Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 4, 875–893

The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in R^{2}

W. Farkas

Abstract. The obtaining of sharp estimates for the asymptotic behaviour of the eigenvalues of the (semi-elliptic) operator acting in the anisotropic Sobolev space

W̊_{2}^{(1,2)}(Ω) = \{u ∈ W_{2}^{(1,2)}(Ω) : u|∂Ω = \frac{∂u}{∂x_{2}} | ∂Ω = 0\}

generated by the quadratic form ∫_{Ω} f(γ) \overline{g(γ)} dμ(γ) is investigated. Here μ is an appropriate self-affine fractal measure on the unit disc Ω ⊂ ℝ^{2}. Keywords: Regular anisotropic fractals, anisotropic function spaces, semi-elliptic differential

operators

AMS subject classification: Primary 35 P 15, secondary 46 E 35, 28 A 90

1. Introduction

Fractal geometry is a very attractive and quickly developing field of modern mathematics. Thank to the work done by A. Jonsson and H. Wallin [18], K. Naimark and M. Solomyak [21, 22], H. Triebel [32], D. E. Edmunds and H. Triebel [9] it turns out that various aspects of this theory are closely connected to Fourier analysis, to the modern theory of function spaces and to spectral theory of partial differential equations. This paper has to be understood as a contribution in this direction. The purpose of our work is to highlight the connection between the asymptotic behaviour of the eigenvalues of some differential operators related to some fractals and the properties of the fractals.

Spectral properties and especially the distribution of the positive eigenvalues of fractal differential operators of the form

U = (−Δ)^{−1} ∘ tr^{Γ}  (1.1)

W. Farkas: Univ. Bundeswehr München, Inst. Theor. Inf. & Math., Werner-Heisenberg-Weg 39, D-85577 Neubiberg. – This work was supported by the DFG Graduiertenkolleg “Analytical and Stochastical Structures and Systems” at the University of Jena and partly by the DFG project Ja522/7-1. It is a pleasure to give our warm thanks to Prof. H. Triebel for stimulating conversations.

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

were studied in [9, 21, 22, 32]. Here (−Δ)^{−1} is the inverse of the Dirichlet Laplacian in the bounded C^{∞} domain Ω ⊂ ℝ^{n}, Γ is a compact fractal with

Γ ⊂ Ω,  |Γ| = 0,  dim_{H}Γ < n,  (1.2)

and tr^{Γ}, closely related to the trace operator tr_{Γ}, is given by

(tr^{Γ}f)(φ) = ∫_{Γ}(tr_{Γ}f)(γ) (φ|Γ)(γ) dμ(γ)  (φ ∈ D(Ω))  (1.3)

where μ is the Radon measure underlying the fractal Γ. The study of operators of type (1.1) is motivated in a natural way by the so-called fractal drums. If λ_{k} (k ∈ ℕ) are the positive eigenvalues of (−Δ)^{−1} ∘ tr^{Γ}, ordered by magnitude and counted with respect to their multiplicities, then η_{k} = λ_{k}^{− \frac{1}{2}} are the eigenfrequencies of a vibrating membrane (interpreted as the bounded domain Ω), fixed at its boundary, having the whole mass concentrated on the fractal Γ ⊂ Ω. More details are given in [32: Subsections 26.2 and 30.1 - 30.5], where one can find a detailed discussion on different aspects concerning fractal drums.

Satisfactory results were obtained if Γ is is a self-similar fractal (see [21, 22]) and if Γ is an (isotropic) d-set with n − 2 < dim_{H}Γ < n (see [32: Theorem 30.2]). Recall Γ ⊂ ℝ^{2} is an isotropic d-set if there are a Radon measure μ in ℝ^{2} and two positive numbers c_{1} and c_{2} such that supp μ = Γ and, for all balls B(γ, r) centred at γ ∈ Γ and of radius 0 < r < 1, c_{1} r^{d} ≤ μ(B(γ, r)) ≤ c_{2} r^{d}. Self-similar fractals are typical examples of (isotropic) d-sets. The notion of (isotropic) d-set occurs both in the theory of function spaces and in fractal geometry (see [10, 11, 18, 32]).

For anisotropic fractals in ℝ^{2} with (1.2) as defined in [32: Definition 5.2] one has for the eigenvalues of U only two-side estimates of type

c_{1} k^{−ω_{1}} ≤ λ_{k}(U) ≤ c_{2} k^{−ω_{2}}  (k ∈ ℕ)  (1.4)

for appropriate positive numbers ω_{1} and ω_{2} with ω_{1} ≥ 1 ≥ ω_{2} (see [32: Theorem 30.7]). Improvements of (1.4) were discussed in [14]. However, the exponents in (1.4) are not equal in the general case (see the precise formulation in [32: Theorem 30.7] and cf. [14: Theorem 3.1]). This fact should be not a big surprise since one can imagine that the different nature of the (isotropic) operator −Δ and the (anisotropic) structure of Γ would cause difficulties (see also the discussion in [32: Subsection 4.16]).

Let 0 < d < 2 and let a = (a_{1}, a_{2}) with 0 < a_{2} ≤ a_{1} < ∞, a_{1} + a_{2} = 2. A regular anisotropic d-set with respect to the anisotropy a is, roughly speaking, a compact set Γ ⊂ ℝ^{2} which can be covered for any j ∈ ℕ_{0} with N_{j} ∼ 2^{jd} disjoint rectangles having sides parallel to the axes and with side lengths r_{1}^{j,l} and r_{2}^{j,l} (l = 1, ..., N_{j}) satisfying r_{i}^{j,l} ∼ 2^{−ja_{i}} for i = 1, 2 (the equivalence constants being independent of j, l and i; cf. [32: Subsection 4.18] and see the precise definition in Section 2). Those fractals are special cases of the anisotropic fractals introduced in [32: Definition 5.2].

Considering Γ a regular anisotropic d-set with respect to the anisotropy a = (a_{1}, a_{2}), our intention is here to replace the Dirichlet Laplacian −Δ in (1.1) with a model semielliptic differential operator A whose “anisotropic structure” coincides with that of the fractal Γ. We will investigate spectral properties of the new operator A^{−1} ∘ tr^{Γ} acting in an appropriate anisotropic Sobolev space W̊_{2}^{(s_{1},s_{2})}(Ω) being interested mainly in the possibility of obtaining estimates of the type λ_{k}(A^{−1} ∘ tr^{Γ}) ∼ k^{−ω} (k ∈ ℕ) (equivalence constants independent of k ∈ ℕ) for its eigenvalues, where ω is an appropriate positive number.

Let Ω = {x = (x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} + x_{2}^{2} < 1} be the unit disc in the plane and let ∂Ω be its boundary. In order to extend the theory of regular elliptic operators to more general classes of operators, H. Triebel considered in [28] the semi-elliptic differential operator

(A_{r}u)(x) = −\frac{∂^{2}u(x)}{∂x_{1}^{2}} + \frac{∂^{4}u(x)}{∂x_{2}^{4}} + \frac{r}{η^{2}(x_{1})}u(x)  (x = (x_{1}, x_{2}) ∈ Ω)  (1.5)

where r ∈ ℝ and η is a C^{∞}-function on the interval [−1, 1] with η(t) > 0 if |t| < 1 and

lim_{t↑1} \frac{η(t)}{1 − t} = lim_{t↓−1} \frac{η(t)}{1 + t} = 1,

and a corresponding boundary value problem A_{r}u(x) = f(x) if x ∈ Ω, u(y) = g_{1}(y) (y ∈ ∂Ω) and \frac{∂u}{∂x_{2}} (y) = g_{2}(y) (y ∈ ∂Ω) where f, g_{1} and g_{2} are given functions. In [28] the author obtained a-priori estimates for the operator A_{r} in the framework of the anisotropic Besov spaces B_{pp}^{(s,2s)}(Ω, x^{0}, x^{1}) and the anisotropic Sobolev spaces W_{p}^{(s,2s)}(Ω, x^{0}, x^{1}) where 1 < p < ∞. These two scales of spaces (our notation here is slightly different from the original one in [28] where they were denoted Ḃ_{pp}^{(s,2s)}(Ω) and Ẇ_{p}^{(s,2s)}(Ω)) were considered in [29] taking into account the singular points x^{0} = (−1, 0) and x^{1} = (1, 0). They are close to the anisotropic spaces on Ω but their elements show (roughly speaking) a peculiar behaviour near the singular points x^{0} and x^{1}. A description of this theory may be found also in [24: Section 4.8].

