RIESZ POTENTIALS AND LIOUVILLE OPERATORS ON FRACTALS by M. Zähle Mathematical Institute, University of Jena, D-07740 Jena, Germany E-mail: [email protected] Abstract An analogue to the theory of Riesz potentials and Liouville operators in Rn for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As an example of related pseudodifferential equations the fractional heat-type equation is solved. Mathematics Subject Classification. Primary 28A80, Secondary 47B07, 35P20 Keywords. fractal d-set, Riesz potential, pseudodifferential operator, fractal Besov space, Dirichlet form 0 Introduction In [17] the Riesz potential of order s of a fractal d-measure µ in Rn with compact support Γ is defined as Z s −(d−s) Iµf(x) := const |x − y| f(y)µ(dy) , f ∈ L2(µ) , where 0 < s < d < n . (Examples for the measure µ are the Haus- dorff measures Hd on arbitrary self-similar sets Γ of dimension d.) Such potentials have a long tradition for the case, where µ is replaced by the Lebesgue measure. For more general µ only a view properties are known. Some references may be found in [17]. We also refer to the fundamental paper [2]. In connection with harmonic analysis and stochastic processes on fractals these potentials appear in a new light. The aim of the present paper is to continue this study in order to get a deeper insight into the interplay between geometry of and analysis on fractal sets and the corresponding properties of the embedding euclidean space. By means of euclidean charts of metric spaces as introduced in [16] these relationships might pay a role also for more general metric spaces. Some basic properties as com- s pactness of Iµ are proved in [17] for general metric spaces. The whole spectral properties are extended in [16] to the case of some equivalent quasi-metrics in arbitrary quasi-metric spaces. Here we focus our attention on pseudodifferential operators, quadratic forms and generated semigroups associated with the above Riesz potentials. In Section 1 we summarize some classical euclidean results which are important for our purposes. This concerns euclidean Riesz potentials, Liouville operators, Fourier representation and related function spaces. s Section 2 contains a brief survey on the properties of Iµ proved in [17]. In Section 3 the method of traces of Besov spaces on fractals is described as it has been introduced by H. Triebel. (This is an equivalent approach to the fractal Besov spaces defined by Jonsson and Wallin.) We concentrate on the Hilbert s + n−d n s spaces H 2 2 (R ) and H 2 (Γ) and prove an isometry property of the operator p s Iµ (Theorem 3.1). s The fractal Liouville operators Dµ defined as the inverses of the Riesz potentials are studied in Section 4. They induce quadratic forms q q s s s Eµ(f, g) := h Dµ f, Dµ giL2(µ) s on L2(µ). In Theorem 4.1, the first main result of the paper, we show that Eµ is s a regular closed quadratic form on L2(µ) with domain H 2 (Γ). Furthermore, it is the trace of its euclidean counterpart s+n−d s + n−d s + n−d 2 2 2 2 n E (f, g) = hD f, D giL2(R ) 1 s + n−d s + n−d n for the euclidean Liouville operator D 2 2 with domain H 2 2 (R ) and a certain extension. s In Section 5 the strongly continuous semigroups (Tt)t≥0 generated by Eµ are considered. As an application we construct the unique solution of the fractal pseudodifferential equation ∂u (t, ·) = −Ds u(t, ·) , t ≥ 0 , ∂t µ with initial condition u(0, ·) = f ∈ L2(µ) . Finally we show in Section 6 that for 0 < s < d ≤ n with s + n − d ≤ 2 the s quadratic form Eµ is a regular Dirichlet form. This leads to associated Markov processes on the fractal set Γ . All results in Sections 1–4 are related to both the complex and real valued cases. The corresponding assertions from [17] extend immediately to the complex case. (The same holds true for the marginal dimension d = n, where the corresponding theory is classical.) In Sections 5 and 6 we restrict to real function spaces. 