Function spaces on the Koch curve

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 Γ be the Koch curve in R 2 .It is an example of a d-set with d = log 4 log 3 .There are two possibilities to introduce Besov spaces on Γ.The first one is to define Besov spaces B s pq (Γ, μ) by traces 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 → Γ, T -1-torus and define B s pq (Γ) by where B s pq (T) are periodic Besov spaces.
The question arises how the function spaces B s pq (Γ, μ) and B s pq (Γ) are interrelated.We concentrate mainly on the case 1 < p = q < ∞, 0 < s < 1.
In particular, we shift the characterization in terms of Daubechies wavelets from (T, ρ = |x − y| 1/d , μ L ), [7, p. 360], to Γ.This paper is organized as follows.In Section 2, we describe the trace method of defining Besov spaces.In Section 3, we present the wavelet characterization of the periodic Besov spaces B s pq (T) and then shift it to Γ.In Section 4, we compare B s pp (Γ, μ) and B s pp (Γ).The main result is contained in Theorem 3.

Trace spaces
2.1.Periodic Besov spaces on T n .Let 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 where the Fourier coefficients {a k } ⊂ C are of at most polynomial growth, κ , for some c > 0, κ > 0 and all k ∈ Z n .
(with the usual modification if q = ∞).Then the Besov space

Trace spaces. Definition 2.
A compact set Γ in R n is called a d-set with 0 < d ≤ n if there is a Radon measure μ in R n with support Γ such that for some positive constants c 1 and c 2 (1) where B(x, r) is a ball in R n centred at x ∈ R n and of radius r > 0.
If Γ is a d-set, then the restriction to Γ of the d-dimensional Hausdorff measure satisfies (1) and any measure μ satisfying (1) When Γ is the Koch curve, it is a d-set in R 2 with d = log 4 log 3 .Moreover Γ is a subset of T 2 .In order to avoid problems in the endpoints (0, 0) and (1, 0) of Γ, we define the Besov spaces as a trace of the periodic Besov spaces B s pq (T 2 ).Suppose that for some where ϕ(γ) denotes the pointwise trace of ϕ ∈ D(T 2 ) on Γ (sometimes we wright ϕ| Γ to denote the pointwise trace of ϕ).D(T 2 ) is dense in B s pq (T 2 ).Then (3) can be extended by completion to any f ∈ B s pq (T 2 ) and the resulting function on Γ is denoted by tr μ f .By standard arguments, it is independent of the approximation of If one has (3) for some s, p, q satisfying (2), then one has also (3) for all spaces B s+ε pv (T n ) with ε > 0 and 0 < v ≤ ∞.From the Corollary 1.175 in [7], it follows that the trace operator This justifies the following definition Here B s pq (Γ, μ) are considered as subsets of L 1 (Γ, μ).Let It was shown in [3] that B s p (Γ) with 1 < p < ∞ and 0 < s < 1 can be equivalently normed by

Wavelets on T and Γ
3.1.Self-similar sets and the snowflaked transform.Let K be a self-similar set in R n with respect to the contractions We can use iterations of the maps F i to give the "address" of a point in K .We introduce the following spaces: let Σ be a set of all infinite sequences Σ = {(ω 1 , ω 2 , . ..) : ω i ∈ {1, 2, . . ., N}}.
We use W m to denote the collection of words of length m: The unit interval I = [0, 1] can be considered as a self-similar set with respect to the similarities T i : R → R, i = 1, 2, The Koch curve Γ is a self-similar set with respect to the similarities .
We denote a mapping π corresponding to I by π I and to Γ by π Γ , The mapping is a homeomorphism between I and Γ.It is called the snowflaked transform.Note that Since the 1-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 1 and T 2 .

Self-similar measures.
Let p 1 , p 2 , . . .p N be numbers such that Then we can define the probability measure μ with the weight (p 1 , p 2 , . . ., p N ) on the hierarchy of sets by repeated subdivision of the measure in the ratio p 1 : p 2 : . . .: p N , so that  If {F i } N i=1 are similarities with factors r i , i = 1, 2, . . ., N and s is the unique number with N i=1 r s i = 1, then μ with weight (r s 1 , r s 2 , . . ., r s N ) is the measure equivalent to the restriction H s | K of the Hausdorff measure H s in R n to K , [4, Theorem 1.5.7].When K is the Koch curve Γ, then the measure μ with the weight 1  2 , 1 2 is equivalent to H ln 4 ln 3 | Γ .When K is the unit interval I, measure ν with the weight 1  2 , 1 2 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 Γ

Wavelet characterization of B s p (T) and B s p (Γ). Let
By B s p (T) and B s p (Γ), 1 < p < ∞, 0 < s < 1 we denote the spaces B s pp (T) and B s pp (Γ) respectively.We are interested in wavelet expansions for the spaces B s p (Γ).We start with the wavelet characterization of B s p (T) and then transfer it with the help of mapping H to Γ.
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 ψ k F and ψ k j by Then ψ k F , ψ k j j∈N0,k∈Z is an orthonormal system in L 2 (R).Let L ∈ N. One can replace ψ F and ψ M by We choose and fix L such that Then one has Given the functions ψ L,k F , ψ L,k j on the real line we can construct their 1 -periodic counterparts by the procedure Function spaces on the Koch curve Define ψ L,k,per F and ψ L,k,per j on the 1 -torus T by Then according to the Proposition 1.34 in [8] It is easy to see that 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 j is fixed, the 1-torus T treated as a self-similar set can be represented as follows Let us introduce the order relation on the set W j+L of words of length j + L .We say that v = (v 1 , . . ., v j+L ) is less than w = (w 1 , . . ., w j+L ) if and only if the first v i which is different from w i is less than w i : The words are ordered in such a way that whenever u follows w in W j+L , the interval T u is the right neighbour of T w .We agree that T 11 . . .Then we notice that When k is fixed, there is a unique sequence of contractions The mapping (8)  2 .Thus there is the following connection between functions ψ L,k,per j and ψ L,0,per j : From (7)   .

1
j+L is the right neighbour of T 22 . . . 2 j+L .
...uj+L , . . .where . . .indicates the procedure of assigning to ψ L,k,per j on each next right neighbour of T u the values of ψ L,0,per j on each next right neighbour of T 11 . . . 1 j+L−1 2 .Let us simplify the notation and denote the functions ψ L,k ,per F and ψ L,k,per j by ψ F,w = ψ F,w1w2...wL , w ∈ W L , ψ w = ψ w1w2...wj+L , j ∈ N 0 , w ∈ W j+L respectively, where w is chosen according to (8).Now we transfer the functions ψ F,w , w ∈ W L , ψ w , w ∈ j∈N0 W j+L from T to the Koch curve Γ. Define ψ F,w and ψ w by