Riesz Potentials and Liouville Operators on Fractals

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 potential theory, Fourier analysis and stochastic processes 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.

3 Traces and Besov spaces on fractals

Traces of euclidean Besov spaces to fractal d-sets Γ have first been introduced by Johnson, Wallin [6] and studied in subsequent papers. We use here the modified approach of Triebel [14], which extends the notion of euclidean tracing in the theory of function spaces to the fractal case. (For 0 < s < d ≤ 1 both versions agree.) The main ideas are as follows: (We need only the case p = 2 from [14].) The trace operator trΓ for Schwartz functions ϕ is defined by the restriction to Γ. As a first basic result the norm estimate

n−d 2 n ||ϕ| L2(µ)|| ≤ const ||ϕ| B2,1 (R )|| is derived by means of atomic decompositions. Then trΓ is continuously extended n−d 2 n to an operator from B2,1 (R ) into L2(µ). In a second step the image is shown to be the whole L2(µ), i.e.,

n−d 2 n trΓ B2,1 (R ) = L2(µ) .

Finally, fractal Besov spaces are introduced by

n−d s s+ 2 n B2,q(Γ) := trΓ B2,q (R ) for s > 0 and 0 < q ≤ ∞. They become quasi-Banach spaces (Banach spaces if q ≥ 1) by n−d s s+ 2 n ||f| B2,q(Γ)|| := inf ||g| B2,q (R )|| n−d s+ s n where the inﬁmum is taken over all g ∈ B2,q (R ) with trΓg = f. Similarly,

s s+ n−d n H (Γ) := trΓ H 2 (R )

s+ n−d n s and the Hilbert space structure of H 2 (R ) generates that of H (Γ). More- over, s+ n−d n n s+ n−d n o s H 2 (R ) = g ∈ H 2 (R ): trΓg = 0 ⊕ H (Γ)

5 (cf. [14], 25.1). For our purposes we slightly modify the construction. In Hσ(Rn) an equivalent σ Hilbert space structure is given when using a modified Riesz-Bessel kernel GR σ instead of the Bessel kernel GB. It is equivalent to the Bessel kernel at infinity and for the points of Γ it is defined by the Riesz kernel, í.e.,

σ σ GR(z) := G (z) if |z| < R , where R > 2 diam(Γ) (cf. [17]). σ R σ σ σ −1 We write IRf(x) := GR(x − y)f(y)dy , DR := (IR) , and σ n ||f| H ( )|| , hf, gi σ n for the equivalent norm and scalar product in R R HR(R ) Hσ(Rn). Furthermore, in the space Hs(Γ) introduced above we use the equivalent norm n−d s s+ 2 n ||f| HR(Γ)|| := inf ||g| HR (R )|| s+ n−d n where the inﬁmum is taken over all g ∈ H 2 (R ) with trΓg = f. The corresponding scalar product is determined by

s 2 s 2 4hf, gi s := ||f + g| H (Γ)|| − ||f − g| H (Γ)|| HR(Γ) R R s 2 s 2 + i||f + ig| HR(Γ)|| − i||f − ig| HR(Γ)|| .

s+ n−d n s H 2 (R ) and H (Γ) provided with these structures are denoted by n−d s+ 2 n s s HR (R ) and HR(Γ), resp. (Below we will show that the norm in HR(Γ) does not depend on the radius R.) The analogue of the above Hilbert space decomposition reads:

n−d n n−d o s+ 2 n s+ 2 n s (3.1) HR (R ) = g ∈ HR (R ): trΓg = 0 ⊕ HR(Γ) .

p s s/2 The operator Iµ plays a basic role for the Hilbert space HR (Γ).

