JENAER SCHRIFTEN MATHEMATIK UND INFORMATIK

Transcription

1 FRIEDRICH-SCHILLER- UNIVERSITÄT JENA JENAER SCHRIFTEN ZUR MATHEMATIK UND INFORMATIK Eingang : 25. November 2005 Math/Inf/09/05 Als Manuskrit gedruckt Aroximation numbers of traces from anisotroic Besov saces on anisotroic fractal d-sets Erika Tamási Abstract This aer deals with aroximation numbers of the comact trace oerator of an anisotroic Besov sace into some L -sace, tr Γ : B s,a R n ) L Γ), s > 0, 1 < <, where Γ is an anisotroic d-set, 0 < d < n. We also rove homogeneity estimates, a homogeneous equivalent norm and the localisation roerty in B s,a. Keywords: anisotroic function saces, fractals, wavelet frames Math. Subject Classification: 46E35, 42B35, 42C40

2 2

3 Introduction The theory of the anisotroic saces has been develoed from the very beginning arallel to the theory of isotroic function saces. We refer in articular to the Russian school and works of S.M. Nikol skiĭ [11], O.V. Besov, V.P.Il in [1]. Let 1 < < and s 1,..., s n ) an n- tule of natural numbers then { n } W s,a R n ) = f S R n ) : f L R n ) + s k f x s L k R n ) < 0.1) k is the classical anisotroic Sobolev sace on R n. It is obvious that unlike in case of the usual isotroic) Sobolev sace s 1 =... = s n ) the smoothness roerties of an element from W s,a R n ) deend on the chosen direction in R n. The number s, defined by 1 s = ), 0.2) n s 1 s n is usually called the mean smoothness, and a = a 1,..., a n ), k=1 a 1 = s s 1,..., a n = s s n 0.3) characterises the anisotroy. Similar to the isotroic situation the more general anisotroic Bessel otential saces fractional Sobolev saces) H s,a R n ), where 1 < <, s R and a = a 1,..., a n ) is a given anisotroy, fit in the scales of anisotroic Besov saces Bq s,a R n ), and anisotroic Triebel-Lizorkin saces Fq s,a R n ), resectively. It is well known that this theory has a more or less comlete counterart to the basic facts definitions, descrition via differences and derivatives, elementary roerties, embeddings for different metrics, interolation) of isotroic saces BqR s n ) and FqR s n ). We shall use the Fourieranalytical definition of Bq s,a R n ), Fq s,a R n ), where any function f S R n ) is decomosed in a sum of entire analytic functions ϕ j f) and this decomosition, measured in l q and L R n ), resectively, is used to introduce the saces. This concet goes back to [16] and [15], see [13, Chater 4]. Our main aim in the resent aer is to rove an anisotroic counterart to the isotroic results, see [21]. As a first goal of this aer we define the anisotroic d-set as follows: Let 0 < d < n, a an anisotroy. Then Γ R n is called an anisotroic d set if there exists a ositive Radon measure µ with suµ = Γ and µb a γ, r)) r d, 0 < r < 1, 0.4) where B a γ, r) = {y R n : y γ a r} is an anisotroic ball and γ Γ. We study the existence and roerties of the trace oerator tr Γ, tr Γ : B s,a R n ) L Γ) 0.5) where Γ is an anisotroic d set. It turns out that tr Γ according to 0.5) exists if, and is comact if 2 j s n ) µ 1 j N 0 j <, = 1, 0.6) where µ j = su µq a jm), and Q a jm are rectangles centered at 2 ja m and with side length 2 ja1,..., 2 jan. m Z n If we can strengthen 0.6) by j J 2 j s n ) µ 1 j 2 J s n ) µ 1 J, J N 0, 0.7) then one obtains for the aroximation numbers a k of the comact oerator tr Γ according to 0.5) ck 1 d n s) 1 ak tr Γ : B s,a R n ) L Γ)) c k 1 d n s) 1, 1 n s > n d, 0.8)

4 as in the isotroic case, see [21]. In order to show the above result we rove, in addition, some imortant roerties of saces B s,a q R n ), which might be of self-contained interest: 1. We obtain the homogeneity estimate fr ) B s,a q R n ) cr s n f B s,a q R n ) j= for all f B s,a q R n ) and R ) 2. We show that the homogeneous norms ϕ ˆf) ) 1/q L + 2 jsq ϕ a ˆf) j L q 0.10) and f L + are equivalent quasi-norms in B s,a q R n ). j= 2 jsq ϕ a j ˆf) L q ) 1/q 0.11) 3. Finally we rove the localisation roerty of B s,a R n ), that is c fj a B s,a R n ) 2 js n ) c k ) 1/ f B s,a R n ) c fj a B s,a R n ), 0.12) k Z n where fj a x) = c k f2 j+1)a x x a j,k)), c k C, j N 0.13) k Z n and f is a roduct of one-dimensional functions n f2 j+1)a x x a j,k)) = f m 2 j+1)am x m 2 jam k m )). 0.14) m=1 The lan of the aer is the following. In the first section we collect definitions, and introduce the anisotroic d-set. Section 2 contains the main results. Proofs are shifted to Section 3 starting with a descrition of the wavelet frames according to [9] and contains also the roerties of Bq s,a R n ) mentioned above. 1 Preliminaries 1.1 General notation As usual, R n denotes the n-dimensional real Euclidean sace, N the collection of all natural numbers, N 0 = N {0}, C stands for the comlex numbers, and Z n means the lattice of all oints in R n with integer-valued comonents. We use the equivalence in ϕx) ψx) always to mean that there are two ositive numbers c 1 and c 2 such that c 1 ϕx) ψx) c 2 ϕx) for all admitted values of x, where ϕ, ψ are non-negative functions. If a R then a + := maxa, 0). Let α = α 1,..., α n ) N n 0 be a multi-index, then α = α α n, α! = α 1! α n!, α N n 0, 1.1) the derivatives D α have the usual meaning, x α means x α = x α 1 1 xαn n αγ = α 1 γ α n γ n, γ R n, stands for the scalar roduct in R n. for x = x 1,..., x n ) R n, and Given two quasi-banach saces X and Y, we write X Y if X Y and the natural embedding of X in Y is continuous. All unimortant ositive constants will be denoted by c, occasionally with additional subscrits within the same formula. We shall mainly deal with function saces on R n ; so for convenience we shall usually omit the R n from their notation, if there is no danger of confusion. 2

