Function spaces on the Koch curve

Transcription

1 JOURNAL OF FUNCTION SPACES AND APPLICATIONS Volume 8, Number 3 (00), c 00, Scientific Horizon Function spaces on the Koch curve Maryia Kabanava (Communicated by Hans Triebel) 000 Mathematics Subect Classification. 4B35, 4C40, 8A80. Keywords and phrases. Periodic Besov spaces; trace spaces; self-similar set; Daubechies wavelets. Abstract. We consider two types of Besov spaces on the Koch curve, defined by traces and with the help of the snowflaked transform. We compare these spaces and give their characterization in terms of Daubechies wavelets.. Introduction Let be the Koch curve in R. It is an example of a d-set with d = log 4 log 3. There are two possibilities to introduce Besov spaces on. The first one is to define Besov spaces Bpq s (,μ) by traces B s pq (,μ)=tr μ B d s+ p pq (T ), <p<, 0 <q<, 0 <s<. We prefer the periodic setting since we are interested to extend the theory to a closed snowflake. The second way is to use the snowflaked transform H : T, T -torus and define B s pq () by B s pq() = { f H : f B s pq(t) } = B s pq(t) H, where Bpq s (T) are periodic Besov spaces.

2 88 Function spaces on the Koch curve The question arises how the function spaces Bpq s (,μ) and Bs pq () are interrelated. We concentrate mainly on the case <p= q<, 0<s<. In particular, we shift the characterization in terms of Daubechies wavelets from (T,ρ= x y /d,μ L ), [7, p. 360], to. This paper is organized as follows. In Section, we describe the trace method of defining Besov spaces. In Section 3, we present the wavelet characterization of the periodic Besov spaces Bpq(T) s andthenshiftitto. In Section 4, we compare Bpp s (,μ) and Bs pp (). The main result is contained in Theorem 3.. Trace spaces.. Periodic Besov spaces on T n. Let T n = {x =(x,...,x n ) R n :0 x i,i=,...,n} x T n and y T n are identified if and only if x y = k, k =(k,...,k n ) Z n. By D(T n ), we denote the collection of all complex-valued infinitely differentiable functions on T n. The topology in D(T n ) is generated by the family of semi-norms sup D α f(x), whereα is an arbitrary multi-index. x T n D (T n ) is defined to be the topological dual of D(T n ). Any f D (T n ) can be represented as f = a k e πikx, x T n, (convergence in D (T n )) k Z n where the Fourier coefficients {a k } C are of at most polynomial growth, a k c ( + k ) κ, for some c>0, κ > 0andallk Z n. Definition. Let ϕ = {ϕ } =0 be a dyadic resolution of unity, s R, 0 <p,0<q and f Bpq(T s n ) = sq q ϕ (k)a k e πikx L p (T n ) q =0 k Z n (with the usual modification if q = ). Then the Besov space B s pq (Tn ) consists of all f D (T n ) such that f B s pq (Tn ) <, [6,Chapter3].

3 M. Kabanava 89.. Trace spaces. Definition. AcompactsetinR n is called a d-set with 0 <d n if there is a Radon measure μ in R n with support such that for some positive constants c and c () c r d μ(b(γ,r)) c r d, γ, 0 <r<, 0 <d n. where B(x, r) is a ball in R n centred at x R n and of radius r>0. If is a d-set, then the restriction to of the d-dimensional Hausdorff measure satisfies () and any measure μ satisfying () is equivalent to H d. When is the Koch curve, it is a d-set in R with d = log 4 log 3.Moreover is a subset of T. In order to avoid problems in the endpoints (0, 0) and (, 0) of, we define the Besov spaces as a trace of the periodic Besov spaces Bpq s (T ). Suppose that for some () s>0, <p<, 0 <q< there is a constant c>0 such that (3) ϕ(γ) μ(dγ) c ϕ Bpq(T s ), for all ϕ D(T ), where ϕ(γ) denotes the pointwise trace of ϕ D(T ) on (sometimes we wright ϕ to denote the pointwise trace of ϕ). D(T ) is dense in B s pq (T ). Then (3) can be extended by completion to any f B s pq (T )and the resulting function on is denoted by tr μ f. By standard arguments, it is independent of the approximation of f in B s pq(t )byd(t ) functions. Any function g tr μ B s pq (T ) L (,μ)isquasi-normedby g tr μ B s pq (T ) =inf { f B s pq (T ) :tr μ f = g }. If one has (3) for some s, p, q satisfying (), then one has also (3) for all spaces Bpv s+ε (T n ) with ε>0and0<v. From the Corollary.75 in [7], it follows that the trace operator tr μ : B s pq(t ) L (,μ) exists if s> d p. This ustifies the following definition

