) (can be extended to cellular domains) be given with 1 p <, s > 1 p and s k p / N 0 for k = 1,..., n, 0 < q < (q 1 for the F -spaces). Then there is an oscillating u-wavelet system with u > s which is a Riesz basis in A s p,q(q). Wavelet bases for function spaces on cellular domains Benjamin Scharf 23/28

38 Wavelets for cubes Theorem (T08 - Theorem 6.30) Let A s p,q(q) (can be extended to cellular domains) be given with 1 p <, s > 1 p and s k p / N 0 for k = 1,..., n, 0 < q < (q 1 for the F -spaces). Then there is an oscillating u-wavelet system with u > s which is a Riesz basis in A s p,q(q). Proof: Use the decomposition B s p,q(q) = B s p,q(q) Ext r,u Γ n 1 N l l=l 0 j=0 α r l s n l α p B p,q (Q l,j ) and the fact that every space on the right hand side has a u-wavelet system which is a Riesz basis. Now one must ensure that wavelet-friendly extension operators are really wavelet-friendly. Wavelet bases for function spaces on cellular domains Benjamin Scharf 23/28

39 The case s 1 p N 0 Let Ω be now a C -domain (so also a cellular domain) and let 1 < p, q <. By a translation and localization argument one always has à s p,q(ω) Å s p,q(ω). The converse is true if s 1 p / N. Using the Hardy inequalities from [T01] one gets f (x) p d sp (x) dx c f F p,q(r s n ) for f F s p,q(ω), Ω where d(x) = dist(x, Γ). Then χ Ω / F s p,q(ω) for s 1 p. Wavelet bases for function spaces on cellular domains Benjamin Scharf 24/28

40 The case s 1 p N 0 Let Ω be now a C -domain (so also a cellular domain) and let 1 < p, q <. By a translation and localization argument one always has à s p,q(ω) Å s p,q(ω). The converse is true if s 1 p / N. Using the Hardy inequalities from [T01] one gets f (x) p d sp (x) dx c f F p,q(r s n ) for f F s p,q(ω), Ω where d(x) = dist(x, Γ). Then χ Ω / F s p,q(ω) for s 1 p. 1 p On the other hand χ Ω F p,q(ω) = Fp,q(Ω) by an atomic decomposition argument. Analogously, it follows F s p,q(ω) F s p,q(ω) if s 1 p no Riesz frames for these F s p,q(ω) (T08 - Prop ). 1 p N. There are Wavelet bases for function spaces on cellular domains Benjamin Scharf 24/28

41 The cases s k p N 0 with k = 2,..., n It is known that not all the conditions are necessary depending on the kind of domain. For planar C -domains (n = 2) the restriction for k = 2 is not necessary, For C -domains with boundaries diffeomorphic to S n the condition for k = n is not necessary, For arbitrary C -domains and 1 p <, 1 p 1 < s < 2 p, s 1 p one finds a Riesz u-wavelet basis without further restrictions since the boundary has a basis, For the torus one needs no further conditions since the boundary is equipped with a u-wavelet basis ([T08] - sect ). So W2 1 (Ω) has a basis if Ω is a torus which is unknown if Ω is a ball and n 3. Wavelet bases for function spaces on cellular domains Benjamin Scharf 25/28

42 Reinforced spaces Let Ω ε = {x Ω, d(x) < ε} and F s,rinf p,q (Ω) = { F s p,q (Ω) if s 1 p / N 0 f F s p,q(ω) : d 1 p r f ν r L p (Ω ε )} if s 1 p = r N 0. If Ω is a bounded C -domain, then { } F s p,q(ω) = f Fp,q s,rinf (Ω) : tr r 1 Γ f = 0. Wavelet bases for function spaces on cellular domains Benjamin Scharf 26/28

43 Reinforced spaces Let Ω ε = {x Ω, d(x) < ε} and F s,rinf p,q (Ω) = { F s p,q (Ω) if s 1 p / N 0 f F s p,q(ω) : d 1 p r f ν r L p (Ω ε )} if s 1 p = r N 0. If Ω is a bounded C -domain, then { } F s p,q(ω) = f Fp,q s,rinf (Ω) : tr r 1 Γ f = 0. This and the observation that the known extension operators also map to Fp,q s,rinf (Ω) instead of Fp,q(Ω) s is the starting point for finding a u-wavelet basis. Wavelet bases for function spaces on cellular domains Benjamin Scharf 26/28

44 Reinforced spaces advanced Roughly speaking: This gives u-wavelet bases for the modified spaces Fp,q s,rinf (Ω) in the cases of domains where no more restrictions than s 1 p / N 0 are necessary. Wavelet bases for function spaces on cellular domains Benjamin Scharf 27/28

45 Reinforced spaces advanced Roughly speaking: This gives u-wavelet bases for the modified spaces Fp,q s,rinf (Ω) in the cases of domains where no more restrictions than s 1 p / N 0 are necessary. The big question: Can one extend this idea to cellular domains with more restrictions? Wavelet bases for function spaces on cellular domains Benjamin Scharf 27/28

46 Reinforced spaces advanced Roughly speaking: This gives u-wavelet bases for the modified spaces Fp,q s,rinf (Ω) in the cases of domains where no more restrictions than s 1 p / N 0 are necessary. The big question: Can one extend this idea to cellular domains with more restrictions? The problem: One has to ensure a composition of type. { } F s p,q(ω) = f Fp,q s,rinf (Ω) : tr r 1 Γ f = 0. and has to check if the extension operators are suitable (which they should be). Furthermore, we need conditions for definiting Fp,q s,rinf (Ω) which handle the lower dimensional boundary faces. This has not been done so far. In [T08], sect , there is some overview given for W2 1(Q). Wavelet bases for function spaces on cellular domains Benjamin Scharf 27/28

47 The end Thank you for your attention Questions? Wavelet bases for function spaces on cellular domains Benjamin Scharf 28/28