Spectral properties of A_{r} acting as an unbounded operator defined in L_{p}(Ω) (1 < p < ∞) having domain of definition D(A_{r}) = {u ∈ W_{p}^{(2,4)}(Ω, x^{0}, x^{1}) : u|∂Ω = \frac{∂u}{∂x_{2}} | ∂Ω = 0\} (see the proof of Theorem 4.2 below for the precise definition of the

space W_{p}^{(2,4)}(Ω, x^{0}, x^{1}) if p = 2) were discussed by V. Shevchik in [25]. In particular, he proved that the asymptotic behaviour of the eigenvalues of A_{r} is ”intermediate” between that one of the homogeneous boundary value problem for the Laplace operator on a domain (λ_{k}(Δ) ∼ k) and the same problem for the biharmonic operator (λ_{k}(Δ^{2}) ∼ k^{2}), namely λ_{k}(A_{r}) ∼ k^{\frac{4}{3}} .

Considering Γ ⊂ Ω a regular anisotropic d-set with respect to the anisotropy a = (\frac{4}{3}, \frac{2}{3}) and tr^{Γ} the trace operator in the interpretation (1.3), the main objective of this work is to show that for sufficiently large r the operator A_{r}^{−1} ∘ tr^{Γ} is compact, non-negative, self-adjoint in the anisotropic Sobolev space

W̊_{2}^{(1,2)}(Ω) = \{u ∈ W_{2}^{(1,2)}(Ω) : u|∂Ω = \frac{∂u}{∂x_{2}} |∂Ω = 0\}

and that it is generated by the quadratic form ∫_{Ω} f(γ) \overline{g(γ)} dμ(γ) in W̊_{2}^{(1,2)}(Ω). Moreover, we prove that there exist constants c > 0 and C > 0 such that its positive eigenvalues λ_{k}(A_{r}^{−1} ∘ tr^{Γ}), repeated according to multiplicity and ordered by their magnitude, can be estimated by

c k^{− \frac{1}{d} (d+ \frac{2}{3} )} ≤ λ_{k}(A_{r}^{−1} ∘ tr^{Γ}) ≤ C k^{− \frac{1}{d} (d+ \frac{2}{3} )}  (k ∈ ℕ).  (1.6)

The precise formulation of (1.6) is contained in Theorem 4.2.

To prove the second estimate in (1.6) we use mainly some mapping properties of the operator A_{r} which were proved by H. Triebel in [28], some recent results from [2, 3] of O. V. Besov concerning interpolation of anisotropic function spaces and the inequality of B. Carl between the eigenvalues and the entropy numbers of a compact operator. The first estimate in (1.6) is proved using the atomic decomposition theorem in anisotropic function spaces and some Hilbert space techniques, in particular the theorem which states that for a compact, non-negative and self-adjoint operator acting in a Hilbert space its eigenvalues coincide with its approximation numbers.

Briefly about the organizing of the manuscript. Regular anisotropic d-sets are presented in Section 2 whereas the L_{p}-spaces on such fractals are presented in Section 3. The main result is formulated in Section 4 and it is proved in Section 5. Our aim was a self-contained work so that we included all necessary basic ingredients (entropy numbers, approximation numbers, atomic decompositions in anisotropic function spaces). The notation is standard. For a normed or quasi-normed space X we denote by ∥x|X∥ the norm of the vector x. The embedding of the quasi-normed space X into the quasinormed space Y is denoted X ↪ Y.

2. Regular anisotropic fractals in ℝ^{2}

Let Λ = {Λ_{1}, ..., Λ_{N}} be a family of contractions of ℝ^{2}. It is well known (see [17] and cf. [10, 11]) that there exists a unique (non-empty) compact set Γ in ℝ^{2} invariant with respect to Λ, i.e. such that Γ = Λ_{1}(Γ) ∪ ... ∪ Λ_{n}(Γ). Γ is called the fractal associated with the system Λ.

Let us assume that the contractions of the system Λ are affinities, i.e. each Λ_{i} has the form Λ_{i}(x) = C_{i}x + h_{i} where C_{i} ∈ GL(2, ℝ) and h_{i} ∈ ℝ^{2}. Let Γ be the fractal associated with Λ. The real number d = d_{Λ}(Γ) uniquely defined by

∑_{i=1}^{N} |det C_{i}|^{\frac{d}{2}} = 1  (2.1)

is called the affine dimension of Γ (see [32: Definition 4.12]). We want to remark that in the case of diagonal affinities the number d_{Λ}(Γ) was already considered by B. Mandelbrot in [19] under the name gap dimension.

A particular case of affinities Λ_{i} is of interest. Let n_{1} ≥ 2 and n_{2} ≥ 2 be integers and let F be the subset of ℝ^{2} given by

F = \{(\frac{t_{1}}{n_{1}} , \frac{t_{2}}{n_{2}} ) : t_{1}, t_{2} integers, 0 ≤ t_{1} < n_{1} and 0 ≤ t_{2} < n_{2}\}.

Suppose that every element Λ_{i} of the system Λ is of the form Λ_{i}(x) = Cx + h_{i} where the translation vectors h_{i} are in F and

C = ( \begin{matrix}\frac{1}{n_{1}} & 0 \\ 0 & \frac{1}{n_{2}} \end{matrix} ) .

Thus each Λ_{i} maps the unit square Q = [0, 1] × [0, 1] onto a rectangle contained in Q. The fractal associated with Λ is a generalisation of the Cantor set in the plane and is called general Sierpinski carpet in [20]. The Hausdorff dimension for this type of fractals was computed in [20] (see also [11: Example 9.11]). The definition of a a regular anisotropic fractal in [32: Subsection 4.18] is slightly more general. We present it below. Assume that every Λ_{i} is of the form Λ_{i}(x) = C_{i}x + h_{i} where

C_{i} = ( \begin{matrix}\frac{±1}{n_{1}} & 0 \\ 0 & \frac{±1}{n_{2}} \end{matrix} )  (2.2)

(the signs depend on i and indicate a possible reflection) and the vectors h_{i} still have as components integer multiples of \frac{1}{n_{1}} and \frac{1}{n_{2}} and are chosen such that Λ_{i}(Q) is still contained in Q and Λ_{i}(Q) is disjoint from Λ_{j}(Q) if i ≠ j. The fractal generated by the affinities Λ_{i} specified above is called a regular anisotropic fractal. In [32] the matrices C_{i} are written in the form

C_{i} = ( \begin{matrix}±2^{−κa_{1}} & 0 \\ 0 & ±2^{−κa_{2}} \end{matrix} )  (2.3)

with

a_{1} = \frac{2 log n_{1}}{log(n_{1}n_{2})},  a_{2} = \frac{2 log n_{2}}{log(n_{1}n_{2})},  κ = \frac{1}{2} log(n_{1}n_{2}).  (2.4)

In particular, a = (a_{1}, a_{2}) is a so-called two-dimensional anisotropy, that is 0 < a_{2}, a_{1} < ∞ and a_{1} + a_{2} = 2 (cf. the next section).