4 90 Function spaces on the Koch curve Definition 3. Let be the Koch curve. Let <p<, 0<q< and s>0. Then s+ d (4) Bpq s (,μ)=tr p μ Bpq (T ). Here B s pq (,μ) are considered as subsets of L (,μ). Let Bp s () = Bs pp (,μ), <p<, 0 <s<. It was shown in [3] that B s p() with <p< and 0 <s< can be equivalently normed by f B s p () with (5) f B s p () p = with μ =H d. f(γ) p μ(dγ)+ f(γ) f(δ) p γ δ d+sp μ(dγ)μ(dδ) 3. Wavelets on T and 3.. Self-similar sets and the snowflaked transform. Let K be a self-similar set in R n with respect to the contractions {F i } N i=, K = N F i (K). We can use iterations of the maps F i to give the address i= of a point in K. We introduce the following spaces: let Σ be a set of all infinite sequences Σ={(ω,ω,...):ω i {,,...,N}}. We use W m to denote the collection of words of length m: Set W = m=0 W m = {(w,w,...,w m ):w i {,,...,N}}. W m.wewritek w = F w (K) forf w F w... F wm (K), where w =(w,w,...,w m ) W. For any ω =(ω,ω,...) Σ define π :Σ K by see [4, Ch..]. π(ω) = m= K ωω...ω m,

5 M. Kabanava 9 The unit interval I = [0, ] can be considered as a self-similar set with respect to the similarities T i : R R, i =,, T (x) = x, T (x) = x +. The Koch curve is a self-similar set with respect to the similarities F i : R R, i =,, ( ) ( ) ( ) x F = 3 x y, 3 y ( ) ( ) ( x ) ( F = 3 x ) y + y. 3 3 We denote a mapping π corresponding to I by π I and to by π, The mapping π I (ω) = m= I ωω...ω m, π (ω) = (6) H = π π I m= ωω...ω m. is a homeomorphism between I and. It is called the snowflaked transform. Note that H(x) H(y) x y d, [7, Proposition 8.6]. Since the -torus T can be identified in the usual way with the unit interval, it can be regarded as a self-similar set with respect to T and T. 3.. Self-similar measures. Let p,p,...p N be numbers such that N p i =, 0 <p i <. i= Then we can define the probability measure μ with the weight (p,p,...,p N ) on the hierarchy of sets by repeated subdivision of the measure in the ratio p : p :...: p N,sothat μ(k ww...w m )=p w p w...p wm and this extends to a Borel measure supported by K, [, Ch..].

6 9 Function spaces on the Koch curve Figure. The snowflaked transform G G G G G G G G H I I I I I I I I If {F i } N i= unique number with are similarities with factors r i, i =,,...,N and s is the N i= r s i =,then μ with weight (rs,rs,...,rs N )isthe measure equivalent to the restriction H s K of the Hausdorff measure H s in R n to K, [4, Theorem.5.7]. When K is the Koch curve, then the measure μ with the weight (, ) ln 4 is equivalent to H ln 3.WhenK is the unit interval I, measure ν with the weight (, ) is the Lebesgue measure. Since the image of the measure ν under a mapping H is the measure μ, one has for a function f defined on (7) f(γ) μ(dγ) = ( f H)(x) ν(dx) = ( f H)(x) dx, 0 0