p s s/2 3.1 Theorem. Iµ is an isometry from L2(µ) onto H (Γ), i.e., q q s s h Iµf, Iµgi s/2 = hf, giL2(µ) . HR (Γ) s Proof. In the proof of Theorem 3.3 in [17] it is shown that Iµ maps L2(µ) into s/2 HR (Γ), s hf, giL2(µ) = hIµf, gi s/2 HR (Γ) s/2 s for f, g ∈ HR (Γ), and Iµ is a compact self-adjoint operator on the Hilbert space s/2 HR (Γ). Moreover, by the above equality it is positive. Therefore we obtain for s/2 any f, g ∈ HR (Γ) q q s s hf, giL2(µ) = h Iµ f, Iµ gi s/2 . HR (Γ)

6 It remains to extend this isometry to all f, g ∈ L2(µ) and to show that q s s/2 Iµ(L2(µ)) = HR (Γ) . s The operator Iµ is also compact, selfadjoint and positive in the Hilbert space L2(µ) (see [17]). From the general theory of such operators it follows that there s exist complete orthogonal sequences of eigenvectors of the operator Iµ in L2(µ) s/2 as well as in HR (Γ). By construction, these systems may be chosen the same. Let now f be an arbitrary function from L2(µ). Then f may be approximated in L2(µ) by linear combinations fn of the eigenvectors e1, e2,... . The above equality implies q q s s hei, eiiL2(µ) = h Iµ ei, Iµ eii s/2 HR (Γ) for any eigenvector ei. Hence, q q s s s/2 ||fn − fm| L2(µ)|| = || Iµfn − Iµfm| HR (Γ)|| .

p s fn converges to f in L2(µ) as n → ∞. Therefore Iµ fn tends to some ϕ in s/2 HR (Γ). On the other hand,

s/2 (3.2) ||h| L2(µ)|| ≤ const||h| HR (Γ)||

s/2 p s for any h ∈ HR (Γ) (see [14], (25.9)). Thus, Iµ fn converges to ϕ in L2(µ), p s p s too. Since Iµ is bounded in L2(µ) we get ϕ = Iµ f. From this we infer q s s/2 ||f| L2(µ)|| = || Iµf| HR (Γ)|| for any f ∈ L2(µ). The equality for the scalar products is a consequence. s/2 Finally, any ϕ from the Hilbert space HR (Γ) is the limit as n → ∞ of

n n ! q X s X −1 ϕiei = Iµ ϕiλi ei i=1 i=1

p s where ϕi := hϕ, eii s/2 and Iµei = λiei. From the above isometry property we HR (Γ) n X −1 p s obtain that ϕiλi ei converges in L2(µ) to some function f. Hence, Iµ f = i=1 ϕ. This yields q s s/2 Iµ(L2(µ)) = HR (Γ) .

7 4 Fractal Liouville operators and quadratic forms associated with Riesz potentials

In analogy to the case euclidean Riesz potentials (cf. Section 1) we introduce fractional pseudodiﬀerential operators with respect to the d-measure µ as above by s s −1 Dµ = (Iµ) s for 0 < s < d ≤ n . Dµ is called fractal Liouville operator of order s. The s space L2(µ) of Riesz potentials of order s may be equipped with the scalar product

s s hf, gi s := hD f, D gi . L2(µ) µ µ L2(µ) s Obviously, this generates a Hilbert space structure. In this way L2(µ) may be σ n considered as fractal analogue of the fractional Sobolev space L2 (R) of L2(R )- σ n functions with derivatives D in L2(R ). Recall that the latter is equivalent to Hσ(Rn). In distinction to the euclidean case we have

s/2 s/2 s Dµ ◦ Dµ 6= Dµ . Therefore we additionally introduce the quadratic form Dq q E s s s Eµ(f, g) := Dµf, Dµg L2(µ) in L2(µ). (For general notions used here and in the sequel see Fukushima, Oshima, Takeda [3].)