If j ∈ ℕ_{0} and N_{j} ∈ ℕ_{0}, we deal with sets of open rectangles {R_{jl} : l = 1, ..., N_{j}} in ℝ^{2} having sides parallel to the axes; the side length of the rectangle R_{jl} with respect to the x_{i}-axis is denoted by r_{i}^{j,l} where i = 1, 2. We will always assume that the side lengths of the rectangles R_{jl} are ordered in the same way, for example r_{1}^{j,l} ≤ r_{2}^{j,l} for any j ∈ ℕ_{0} and any l = 1, ..., N_{j}.

Theorem 2.1 [13: Theorem 3.1]. Let Γ be the fractal generated by the system Λ of affinities specified by (2.2) − (2.4), having affine dimension d = d_{Λ}(Γ) according to (2.1). There exist numbers c_{1}, c_{2} > 0 such that for any j ∈ ℕ_{0}: there exists a natural number N_{j} with

c_{1} 2^{jd} ≤ N_{j} ≤ c_{2} 2^{jd} if j ∈ ℕ, N_{0} = 1,

and there exists a finite sequence of open rectangles {R_{jl} : l = 1, ..., N_{j}} having sides parallel to the axes, R_{01} = Q̊, satisfying the following conditions:

(i) There exists a constant 0 < c_{0} ≤ 1 such that for all j ∈ ℕ_{0} and all l = 1, ..., N_{j}

(c_{0}2^{−j})^{a_{i}} ≤ r_{i}^{j,l} ≤ 2^{−ja_{i}} (i = 1, 2).

(ii) If l ≠ l′, then R_{jl} ∩ R_{jl′} = ∅.

(iii) For any rectangle R_{j+1,k} there is a rectangle R_{jl}, l = l(k), such that R_{j+1,k} ⊂ R_{jl}.

(iv) For any j ∈ ℕ_{0} and any l = 1, ..., N_{j}, (vol R_{jl})^{\frac{d}{2}} = ∑_{R_{j+1,k}⊂R_{jl}}(vol R_{j+1,k})^{\frac{d}{2}} .

(v) Γ = ⋂_{j=0}^{∞} ⋃_{l=1}^{N_{j}} \overline{R_{jl}}.

Taking into account Theorem 2.1 and the terminology from [32: Subsection 4.18] the fractal Γ is called a regular anisotropic d-set with respect to the anisotropy a = (a_{1}, a_{2}) (see [13]).

As remarked in [32: Subsections 4.14 and 4.22], if the anisotropy a = (a_{1}, a_{2}) is non-trivial, that means a = (a_{1}, a_{2}) ≠ (1, 1), then the number d from (2.1) and the Hausdorff dimension dim_{H}Γ of the fractal Γ, are completely unrelated. Furthermore, it can happen that the regular anisotropic d-set Γ is also an isotropic d′-set with d ≠ d′ (see [32: Subsection 4.21]).

For a detailed discussion concerning dimension problems for self-affine fractals the reader is referred to the recent work [1].

Theorem 2.2 [32: Theorems 5.5 and 4.15]. Let Γ be a regular anisotropic d-set with respect to the anisotropy a = (a_{1}, a_{2}) and let {R_{jl} : j ∈ ℕ_{0}, l = 1, ..., N_{j}} be the rectangles from Theorem 2.1. Then there exists a Radon measure μ in ℝ^{2} uniquely determined with supp μ = Γ and

μ(Γ ∩ R_{jl}) = (vol R_{jl})^{\frac{d}{2}}  for all j ∈ ℕ_{0} and l = 1, ..., N_{j}.  (2.5)

As it was remarked by M. Bricchi (cf. [5]), following the proofs in [32: Theorems 5.5 and 4.15] and [17] it is not difficult to see that the measure μ from Theorem 2.2 coincides with the unique Borel regular (outer) measure μ constructed in [17], with supp μ = Γ and of total mass 1 which is invariant with respect to Λ, that is

μ(E) = \frac{1}{N} ∑_{i=1}^{N} μ(Λ_{i}^{−1}(E)) for all E ⊂ ℝ^{2}.

We know that if Γ is an isotropic d-set with underlying measure μ and if 0 < κ < 1, then

μ(B(γ, κr) ∩ Γ) ∼ μ(B(γ, r) ∩ Γ) ∼ r^{d}  (2.6)

where the equivalence constants depend on κ but not on γ ∈ Γ and 0 < r ≤ 1. For a regular anisotropic d-set Γ we have (2.5) but no counterpart of (2.6). At least a weak version of (2.6) will be needed. If 0 < κ < 1, then κR_{jl} denotes the rectangle concentric with R_{jl} and with side lengths respectively κr_{1}^{j,l} and κr_{2}^{j,l}.

Definition 2.3. A regular anisotropic d-set with respect to the anisotropy a = (a_{1}, a_{2}) equipped with measure μ according to Theorem 2.2 is called proper if there exist two numbers 0 < κ < 1 and 0 < c ≤ 1 such that

μ(Γ ∩ κR_{jl}) ≥ c (vol R_{jl})^{\frac{d}{2}}  (j ∈ ℕ_{0}, l = 1, ..., N_{j})  (2.7)

where {R_{jl} : j ∈ ℕ_{0} and l = 1, ..., N_{j}} are the rectangles from Theorem 2.1.

This is a slightly different version of [32: Definition 5.11]. Since Γ is generated by linear contractions, following the lines of the proof of [32: Proposition 5.13] it turns out that if Γ ∩ Q̊ ≠ ∅, then Γ is proper. Hence condition (2.7) is rather natural.

3. L_{p}-spaces on regular anisotropic fractals

3.1 Some preliminaries on anisotropic function spaces. If 1 < p < ∞ and (s_{1}, s_{2}) is a pair of natural numbers, then the classical anisotropic Sobolev space W_{p}^{(s_{1},s_{2})}(ℝ^{2}) is the collection of all tempered distributions f ∈ S′(ℝ^{2}) such that

∥f | W_{p}^{(s_{1},s_{2})}(ℝ^{2})∥ = ∥f | L_{p}(ℝ^{2})∥ + ∥\frac{∂^{s_{1}}f}{∂x_{1}^{s_{1}}} | L_{p}(ℝ^{2})∥ + ∥\frac{∂^{s_{2}}f}{∂x_{2}^{s_{2}}} | L_{p}(ℝ^{2})∥

is finite. In contrast to the usual (isotropic) Sobolev space (s_{1} = s_{2}) the smoothness properties of an element from W_{p}^{(s_{1},s_{2})}(ℝ^{2}) depend on the chosen direction in ℝ^{2}. The number s defined by

\frac{1}{s} = \frac{1}{2} (\frac{1}{s_{1}} + \frac{1}{s_{2}} )  (3.1)

is usually called the ”mean smoothness” and a = (a_{1}, a_{2}), where a_{1} = \frac{s}{s_{1}} and a_{2} = \frac{s}{s_{2}} , characterises the anisotropy. Sometimes we will use the notation W_{p}^{s,a}(ℝ^{2}) for these spaces.

We recall now the definition of anisotropic Besov spaces on ℝ^{2}. First we fix some notation. A pair a = (a_{1}, a_{2}) with 0 < a_{1}, a_{2} < ∞ and a_{1} + a_{2} = 2 is called an anisotropy. If a = (1, 1), then we speak about the ”isotropic case”. The action of t ∈ [0, ∞) on x = (x_{1}, x_{2}) ∈ ℝ^{2} is defined by the formula

t^{a}x = (t^{a_{1}}x_{1}, t^{a_{2}}x_{2}).

For t > 0 and s ∈ ℝ let t^{sa}x = (t^{s})^{a}x. In particular, t^{−a}x = (t^{−1})^{a}x and 2^{−ja}x = (2^{−j})^{a}x.

