Potential spaces and traces of Lévy processes on h-sets

M. Zähle

(University of Jena, Mathematical Institutee-mail: zaehle@math.uni-jena.de)

Keywords: Subordinate Lévy processes, potential spaces, traces on fractals, time change

Abstract

Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brow-nian motion in Rn with generator f(−∆) are investigated. Estimates for the related Riesz-and Bessel-type kernels of order s are derived which include the classical case f(r) = rα/2

with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernsteinfunctions f potential representations of the trace spaces, the trace Dirichlet forms, andthe trace processes on fractal h-sets are derived. Here we suppose the trace condition∫ 1

0r−(n+1)f(r−2)−1h(r) dr <∞ on f and the gauge function h.

0 Introduction

Let h be a continuous non-decreasing positive function on (0, 1] satisfying the doubling conditionh(2r) ≤ consth(r). µ is called an h-measure on Rn if it is a Borel measure with compact supportF such that

c−1h(r) ≤ µ(B(x, r)) ≤ ch(r) , x ∈ F, 0 < r ≤ 1,

for some c > 0. It can be shown that for given F all associated h-measures are equivalent to theHausdorff measure Hh with gauge function h restricted to F . The latter has been introduced in[17] extending Hausdorff’s classical version Hd, where h(r) = rd. Therefore F is called an h-set(or a d-set, respectively). A more detailed investigation of h-sets may be found in [2].

We are interested in traces of certain potential spaces and stochastic processes in Rn on the (frac-tal) set F . Such tracing procedures have been studied in the literature from the analytical as wellas from the probabilistic point of view. Here we continue the papers [23], [24], [9], [14] combiningboth the analytical and probabilistic side.Our starting point is a Lévy process Xf in Rn subordinate to Brownian motion according to aBernstein function f , i.e., the pseudodifferential operator f(−∆) with Laplace operator ∆ is itsinfinitesimal generator. The domain of the corresponding Dirichlet form is a Bessel-type potentialspace in Rn in the sense of [4], [10]. For the special case of symmetric α-stable Lévy processes theseare the classical Bessel potential spaces which agree with the Besov spaces Bα/22,2 (Rn). In [14] wehave interpreted the above Bessel-type potential spaces as Besov spaces of generalized smoothnessand proved some properties well-known for the classical Besov spaces Bsp,q(Rn). In particular, aquarkonial representation in the sense of [21] holds true.Under some trace condition on the Bernstein function f and the gauge function h of the fractalmeasure µ the quarkonial approach was used in order to obtain the trace spaces on F of the aboveEuclidean function spaces. Moreover, under some additional conditions, this is equivalent to an

1

extension of the trace procedure given in [12] for classical Besov spaces. The analytical approachwas applied to determining the domain of the Dirichlet form of the trace of the Lévy process Xf

on F via time change in the sense of [7], [6].The aim of the present paper is to complete some results in [14] and and to work out the corre-sponding potential theory.The related notions and results in Rn are presented in section 1.In 1.1 the subordinate process Xf and some potential properties of independent interest are stud-ied. In particular, we show that the associated potential operator Uf has a Riesz-type kernelKf satisfying Kf (x) ≤ const |x|−n f(|x|−2)−1, x 6= 0 (under an additional growth condition on ffor n ≤ 2). If f satisfies some lower scaling condition then the opposite estimate for a differentconstant holds true. (See Theorem 1.1.2.)In 1.2 similar estimates for Riesz-type potentials of arbitrary order associated with complete Bern-stein functions f are derived.These results are applied in 1.3 in order to estimate the corresponding Bessel-type potentials.Moreover, for tame Bernstein functions an f -Riesz-Bessel kernel equivalent to the f -Bessel kernelis constructed which is more appropriate for our purposes.The relationship between the Dirichlet form of Xf and the Bessel-type potential space Hf,1(Rn)(or its Riesz-Bessel-type variant) in the sense of [10], [11] is presented in 1.3.

In Section 2 we consider the traces on h-sets. Here we introduce the trace condition1∫

0

h(r)f(r−2)r−(n+1)dr <∞ .