5 1.2 Anisotroic function saces Let a = a 1,..., a n ) be a fixed n tule of ositive numbers with a a n = n, then we call a an anisotroy. We shall denote a min = min{a i : 1 i n} and a max = max{a i : 1 i n}. If a = 1,..., 1) we seak about the isotroic case. The action of t [0, ) on x R n is defined by the formula t a x = t a1 x 1,..., t an x n ). 1.2) For t > 0 and s R we ut t sa x = t s ) a x. In articular we write t a x = t 1 ) a x and 2 ja x = 2 j ) a x. Definition 1.1 An anisotroic distance function is a continuous function u : R n R with the roerties ux) > 0 if x 0 and ut a x) = tux) for all t > 0 and all x R n. Remark 1.2 It is easy to see that u λ : R n R defined by n ) 1/λ u λ x) = x i λ a i 1.3) i=1 is an anisotroic distance function for every 0 < λ <, u 2 is usually called the anisotroic distance of x to the origin, see [13, 4.2.1]. It is well known, see [3, 1.2.3] and [22, 1.4], that any two anisotroic distance functions u and u are equivalent in the sense that there exist constants c, c > 0 such that cux) u x) c ux) for all x R n ) and that if u is an anisotroic distance function there exists a constant c > 0 such that ux + y) cux) + uy)) for all x, y R n. We are interested to use smooth anisotroic distance functions. Note that for aroriate values of λ one can obtain arbitrary finite) smoothness of the function u λ from 1.3), cf. [3, 1.2.4]. A standard method concerning the construction of anisotroic distance functions in C R n \{0}) was given in [14]. For x = x 1,..., x n ) R n, x 0, let x a be the unique ositive number t such that x 2 1 t 2a1 + + x2 n = 1 1.4) 2an t and let 0 a = 0; then a is an anisotroic distance function in C R n \{0}), see [22, 1.4/3,8]. Plainly, x a is in the isotroic case the Euclidean distance of x to the origin. Before introducing the function saces under consideration we need to recall some notation. By S we denote the Schwartz sace of all comlex-valued, infinitely differentiable and raidly decreasing functions on R n and by S the dual sace of all temered distributions on R n. Furthermore, L with 0 <, stands for the usual quasi-banach sace with resect to the Lebesgue measure, quasi-normed by ) 1/, f L := fx) dx R n with the obvious modification if =. If ϕ S then ϕξ) Fϕ)ξ) := 2π) n/2 R n e ixξ ϕx)dx, ξ R n, 1.5) denotes the Fourier transform of ϕ. As usual, F 1 ϕ or ϕ, stands for the inverse Fourier transform, given by the right-hand side of 1.5) with i in lace of i. Here xξ denotes the scalar roduct in R n. Both F and F 1 are extended to S in the standard way. Let ϕ S be such that and for each j N let ϕx) = 1 if x a 1 and su ϕ {x R n : x a 2}, 1.6) ϕ a j x) := ϕ2 ja x) ϕ2 j+1)a x), x R n. 1.7) Then the sequence ϕ a j ) j=0, with ϕ 0 = ϕ, forms a smooth anisotroic dyadic resolution of unity, cf. [13, 4.2]. Let f S, then the comact suort of ϕ a j f imlies by the Paley - Wiener - Schwartz theorem that ϕ a j f) is an entire analytic function on R n. 3

6 Definition 1.3 Assume 0 <, 0 < q, s R, a an anisotroy, and ϕ a j ) j=0 a smooth anisotroic dyadic resolution of unity. Then { Bq s,a = f S : ) 1/q } f Bq s,a ϕ = 2 jsq ϕ a f) j L q < with the usual modification if q = ). in the sense of equivalent quasi-norms); therefore we omit in our notation the subscrit ϕ in the sequel. It is well-known that Bq s,a Note that there is a arallel definition for saces of tye Fq s,a, 0 < <, 0 < q, s R, a an anisotroy, when interchanging the order of l q - and L - quasi-norms in 1.8). It is obvious, that the quasinorm 1.8) deends on the chosen system ϕ a j ) j N 0, but not the sace B s,a q are quasi-banach saces Banach saces if 1 and q 1), and, as in the isotroic case, S Bq s,a S for all admissible values of, q, s, see [17, 2.3.3]. If s R and 0 < <, 0 < q < then S is dense in Bq s,a, see [22, 3.5] and [3, ]. Note that we indicated the only formal) difference to the isotroic counterarts of 1.8) by the additional suerscrit at the smooth anisotroic dyadic resolution of unity ϕ a j ) j=0. Remark 1.4 A systematic treatment of the theory of isotroic) Bq s and Fq) s saces may be found in the monograhs [17], [18], [19] and [20]; see also [4] and [12]. A survey on the basic results for the anisotroic) saces Bq s,a and Fq s,a ) is given in [13, ] and [10, ]. In addition to the literature mentioned in our introduction we essentially rely on [8] and [7] in the sequel. j=0 For convenience, in case of = q we shall stick to the notation in the sequel. B s,a 1.8) = B s,a where 0 <, s R, 1.9) 1.3 Anisotroic d sets in R n We assume that µ is a ositive Radon measure in R n with comact suort Γ = su µ, 0 < µr n ) <, Γ = 0, 1.10) where Γ denotes the Lebesgue measure of Γ. For 1 < we consider the usual comlex Banach sace L Γ, µ), normed by ) 1/ f L Γ, µ) = fx) µdx) = fγ) µdγ) R n Γ ) 1/ where we shall use both notations. Let again a = a 1,..., a n ) be a given anisotroy. Definition 1.5 Let 0 < d < n. Then Γ R n is called an anisotroic d set if where B a γ, r) = {y R n : y γ a r} and γ Γ. µb a γ, r)) r d, 0 < r < 1, 1.11) In the following roosition we rove the existence of anisotroic d sets. Proosition 1.6 For every 0 < d < n there exists an anisotroic d set. 4