For x = (x_{1}, x_{2}) ∈ ℝ^{2} and x ≠ 0 let |x|_{a} be the unique positive number t such that

\frac{x_{1}^{2}}{t^{2a_{1}}} + \frac{x_{2}^{2}}{t^{2a_{2}}} = 1

and let |0|_{a} = 0. By M. Yamazaki [34: Theorem 1.4/3,8], | · |_{a} is an anisotropic distance function in C^{∞}(ℝ^{2}\{0}). Remark that in the isotropic case |x|_{a} is the Euclidean distance of x to the origin.

Let φ_{0} a C^{∞} function on ℝ^{2}, φ_{0}(x) = 1 if |x|_{a} ≤ 1, supp φ_{0} ⊂ {x ∈ ℝ^{2} : |x|_{a} ≤ 2} and φ_{j}(x) = φ_{0}(2^{−ja}x) − φ_{0}(2^{(−j+1)a}x) if j ∈ ℕ. Then ∑_{j=0}^{∞} φ_{j}(x) = 1 if x ∈ ℝ^{2} and (φ_{j})_{j∈ℕ_{0}} is a smooth anisotropic dyadic resolution of unity (cf. [24: Section 4.2]).

Let 0 < p ≤ ∞, 0 < q ≤ ∞ and s ∈ ℝ. The anisotropic Besov space B_{pq}^{s,a}(ℝ^{2}) consists of all tempered distributions f ∈ S′(ℝ^{2}) for which the quasi-norm

∥f | B_{pq}^{s,a}(ℝ^{2})∥ = (∑_{j=0}^{∞} 2^{jsq}∥(φ_{j}f̂)^{∨} | L_{p}(ℝ^{2})∥^{q})^{\frac{1}{q}}

(with usual modification if q = ∞) is finite. Here f̂and g^{∨} denote the Fourier transform of f respectively the inverse Fourier transform of g. These are quasi-Banach spaces (Banach spaces if p ≥ 1 and q ≥ 1) which are independent of the choice of (φ_{j})_{j∈ℕ_{0}}. The space H_{2}^{s,a}(ℝ^{2}) = B_{22}^{s,a}(ℝ^{2}) is the anisotropic fractional Sobolev space. In particular, if s_{1} = \frac{s}{a_{1}} and s_{2} = \frac{s}{a_{2}} are natural numbers, then B_{22}^{s,a}(ℝ^{2}) = W_{2}^{s,a}(ℝ^{2}).

Directly from the definition we have

B_{pq_{0}}^{s+ε,a}(ℝ^{2}) ↪ B_{pq_{1}}^{s,a}(ℝ^{2})  if ε > 0  (3.2)

for any s ∈ ℝ, any 0 < p ≤ ∞, and any 0 < q_{0}, q_{1} ≤ ∞ (see [27: Proposition 2.3.2/2]). Anisotropic function spaces of F_{pq}^{s,a}(ℝ^{2}) type, 0 < p < ∞, are defined changing the roles of the spaces L_{p}(ℝ^{2}) and l_{q} in the above definition but we do not stress this point here. The above spaces are denoted B_{pq}^{s}(ℝ^{2}) in the isotropic case. A systematic treatment of the theory of isotropic B_{pq}^{s}(ℝ^{2}) and F_{pq}^{s}(ℝ^{2}) spaces may be found in the books of H. Triebel [27, 30].

Anisotropic function spaces have been studied in great detail by S. M. Nikol’skij (see [23]) and by O. V. Besov, V. P. Il’in and S. M. Nikol’skij (see [4]). For a list of contributions to the theory of anisotropic function spaces we refer to [12].

If s ∈ ℝ, 1 < p < ∞, 1 ≤ q ≤ ∞, and Ω ⊂ ℝ^{2} is a domain, then B_{pq}^{s,a}(Ω) is the restriction of B_{pq}^{s,a}(ℝ^{2}) to Ω normed by

∥f | B_{pq}^{s,a}(Ω)∥ = inf ∥g | B_{pq}^{s,a}(ℝ^{2})∥  (3.3)

where the infimum is taken over all g ∈ B_{pq}^{s,a}(ℝ^{2}) with g|Ω = f|Ω (in the sense of distributions on Ω). Of course, this definition works for all (bounded or unbounded) domains Ω but we will be interested later only on the unit disc. Let us recall that

W_{2}^{s,a}(Ω) = B_{22}^{s,a}(Ω)  if s > 0.  (3.4)

For s ∈ ℝ, 1 < p < ∞, and 1 ≤ q ≤ ∞ we will use the (non-standard) notation

B̊_{pq}^{s,a}(Ω) = \{f ∈ B_{pq}^{s,a}(Ω) : f | ∂Ω = \frac{∂f}{∂x_{2}} | ∂Ω = 0\} .  (3.5)

3.2 Embeddings of L_{p}(Γ) in anisotropic Besov spaces. Let Γ be a regular anisotropic d-set with respect to the given anisotropy a = (a_{1}, a_{2}). The L_{p}-spaces on Γ (0 < p ≤ ∞) are introduced in the usual way with respect to the underlying Radon measure μ on Γ according to Theorem 2.2. If 1 ≤ p ≤ ∞ any f_{Γ} ∈ L_{p}(Γ) will be interpreted as a tempered distribution f ∈ S′(ℝ^{2}) given by

f(φ) = ∫_{Γ} f_{Γ}(γ) (φ|Γ)(γ) dμ(γ)  (φ ∈ S(ℝ^{2})),  (3.6)

where φ|Γ is the restriction of φ to Γ.

Theorem 3.1. Let 0 < d < 2 and let Γ be a regular anisotropic d-set in ℝ^{2} with respect to the anisotropy a = (a_{1}, a_{2}). If 1 < p ≤ ∞ and \frac{1}{p} + \frac{1}{p′} = 1, then (in the sense of (3.6))

L_{p}(Γ) = \{f ∈ B_{p∞}^{− \frac{2−d}{p′} ,a}(ℝ^{2}) : f(φ) = 0 if φ ∈ S(ℝ^{2}) and φ|Γ = 0\} .  (3.7)

The above theorem was proved in [13: Theorem 3.6] and it is the anisotropic counterpart of [33: Theorem 2] (see also [32: Theorem 18.2]).

3.3 Traces of anisotropic Besov spaces on fractals. Assume that Γ is a regular anisotropic d-set in ℝ^{2} with respect to the anisotropy a = (a_{1}, a_{2}). If φ ∈ S(ℝ^{2}), then tr_{Γ}φ = φ|Γ makes sense pointwise. If 0 < p, q < ∞ and s ∈ ℝ, then the embedding tr_{Γ}B_{pq}^{s,a}(ℝ^{2}) ↪ L_{p}(Γ) must be understood as follows: there exists a positive number c > 0 such that, for any φ ∈ S(ℝ^{2}), ∥tr_{Γ}φ | L_{p}(Γ)∥ ≤ c ∥φ | B_{pq}^{s,a}(ℝ^{2})∥. Since S(ℝ^{2}) is dense in B_{pq}^{s,a}(ℝ^{2}) for 0 < p, q < ∞ this inequality can be extended by completion to any f ∈ B_{pq}^{s,a}(ℝ^{2}) and the resulting function is denoted tr_{Γ}f. In addition, the equality tr_{Γ}B_{pq}^{s,a}(ℝ^{2}) = L_{p}(Γ) means that any f_{Γ} ∈ L_{p}(Γ) is the trace of a suitable g ∈ B_{pq}^{s,a}(ℝ^{2}) on Γ and ∥f_{Γ} | L_{p}(Γ)∥ ∼ inf{∥g | B_{pq}^{s,a}(ℝ^{2})∥ : tr_{Γ}g = f_{Γ}}.