In 2.1 the trace Hilbert spaces are determined. In particular, we show that the trace operatortrµ makes sense in the L2(F, µ)-setting. For the case of classical Besov spaces this goes back to[22]. Similar tools may be used in order to prove that the above trace condition is also necessary.Moreover, the corresponding extension operators for both the f-Bessel potential spaces and thef -Riesz-Bessel potential spaces are determined.In 2.2 for tame Bernstein functions f the Riesz-Bessel version of Hf,1(Rn) discussed in 1.2 and1.3 enables us to derive a similar potential representation for the trace space Hf,1(µ) on the h-set F . We show that the associated potential function Ufµ (x) =

∫Kf (x − y)µ(dy) is bounded

and uniformly continuous on the whole Rn. This leads to a pointwise representation of the tracepotential operator Ufµ . The main results are formulated in Theorem 2.2.5. In particular, the scalarproduct in Hf,1(µ) satisfies

〈u, v〉Hf,1(µ) =⟨√

Dfµu,

√Dfµv

⟩L2(F,µ)

=: Efµ (u, v),

for Dfµ := (Ufµ )−1. This is the fractal counterpart to the Euclidean version for the f -Bessel po-

tential space Hf,1(Rn).

Relationships to Besov spaces of generalized smoothness and auxiliary tools for sections 1 and 2are presented in the Appendix. In particular, the compactness of the operators trµ and Ufµ onappropriate function spaces is shown. Furthermore, we prove global versions of Tauberian-typetheorems for the Laplace transform and estimates for the Fourier transform of Riesz-Bessel-typekernels.

In section 3 we infer that Ufµ may be interpreted as the potential operator of the trace of the Lévyprocess Xf on the h-set F : A trace process Xf

µ is obtained from Xf via time change accordingto the positive continuous additive functional with Revuz measure µ (in the sense of [7]). For thespecial case of symmetric α-stable Lévy processes and d-sets F such a time change was considered

2

in [15], [8], [9]. Here we extend the latter approach. Our main result is that the above quadraticform Efµ is the Dirichlet form of the time changed process and therefore Xf

µ has a version Xfµ which

is a Hunt process. By construction, Ufµ is the potential operator of the trace process Xfµ and the

Dirichlet form satisfies the Poincaré inequality. (This also refines the corresponding results fromthe above papers.)Note that in [3] the Beurling-Deny decomposition of the time changed Dirichlet form for arbitrarysymmetric Markov processes X and rather general Revuz measures µ with support F is given interms of the Lévy system and some Feller measures of X. Applied to our situation this leads to arepresentation of the Dirichlet form Efµ by means of those Feller measures of Xf .

1 Lévy processes subordinate to Brownian motion and asso-ciated potential spaces

1.1 Subordination and Riesz-type potentials

More details on the following probabilistic notions and results as well as references to the originalliterature may be found in Bertoin [1] and Jacob [10], [11]. The reader not familiar with stochasticprocesses can switch to formulas (4) and (6) below considered as definitions for the potentialmeasure Vf and the potential kernel Kf associated with the Bernstein function f . The classicalanalytical counterpart concerning Riesz and Bessel potentials may be found in many textbookson potential analysis.Let (Xf (t))t≥0 be a Lévy process subordinate to Brownian motion (B(t))t≥0 in Rn. This meansthat the characteristic function is given by

E exp(i〈Xf (t), ξ〉) = exp(−tψ(ξ)) (1)

with ψ(ξ) := f(|ξ|2) for a Bernstein function f (f ∈ C∞(0,∞), f ≥ 0, (−1)nf (n) ≤ 0, n ∈ N).Throughout the paper we exclude the trivial case f ≡ 0. In the analytical context the infinitesimalgenerator of the Markov semigroup (T ft )t≥0 of Xf is given by f(−4) (for the Laplace operator4) interpreted as a pseudodifferential operator with symbol ψ = f(| · |2).Let (ηft )t≥0 be the convolution semigroup of measures on [0,∞) associated with f , i.e. its Laplacetransform is given by ∫

e−rsηft (ds) = e−tf(r) . (2)