7 P r o o f. For simlicity we rove this roosition for the case n = 2. If n > 2 this can be done in a similar way. Let Q = [0, 1] 2 be the closed cube with side-length 1, we take the affine contractions A m ) N m=1 on R 2 which ma the unit square to the rectangles A m Q) N m=1 with side-lengths r a 1 and r a2 where 0 < a 1 < a 2 and a 1 +a 2 = 2 as in Figure 1, so that they are disjoint and µa m Q) = N 1. Let Fig.1 N AQ = AQ) 1 = A m Q, AQ) 0 = Q, m=1 AQ) k = AAQ) k 1 ). The sequence of sets is monotonically decreasing and by [6, Theorem 8.3] Γ = AQ) = k NAQ) k = lim k AQ)k is the uniquely determined fractal generated by the contractions A m ) N m=1. But on the other hand we assume that µa m Q) = r d, where m = 1,..., N, and from here we get that d = log N log r, for arbitrary N N and 0 < r < 1. Remark 1.7 Our definition for the anisotroic d set is a generalization of Farkas definition [7, 3.1]. In the following we recall his definition. If j N 0 and N j N 0 we deal with sets of oen rectangles {R jl : l = 1,..., N j } in R n having sides arallel to the axes, the side length of the rectangle R jl with resect to the x i axis is denoted by r j,l i where i = 1,..., n. We will always assume that the side lengths of the rectangles R jl are ordered in the same way, for examle r j,l 1... rn j,l for any j N 0 and any l = 1,..., N j. Let Q be a cube in R n with side length 1, let 0 < d < n, let a = a 1,..., a n ) a given anisotroy and let c 1, c 2 > 0 given numbers. Let N 0 = 1 and for any j N let N j be a natural number satisfying c 1 2 jd N j c 2 2 jd. A comact set Γ R n is called a regular anisotroic d set with resect to the anisotroy a) if for any j N 0 there exists a finite sequence of oen rectangles {R jl : l = 1,..., N j } having sides arallel to the axes, R 01 = Q, the interior of Q, such that: i) there exists a constant 0 < c 0 1 such that for all i = 1,..., n, all j N 0 and all l = 1,..., N j ii) if l l then R jl R jl = c 0 2 j ) ai r j,l i 2 jai 1.12) iii) for any rectangle R j+1,k there exists a rectangle R jl, l = lk), such that R j+1,k R jl iv) for any j N 0 and any l = 1,..., N j vol R jl ) d n = vol R j+1,k ) d n 1.13) R j+1,k R jl 5

8 v) N j Γ = R jl. j=0 l=1 Let n = 2, let a = a 1, a 2 ) a 2 dimensional anisotroy and let 0 < d < 2, then Γ is also an anisotroic d set in the sense of Triebel [19, 5.2]. Let 0 < d < n and let Γ be the regular anisotroic d set with resect to the given anisotroy a = a 1,..., a n )) introduced above. Then there exists a Radon measure µ in R n uniquely determined with su µ = Γ and µγ R jl ) = vol R jl ) d n, j N0 and l = 1,..., N j, 1.14) see [7, Theorem 3.5]. Let B a x, 2 j ) = {y R n : y x a 2 j } be an anisotroic ball like in Definition 1.5 with r = 2 j. It is easy to see that B a x, 2 j ) {y R n : y i x i c2 ja i, i = 1,..., n}. By 1.12) B a x, 2 j ) has a nonemty intersection with at most N rectangles R jl, l = 1,..., N j ), where N is indeendent of j so that using 1.14) we get µb a x, 2 j ) Γ) c 2 jd where c > 0 is indeendent of j, so it is easy to see that Farkas regular anisotroic d-set is also an anisotroic d-set according to Definition 1.5. By [19, Definition 3.1] if Γ is an isotroic d set with underlying measure µ and if 0 < κ < 1 then µbγ, κr) Γ) µbγ, r) Γ) r d 1.15) where the equivalence constants deend on κ but not on γ Γ and 0 < r 1. For a regular anisotroic d set Γ, in the above sense of Farkas, we have 1.14) but no counterart of 1.15). If 0 < κ < 1 then κr jl denotes the rectangle concentric with R jl and with side lengths resectively κr j,l 1,..., κrj,l n. The regular anisotroic d set introduced above equied with the measure µ according to 1.14) is called roer if there exist two numbers 0 < κ < 1 and 0 < c 1 such that µγ κr jl ) cvol R jl ) d n, j N0, l = 1,..., N j. 1.16) Following the lines of the roof of [19, 5.13] it turns out that if Γ is generated by linear contractions and if Γ Q then Γ is roer. The Definition 1.5 covers also the feature of Γ to be roer, because by 1.11) we have that µb a x, 2 j ) Γ) c 2 jd, where c > 0. Examle 1.8 Let Q = [0, 1] 2 and let log be taken with resect to the base 2, let 1 < K 1 < K 2 be natural numbers so that K 1 K 2 = 2k + 1 for some k N, and let a 1 = 2 log K 1 logk 1 K 2 ), a 2 = 2 log K 2 logk 1 K 2 ), κ = 1 2 logk 1K 2 ). We can see that a 1, a 2 > 0 and a 1 + a 2 = 2. Let A m ) N m=1 be N 2 contractions of R 2 into itself secified by A m : x = x 1, x 2 ) η m 1 2 κa1 x 1, η m 2 2 κa2 x 2 ) + x m 1.17) for every m = 1,..., N where η2 m -is always 1 and we choose η1 m = 1 in the first K 2 columns, η1 m = 1 in the second K 2 columns, then again η1 m = 1 in the third K 2 columns and so on, and x m in 1.17) is chosen such that we have the situation as deicted in Fig.2. We assume A m Q Q for all m = 1,..., N, A m Q Am Q = if m m, and we suose that the rectangles A m Q are located in the columns as indicated in Fig.2. Let N AQ = AQ) 1 = A m Q, AQ) 0 = Q, m=1 6

9 AQ) k = AAQ) k 1 ). The sequence of sets is monotonically decreasing and by [19, Theorem 4.2] Γ = AQ) = k NAQ) k = lim k AQ)k is the uniquely determined fractal generated by the contractions A m ) N m=1. Under these assumtion the resulting Γ is the grah of a continuous function, see [19, 4.21]. Moreover, Γ is an isotroic d-set where d = dim H Γ = 2 min1, a 2 a 1 ), see [19, 4.22, 4.23], and in the Farkas sense it is a regular anisotroic d-set with d = a 1. In our examle in Fig.2 with N = 7 we calculated d in the same way like in the roof of Proosition 1.6 and we get that d = log 7 κ. Examle 1.9 Let A 1, A 2 be the affine contractions on R n which ma the unit square onto the rectangles R 1, R 2 of sides 2 a 1 and 2 a 2 where 0 < a 2 < a 1 and a 1 + a 2 = 2 as in Figure 3. In the same way like in Examle 1.8 we have that d = 1, in the sense of our definition. In [7, 3.1] we can see that Farkas also has for this examle d = 1. Fig Traces Assume that Γ is an anisotroic d set in R n with resect to the anisotroy a = a 1,..., a n ). If ϕ S then tr Γ ϕ = ϕ Γ makes sense ointwise. If 0 <, q < and s R then the embedding tr Γ Bq s,a L Γ) must be understood as follows: there exists a ositive number c > 0 such that for any ϕ S Since S is dense in B s,a q tr Γ ϕ L Γ) c ϕ B s,a q. Fig.3 for 0 <, q < this inequality can be extended by comletion to any f B s,a q and the resulting function is denoted by tr Γ f and the indeendence of tr Γ f from the aroximating sequence is shown in the standard way. 7