Theorem 3.2. Let 0 < d < 2 and let Γ be a regular anisotropic d-set in ℝ^{2} with respect to the anisotropy a = (a_{1}, a_{2}). If \frac{d}{2} < p < ∞ and 0 < q ≤ min(1, p), then

tr_{Γ}B_{pq}^{\frac{2−d}{p} ,a}(ℝ^{2}) = L_{p}(Γ).  (3.8)

The above theorem was proved in [13: Theorem 3.7] and it is the anisotropic counterpart of [33: Theorem 3] (see also [32: Theorem 18.6]).

4. The main result

Let Ω = {x = (x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} + x_{2}^{2} < 1} be the unit disc in the plane and let ∂Ω be its boundary. In what follows we will consider Γ ⊂ Ω a regular anisotropic d-set with respect to the anisotropy a = (\frac{4}{3}, \frac{2}{3}). One can take, for example, n_{1} = 16 and n_{2} = 4 in (2.4) and (2.2). In the sequel we shall not distinguish between f_{Γ} as an element of some L_{p}(Γ) and as the distribution belonging to some B_{p∞}^{−s,a}(Ω) according to (3.7).

To avoid any misunderstanding we emphasise that the trace operator has two different meanings which we distinguish by tr_{Γ} and tr^{Γ} if extra clarity is desirable. If 1 < p < ∞, then by (3.8)

tr_{Γ} : B_{p1}^{\frac{2−d}{p} ,a}(Ω) → L_{p}(Γ),  (4.1)

and if one applies in addition (3.7)

tr^{Γ} : B_{p1}^{\frac{2−d}{p} ,a}(Ω) → B_{p∞}^{− \frac{2−d}{p′} ,a}(Ω).  (4.2)

The latter can be rephrased asking for an optimal extension of tr^{Γ} considered as a mapping from D(Ω) into D′(Ω) given by (1.3) (cf. [32: Subsection 28.1]).

We consider the anisotropic Sobolev space

W_{2}^{(1,2)}(Ω) = \{u ∈ S′(Ω) : ∥u | L_{2}(Ω)∥ + ∥\frac{∂u}{∂x_{1}} | L_{2}(Ω)∥ + ∥\frac{∂^{2}u}{∂x_{2}^{2}} | L_{2}(Ω)∥ < ∞\} .

Clearly (see (3.1)), the mean smoothness is s = \frac{4}{3} and a = (\frac{4}{3}, \frac{2}{3}) characterises the anisotropy. Using the notation from the previous section we have

W_{2}^{\frac{4}{3} ,a}(Ω) = W_{2}^{(1,2)}(Ω).  (4.3)

Remark 4.1. Let

W̊_{2}^{(1,2)}(Ω) = \{u ∈ W_{2}^{(1,2)}(Ω) : u | ∂Ω = \frac{∂u}{∂x_{2}} | ∂Ω = 0\}

and let A_{r} be the operator defined in (1.5). We know from [28: Proposition 1] that there exists an r_{0} ∈ ℝ such that for any r > r_{0} the operator Ar^{\frac{1}{2}} maps W̊_{2}^{(1,2)}(Ω) isomorphically onto L_{2}(Ω); consequently, we may fix the norm on W̊_{2}^{(1,2)}(Ω) by

∥u | W̊_{2}^{(1,2)}(Ω)∥ = ∥Ar^{\frac{1}{2}} u | L_{2}(Ω)∥

and a corresponding scalar product.

We are able now to present our main result.

Theorem 4.2. Let 0 < d < 2 and let Ω be the unit disc in the plane. Let Γ ⊂ Ω be a regular anisotropic d-set with respect to the anisotropy a = (\frac{4}{3}, \frac{2}{3}). Let tr^{Γ} be the trace operator in the interpretation (4.2) and (1.3) whereas tr_{Γ} stands for the trace operator according to (4.1). Let A_{r} be the operator from (1.5).

(i) There exists r_{0} ∈ ℝ such that for any r > r_{0} the operator

T = A_{r}^{−1} ∘ tr^{Γ}  (4.4)

is compact, non-negative, self-adjoint in W̊_{2}^{(1,2)}(Ω) and has null space

N(T) = {f ∈ W̊_{2}^{(1,2)}(Ω) : tr_{Γ}f = 0}.  (4.5)

Furthermore, T is generated by the quadratic form in W̊_{2}^{(1,2)}

∫_{Γ} f(γ) \overline{g(γ)} dμ(γ) = (T f, g)_{W_{2}^{(1,2)}(Ω)}  where f, g ∈ W̊_{2}^{(1,2)}(Ω)  (4.6)

and μ is the Radon measure according to Theorem 2.2.

(ii) Let r > r_{0}. There exists a constant C > 0 such that positive eigenvalues λ_{k}(T) of T, repeated according to the multiplicity and ordered by their magnitude, can be estimated by

λ_{k}(T) ≤ C k^{− \frac{1}{d} (d+ \frac{2}{3} )} (k ∈ ℕ). (4.7)

If, in addition, Γ is proper according to Definition 2.3, then there exists a constant c > 0 such that

c k^{− \frac{1}{d} (d+ \frac{2}{3} )} ≤ λ_{k}(T) (k ∈ ℕ).

The detailed proof is given in the next section.

Remark 4.3. The additional assumption on Γ to be proper in the estimate (4.8) excludes by the considerations after Definition 2.3 only pathological cases where the whole fractal retreats in the boundary of the starting square.

5. Proof of Theorem 4.2

Proof of Theorem 4.2/(i). Let a = (\frac{4}{3}, \frac{2}{3}). Using the elementary embedding

W_{2}^{(1,2)}(Ω) = W_{2}^{\frac{4}{3} ,a}(Ω) ↪ B_{22}^{\frac{2−d}{2} ,a}(Ω) (see (3.2)) and applying (3.8) there exists a constant c > 0 such that

∥tr_{Γ}f | L_{2}(Γ)∥ ≤ c ∥f | W̊_{2}^{(1,2)}(Ω)∥  for any f ∈ W̊_{2}^{(1,2)}(Ω).  (5.1)

Defining

q(f, g) = ∫_{Γ} f(γ) \overline{g(γ)} dμ(γ) for any f, g ∈ W̊_{2}^{(1,2)}(Ω),

it is clear that q(·, ·) is a non-negative quadratic form in W̊_{2}^{(1,2)}(Ω). By [31: page 91] there exists a non-negative and self-adjoint operator T uniquely determined such that

q(f, g) = (T f, g)_{W_{2}^{(1,2)}(Ω)} for any f, g ∈ W̊_{2}^{(1,2)}(Ω).

Furthermore,

∥tr_{Γ}f | L_{2}(Γ)∥ = ∥\sqrt{T} f | W̊_{2}^{(1,2)}(Ω)∥  (5.2)

where \sqrt{T} = T^{\frac{1}{2}} and this proves (4.5).

So it remains to prove that the above operator is the same as in (4.4). Let f ∈ W̊_{2}^{(1,2)}(Ω) and φ ∈ D(Ω). Then

∫_{Γ} f(γ) \overline{φ(γ)} dμ(γ) = (T f, φ)_{W_{2}^{(1,2)}(Ω)} = (Ar^{\frac{1}{2}} T f, Ar^{\frac{1}{2}} φ)_{L_{2}(Ω)} = (A_{r}T f, φ)_{L_{2}(Ω)} (5.3)

the second equality in (5.3) being justified by the fact that for r > r_{0} we fixed the norm in W̊_{2}^{(1,2)}(Ω) by ∥Ar^{\frac{1}{2}} (·) | L_{2}(Ω)∥ and a corresponding scalar product (see Remark 4.1). Considered as a dual pairing in (D(Ω), D′(Ω)) we obtain A_{r}T f = tr^{Γ}f and (4.6) follows by the same arguments as in [32: Theorem 27.15/Step 1]. This completes the first part of the proof of Theorem 4.2.