Then we have for u ∈ L2(Rn) the subordination formula

T ft u(x) =

∞∫0

∫Rn

u(y) ps(x− y) dy ηft (ds) (3)

wherept(x) := π−n/2 t−n/2 e−|x|

2/2t

denotes the density of the distribution of B(t).Let Rfλ, R

ηf

λ and Uf := Rf0 , Vf := Rηf

0 be the resolvents and the potentials operators of the Lévyprocess Xf and the convolution semigroup ηf , respectively, i.e.,

Rfλu(x) =

∞∫0

e−λt T ft u(x) dt

Rηf

λ u(r) =

∞∫0

e−λt∫

u(r − s) ηft (ds) dt, λ ≥ 0,

3

for all nonnegative measurable functions u. The operator Vf may be interpreted as a Radonmeasure - the potential measure of the semigroup ηf . It is determined by the Laplace transform

∞∫0

e−rs Vf (ds) =1

f(r). (4)

The distribution function of Vf will be denoted by

V f (s) :=∫

1[0,s] dVf , s ≥ 0 .

Furthermore, the potential operator of Xf is representable as

Ufu(x) =∫

Rn

u(y)Kf (x− y) dy

with density kernel

Kf (x) =

∞∫0

ps(x) dV f (s) . (5)

Convergence of the integral (under an additional condition on f if n ≤ 2) follows from Theorem1.1.2 below.The analogue of equation(4) then reads:

FKf (ξ) = f(|ξ|2)−1 (6)

where F denotes the Fourier transform, in general, in the distributional sense. Therefore Kf hasa radial representation:

Kf (x) =: kfn(|x|).

Recall that f : (0,∞) → R is said to be a complete Bernstein function if there exists a Bernsteinfunction g such that f(r) = r2L(g)(r) where L denotes the Laplace transform. (For related resultsand references cf. [10, 4.9].) In particular, f(r) is then a Bernstein function itself. Moreover, afunction f(r) is a complete Bernstein function if and only if rf(r)−1 is so.

Below we will need the following additional lower scaling condition on a Bernstein function f(two versions):

f(λr) ≥ constλδf(r) , λ ≥ 1 , r > 1/r0 (7)

(or r < r0) for some constants 0 < δ < 1 and 0 < r0 ≤ ∞ .

Note that for an arbitrary Bernstein function f we have the opposite estimate

f(λr) ≤ λf(r) , r > 0 , λ ≥ 1 . (8)

1.1.1 Remark. (i) Complete Bernstein functions are characterized by the representation

f(r) = a+ br +

∞∫0

(1− e−rs)

∞∫0+

e−stτ(dt) ds (9)

for some a, b > 0 and a measure τ with∫ 1

0+t−1τ(dt) +

∫∞1t−2τ(dt) < ∞ (cf. Schilling [18]

or [10, Theorem 4.9.29]).Then a sufficient condition for (7) is that τ has a density τ satisfyingτ(λt) ≥ const λδ τ(t) , t > 0 , λ ≥ 1, for some 0 < δ ≤ 1.

4

(ii) Condition (7) in terms of f(r) := rf(r)−1 reads as follows:

f(λr) ≤ constλ1−δ f(r) , λ ≥ 1 ,

in a neighborhood of 0 or of ∞, respectively.

Our first result are the following potential kernel estimates (which include those of Rao, Song,Vondracek [16] for special Bernstein functions f).

1.1.2 Theorem. (i) For any Bernstein function f and n ≥ 3 we have

limr→∞

r−nf(r−2)−1 = 0 . (10)

(ii) Under the condition (10) (which is restrictive only for n ≤ 2) the potential density kfn of theassociated subordinate Lévy process Xf in Rn, i.e. of the potential operator Kf given by (6)possesses the following properties:

kfn(r) ≤ const r−nf(r−2)−1 , r > 0 . (11)

kfn is differentiable and(kfn)(r) = −2πr kfn+2(r) . (12)

(iii) If the Bernstein function f satisfies the lower scaling condition (7) for r > 1/r0 (or r < r0)and the associated potential measure Vf determined by (4) has a monotone density V f thenthe opposite estimate is valid:

kfn(r) ≥ const r−nf(r−2)−1 , r < r0 , (13)

(resp. r > 1/r0). In particular, if (7) is given on (0,∞), i.e., r0 = ∞, then (13) holds forall r > 0.