1 Survey on related euclidean results from poten- tial theory, Fourier analysis and stochastic pro- cesses n X ∂2 Let ∆ = be the classical Laplace operator in n. The inverse operator ∂x2 R i=1 i of −∆ is given by the Newtonian potential Z 1 I2f(x) := const f(y)dy |x − y|n−2 n R n for f ∈ L2(R ) if n > 2 . The Riesz potential of order 1 Z 1 I1f(x) := const f(y) dy |x − y|n−1 n R corresponds to (−∆)1/2 . In general, the Riesz potential of order 0 < σ < n Z 1 Iσf(x) := const f(y) dy , |x − y|n−σ n R n f ∈ L2(R ), is the inverse of the pseudodifferential operator of order σ Dσ := (−∆)σ/2 2 which is also called Liouville operator. The constant is equal to n − σ σ c (σ) := Γ (2σπn/2Γ ) n 2 2 and σ −(n−σ) G (z) := cn(σ) |z| is called Riesz kernel of order σ in Rn. Up to certain exceptional orders σ the operators Dσ can be realized by hypersingular integrals (cf. Stein [12] or Samko, Kilbas and Marichev [11]). In the Fourier analytical approach for arbitrary σ ≥ 0, Dσ := F −1(|ξ|σ F ) Iσ := F −1(|ξ|−σ F ) are defined on distributions, where F means the Fourier transform in Rn. Simi- larly, fractional powers of the resolvent (Id − ∆)−1 are introduced: Dσ := F −1((1 + |ξ|2)σ/2F ) with inverse operators Iσ := F −1((1 + |ξ|2)−σ/2F ) i. e., Iσ = (Id − ∆)σ/2 . The latter are representable as Z σ σ I f(x) := GB(x − y) f(y) dy n R σ for the Bessel kernel GB which has a better behavior at infinity than the Riesz kernel Gσ. For σ ≥ 0 one also denotes I−σ := Dσ . Related distribution spaces are σ n σ n H (R ) := I (L2(R )) , σ ∈ R , which may be provided with a Hilbert space structure by R 2 σ σ n hf, giH (R ) := (1 + |ξ| ) F (f)(ξ) F (g)(ξ) dξ n R σ σ n = hD f, D giL2(R ). 3 Up to (semi) norm equivalence the space of Bessel potentials Hσ(Rn) agrees σ n with the Sobolev-Slobodeckij space W2 (R ) (for σ ≥ 0) and with the Besov σ n σ n space B2,2(R ). Moreover, for σ > 0 the space H (R ) consists of those functions n σ n f ∈ L2(R ) which possess fractional derivatives D f (in the first sense) in L2(R ). (General references are, e. g. [12], [15], [11] and the literature cited there.) We also consider the quadratic forms Eσ(f, g) := R |ξ|σ F (f)(ξ) F (g)(ξ) dξ n R σ/2 σ/2 n = hD f, D giL2(R ) n σ/2 n in L2(R ) with domain H (R ), which are closed and regular. Therefore the operator −Dσ related to this form is the infinitesimal generator of a strongly continuous contraction semigroup in Rn. For 0 < σ ≤ 2 this corresponds to the Dirichlet form and the Markov semigroup of a σ-stable symmetric Levy process in Rn (with characteristic function exp(−t|ξ|σ)). (More details on the last notions and results may be found in [3] and [5].) 2 Riesz potentials of fractal d-measures in Rn The Lebesgue measure as reference measure is now replaced by a finite Borel measure µ in Rn with compact support Γ. We suppose that for all x ∈ Γ and 0 < r ≤ 1, µ(B(x, r)) C−1 ≤ ≤ C rd for some constant C > 0, where B(x, r) denotes the closed ball with centre x and radius r. Such measures are called d-measures with scaling exponent d ∈ (0, n]. In [17] the fractal Riesz potentials of order s < d are studied: Z 1 Isf(x) := c (s + n − d) f(y) dµ(y) , µ n |x − y|d−s f ∈ L2(µ). They are shown to be closely related to their euclidean counterpart by s σ Iµf = trΓ ◦ I ◦ (fµ) for σ := s + n − d, where n − d appears as a fractal defect. fµ stands for the distribution given by integrating with respect to fdµ and the trace operator trΓ agrees here with the restriction to Γ. (Its general construction going back to Triebel [14] will be summarized below.) In [17] it is shown (for the case of s general metric spaces) that Iµ is a compact self-adjoint operator in the Hilbert n space L2(µ) and in the case of R it is positive. The method of proving the last fact (Theorem 3.2 in [17]) is based on an isometry property which is essential 4 for the aim of the present paper. Furthermore, the asymptotic behavior of the s n eigenvalues λk of Iµ in R is determined by −s/d λk k , k −→ ∞ , with methods of [14] (Theorem 3.3 in [17]). This corresponds to the Weyl spec- trum for the fractional powers of the Laplace operator in the classical euclidean case. In Triebel, Yang [16] these asymptotics are extended to d-measures in rather general quasi-metric spaces.