10 Let Q a jm be the rectangles in Rn with side length 2 ja1,..., 2 jan and centered at 2 ja m where m Z n and j N 0. Let µ j = su µq a jm), j N ) m Z n Proosition 1.10 Let Let µ be the Radon measure in R n with If then exists. 1 < <, = 1, s > 0. Γ = su µ comact, 0 < µr n ) <, Γ = ) P r o o f of Proosition 1.10). 2 j s n ) µ 1 j N 0 j < where µ j = su m Z n µq a jm), 1.20) tr Γ : B s,a R n ) L Γ) 1.21) We rove the roosition like in isotroic case, see [21, Pro. 2], [20, Theorem 9.3, Corollary 9.8]. In our case we use the anisotroic local means and the equivalent norm in anisotroic function saces, see [7, 2.2]. In comarison with [20, Theorem 9.3] we need only a secial case where u = v = and σ = s. Remark 1.11 If Γ is a regular anisotroic d set in Farkas sense, with resect to the anisotroy a = a 1,..., a n ), and if d n < < and 0 < q < min1, ) then see [7, Theorem 3.12]. In addition, the equality tr Γ Bq s,a on Γ and 2 Main results tr Γ B n d,a q = L Γ), 1.22) = L Γ) means that any f Γ L Γ) is the trace of a suitable g B s,a q f Γ L Γ) inf{ g B s,a q : tr Γ g = f Γ }. 1.23) In the sequel, we only consider the case = q. We roceed similar to [21], dealing with the isotroic case. Let Q a jm be the rectangles defined above. By 1.11) with r = 2 j we have that µγ Q a jm ) 2 jd, and µγ Q a jm ) c2 jd. Proosition 2.1 Let 1 < <, Let µ be the Radon measure in R n with and = 1, s > 0, 0 < d < n. Γ = su µ comact, 0 < µr n ) <, Γ = 0, 2.1) 2 j s n ) µ 1 j N 0 j < where µ j = su m Z n µq a jm). 2.2) Then tr Γ tr Γ : B s,a R n ) L Γ) 2.3) is comact. Furthermore there is a constant c deending on and s) such that for all measures µ with 2.1), 2.2), ) 1 tr Γ c 2 j s n ) µ 1 j. 2.4) j N 0 The result above is the anisotroic version of [21, Proosition 3.]. 8

11 2.1 Aroximation numbers Let A and B be two Banach saces and let T LA, B). Then given any k N the kth aroximation number a k T ) of T is given by a k T ) = inf{ T L : L LA, B), rank L < k}, 2.5) where rank L is the dimension of the range of L. These numbers have various roerties given in the following lemma. Lemma 2.2 Let A and B be two Banach saces and let T, S LA, B). i) T = a 1 T ) a 2 T ) 0 ii) for all n, m N, iii) for all n, m N, and R LB, C) iv) a n T ) = 0 rank T < n. a m+n 1 S + T ) a m S) + a n T ) a m+n 1 RT ) a m R)a n T ) This is a well-known concet and can be found for instance in [4, 1.3.1] and [5, II]. Some roof of the above lemma is given in [4, 1.3.1] and [5, II] for examle. Let T = tr Γ according to Proosition 2.1. We strengthen 2.2) by j J 2 j s n ) µ 1 j 2 J s n ) µ 1 J, J N 0, 2.6) where only the cases s n are of interest, otherwise 2.6) is always satisfied. Proosition 2.3 Let 1 1 < <, + 1 = 1, s > 0. Let µ be a Radon measure in R n with 2.1) and 2.6). Let a k = a k tr Γ ) be the aroximation numbers of the comact oerator tr Γ in 2.3). There are two ositive numbers c and c such that a c2 nj where c2 nj is always assumed to be a natural number. c 2 Js n ) µ 1 J, J N 0 2.7) Theorem 2.4 Let the anisotroic d-set Γ and µ be given according to 1.11), and 0 < d < n, 1 < <, = 1, n s > n d. Let a k = a k tr Γ ) be the aroximation numbers of the comact oerator tr Γ according to 2.3). Then there exist numbers c, c > 0 so that for all k N ck 1 d n s) 1 ak tr Γ : B s,a R n ) L Γ)) c k 1 d n s) ) Remark 2.5 Let Γ be the regular anisotroic d-set according to Remark 1.7. Farkas roved in [7, 3.2.3] that e k tr Γ : B n d δ+,a 1 1 q R n ) L 2 Γ)) ck δ d where 0 < 1, 2, q, and δ > 0. Recall that Γ is also an anisotroic d-set according to Definition 1.5. Let 1 = 2 = q = and s = δ + n d we get that e k tr Γ : B s,a R n ) L Γ)) ck 1 d n s) 1. So we have the same results for the entroy and aroximation numbers in the secial case 1 = 2 which is not surrising, but cannot be exected for 1 2. In view of the isotroic result [21, Theorem 2, Remark 9], if we restrict the outcome [21] to the classical examle of a comact d-set with 0 < d < n, then we have the same result like in the anisotroic setting. 9