(iv) If f is a complete Bernstein function then its potential distribution function V f is also aBernstein function, and therefore the derivative V f is monotone decreasing.

(v) If V f has a monotone decreasing derivative V f then the function rn−2kfn(r) is monotonedecreasing.

Proof. (i): We use the representation (5),

πn/2kfn(r) =

∞∫0

s−n/2 e−r2/2s dV f (s).

It is easy to see (cf. Bertoin [1, III. 1, Proposition 1] or proof of Theorem B1 in the Appendix)that

(const)−1f(s−1)−1 ≤ V f (s) ≤ const f(s−1)−1 . (14)

Furthermore, substituting in (8) r by s−1 and λ by sr−2 we obtain for s ≥ r2 ,

r2 s−1 f(s−1)−1 ≤ f(r−2)−1 . (15)

In particular,lim sups→∞

s−1f(s−1)−1 <∞ (16)

which proves (i).

5

(ii): (14) and (15) imply∞∫r2

s−n/2−1V f (s) ds ≤ const

∞∫r2

s−n/2−1 f(s−1)−1 ds

≤ const f(r−2)−1 r−2

∞∫r2

s−n/2 ds = const r−n f(r−2)−1 .

Next we split off the above integral for kfn(r),

∞∫0

s−n/2 e−r2/2s dV f (s) =

r2∫0

. . .+

∞∫r2

. . . =: I1 + I2

and estimate as follows:

I2 ≤ const

∞∫r2

s−n/2 dV f (s)

= V f (s) s−n/2∞r2

+n

2

∞∫r2

s−n/2−1 V f (s) ds

≤ const r−n f(r−2)−1

because of (14), (15), (10) and the above integral estimate. (The latter also justifies integration-by-parts.)

In I1 we denote ϕr(s) := s−n/2 e−r2/2s and get

I1 =

r2∫0

ϕr(s) dV f (s) = V f (s)ϕr(s)r2

0

−r2∫0

V f (s) ϕr(s) ds

In view of (14) the first summand does not exceed const r−n f(r−2)−1. Moreover,

∣∣∣ r2∫0

V f (s) ϕr(s) ds∣∣∣ ≤ V f (r2)

r2∫0

|ϕr(s)| ds

≤ const f(r2)−1 r2r2∫0

s−n/2−2 e−r2/2s ds

≤ const r−n f(r−2)−1

since |ϕr(s)| ≤ const r2s−n/2−2e−r2/2s.

Thus,kfn(r) ≤ const r−n f(r−2)−1 .

Using the derivative of the function exp(−(·)2/2s) one can prove (12) by means of similar estimatesfor the integrals and Lebesgue’s dominated convergence theorem.

(iii) and (v): Under the additional assumptions on the Bernstein function f the functions v(r) :=V f (r) and w(r) := 1/f(r) satisfy the conditions of Theorem B1,(ii) in the Appendix. This yieldsthe estimates

(const)−1r−1f(r−1)−1 ≤ V f (r) ≤ const r−1f(r−1)−1 (17)

6

for r < r0 (resp. r > 1/r0) on the left side and r > 0 on the right side. The assertion (13) is nowa consequence of (17) and (5):

const rn−2kfn(r) = rn−2

∞∫0

s−n/2 e−r2/2s dV f (s)

≥ rn−2

r0∫0

s−n/2 e−r2/2s V f (s) ds

=

r0/r2∫

0

s−n/2 e−1/2s V f (r2s) ds .

In particular, in the case r0 = ∞ we obtain the equality for all r > 0 which implies (v).Moreover, using the left side of (17) for r < r0 we infer for the case r2 < r0,

const rn−2kfn(r) ≥ const

1∫0

s−n/2 e−1/2s V f (r2s) ds

≥1∫

0

s−n/2 e−1/2s r−2s−1f(r−2s−1)−1 ds

≥ r−2f(r−2)−1

1∫0

s−n/2 e−1/2s ds .

In the last inequality we have used f(r−2s−1) ≤ s−1f(r−2) for s ≤ 1. Hence,

kfn(r) ≥ const r−2f(r−2) , r2 < r0 .

