Function spaces on the Koch curve

Transcription

1 Function spaces on the Koch curve Maryia Kabanava Mathematical Institute Friedrich Schiller University Jena D Jena, Germany 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. Key words: periodic Besov spaces, trace spaces, self-similar set, Daubechies wavelets 000 Mathematics Subect Classification: 4B35, 4C40, 8A80 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 B s pq(, µ) by traces s+ d p Bpq(, s µ) = tr µ Bpq (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 B s pq(t) are periodic Besov spaces.

2 The question arises how the function spaces B s pq(, µ) and B s 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 we present the wavelet characterization of the periodic Besov spaces B s pq(t) and then shift it to. In Section 3 we compare B s pp(, µ) and B s 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), where α is an arbitrary multi-index. D (T n ) x T n is defined to be the topological dual of D(T n ). Any f D (T n ) can be represented as f = k Z n a k e πikx, x T n, (convergence in D (T n )) where the Fourier coefficients {a k } C are of at most polynomial growth, a k c ( + k ) κ, for some c > 0, κ > 0 and all k 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 k Z n =0 (with the usual modification if q = ). Then the Besov space B s pq(t n ) consists of all f D (T n ) such that f B s pq(t n ) <, [6, Chapter 3].

3 . Trace spaces Definition. A compact set in R 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, holds 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. Moreover log 3 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(T s ). Suppose that for some s > 0, < p <, 0 < q < () there is a constant c > 0 such that ϕ(γ) µ(dγ) c ϕ Bpq(T s ), for all ϕ D(T ), (3) 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 Bpq(T s ). Then (3) can be extended by completion to any f Bpq(T s ) and the resulting function on is denoted by tr µ f. By standard arguments, it is independent of the approximation of f in Bpq(T s ) by D(T ) functions. Any function g tr µ Bpq(T s ) L (, µ) is quasi-normed by 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 > 0 and 0 < v. From the Corollary.75 in [7] follows that the trace operator tr µ : B s pq(t ) L (, µ) exists if s > d. This ustifies the following definition p 3

4 Definition 3. Let be the Koch curve. Let < p <, 0 < q < and s > 0. Then Bpq(, s d s+ p µ) = tr µ Bpq (T ). (4) B s pq(, µ) considered as subsets of L (, µ). Let B s p() = B s 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 f Bp() s p = with µ = H d. f(γ) p µ(dγ) + f(γ) f(δ) p γ δ d+sp µ(dγ)µ(dδ) (5) Wavelets on T and. 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 of i= 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. We write K w = F w (K) for F w F w... F wm (K), where w = (w, w,..., w m ) W. For any ω = (ω, ω,...) Σ define π : Σ K by π(ω) = K ω ω...ω m, [4, Ch..]. m= 4

5 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, y 3 ( ) ( ) ( x ) ( F = 3 x ) y +. y 3 3 We denote a mapping π corresponding to I by π I and to by π, π I (ω) = I ω ω...ω m, π (ω) = ω ω...ω m. m= The mapping H = π π I (6) 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.. Self-similar measures Let p, p,... p N be numbers such that m= 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, so that µ(k w w...w m ) = p w p w... p wm 5

6 Figure : The snowflaked transform and this extends to a Borel measure supported by K, [, Ch..]. If {F i } N i= are similarities with factors r i, i =,,..., N and s is the unique number with N ri s =, then µ with weight (r, s r, s..., rn s ) is the i= 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 H ln 3. When K 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, 6

7 one has for a function f defined on f(γ) µ(dγ) = ( f H)(x) ν(dx) = ( f H)(x) dx, (7) [5, Theorem.9] 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 and B s p(), < p <, 0 < s < we denote the spaces Bpp(T) s and B s pp() correspondingly. We are interested in wavelet expansions for the spaces B s p(). We start with the wavelet characterization of Bp(T) s 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) and ψ M C u (R) be a father and a mother Daubechies wavelet on R respectively. Define ψf k and ψk by ψf k = 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 ψ L,k F (x) = L ψf ( L x k), ψ L,k (x) = ψm ( x k), k Z. 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 < }. + 7

8 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 Define ψ L,k,per F ψ L,k F,per (x) = l= and ψ L,k,per ψ L,k F ψ L,k,per F (x) = ψ L,k F,per (x), (x + l), ψl,k,per (x) = on the -torus T by 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 l= ψ L,k (x + l). ψl,k,per (x) = ψ L,k,per (x), x T., 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 ) is less than w = (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.... 8