5.2 Proof of the estimate (4.7). Step 1: Basic facts on entropy numbers and their relation to the eigenvalues. Let B_{1} and B_{2} two quasi-Banach spaces. The family of all linear bounded operators U : B_{1} → B_{2} is denoted by L(B_{1}, B_{2}) or L(B_{1}) if B_{1} = B_{2}. We will assume that the reader is familiar with the definition of the entropy numbers e_{k}(U) (k ∈ ℕ) of a compact map U ∈ L(B_{1}, B_{2}). This definition is given, for example, in [8: Subsection 1.3.1] where one can find also comments and historical references. We only want to mention here that if B_{1}, B_{2}, B_{3} are quasi-Banach spaces, if X ∈ L(B_{1}, B_{2}) and R ∈ L(B_{2}, B_{3}), then for all k, l ∈ ℕ,

e_{k+l−1}(R ∘ X) ≤ e_{k}(R) e_{l}(X)  (5.4)

(see [8: Lemma 1.3.1/1]).

Let B be a complex quasi-Banach space and U ∈ L(B) a compact map. We know from [8: Theorem 1.2] that the spectrum of U, apart from the point 0, consists solely of eigenvalues of finite algebraic multiplicity: let {λ_{k}(U) : k ∈ ℕ} be the sequence of all non-zero eigenvalues of U, repeated according to algebraic multiplicity and ordered so that

|λ_{1}(U)| ≥ |λ_{2}(U)| ≥ ... ≥ 0.  (5.5)

If the operator U has only m (< ∞) distinct eigenvalues and M is the sum of their algebraic multiplicities, we put λ_{k}(U) = 0 for k > M. Due to B. Carl (see [6]) we know the following connection between the eigenvalues of the operator U and its entropy numbers: Let U and {λ_{k}(U) : k ∈ ℕ} as above. Then

|λ_{k}(U)| ≤ \sqrt{2} e_{k}(U).  (5.6)

Step 2: Entropy numbers for traces on regular anisotropic d-sets.

Theorem 5.1. Let 0 < d < 2 and let Γ be a regular anisotropic d-set in ℝ^{2} with respect to the anisotropy a = (a_{1}, a_{2}). Let 0 < p_{1} ≤ ∞, 0 < p_{2} ≤ ∞, 0 < q ≤ ∞ and s ∈ ℝ such that

δ_{+} = s − d (\frac{1}{p_{1}} − \frac{1}{p_{2}} )_{+} > 0.  (5.7)

Then the trace operator

tr_{Γ} : B_{p_{1}q}^{s+ \frac{2−d}{p_{1}} ,a}(ℝ^{2}) → L_{p_{2}}(Γ)  (5.8)

is compact and there exists a constant C > 0 such that, for all k ∈ ℕ,

e_{k}(tr_{Γ} : B_{p_{1}q}^{s+ \frac{2−d}{p_{1}} ,a}(ℝ^{2}) → L_{p_{2}}(Γ)) ≤ C k^{− \frac{s}{d}} .

The above theorem was proved in [13: Theorem 6.1] and it is the anisotropic counterpart of [32: Theorem 20.6] (cf. also [32: Theorem 22.2]). If, in addition, Γ is proper according to Definition 2.3, then there exists a constant c > 0 such that, for all k ∈ ℕ,

c k^{− \frac{s}{d}} ≤ e_{k}(tr_{Γ} : B_{p_{1}q}^{s+ \frac{2−d}{p_{1}} ,a}(ℝ^{2}) → L_{p_{2}}(Γ)).

But the last estimate will not be used later. We want to remark also that assumption (5.7) is crucial for the compactness of the operator in (5.8) and that this assumption cannot be weakened by δ_{+} = 0 (see [32: Subsection 20.7]).

Step 3: Let a = (\frac{4}{3}, \frac{2}{3}) and let, according to (4.3), W̊_{2}^{\frac{4}{3} ,a}(Ω) = W̊_{2}^{(1,2)}(Ω). Let also r > r_{0} and recall the notation (3.5). We prove now that the operator T = A_{r}^{−1} ∘ tr^{Γ} can be factorised by

T = id_{2} ∘ A_{r}^{−1} ∘ id_{1} ∘ tr_{Γ}

where

tr_{Γ} : W̊_{2}^{\frac{4}{3} ,a}(Ω) → L_{2}(Γ)
id_{1} : L_{2}(Γ) → B_{2∞}^{− \frac{2−d}{2} ,a}(Ω)
A_{r}^{−1} : B_{2∞}^{− \frac{2−d}{2} ,a}(Ω) → B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω)
id_{2} : B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω) → W̊_{2}^{\frac{4}{3} ,a}(Ω).

(5.9)

The boundedness of tr_{Γ} in the first line of (5.9) was discussed in the previous section (see (5.1)). According to (3.3) the embedding id_{1} is (3.7) whereas the embedding id_{2} is a simple consequence of the inequality \frac{8}{3} − \frac{2−d}{2} > \frac{4}{3} and of the elementary embedding (3.2) between anisotropic function spaces.

So it remains only to justify the boundedness of A_{r}^{−1} as indicated in the third line of (5.9). Let us denote x^{0} = (−1, 0) and x^{1} = (1, 0) and let W_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}) = W_{2}^{(2,4)}(Ω, x^{0}, x^{1}) be the collection of all f ∈ W_{2}^{\frac{8}{3} ,a}(Ω) such that

\frac{∂^{m_{1}+m_{2}}f}{∂x_{1}^{m_{1}}∂x_{2}^{m_{2}}} (x^{0}) = \frac{∂^{m_{1}+m_{2}}f}{∂x_{1}^{m_{1}}∂x_{2}^{m_{2}}} (x^{1}) = 0 if 2m_{1} + m_{2} + \frac{3}{2} < 4

(see [28: Formula (17)]). Let also

W̊_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}) = \{f ∈ W_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}) : f | ∂Ω = \frac{∂f}{∂x_{2}} | ∂Ω = 0\} .  (5.10)

Recall that there exists a number r_{0} ∈ ℝ such that for any r > r_{0} the operator

A_{r} maps W̊_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}) isomorphically onto L_{2}(Ω)  (5.11)

(this was proved by H. Triebel in [28: Theorem 4]; see also [25: Theorem 2.1]). Furthermore, by [28: Proposition 2 and Remark 6] the operator

A_{r} maps W̊_{2}^{\frac{4}{3} ,a}(Ω) isomorphically onto B_{22}^{− \frac{4}{3} ,a}(Ω).  (5.12)

Let

θ = \frac{3(2 − d)}{8}.  (5.13)

Then, clearly, we have 0 < θ < 1 and (1 − θ) · 0 + θ · (−\frac{4}{3}) = −\frac{2−d}{2}. Since the unit disc Ω is a domain in ℝ^{2} having the so-called C^{m} flexible (\frac{4}{3}, \frac{2}{3}) horn condition for any m ∈ ℕ_{0}^{2} (see [2: Definition 1.2]), by [3] with the above θ we have the interpolation result

(L_{2}(Ω), B_{22}^{− \frac{4}{3} ,a}(Ω))_{θ,∞} = B_{2∞}^{− \frac{2−d}{2} ,a}(Ω).  (5.14)

From (5.11), (5.12) and (5.14) it follows that for r > r_{0} sufficiently large the operator

A_{r}^{−1} maps B_{2∞}^{− \frac{2−d}{2} ,a}(Ω) bounded into (W̊_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}), W̊_{2}^{\frac{4}{3} ,a}(Ω))_{θ,∞} . (5.15)

Since

\{f ∈ W_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}) + W_{2}^{\frac{4}{3} ,a}(Ω) : f | ∂Ω = \frac{∂f}{∂x_{2}} | ∂Ω = 0\}

is a complemented subspace of W_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}) + W_{2}^{\frac{4}{3} ,a}(Ω) with the same projection operator, we may use [26: Theorem 1.17.1/1] and have

\begin{matrix}(W̊_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}), W̊_{2}^{\frac{4}{3} ,a}(Ω)) & θ,∞ \\ = \{f ∈ (W_{2}^{\frac{8}{3} ,a}(Ω, x^{0}, x^{1}), W_{2}^{\frac{4}{3} ,a}(Ω))_{θ,∞} : f | ∂Ω = \frac{∂f}{∂x_{2}} | ∂Ω = 0\} . \end{matrix}  (5.16)

On the other hand, using (3.4), by [2: Theorem 4.2/a] we have

(W_{2}^{\frac{8}{3} ,a}(Ω), W_{2}^{\frac{4}{3} ,a}(Ω))_{θ,∞} = B_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω)  (5.17)

where θ is again the number defined in (5.13). It follows from (5.15) - (5.17) and elementary properties of real interpolation that

A_{r}^{−1} maps B_{2∞}^{− \frac{2−d}{2} ,a}(Ω) bounded into B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω)

and this completes the proof of the boundedness from the third line of (5.9).