By continuity of the functions we get this estimate for general r < r0.(The case r > 1/r0 may be treated similarly using

∫∞r0

instead of∫ r00

and f(r−2s−1) ≤ f(r−2) fors ≥ 1.)

(iv): Recall that rf(r)−1 is a complete Bernstein function if and only if f is so. Therefore in thiscase we have rf(r)−1 = r2L(g)(r) for some Bernstein function g, i.e.,

f(r)−1 = r L(g)(r) = r

∞∫0

e−rsg(s) ds .

On the other hand,

f(r)−1 =

∞∫0

e−rs dV f (s) = r

∞∫0

e−rs V f (s) ds .

By the uniqueness property of the Laplace transform we get V f (s) = g(s). Moreover, g is monotonedecreasing, since g ≤ 0 for any Bernstein function g.

1.1.3 Remark. In the special case f(r) = rα/2 for 0 < α < 2 ∧ n we obtain the symmetricα-stable Lévy process Xα. Here the potential density is given by the Riesz kernel

Kα(x) =const|x|n−α

. (18)

7

Its generator is equal to (−4)α/2. In analogy to this case we call for a Bernstein function f thefunction Kf the f-Riesz kernel (with generator f(−4)). Note that the generator of the Markovsemigroup is the inverse of the potential operator.

Below we will need some estimates for the derivative hfn of the auxiliary function

hfn(r) := rn−2kfn(r) (19)

for the f -Riesz-kernel kfn(|x|) = Kf (x). Here we suppose the additional condition

f(r) ≤ θ r−1f(r) , r > 1/r0 , (20)

for some 0 < θ < 1.

For an arbitrary Bernstein function f we have only f(r) ≤ r−1f(r).In the above special case f(r) = rα/2 with 0 < α < 2 condition (20) is fulfilled for θ = α/2.More generally, let f be a complete Bernstein function with representation (9). Then a sufficientcondition for (20) is that b = 0 and the measure τ has a C1-density of the form τ(t) = tα/2ψ(t),where 0 < α < 2, 0 ≤ ψ(t) ≤ const(1 + t)−1 and limt→∞ ψ(t) = ∞, or ψ = const.

Recall that for a complete Bernstein function f the potential distribution function V f is a Bern-stein function. In particular, −V f is nonnegative and monotone decreasing.

We now will formulate our main conditions on the Bernstein function f needed for associatedpotential spaces and traces on h-sets.

1.1.4 Definition. We say that a Bernstein function f is tame if it satisfies the lower scaling con-dition (7), the upper scaling condition (20), and −V f is nonnegative and monotone decreasing.For n ≤ 2 we additionally assume r−nf(r−2)−1 → 0 as r →∞.

Note that the last example of complete Bernstein functions for α < n∧ 2 and ψ as above providesa class of tame Bernstein functions.

1.1.5 Lemma. If f is a tame Bernstein function, then the derivative of the auxiliary functionhfn(r) = rn−2kfn(r) satisfies

(const)−1r−3f(r−2)−1 ≤ −hfn(r) ≤ const r−3f(r−2)−1 (21)

for r > 0 on the right side and for r < r0 on the left side, where r0 is from condition (20).Moreover, −hfn is monotone decreasing.

Proof. Recall that

f(r)−1 =

∞∫0

e−rs V f (s) ds .

Therefore we choose in Theorem B1,(iii) in the Appendix w := 1/f and v := V f . The conditionthat −V f is nonnegative and monotone decreasing implies the same properties for −v. Further-more, the lower scaling condition (7) on f yields that on w, and w = −f/f2 together with theupper scaling property (20) of f leads to that for w.. Thus, we obtain

(const)−1r−2f(r−1)−1 ≤ −V f (r) ≤ const r−2f(r−1)−1 (22)

8

for r > 0 on the right side and r < r0 on the left side.Now we use the representation

hfn(r) = rn−2kfn(r) = const rn−2

∞∫0

s−n72 e−r2/2s dV f (s)

= const

∞∫0

s−n/2 e−1/2s V f (r2s) ds .

Then the derivative of hfn equals

hfn(r) = const r

∞∫0

s−n/2 e−1/2s V f (r2s) ds . (23)

Consequently, −hfn is positive and monotone decreasing. Moreover, completely analogous argu-ments as in the proof of Theorem 1.1.2, (ii) and (iii), show that (22) and (23) lead to the desiredestimates.