9 Then we notice that x k = ( () x k ). When k is fixed, there is a unique sequence of contractions such that T w, T w,..., T w, w = (w, w,..., w ) W ( () x k ) = T Tw... Tw Tw (x). (8) The mapping (8) transfers T w w...w to T.... The values of ψ L,k,per T u, where u = (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 and ψ L,0,per : ψ L,k,per (x) = = ψ L,0,per ψ L,0,per... T Tw... Tw Tw (x), if x T w w...w, T T Tu... Tu Tu (x), if x T u u...u, where... indicates the procedure of assigning to ψ L,k,per neighbour of T u the values of ψ L,0,per by on on each next right on each next right neighbour of T.... Let us simplify the notation and denote the functions ψ L,k,per F ψ F,w = ψ F,w w...w L, w W L, ψ w = ψ w w...w, N 0, w W correspondingly, where w is chosen according to (8). Now we transfer the functions ψ F,w, w W L, ψ w, w to the Koch curve. Define ψ F,w and ψ w by ψ F,w (γ) = ψ F,w H (γ), ψ w (γ) = ψ w H (γ). 9 and ψ L,k,per N 0 W from T

10 { From (7) follows that the system ψf,v, ψ } w v W L,w S N 0 W in L (, µ). Let γ w w...w, then there is an ω Σ such that is orthonormal γ = w w...w ω ω...ω m. m= Recall that γ corresponds to x = H (γ) T w w...w given by x = T w w...w ω ω...ω m. m= The connection between the values of ψ w w...w and ψ... on w w...w is the following: ψ w w...w (γ) = ψ w w...w H (γ) = = ψ... = ψ... = ψ... T Tw... Tw Tw ( m= ( T m= ( m= ) ( ω ω...ω m = ψ... H m= ω ω...ω m ) = = ψ... F Fw... Fw Fw ( = ψ... F Fw... Fw Fw (γ). m= T w w...w ω ω...ω m ) =... ω ω...ω m ) = w w...w ω ω...ω m ) = In the same way we can follow the connection between the values of ψ w w...w 0

11 and ψ... on other v, v W. So we get ψ w (γ) = ψ... F Fw... Fw Fw (γ), if γ w w...w, = ψ... F F Fu... Fu Fu (γ), if γ u u...u,... Similarly for ψ F,w. According to [8, Ch..3.3] the following theorem holds. Theorem. Let < p <, 0 < s < and u > s. Let f L p (T). Then f B s p(t) if, and only if, it can be represented as f = a w L w W L ψf,w + =0 w W b w ψw, unconditional convergence being in L p (T). Furthermore this representation is unique, a w = L (f, ψ F,w ) T, w W L, b w = (f, ψ w ) T, w W and I : f {a w, w W L, b w, w W, N 0 } is an isomorphic map of B s p(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 L p (). 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,

12 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 B s p() onto the sequence space such that w W L a w p + (s p )p =0 w W b w p <. 3 Comparison of B s p() and B s p() The spaces B s p(t) can be normed by f Bp(T) s p = f(x) p dx f(x) f(y) p x y +sp dx dy. (9) 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 f B s p() p = f(γ) p µ(dγ) + Together with (5) this leads to f(γ) f(δ) p γ δ d+sdp µ(dγ)µ(dδ). (0) B s p() = B s d p (). () The analogue of Theorem for the spaces B s p() reads as follows

13 Theorem 3. Let < p <, 0 < s < and u > s. Let f L p (). 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, 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 B s p() onto the sequence space such that w W L a w p + =0 ( 3 ) (s d p )p Proof. This follows from the observation () and w W b w p <. ( s d p) = d (s d p) = ( 3 ) (s d p ). References [] Edmunds, D.E., Triebel, H., Function spaces, entropy numbers, differential operators, Cambridge Univ. Press, 996. [] Falconer, K., Techniques in fractal geometry, Chichester, Wiley, 997. [3] Jonsson, A., Wallin, H., Function spaces on subsets of R n, Math. reports,, London, Harwood acad. publ., 984. [4] Kigami, J., Analysis on fractals, Cambridge Univ. Press, 003 [5] Mattila, P., Geometry of sets and measures in Euclidean spaces, Cambridge Univ. Press,

14 [6] Schmeisser, H.-J., Triebel, H., Topics in Fourier analysis and function spaces, Chichester, Wiley, 987. [7] Triebel, H., Theory of function spaces III, Basel, Birkhäuser, 006. [8] Triebel, H., Function spaces and wavelets on domains, Zürich, European Math. Soc. Publishing House,