4.1 Theorem. (i) s Eµ(f, g) = hf, gi s/2 , HR (Γ) s/2 s/2 f, g ∈ H (Γ) . In particular, the scalar product in HR (Γ) does not depend on the choice of the radius R.

s s/2 (ii) Eµ is a closed and regular quadratic form in L2(µ) with domain H (Γ).

s (iii) Eµ is the trace of its euclidean extension:

s s+n−d s + n−d s + n−d ˜ 2 2 ˜ 2 2 n Eµ(f, g) = E (f, g˜) = hD f, D g˜iL2(R )

where the Riesz extension f˜ of f ∈ Hs/2(Γ) is determined by f˜ = s s+n−d s extµ f := I (ϕµ), if f = Iµϕ, and by continuity w.r.t. the seminorm (Es+n−d(·, ·))1/2 .

8 s Proof. (i) is a consequence of the deﬁnition of Eµ and Theorem 3.1. In particular,

s s/2 dom Eµ = H (Γ) which is dense in L2(µ). Closedness of a quadratic form E in L2(µ) means that the domain of E is complete 2 1/2 w.r.t. the E1-norm (E(f, f) + kf| L2(µ)k ) . By (3.2) the latter in our case is equivalent to kf| Hs/2(Γ)k. Furthermore, by construction the restriction C∞(Γ) ∞ n s/2 of C0 (R ) to the fractal d-set Γ is dense in the Hilbert space HR (Γ). It is also dense in the space of continuous functions with the uniform norm. Hence, ∞ s C (Γ) forms a core of the quadratic form Eµ, i.e., the latter is regular. Thus, (ii) is proved. For (iii) we use Theorem 3.2 in [17] which implies

Dq q E s + n−d s + n−d s s 2 2 2 2 n Iµ ϕ, Iµ ψ = hI (ϕµ),I (ψµ)iL2(R ) L2(µ)

˜ s+n−d s+n−d for any ϕ, ψ ∈ L2(µ). Suppose now that f = I (ϕµ), g˜ = I (ψµ), i.e., s s their traces on Γ are f = Iµϕ and g = Iµψ, respectively. Then we infer Dq q E s s s Eµ(f, g) = Dµ f, Dµ g L2(µ) Dq q E s s = Iµ ϕ, Iµ ψ L2(µ) s + n−d s + n−d 2 2 2 2 n = hI (ϕµ),I (ψµ)iL2(R ) s + n−d s + n−d 2 2 ˜ 2 2 n = hD f, D g˜iL2(R ) = Es+n−d(f,˜ g˜) .

˜ s We call f the corresponding Riesz extension of f denoted by extµf. Arbitrary s/2 f ∈ H (Γ) may be approximated in the given norm by fn of the above type. According to (i) and the equality

s s+n−d ˜ ˜ ˜ ˜ Eµ(fn − fm, fn − fm) = E (fn − fm, fn − fm)

s + n−d ˜ ˜ the derivatives D 2 2 fn of the Riesz extensions fn of fn possess a limit h in n ˜ s + n−d L2(R ). We set f := I 2 2 h and obtain s s+n−d ˜ ˜ Eµ(f, f) = E (f, f) , and f˜is the Riesz extension of f. The equality for f, g ∈ Hs/2(Γ) is a consequence.

9 5 Generated semigroups and fractal pseudodiﬀer- ential equations

By the general theory (see [3], Section 1.3) any closed quadratic form E in a real Hilbert space H generates a strongly continuous semigroup (Tt)t≥0 (of contractive symmetric operators) on H. (Recall that this means the following: dom(Tt) = H, kTtk ≤ 1, Tt is symmetric, lim kTtf − fk = 0 for f ∈ H t&0 −1 and lim t (Ttf − f) = Af , where E(f, f) = hf, −AfiH on the domain of the t&0 operator A which is called inﬁnitesimal generator of the semigroup.) In our s case H = L2(µ), E = Eµ with Dq q E s s s Eµ(f, g) = Dµ f, Dµ g L2(µ)

s according to Theorem 4.1. This implies the following for A = −Dµ. (Recall that 0 < s < d ≤ n .)

s s 5.1 Corollary. The operator −Dµ corresponding to the quadratic form Eµ is the inﬁnitesimal generator of a strongly continuous semigroup (Tt)t≥0 on the real Hilbert space L2(µ).