12 3 Proofs Now we collect all roofs but first we describe wavelet frames which are an effective instrument to estimate aroximation numbers. Let k be a non-negative C function in R n with su k {y R n : y a < 2 J, y j > 0}, 3.1) for some J N, and m Z n kx m) = 1, x R n. Recall x β = x β1 1 xβ n n where x = x 1,..., x n ) R n and β N n 0, and ut k β x) = 2 Ja x) β kx) 0, x R n, β N n 0. Let ω S, su ω π, π) n, ωx) = 1 if x a 2, and let and ω β x) = i β 2 Jaβ 2π) n β! xβ ωx) for x R n, β N n 0, Ω β x) = m Z n ω β ) m)e imx, x R n, β N n 0, where β = β β n, and β! = β 1! β n! and aβ = a 1 β 1 + a n β n. Let ϕ 0 be a C function in R n with ϕ 0 x) = 1 if x a 1 and ϕ 0 x) = 0 if x a 3 2, 3.2) and let ϕx) = ϕ 0 x) ϕ 0 2 a x). Then { Φ β jm x) = Φ β F x m), if j = 0, Φ β M 2ja x m), if j N, 3.3) are analytic wavelets where the father wavelets Φ β F x) and the mother wavelets Φβ M x) are given by their inverse Fourier transforms ) Φ β F ξ) = ϕ0 ξ)ω β ξ), ξ R n, 3.4) Φ β M ) ξ) = ϕξ)ω β ξ), ξ R n. 3.5) For the sequence λ = {λ β jm C : j N 0, m Z n, β N n 0 }, 3.6) s R, 0 <, and ϱ 0, we ut For f L R n ), λ b s,ϱ = β N n 0 j=0 m Z n 2 ϱaβ+js n/) λ β jm ) 1/. 3.7) λ β jm f) = 2jn R n fx)φ β jm x)dx, j N 0, m Z n, β N n ) 10

13 Theorem 3.1 [9, Theorem 2.1] Let 0 <, s > σ where σ = n 1 1), ϱ 0, and a an + anisotroy. Then f S is an element of B s,a if, and only if, it can be reresented as f = β N n 0 j=0 m Z n λ β jm kβ 2 ja x m), x R n, 3.9) with λ b s,ϱ <, absolute convergence being in L max1,). Furthermore, f B s,a inf λ b s,ϱ, 3.10) where the infimum is taken over all admissible reresentations 3.9). In addition, any f B s,a can be otimally reresented by f = λ β jm f) kβ 2 ja x m), 3.11) j=0 m Z n with β N n 0 Remark 3.2 In the sequel we shall stick to the notation P r o o f of Proosition 2.1). f B s,a λf) b s,ϱ. 3.12) k β jm x) = kβ 2 ja x m), β N n 0, j N 0, m Z n. 3.13) Note that the comactness of tr Γ with Γ in Farkas sense is already covered by [7, Theorem 3.13]; for more general anisotroic d-sets Γ, see Definition 1.5, we resent a new roof, we follow ideas in [21]. Ste 1. Let f B s,a be given by 3.11), 3.12) we use the notation 3.13)). For any fixed β N n 0 we have λ β jm f)kβ jm L Γ) λ β jm f)kβ jm L Γ) j=0 m Z n j=0 m Z n ) 1/ c λ β jm f) k β jm x) µdx) j=0 R n m Z n ) 1/ c λ β jm f) k β jm x) µdx) j=0 m Z n cq a jm c µ 1 ) 1/ j λ β jm f) 3.14) j=0 m Z n where we used the boundedness of k and 1.18). We aly the Hölder inequality, recall = 1, and so we can continue λ β jm f)kβ jm L Γ) ) c 2 j s n ) 1/ ) 1/ µ j 2 js n ) λ β jm f). 3.15) j=0 m Z n j=0 j,m We choose ϱ > 0. Then it follows by 3.7) and 3.12) that tr Γ f L Γ) c j=0 ) 1/ 2 j s n ) µ 1 j f B s,a 3.16) where c is indeendent of µ. This roves 2.4). Ste 2. We rove that tr Γ is comact. Let B N, J N, [aβ] = max{r Z : r aβ}, and let tr B,J Γ be given by tr B,J Γ f = Γ λ β jm f)kβ jm, 3.17) j J m Z n [aβ] B 11

14 where again f B s,a is given by 3.11),3.12) and where the sum Γ m Z is restricted to those m Z n n such that the rectangles Q a jm have a non-emty intersection with Γ. For given δ > 0 and suitably chosen ϱ > 0 it follows by the above arguments for f B s,a having norm of at most 1 that tr Γ tr B,J Γ )f L Γ) c [aβ] B ) 2 δaβ + c [aβ] B 2 δaβ) j J ) 1/ 2 j s n ) µ 1 j, 3.18) see 3.7) and 3.15), 3.16). By 2.2) we find for any given ε > 0 sufficiently large numbers B and J such that tr Γ tr B,J Γ ε. 3.19) Then tr Γ is comact, as tr B,J Γ are finite rank oerators. P r o o f of Proosition 2.3). Note that 2.6) imlies 2.2), thus by Proosition 2.1 the oerator tr Γ is comact. We refine 3.17) by tr J Γf = [aβ] J j J [aβ] Γ λ β jm f)kβ jm, J N, 3.20) m Z n where again f B s,a is given by 3.11),3.12) and the last sum has the same meaning as the last sum in 3.17). As µ is a measure in R n we have that µ K c2 J K)n µ J, K J, 3.21) also in the anisotroic case, recall a a n = n. Let δ > 0 be sufficiently large. By 3.21) we obtain for f B s,a having norm of at most 1 in analogy to 3.18) that tr Γ tr J Γ)f L Γ) c2 δj + c [aβ] J c2 δj + c [aβ] J c2 δj + cµ 1 J 2 Js n ) 2 δaβ j J [aβ] 2 j s n ) µ 2 δaβ 2 J [aβ])s n ) µ 1 J [aβ] [aβ] J 2 δaβ+aβs n )+aβ n ) 1/ j c 2 Js n ) µ 1 J. 3.22) In the second estimate we used assumtion 2.6) and in the next one 3.21). For the rank of trγ J the estimate ranktrγ) J c 2 nj [aβ]) c 2 nj. This roves 2.7). [aβ] J we have P r o o f of the right-hand side of the estimate 2.8) in Theorem 2.4). Again we use the wavelet exansion 3.11), 3.12). For fixed β N n 0 we ut tr β Γ f = j N 0 m Z n λ β jm f)kβ jm 3.23) and tr β,j Γ f = Γ λ β jm f)kβ jm, 3.24) j J m Z n 12