Step 4: Let f ∈ W̊_{2}^{(1,2)}(Ω) be an eigenfunction of T. Then it follows from (5.9) that f belongs also to

B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω)

and so it is an eigenfunction of the operator T restricted to this space. Obviously, the converse is also true. Hence the root systems considered in W̊_{2}^{(1,2)}(Ω) and in

B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω) coincide. Then the eigenvalues of T considered in these spaces also coincide, inclusively their multiplicities. Using the multiplicity property (5.4) for entropy numbers and using (5.9) there exists a constant c > 0 such that, for all k ∈ ℕ,

e_{k}(T : B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω) → B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω)) ≤ c e_{k}(tr_{Γ} : B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω) → L_{2}(Γ)). (5.18)

Inserting in Theorem 5.1 a = (\frac{4}{3}, \frac{2}{3}), p_{1} = p_{2} = 2, q = ∞ and s + \frac{2−d}{2} = \frac{8}{3} − \frac{2−d}{2} we have from (5.18)

e_{k}(T : B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω) → B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω)) ≤ c k^{− \frac{1}{d} (d+ \frac{2}{3} )}.  (5.19)

Estimate (4.7) is now a simple consequence of (5.19) using Carl’s inequality (5.6).

5.3 Proof of estimate (4.8). Step 1: Atomic decompositions in anisotropic Besov spaces. Let a = (a_{1}, a_{2}) be a given two-dimensional anisotropy and let ℤ^{2} be the lattice of all points in ℝ^{2} with integer-valued components. If ν ∈ ℕ_{0} and m = (m_{1}, m_{2}) ∈ ℤ^{2}, we denote Q_{νm}^{a} the rectangle in ℝ^{2} centred at 2^{−νa}m = (2^{−νa_{1}}m_{1}, 2^{−νa_{2}}m_{2}) which has sides parallel to the axes and side lengths respectively 2^{−νa_{1}} and 2^{−νa_{2}}. Remark that Q_{0m}^{a} is a square with side length 1. If Q_{νm}^{a} is such a rectangle in ℝ^{2} and c > 0, then cQ_{νm}^{a} is the rectangle in ℝ^{2} concentric with Q_{νm}^{a} and with side lengths respectively c2^{−νa_{1}} and c2^{−νa_{2}}.

If β = (β_{1}, β_{2}) ∈ ℕ_{0}^{2}, the derivatives D^{β} have the usual meaning, and if x = (x_{1}, x_{2}) ∈ ℝ^{2}, then x^{β} = x_{1}^{β_{1}}x_{2}^{β_{2}}. The scalar product between the anisotropy a = (a_{1}, a_{2}) and β is aβ = a_{1}β_{1} + a_{2}β_{2}.

Definition 5.2. Let s > 0, 1 ≤ p ≤ ∞, K ∈ ℝ and c > 1. A function ρ : ℝ^{2} → ℂ for which there exist all derivatives D^{β}ρ if aβ ≤ K (continuous if K ≤ 0) is called an anisotropic (s, p)_{K}-atom or simply (s, p)-atom, if

supp ρ ⊂ cQ_{νm}^{a}  for some ν ∈ ℕ and m ∈ ℤ^{2}  (5.20)
|D^{β}ρ(x)| ≤ 2^{−ν(s− \frac{2}{p})}2^{νaβ}  if aβ ≤ K.  (5.21)

If conditions (5.20) and (5.21) are satisfied for ν = 0, then ρ : ℝ^{2} → ℂ is called an anisotropic 1_{K}-atom.

If the atom ρ is located at Q_{νm}^{a} (that means supp ρ_{νm}^{a} ⊂ cQ_{νm}^{a} with ν ∈ ℕ_{0}, m ∈ ℤ^{2} and c > 1), then we will write it ρ_{νm}^{a}.

We give some technical explanations. The value of the number c > 1 in (5.20) is unimportant. It simply makes clear that at the level ν some controlled overlapping of the supports of ρ_{νm}^{a} must be allowed. If K ≤ 0, then (5.21) is |ρ(x)| ≤ 2^{−ν(s− \frac{2}{p})}. The reason for the normalising factor in (5.21) is that there exists a constant c > 0 such that for all these atoms we have ∥ρ | B_{pq}^{(s,a)}(ℝ^{2})∥ ≤ c. Hence, as in the isotropic case, atoms are normalised building blocks. This construction generalizes isotropic atoms as they are in the works of M. Frazier and B. Jawerth (see [15, 16]).

If 0 < p, q ≤ ∞, then b_{pq} is the collection of all sequences λ = {λ_{νm} ∈ ℂ : ν ∈ ℕ_{0} and m ∈ ℤ^{2}} such that

∥λ | b_{pq}∥ = (∑_{ν=0}^{∞} ( ∑_{m∈ℤ^{2}} |λ_{νm}|^{p})^{\frac{q}{p}} )^{\frac{1}{q}}

(with the usual modification if p = ∞ and/or q = ∞) is finite. Clearly b_{pq} is a quasi-Banach space.

Theorem 5.3. Let a = (a_{1}, a_{2}) a given anisotropy with a_{2} ≤ a_{1}. Let s > 0, 1 ≤ p ≤ ∞, 0 < q ≤ ∞, and let K ∈ ℝ such that K ≥ a_{1} + s. Then g ∈ S′(ℝ^{2}) belongs to B_{pq}^{s,a}(ℝ^{2}) if and only if it can be represented as

g = ∑_{ν=0}^{∞} ∑_{m∈ℤ^{2}} λ_{νm}ρ_{νm}^{a}  (convergence being in S′(ℝ^{2}))  (5.22)

where ρ_{νm}^{a} are anisotropic 1_{K}-atoms (ν = 0) or anisotropic (s, p)_{K}-atoms (ν ∈ ℕ) and λ ∈ b_{pq} where λ = {λ_{νm} : ν ∈ ℕ_{0} and m ∈ ℤ^{2}}. Furthermore, inf ∥λ | b_{pq}∥ where the infimum is taken over all admissible representations (5.22) is an equivalent quasi-norm in B_{pq}^{s,a}(ℝ^{2}).

This is a weak version of [12: Theorem 3.3] which will be needed below. We will refer to the above theorem as to the atomic decomposition theorem in anisotropic function spaces since it generalises to the anisotropic case the well known results of M. Frazier and B. Jawerth from [15, 16].