This semigroup (Tt)t≥0 is also denoted by

s −tDµ (e )t≥0 .

As an application we consider the following pseudodiﬀerential equation on the fractal support Γ of µ. For s + n − d = 2 it may be interpreted as the trace on Γ of the euclidean heat equation. Denote R+ := [0, ∞) . 5.2 Theorem. The equation ∂u (t, ·) = −Ds u(t, ·) , ∂t µ t ∈ R+ , with initial condition u(0, ·) = f for some f ∈ L2(µ) has the unique solution s u(t, ·) = e−tDµ f within the class C(R+,L2(µ)) .

∂u 5.3 Remark. The partial derivative ∂t (t, ·) on the left side of the equation is deﬁned as the L2(µ)-limit 1 lim (u(t + s, ·) − u(t, ·)) . s→0 s

10 Proof. of Theorem 5.2. It follows from Corollary 5.1 and the semigroup property s that e−tDµ f is a solution with the above property. Uniqueness in our case also follows from the general theory of strongly continuous semigroups. The following alternative procedure, which is similar as in the euclidean case, may be applied to more general linear pseudodiﬀerential equations than that under consideration. Let e1(x), e2(x),... be a complete orthonormal s system of eigenfunctions of the Riesz potential Iµ in L2(µ) . Then ei is also s an eigenfunction of the inverse operator Dµ with eigenvalue, say, λi . If u is a solution of the above problem then it may be decomposed by

∞ X u(t, x) = ui(t) ei(x) i=1

∂u with continuous ui(t) = hu(t, ·), eiiL2(µ) . Moreover, existence of ∂t (t, ·) implies 0 that of the derivatives ui(t) and ∞ ∂u X (t, ·) = u0 (t) e (x) . ∂t i i i=1

s On the other hand, u(t, ·) belongs to the space of Riesz potentials (Dµu(t, ·) is determined) which yields

∞ ∞ s X s X −Dµu(t, ·) = ui(t)(−Dµei) = ui(t)(−λi) ei . i=1 i=1

Comparing the coeﬃcients of ei in the pseudodiﬀerential equation we obtain

0 ui(t) = −λi ui(t) , t ∈ R+ , and from this −λit ui(t) = ui(0) e . Consequently, ∞ X −λit u(t, x) = ui(0) e ei(x) . i=1

The initial condition leads to ui(0) = hf, eiiL2(µ) . Thus,

∞ X −λit u(t, x) = e hf, eiiL2(µ) ei(x) i=1 which agrees with the spectral representation of the above solution. 5.4 Remark. Similarly as in the last proof instead of fractional heat equations the corresponding fractional wave equations may be solved. This will be shown elsewhere.

11 6 Dirichlet forms

s In Theorem 4.1 we have shown that Eµ is a regular closed quadratic form on the Hilbert space L2(µ) which is again assumed to be real. Such a form is called a regular Dirichlet form if it additionally possesses the Markov property. We will work with the ﬁrst of the following deﬁnitions (cf. [3]):

6.1 Deﬁnition. Let (X, X, m) be a σ-ﬁnite measure space and E be a closed symmetric form on L2(X m) with domain D(E). Then E is said to be Markovian if one of the following equivalent conditions holds:

(i) If f ∈ D(E) and f˜ := (0 ∨ f) ∧ 1 then f˜ ∈ D(E) and E(f,˜ f˜) ≤ E(f, f).

(ii) If f ∈ D(E) and g is a normal contraction of f then g ∈ D(E) and E(g, g) ≤ E(f, f).