15 where the second sum has the same meaning as the last sum in 3.17). By the same reasoning as in 3.20) and 3.22) but now for fixed β we have for f B s,a with norm of at most 1, tr β Γ trβ,j Γ )f L Γ) c2 δaβ 2 J n s) µ 1 J 3.25) By the Definition 1.5 there exists a constant c > 0 indeendent of j N 0 with µq a jm Γ) c2 jd and we obtain that tr β Γ trβ,j Γ )f L Γ) c2 δaβ 2 J n s) 2 J d. 3.26) In definition 2.5) ut L = tr β,j Γ, T = trβ Γ, and note that for j N 0, Γ ) rank λ β jm f)kβ jm c2 jd. 3.27) m Z n Thus we obtain by 3.24) that ranktr β,j Γ ) c j J 2 jd c 2 Jd. 3.28) Then 3.27) imlies that there are two ositive numbers c and c such that For k N there are numbers J k N such that inserted in 3.29) this leads to a c2 Jdtr β Γ ) c 2 δaβ 2 J n s) 2 J d. 3.29) 2 J kd k with J 1 J 2 J n ; 3.30) a ck tr β Γ ) c2 δaβ 2 J k n s) k ) Let ε > 0, for given k N we aly 3.31) to k β N with k β 2 εaβ k. Then it follows by the additivity roerty of aroximation numbers and from 3.31) that a ck tr β Γ ) a kβ tr β Γ ) β N n 0 c 2 δaβ 2 J k β n s) 2 εaβ k) 1 β N n 0 c 2 J k n s) k 1 2 aβδ ε ) β N n 0 c 2 J k n s) k ) for ε > 0 small. We used s n, such that J k β n s) J k n s). Finally 3.30) imlies a ck tr β Γ ) c k 1 d n s) ) and so we finished the roof of the right-hand side of the estimate 2.8). In order to rove the left-hand side of the estimate 2.8) in Theorem 2.4 we need some rearation and first exlain the main idea and structure of this argument. We construct secial functions f a j x) = k Z n c k f2 j+1)a x x a j,k)), c k C, j N 13

16 as given in Theorem 3.6 below that satisfy c fj a B s,a 2 js n ) c k ) 1/ f B s,a c fj a B s,a. k Z n They will serve us to obtain lower bounds for tr Γ T for any finite rank oerator; this leads to a lower estimate for a k tr Γ ) corresondingly. To obtain Theorem 3.6 we need in addition some homogeneity estimate fr ) Bq s,a cr s n f B s,a q for all f Bq s,a and R 1, see Pro. 3.5, and an equivalent homogeneous norm in Theorem 3.3 below. Let ϕ S as in Section 1.2, in articular we have 1.6). We extend the definition of ϕ a j from 1.7) to all integers j. It should be noted that ϕ a 0 has now a different meaning as in 1.2, i.e. for f S then we have that f = ϕ ˆf) + ϕ a ˆf) j convergence in S ). 3.34) j=1 Theorem 3.3 Let 0 <, 0 < q, s > σ and a an anisotroy, then ϕ ˆf) ) 1/q L + 2 jsq ϕ a ˆf) j L q 3.35) and f L + j= j= modification if q = ) are equivalent quasi-norms in B s,a q. P r o o f of Theorem 3.3). 2 jsq ϕ a j ˆf) L q ) 1/q 3.36) We closely follow the roof in [18, 2.3.3] for the isotroic case. Ste 1. We rove that 3.35) is an equivalent quasi-norm in Bq s,a. It is sufficient to show that there exists a constant c > 0 such that holds, because we need to rove that 1 ϕ a j ˆf) L c2 jσ ϕ ˆf) L, j N, 3.37) j= 2 jsq ϕ a j ˆf) L q ) 1/q c ϕ ˆf) L and this is satisfied if 3.37) is true. For those j s we have that ϕ a j x) = ϕa j x)ϕx) by the suort condition 1.6) and 1.7) with j N, and hence ϕ a j ˆf) L = ϕ a j ϕ ˆf) ) ) L c ˇϕ a j L r ϕ ˆf) L, r =min1, ), 3.38) where the inequality comes from the Fourier multilier assertion for entire analytic functions, F 1 MF f L F 1 M L f L where = min1, ), roved in [17, Proosition 1.5.1]. Elementary calculations show that ˇϕ a j x) = 2jn ˇϕ 0 2 ja x) such that ˇϕ a j L r = 2 j n r ˇϕ 0 2 ja ) L r c2 j n r +jn as a a n = n. By 3.38) we thus have that ϕ a j ˆf) L 2 jn 1 r 1) ϕ ˆf) L and we obtain 3.37). Ste 2. We rove that 3.36) is an equivalent quasi-norm in Bq s,a. By our assumtion s > σ, we may assume that 3.34) converges not only in S, but also, say almost everywhere in R n. Then we have f L c ϕ ˆf) ) 1/ L + c ϕ a ˆf) j L 3.39) j=1 14

17 if 0 < 1 and a corresonding estimate if 1 < <. Now 3.35) and 3.39) rove that 3.36) can be estimated from above by c f Bq s,a. We consider the converse inequality. Because f is a regular distribution we have a.e. that ϕ ˆf) x) = fx) + 1 ϕ )) ˆf) x) = fx) + By the above-mentioned Fourier multilier assertion we have 1 ϕ ))ϕ a j ) ˆf) x). 3.40) j=0 ϕ ˆf) ) 1/ L c f L + c ϕ a ˆf) j L 3.41) if 0 < 1 and a corresonding estimate if 1 < <. Now 3.35) and 3.41) rove that f Bq s,a can be estimated from above by the quasi-norm 3.36). Remark 3.4 The quasi-norms of tye 3.35), 3.36) have a continuous counterart. We introduce ρ a tξ) = ϕt a ξ) ϕ2t) a ξ) where t > 0. Then the counterart of 3.35) reads as follows: Let 0 <, 0 < q, s > σ and a an anisotroy, then j=0 f L + t sq ρ a t ) ˆf) L q dt ) 1/q 3.42) 0 t modification if q = ) is an equivalent quasi-norm in B s,a q. Now we can extend the well-known homogeneity estimate for B s,q see [17, Pro ]) to anisotroic saces. Proosition 3.5 Let 0 <, 0 < q, s > σ and a an anisotroy. There exists a constant c > 0 such that for all R 1, fr ) B s,a q cr s n f B s,a q for all f B s,a q. 3.43) P r o o f of Pro. 3.5). We closely follow the roof in [4, Pro ] for the isotroic case. Let ψ = ϕ 1 be the same function as in 1.7). We have by 3.42) f L + t sq ψt ) ˆf) L q dt ) 1/q 3.44) 0 t is an equivalent quasi-norm on B s,a q. Elementary calculation shows that ψt )fr ) )) x) = ψt ) ˆfR 1 )) x)r n = ψtr )) ˆf )) Rx). 3.45) also in the anisotroic case, where a a n = n. From 3.44), with frx) in lace of fx), and 3.45) we obtain ) 1/q fr ) Bq s,a c 1 fr ) L + c 1 t sq F 1 ψt )F[fR )]) L q dt 0 t c 2 R n f L + c 3 R s n t sq F 1 ψtr ))Ff) L q dt t 0 ) 1/q and from here follows 3.43) for R 1, c 1, c 2, c 3 > 0 and s > σ. 15