Step 2: Approximation numbers and eigenvalues of operators acting in Hilbert spaces. We recall the definition of the approximation numbers. Let B_{1} and B_{2} be two quasi-Banach spaces and let U ∈ L(B_{1}, B_{2}). Then given any k ∈ ℕ, the k-th approximation number α_{k}(U) of U is defined by

α_{k}(U) = inf {∥U − L∥ : L ∈ L(B_{1}, B_{2}) and rank L < k}

where rank L is the dimension of the range of L. Usually the approximation numbers are denoted a_{k}(U). The above notation is used only to avoid any possible confusion between these numbers and the anisotropy a = (a_{1}, a_{2}). We do not want to discuss here properties of approximation numbers, this is done in [8: Lemma 1.3.1/2 and Remark 1.3.1/6] and [7: Section II.2.3]. We only want to mention that approximation numbers have important connections with eigenvalues, the picture being clearest in a Hilbert space setting.

Theorem 5.4. Let H be a Hilbert space and let U ∈ L(H) be a compact, nonnegative and self-adjoint operator. Then the approximation numbers α_{k}(U) of U coincide with its eigenvalues (ordered as in (5.5)).

A proof can be found in [7: Section II.5.10] (see also [8: page 21]).

Step 3: If one applies Theorem 5.4, then estimate (4.8) is a simple consequence of the next proposition.

Proposition 5.5. Let Ω and d as above, let Γ be proper and let T be the compact, non-negative, self-adjoint operator in W̊_{2}^{(1,2)}(Ω) defined in (4.4). There exists a constant c > 0 such that the approximation numbers α_{k}(\sqrt{T}) of \sqrt{T} = T^{\frac{1}{2}} can be estimated by

α_{k}(\sqrt{T}) ≥ c k^{− \frac{1}{d} ( \frac{d}{2} + \frac{1}{3} )}  (k ∈ ℕ).  (5.23)

Step 4: Proof of Proposition 5.5. Let x^{j,l} be the centre of the rectangle R_{jl} of side lengths r_{1}^{j,l} and r_{2}^{j,l}, and let N_{j} ∼ 2^{jd} having the same meaning as in Theorem 2.1. Since Γ ⊂ Ω, it is clear that there exists a j_{0} ∈ ℕ_{0} such that for any j ≥ j_{0} the rectangles R_{jl} (l = 1, ..., N_{j}) are contained in Ω. Let φ a non-negative C^{∞} function on ℝ^{2} with support in {x ∈ ℝ^{2} : |x_{1}| < 1 and |x_{2}| < 1}. We may assume |φ(x)| ≥ δ > 0 if |x_{1}| ≤ κ and |x_{2}| ≤ κ where 0 < κ < 1 is the number from Definition 2.3 of a proper set. If

φ_{jl}(x) = φ (\frac{2(x_{1} − x_{1}^{j,l})}{r_{1}^{j,l}}, \frac{2(x_{2} − x_{2}^{j,l})}{r_{2}^{j,l}}) ,

then supp φ_{jl} ⊂ R_{jl}. Furthermore, there exist two constants c_{1}, c_{2} > 0 such that

c_{1} 2^{−j \frac{d}{2}} (∑_{l=1}^{N_{j}} |c_{jl}|^{2})^{\frac{1}{2}} ≤ ∥∑_{l=1}^{N_{j}} c_{jl} φ_{jl} | L_{2}(Γ)∥ ≤ c_{2} 2^{−j \frac{d}{2}} (∑_{l=1}^{N_{j}} |c_{jl}|^{2})^{\frac{1}{2}}  (5.24)

for any complex numbers c_{jl} and for any j ≥ j_{0} and l = 1, ..., N_{j}. Indeed, using (2.7) we have

∥∑_{l=1}^{N_{j}} c_{jl} φ_{jl} | L_{2}(Γ)∥^{2} = ∑_{l=1}^{N_{j}} ∫_{Γ∩R_{jl}} |c_{jl}|^{2}|φ_{jl}(γ)|^{2}dμ(γ)

≥ ∑_{l=1}^{N_{j}} |c_{jl}|^{2}δ^{2}μ(Γ ∩ κR_{jl})

≥ c 2^{−jd} ∑_{l=1}^{N_{j}} |c_{jl}|^{2}

and this is the first inequality in (5.24). The second part of (5.24) is a simple consequence of (2.5).

Remark now that if j ≥ j_{0} is fixed and l ∈ \{1, ..., N_{j}\}, then 2^{−j \frac{1}{3}} φ_{jl} is an anisotropic (\frac{4}{3}, 2) atom in W̊_{2}^{(1,2)}(Ω). Hence, using (4.3) and the atomic decomposition theorem in anisotropic Besov spaces with s = \frac{4}{3} and p = q = 2 we have

∥g_{j} | W̊_{2}^{(1,2)}(Ω)∥ ≤ c 2^{j \frac{1}{3}} (∑_{l=1}^{N_{j}} |c_{jl}|^{2})^{\frac{1}{2}}  (5.25)

for any function g_{j} of the type

g_{j} = ∑_{l=1}^{N_{j}} c_{jl} φ_{jl} = 2^{j \frac{1}{3}} ∑_{l=1}^{N_{j}} c_{jl}(2^{−j \frac{1}{3}} φ_{jl}),  (5.26)

the constant c > 0 in (5.25) being independent of j, l and of the complex numbers c_{jl}. By (5.2), (5.24) and (5.25) we find a constant c_{0} such that

∥\sqrt{T} g_{j} | W̊_{2}^{(1,2)}(Ω)∥ ∼ 2^{−j \frac{d}{2}} (∑_{l=1}^{N_{j}} |c_{jl}|^{2})^{\frac{1}{2}} ≥ c_{0} 2^{−j( \frac{d}{2} + \frac{1}{3} )}∥g_{j} | W̊_{2}^{(1,2)}(Ω)∥. (5.27)

There exists an operator L = L(N_{j}) in W̊_{2}^{(1,2)}(Ω) with rank L < N_{j} such that

α_{N_{j}}(\sqrt{T}) ≥ ∥\sqrt{T} − L∥ − \frac{1}{2}c_{0} 2^{−j( \frac{d}{2} + \frac{1}{3} )}

where c_{0} is the number from the last line in (5.27). We may assume that the dimension of the span of the admitted functions g_{j} in (5.26) is larger than N_{j} ∼ 2^{jd}. Then we find a function g_{j} of type (5.26) in W̊_{2}^{(1,2)}(Ω) such that ∥g_{j} | W̊_{2}^{(1,2)}(Ω)∥ = 1 and Lg_{j} = 0. It follows

α_{N_{j}}(\sqrt{T}) ≥ ∥\sqrt{T} g_{j} − Lg_{j} | W̊_{2}^{(1,2)}(Ω)∥ − \frac{1}{2}c_{0} 2^{−j( \frac{d}{2} + \frac{1}{3} )} ≥ c_{0} 2^{−j( \frac{d}{2} + \frac{1}{3} )}  (5.28)

where c_{0} > 0 is independent of j. Using elementary properties of approximation numbers it is easy to see that (5.28) implies (5.23) and this completes the proof of the theorem

A final remark. Probably the most important result stated in Theorem 4.2 is estimate (4.7). We want to point out that in the proof of (4.7) the key role is played by the mapping property in the third line of (5.9), more precisely by the fact that for sufficiently large r the operator A_{r}^{−1} maps the anisotropic Besov space B_{2∞}^{− \frac{2−d}{2} ,a}(Ω) bounded into

B̊_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω) = \{f ∈ B_{2∞}^{\frac{8}{3} − \frac{2−d}{2} ,a}(Ω) : f | ∂Ω = \frac{∂f}{∂x_{2}} | ∂Ω = 0\} .

This was obtained as a consequence of the results from [28] using some interpolation results from [2, 3].

We think this remark paves the way to further investigations. One can try to replace the model operator A_{r} by a more general (semi-elliptic) operator, the space W̊_{2}^{(1,2)}(Ω) by another anisotropic function space but the considered anisotropic fractal should be chosen having the same anisotropy as the differential operator. Some steps in this direction were done in [13] were the surrounding domain of the fractal was the whole ℝ^{2}.

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Received 29.01.1999