Here a function g is called a normal contraction of f if |g(x)| ≤ |f(x)| and |g(x) − g(y)| ≤ |f(x) − f(y)|, x, y ∈ X. Note that the truncation operator

T (f) := (0 ∨ f) ∧ 1 is a normal contraction. Moreover, it is a so called composition operator, i.e., T (f) = c(f) for a continuous function c : R → R (which is given here by c(r) := (0 ∨ r) ∧ 1).

σ n It is well-known that the regular and closed quadratic form E on L2(R ) with domain Hσ/2(Rn) considered above has the Markov property if and only if 0 < σ ≤ 2. This is shown by means of a close interplay with associated Markov semigroups and Fourier analysis (see, e.g. Jacob [5], Examples 4.5.23 and 5.7.28 and [3], Example 1.5.2). Using the method of traces we will derive now the Markov property for the s corresponding forms Eµ if s + n − d ≤ 2. (This further restriction is due to the techniques in the present paper .) We need the following auxiliary result. Let trΓ be the trace operator described in Section 3.)

α n n−d 6.2 Lemma. For any f ∈ H (R ) and any d-set Γ as above with 0 ≤ 2 < α ≤ 1 and the truncation operator T we have

trΓT (f) = T (trΓf) .

Proof. For 0 < α ≤ 1 the space Hα(Rn) has an equivalent norm given by the sum of the L2-norm and a seminorm determined by diﬀerences of the functions. Since the truncation operator is a normal contraction, it follows that it is bounded in Hs(Rn). Every bounded composition operator in Hs(Rn) is continuous. For α = 1 this is proved in [8]. For the case 0 < α < 1 it is derived in [10], Theorem

12 3, p. 377, by interpolation methods for nonlinear operators. (Continuity of the truncation operator can also be shown directly using the representation

1 Z c(x) − c(y) = c0(λx + (1 − λ)y) dλ (x − y)

0 1 Z = 1[0,1](λx + (1 − λ)y) dλ (x − y) 0 for the associated composition function c together with the above equivalent norm in Hα(Rn).) Let κε, ε → 0, be a usual family of smoothing kernels such that f ∗ κε converges α n to f in H (R ). Then by continuity, T (f ∗ κε) tends to T (f) in the same sense. n−d α n 2 n−d H (R ) is embedded into B2,1 for 2 < α. Therefore the deﬁnition of the trace operator trΓ implies the L2(µ)-convergences

lim f ∗ κε|Γ = trΓf ε→0 and lim T (f ∗ κε)|Γ = trΓT (f) ε→0 for any d-measure µ with support Γ. Since T is the truncation operator the ﬁrst convergence yields also

lim T (f ∗ κε)|Γ = T (trΓf) ε→0 in L2(µ). Hence, T (trΓf) = trΓT (f) . From this and Theorem 4.1 we conclude the following.

6.3 Theorem. Let 0 < s < d ≤ n and s + n − d ≤ 2. Then the quadratic form Dq q E s s s Eµ(f, g) = Dµ f, Dµ g L2(µ) s/2 on L2(µ) is a regular Dirichlet form with domain H (Γ).

s Proof. According to Theorem 4.1 it remains to show the Markov property Eµ. For, we will prove that f ∈ Hs/2(Γ) implies

s/2 s s T (f) ∈ H (Γ) and Eµ(T (f),T (f)) ≤ Eµ(f, f) for the truncation operator T . The ﬁrst assertion follows from Lemma 6.2.