18 Our next aim is to extend the localisation roerty, see [4, 2.3.2], to the anisotroic saces. Let x a j,k = 2 ja k with k Z n and j N. Let f S with where b > 0 and b 1 4 2a min + 1). Let su f Q a b = {x R n : x = x 1, x 2,..., x n ), x a < b} 3.46) f a j x) = where f is a roduct of one-dimensional functions, k Z n c k f2 j+1)a x x a j,k)), c k C, j N 3.47) f2 j+1)a x x a j,k)) = and f 1 y) = = f n y) where y R. n f m 2 j+1)a m x m 2 ja m k m )) 3.48) m=1 Theorem 3.6 Let s > σ, 0 <, 0 < q, a an anisotroy and 0 < b 1 4 2a min + 1). There exist two constants c > 0 and c > 0 such that for all f B s,a with su f Q a b and all j N and all given by 3.47) f a j P r o o f of Theorem 3.6). Ste 1. where f B s,a c fj a B s,a 2 js n ) c k ) 1/ f B s,a c fj a B s,a. 3.49) k Z n At first we rove the left-hand side of 3.49). By 3.47) we have f a j 2 j+1)a x) = and 3.46) is true. We would like to rove that k Z n c k fx 2 a k), c k C, j N, 3.50) ) 1/ c k f 2 a k) B s,a c k f B s,a. 3.51) k Z n k Z n We use the characterisation of B s,a via local means; see [8, 4.4]. Recall notation 1.2). Let k C so that su k B a = {y R n : y a 1} and kt, f)x) = ky)fx + t a y)dy, t > ) R n Let k 0 C such that su k 0 B a, and s 1 > maxs, σ ) + σ then ) 1/q f B s,a k 0 1, f) L + 2 jsq k2 j, f) L q 3.53) is an equivalent quasi-norm in B s,a ; see [8, 4.4]. We insert 3.50) in 3.52) and obtain k t, ) ) c m f 2 a m) x) = ky) c m fx + t a y 2 a m) dy m Z n R n m Z n = ky)fx 2 a m + t a y)dy R n j=1 m Z n c m = m Z n c m kt, f)x 2 a m) 3.54) 16

19 and it follows c m f 2 a m) B s,a m Z n k 0 1, c m f 2 a m)) L + 2 jsq k2 j, c m f 2 a m)) L q) 1 q m Z n j=1 m Z n ) 1/ c m k 0 1, f) L + 2 jsq k2 j, f) L q) ) 1 q m Z n j=1 ) 1/ c m f B s,a. m Z n Now the left-hand side of inequality of 3.49) is an easy consequence of Proosition 3.5, 3.50) and 3.51): fj a B s,a c2 js n/) c m f 2 a m) B s,a m Z n ) 1/ c 2 js n/) c m f B s,a. 3.55) m Z n Ste 2. In this ste we rove the right-hand side of 3.49). For this we would like to use the localization roerty given in [4, 2.3.2] if n = 1 and for the functions f jα x) = m Z c m f2 j+1)α x 2 α m), c m C, j, α N, 3.56) where f S R). By [4, 2.3.2/4] we know that there exist two constants c > 0 and c > 0 such that for all f B s c f jα B s 2 jαs 1 ) c k ) 1/ f B s c f jα B, s 3.57) k Z as for n = 1 isotroic and anisotroic results coincide. For the functions fj a given in 3.47) we use the Fubini roerty of B s,a ; see [2, 6.], i.e. n fj a B s,a R n ) fj a x 1,..., x m 1,, x m+1,..., x n ) B sm R) xm L R n 1 ) 3.58) m=1 where x = x 1,..., x m 1, x m+1,..., x n ) and s m = s a m. By 3.47) and 3.48) fj a x 1,..., x m 1,, x m+1,..., x n ) B s m R) xm = = f m 2 j+1)a m x m 2 a m k m )[ c f k1,...,k n ) ] B sm R) 3.59) xm k Z n 1 k m = where k = k 1,..., k m 1, k m+1,..., k n ) and Let d km = that l Z n lm=km f = f 1 2 j+1)a1 x 1 2 a1 k 1 ) f m 1 2 j+1)am 1 x m 1 2 am 1 k m 1 ) f m+1 2 j+1)am+1 x m+1 2 am+1 k m+1 ) f n 2 j+1)an x 1 2 an k n ). 3.60) c l ) 1/ and without restriction of generality we may assume that d km > 0 we have fj a x 1,..., x m 1,, x m+1,..., x n ) B s m R) xm = = f m 2 j+1)a m x m 2 a m c k1,...,kn) k m )[ d km d km k Z n 1 k m= f ] B s m x R). 3.61) xm 17

20 Let c k = c k 1,...,kn) d k m and by 3.61) we get that fj a x 1,..., x m 1,, x m+1,..., x n ) B s m R) xm = = d km f m 2 j+1)a m x m 2 a m k m )[ k m= = [ c k f ] k Z n 1 k m = k Z n 1 c k f ] B sm d km f m 2 j+1)a m x m 2 a m k m ) B s m R) xm R). 3.62) xm By 3.62) fj a x 1,..., x m 1,, x m+1,..., x n ) B s m R) xm L R n 1 ) = x = [ c k f ] d km f m 2 j+1)am x m 2 am k m ) B sm R) L R n 1 ) xm k Z n 1 k m = x = d km f m 2 j+1)a m x m 2 a m k m ) B s m R) c k f L R n 1 ). 3.63) xm x k Z n 1 k m= Note that c k f L R n 1 ) k Z n 1 x = ) 1/ c k 2 j+1)am n)/ k Z n 1 f 1 f m 1 f m+1 f n L R n 1 ) x, 3.64) recall a a m 1 + a m a n ) = a m n. Now we use 3.57) for the saces B s m R) and by 3.63), 3.64) fj a x 1,..., x m 1,, x m+1,..., x n ) B s m R) xm L R n 1 ) On the other hand, c 2 ja ms m 1 ) k m = x d km ) 1/ f m B sm xm ) 1/ 2 j am n) c k f 1 f m 1 f m+1 f n L R n 1 ) x. 3.65) k Z n 1 and k m = d km ) 1/ = = k m = l Z n lm=km c l ) 1/ ) 1/ c l 3.66) l Z n ) 1/ ) c k c 1/ = k1,...,kn) d k Z n 1 k Z n 1 k m 1 ) 1/ = c k1,...,k d n ) km k Z } n 1 {{} d k m ) 18