7 M. Kabanava 93 [5, Theorem.9]. 3.3 Wavelet characterization of B s p(t) and B s p(). Let B s pq() = { f H : f B s pq(t) } = B s pq(t) H. By Bp(T) s andb s p(), <p<, 0<s< we denote the spaces Bpp(T) s and B s pp() respectively. We are interested in wavelet expansions for the spaces B s p (). We start with the wavelet characterization of Bs p (T) and then transfer it with the help of mapping H to. Let C u (R), u N denote the collection of all complex-valued continuous functions on R having continuous bounded derivatives up to order u inclusively. Let ψ F C u (R) andψ M C u (R) be a father and a mother Daubechies wavelet on R respectively. Define ψf k and ψk by ψ k F = ψ F (x k), ψ k (x) = ψm ( x k), N 0,k Z. Then { } ψf k,ψk is an orthonormal system in L N 0,k Z (R). Let L N. One can replace ψ F and ψ M by ψ L F ( ) =ψ F ( L ), ψ L M ( ) =ψ M ( L ), ψ k F and ψk by F (x) = L ψf ( L x k), ψ L,k (x) = ψm ( x k), k Z. ψ L,k We choose and fix L such that { supp ψf L x : x < } {, supp ψm L x : x < }. Then one has supp ψ L,0 = supp ψ L M ( ) { x : x < } +. Let P = { m Z :0 m< }, N 0. Given the functions ψ L,k F, ψl,k on the real line we can construct their -periodic counterparts by the procedure ψ L,k F,per (x) = l= ψ L,k F (x + l), ψl,k,per (x) = l= ψ L,k (x + l).

8 94 Function spaces on the Koch curve Define ψ L,k,per F and ψ L,k,per on the -torus T by ψ L,k,per F (x) =ψ L,k F,per (x), ψl,k,per (x) =ψ L,k,per (x), x T. Then according to the Proposition.34 in [8] { ψ L,k,per F is an orthornomal basis in L (T). It is easy to see that,ψ k,l,per,k P 0, N 0,k P } ψ L,k,per (x) =ψ L,0,per ( x k ), k P, on T with the usual interpretation. The shift operation is well-defined on the real line, but it can not be defined on the Koch curve. Therefore we would like to replace this operation in order to be able to construct its counterpart on the Koch curve. First of all when is fixed, the -torus T treated as a self-similar set can be represented as follows T = T w. w W Let us introduce the order relation on the set W of words of length + L. We say that v =(v,...,v )islessthanw =(w,...,w )if and only if the first v i which is different from w i is less than w i : v<w v min{i:wi v i} <w min{i:wi v i}. The words are ordered in such a way that whenever u follows w in W, the interval T u is the right neighbour of T w. We agree that T... is the }{{} right neighbour of T.... Then we notice that } {{ } x k = () ( x k ). When k is fixed, there is a unique sequence of contractions T w,t w,...,t w, w =(w,w,...,w ) W such that (8) () ( x k ) = T Tw... Tw Tw (x).

9 M. Kabanava 95 The mapping (8) transfers T ww...w to T....Thevaluesofψ L,k,per }{{} on T u,whereu =(u,u,...u ) follows w, coincide with the values of ψ L,0,per on T.... Thus there is the following connection between functions ψ L,k,per ψ L,k,per (x) = = ψ L,0,per ψ L,0,per... }{{} and ψ L,0,per : T Tw... Tw Tw (x), if x T ww...w, T T Tu... Tu Tu (x), if x T uu...u, where... indicates the procedure of assigning to ψ L,k,per on each next right neighbour of T u the values of ψ L,0,per on each next right neighbour of T... }{{}. Let us simplify the notation and denote the functions ψ L,k,per F by ψ L,k,per and ψ F,w = ψ F,ww...w L, w W L, ψ w = ψ ww...w, N 0,w W respectively, where w is chosen according to (8). Now we transfer the functions ψ F,w, w W L, ψ w, w from T to the Koch curve N 0 W. Define ψ F,w and ψ w by ψ F,w (γ) =ψ F,w H (γ), ψ w (γ) =ψ w H (γ). { From (7) follows that the system ψf,v, ψ } w v W L,w N 0 W is orthonormal in L (,μ). Let γ ww...w. Then there is an ω Σ such that γ = ww...w ω ω...ω m. m=