13 Let f˜ be the Riesz extension of f as in Theorem 4.1 (iii). Below we will show that

s s+n−d ˜ ˜ (6.1) Eµ(T (f),T (f)) ≤ E (T (f),T (f)) .

s+n−d n s + n−d n Recall that E is a Dirichlet form in L2(R ) with domain H 2 2 (R ), but it s + n−d 2 is deﬁned for all functions g from the spaces of Riesz potentials I 2 2 (L2(R )) . Moreover, by Fourier analytical methods and the Levy-Chintchin representation one obtains for such g and s + n − d < 2

s+n−d s + n−d s + n−d 2 2 2 2 n E (g, g) = hD g, D giL2(R ) (6.2) ZZ (g(x) − g(y))2 = const dx dy |x − y|s+n−d+n

(cf. [3], Example 1.4.1, [5], 5.7.28). Since

(T (g)(x) − T (g)(y))2 ≤ (g(x) − g(y))2

s+n−d s + n−d n this implies the Markov property of E on the larger space I 2 2 (L2((R )). In particular, Es+n−d(T (f˜),T (f˜)) ≤ Es+n−d(f,˜ f˜) . s The right side equals Eµ(f, f) in view of Theorem 4.1 (iii). Hence,

s s Eµ(T (f),T (f)) ≤ Eµ(f, f) .

s + n−d n It remains to prove the inequality (6.1). The trace operator trΓ : H 2 2 (R ) → s/2 s + n−d n H (Γ) may be extended to I 2 2 (L2(R )) as follows: By construction we have s + n−d n for g ∈ H 2 2 (R )

s s n−d s 2 2 + 2 n Eµ(trΓg, trΓg) = ||trΓg| HR (Γ)k ≤ kg| HR (R )k . Taking the limit as R → ∞ on the right-hand side, which agrees with

Z s + n−d Z s n−d 2 2 2 2 + 2 2 s+n−d lim (DR g(x)) dx = (D g(x)) dx = E (g, g) , R→∞ we obtain s s+n−d Eµ(trΓg, trΓg) ≤ E (g, g) .

s + n−d n s + n−d n Since H 2 2 (R ) is dense in the space of Riesz potentials I 2 2 (L2(R )), the continuous extension

s + n−d n s/2 trΓ : I 2 2 (L2(R )) → H (Γ)

14 is determined. Furthermore, as in the proof of Lemma 6.2 we get from (6.2) for any g ∈ s + n−d n I 2 2 (L2(R )) , T (trΓg) = trΓT (g) . This yields s s+n−d Eµ(T (trΓg),T (trΓg)) ≤ E (T (g),T (g)) and, in particular, (6.1). The case s + n − d = 2 may be treated analogously when replacing (6.2) by Z E2(g, g) = |grad g(x)|2 dx .

6.4 Remark.

(i) The above Dirichlet forms generate fractal variants of symmetric stable Levy processes. Other stable-like jump processes on Γ (and in more general metric spaces) are discussed in Kumagai [7] and Stós [13].

dw (ii) In Hu and Lau [4] it is shown that the Green’s kernel of Iµ for the ”walk dimension” dw of certain generalized Sierpinski carpets Γ is comparable to that of the potential operator of the Brownian motion constructed by Barlow and Bass [1].(See Mosco [9] for the corresponding Laplace operator w.r.t. an appropriate inner quasimetric. )

(iii) A survey on [17] and the present topic may be found in [18].

Acknowledgement I would like to thank my colleague H. Triebel for some useful discussions on the topic of Section 3.

References

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[11] Samko, S.G., Kilbas, A.A., and Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach 1993.

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[13] Stós, A., Symmetric α-stable processes on d-sets, Bull. Polish Acad. Sci. Math. 48 (2000), 237-245.

[14] Triebel, H.: Fractals and Spectra, Birkhäuser 1997.

[15] Triebel, H.: The Structure of Functions, Birkhäuser 2001.

[16] Triebel, H., and Yang, D.: Spectral theory of Riesz potentials on quasi-metric spaces, Math. Nachr. 238 (2002), 160-184.

[17] Zähle, M.: Harmonic calculus on fractals - A measure geometric approach II, Preprint.

[18] Zähle, M.: Riesz potentials and Besov spaces on fractals, in: Fractals in Graz 2001 (Eds. P. Grabner and W.Woess), Birkhäuser, Trends in Math. 2002, 271-276.