21 Such that by 3.65), 3.66) and 3.67) and s m a m = s, we conclude fj a x 1,..., x m 1,, x m+1,..., x n ) B sm R) xm L R n 1 ) x ) 1/ c 2 js n ) c k f m B s m R) xm f 1 f m 1 f m+1 f n L R n 1 ) x k Z n ) 1/ f1 c 2 js n ) c k f n B sm R) xm L R n 1 ). x k Z n 3.68) By 3.68) and the Fubini roerty 3.58) we obtain the right-hand side of inequality of 3.49) ) 1/ n fj a B s,a c2 js n ) c k k Z n m=1 c 2 js n ) k Z n c k f 1 f n B s m We finally can comlete the roof of Theorem 2.4. P r o o f of the left-hand side of the estimate 2.8) in Theorem 2.4). xm L R n 1 ) x ) 1/ f B s,a. 3.69) We closely follow the argument in [21, 4.4] for the isotroic case. Let J N and c > 0 be suitably chosen numbers such that there are lattice oints with γ j,l = 2 j J)a m with m Z n, l = 1,..., M j where M j 2 jd 3.70) distγ j,l, Γ) c2 j and disjoint anisotroic balls B a γ j,l, c2 j+1) ). 3.71) With k as in 3.1) we ut for j N 0, M j fj a x) = c jl 2 js n ) k2 ja x γ j,l )), c jl C, x R n. 3.72) l=1 Then we obtain by the Theorem 3.6 and f a j B s,a f a j L Γ) = Mj 2 js n ) l=1 fj a x) µdx) Γ M j 2 js n ) c jl l=1 ) 1 M 2 js n ) c jl j = ) 1/ M j 2 js n ) c jl c2 js n ) 2 j d l=1 M j Γ l=1 c jl ) 1 ) 1/ k 2 ja x γ j,l ))µdx) Γ B a γ j,l,c2 j ) ) 1/ k 2 ja x γ j,l ))µdx) 3.73) ) 1/ c jl 3.74) l=1 using our assumtion 1.11) in the last estimate. Hence f a j L Γ) c2 js n ) 2 jd if f a j B s,a ) 19

22 Now let T be an arbitrary linear oerator, T : B s,a L Γ) with rank T M j ) Then we can find a function fj a 3.74) and 3.75), according to 3.72) with norm 1 in Bs,a and T fj a { } tr Γ T = su tr Γ T )f L Γ) : f B s,a 1 tr Γ T )f a j L Γ) = f a j L Γ) As this is true for all T according to 3.76), we obtain For k N there are numbers j k N such that inserted in 3.75) we obtain = 0. Consequently, by c2 js n ) j d. 3.77) a Mj tr Γ ) = inf{ tr Γ T : rank T M j 1 } c2 js n ) j d. 3.78) 2 j kd k with j k1 j k2 j kn a k tr Γ ) c2 j ks n ) k 1 c k 1 d n s) 1, 3.79) i.e. the left-hand side of the estimate 2.8). Acknowledgement I wish to exress my deeest areciation to Professor Hans Triebel and PD Dr. Dorothee D. Haroske for many insiring discussion. References [1] O.V. Besov, V.P. Il in, and S.M. Nikol skiĭ. Integral reresentations of functions, and embedding theorems. Nauka, Moscow, [2] S. Dachkovski. Anisotroic function saces and related semi-linear hyoellitic equations. Math. Nachr., ):40 61, [3] P. Dintelmann. On Fourier multiliers between anisotroic weighted function saces. PhD thesis, TH Darmstadt, German. [4] D.E. Edmunds and H. Triebel. Function Saces, Entroy Numbers, Differential Oerators. Cambridge Univ. Press, Cambridge, [5] D.E. Edmunds and W.D. Evans. Sectral theory and differential oerators. Oxford Univ. Press, Oxford, [6] K.J. Falconer. The geometry of fractal sets. Cambridge University Press, Cambridge, [7] E.W. Farkas. Anisotroic function saces, fractals, and sectra of some ellitic and semi-ellitic oerators PhD thesis, Friedrich-Schiller-Universität Jena, Germany, [8] W. Farkas. Atomic and subatomic decomositions in anisotroic function saces. Math. Nachr., 209:83 113,

23 [9] D.D. Haroske and E. Tamàsi. Wavelet frames in anisotroic Besov saces. Georgian Math. J., to aear. [10] J. Johnsen. Pointwise multilication of Besov and Triebel-Lizorkin saces. Math. Nachr., 175:85 133, [11] S.M. Nikol skiĭ. Aroximation of functions of several variables and embedding theorems. Nauka, Moscow, [12] Th. Runst and W. Sickel. Sobolev saces of fractional order, Nemytskij oerators, and nonlinear artial differential equations. De Gruyter, Berlin, [13] H.-J. Schmeißer and H. Triebel. Toics in Fourier Analysis and Function Saces. Wiley, Chichester, [14] E.M. Stein and S. Wainger. Problems in harmonic analysis related to curvature. Bull. Amer. Math. Soc., 846): , [15] B. Stöckert and H. Triebel. Decomosition methods for function saces of B s,q tye and F s,q tye. Math. Nachr., 89: , [16] H. Triebel. Fourier analysis and function saces. Teubner, Leizig, [17] H. Triebel. Theory of Function Saces. Birkhäuser, Basel, [18] H. Triebel. Theory of Function Saces II. Birkhäuser, Basel, [19] H. Triebel. Fractals and Sectra. Birkhäuser, Basel, [20] H. Triebel. The structure of functions. Birkhäuser, Basel, [21] H. Triebel. Aroximation numbers in function saces and the distribution of eigenvalues of some fractal ellitic oerators. Journal of Aroximation Theory, )1-27. [22] M. Yamazaki. A quasihomogeneous version of aradifferential oerators. I. Boundedness on saces of Besov tye. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 331): , Erika Tamási Mathematical Institute Friedrich-Schiller-University Jena D Jena Germany tamasi@minet.uni-jena.de 21