10 96 Function spaces on the Koch curve Recall that γ corresponds to x = H (γ) T ww...w given by x = T ww...w ω ω...ω m. m= The connection between the values of ψww...w and ψ... }{{} is the following: ww...w on ψ ww...w (γ) =ψ ww...w H (γ) = ( ) = ψ... T }{{} Tw... Tw Tw T ww...w ω ω...ω m = m= = ψ... T... ω }{{}}{{} ω...ω m = ψ... H... ω }{{}}{{} ω...ω m = m= m= = ψ = } {{ } m= } {{ } ω ω...ω m ( = ψ... F }{{} Fw... Fw Fw = ψ... F }{{} Fw... Fw Fw (γ). m= ww...w ω ω...ω m ) = In the same way we can follow the connection between the values of ψ ww...w and ψ... on other v, v W.Soweget }{{} ψ w (γ) = ψ... F }{{} Fw... Fw Fw (γ), if γ ww...w, = ψ... F }{{} F Fu... Fu Fu (γ), if γ uu...u,... Similarly for ψ F,w. According to [8, Ch..3.3] the following theorem holds.

11 M. Kabanava 97 Theorem. Let <p<, 0 <s< and u>s. Let f L p (T). Then f Bp s (T) if, and only if, it can be represented as f = w W L a w L ψf,w + =0 w W b w ψw, unconditional convergence being in L p (T). Furthermore this representation is unique, and a w = L (f,ψ F,w ) T, w W L, b w = (f,ψ w ) T, w W I : f {a w,w W L,b w,w W, N 0 } is an isomorphic map of Bp s (T) onto the sequence space such that w W L a w p + (s p )p =0 Similar theorem holds for the spaces B s p(). w W b w p <. Theorem. Let <p<, 0 <s< and u>s. Let f Lp (). Then f B s p () if, and only if, it can be represented as f = w W L a w L ψf,w + =0 w W b w ψw, unconditional convergence being in L p (). Furthermore this representation is unique, and a w = L ( f, ψ F,w ), w W L, b w = ( f, ψ w ), w W I : f {a w,w W L,b w,w W, N 0 } is an isomorphic map of B s p () onto the sequence space such that w W L a w p + (s p )p =0 w W b w p <.

12 98 Function spaces on the Koch curve 4. Comparison of B s p () and Bs p () The spaces B s p(t) canbenormedby (9) f Bp s (T) p = 0 f(x) p dx f(x) f(y) p x y +sp dx dy. Since (9) is equivalent to f(γ) p μ(dγ)+ f(γ) f(δ) p γ δ d+sp μ(dγ)μ(dδ), where f = f H, we endow the spaces B s p () with the norm (0) f B s p() p = f(γ) p μ(dγ)+ Together with (5) this leads to () B s p () = B s d p (). f(γ) f(δ) p γ δ d+sdp μ(dγ)μ(dδ). The analogue of Theorem for the spaces B s p() reads as follows Theorem 3. Let <p<, 0 <s< and u>s. Let f Lp (). Then f B s p() if, and only if, it can be represented as f = a w L w W L ψf,w + =0 b w w W ψw, unconditional convergence being in L p (). Furthermore this representation is unique, a w = L ( f, ψ F,w ), w W L, b w = ( f, ψ w ), w W and I : f {a w,w W L,b w,w W, N 0 } is an isomorphic map of Bp s () onto the sequence space such that ( ) (s a w p d p + )p 3 b w p <. w W L =0 w W

13 M. Kabanava 99 Proof. This follows from the observation () and ( s d p) = d (s d p) = ( 3 ) (s d p ). References [] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, 996. [] K. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, 997. [3] A. Jonsson and H. Wallin, Function Spaces on Subsets of R n,math. Reports,, London, Harwood Acad. Publ., 984. [4] J. Kigami, Analysis on Fractals, Cambridge University Press, 003. [5] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, 995. [6] H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley, Chichester, 987. [7] H. Triebel, Theory of Function Spaces III, Birkhäuser, Basel, 006. [8] H. Triebel, Function Spaces and Wavelets on Domains, European Math. Soc. Publishing House, Zürich, 008. Mathematical Institute Friedrich Schiller University Jena D Jena, Germany ( maryia.kabanava@uni-ena.de) (Received : February 009 )

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