1.2 Riesz-type potentials of arbitrary order

We now turn to Riesz-type potentials where the Fourier transform of the associated kernelKf,σ(x) =kf,σn (|x|) is given by

FKf,σ(ξ) = f(|ξ|2)−σ/2

for an arbitrary 0 < σ < n. Here we suppose that f is a complete Bernstein function with thelower scaling property (7). Note that in the case 0 < σ ≤ 2, f(r)σ/2 is also a complete Bernsteinfunction (see [10, Theorem 4.9.29,3]). Obviously, this function fulfills (7) if f does so. ThereforeTheorem 1.1.2 is applicable and we obtain upper and lower estimates for the kernel kf,σn . We willshow now that under the condition (7) on f these estimates remain true for arbitrary 0 < σ < n.

1.2.1 Theorem. Let f be a complete Bernstein function satisfying

f(λr) ≥ constλδf(r) , λ ≥ 1 , r > 0 ,

for some 0 < δ ≤ 1. If n ≤ 2 we suppose additionally that limr→∞ r−nf(r−2)−1 = 0. Then forany 0 < σ < n the function f(|ξ|2)−σ/2 is the Fourier transform of a radial function Kf,σ(x) =kf,σn (|x|) such that

(const)−1r−nf(r−2)−σ/2 ≤ kf,σn (r) ≤ const r−nf(r−2)−σ/2 . (24)

kf,σn is differentiable and kf,σn (r) = −2πr kf,σn+2(r).1.2.2 Remark. (i) An analysis of the following proof shows that we can replace the global

lower scaling condition on the Bernstein function f by one of its local versions from (7) andobtain the estimate (23) in a neighborhood of 0 or of ∞, respectively.

(ii) In view of the classical case f(r) = r, where Kf,σ = Kσ (cf. Remark 1.1.3) we call, ingeneral, Kf,σ the Riesz-type kernel of order σ associated with f , or briefly, the (f, σ)-Rieszkernel.

Proof. As mentioned above, for 0 < σ ≤ 2, Theorem 1.1.2 implies the assertion. Thus, for thefirst part of the theorem it suffices to consider the case σ = 2mα for 0 < α ≤ 1, m ∈ N, 2mα < n,and to prove the existence of kf,2mαn and the estimate

(const)−1r−nf(r−1)−mα ≤ kf,2mαn (r) ≤ const r−nf(r−1)−mα .

9

Consider the kernel kf,2αn = kfα

n associated with fα. According to Theorem 1.1.2 it exists andsatisfies (24). Furthermore,

FKf,2mα(ξ) = f(|ξ|2)−αm =( ∞∫

0

e−|ξ|2s Vα(ds)

)mwhere for brevity Vα here denotes the potential measure Vfα

of fα. Let Vα,m be the convolutionof order m of the measure Vα. Then we obtain

f(r)−αm =

∞∫0

e−rs Vα,m(ds) (25)

and

Kf,2mα(x) = π−n/2∞∫0

s−n/2e−|x|2/2s Vα,m(ds) . (26)

(The finiteness of the integral will be shown below. Calculating its distributional Fourier transformleads to the above expression for FKf,2mα.) Recall that Vα has a density V α. Hence,

πn/2 kf,2mαn (r) =

∞∫0

. . .

∞∫0

(s1 + . . .+ sm)−n/2 exp(−r2/2(s1 + . . .+ sm))

V α(sm) . . . V α(s1) dsm . . . ds1

= r2m−n∞∫0

. . .

∞∫0

(s1 + . . .+ sm)−n/2 exp(−1/2(s1 + . . .+ sm))

V α(r2sm) . . . V α(r2s1) dsm . . . ds1 .

By the convolution property of the Gauss kernel we obtain for any vector e with |e| = 1,

(s1 + . . .+ sm)−n/2 exp{−(1/2(s1 + . . .+ sm) |e|2}

=∫

Rn

. . .

∫Rn

s−n/21 exp{−|x1|2/2s1} . . . s

−n/2m−1 exp{−|xm−1|2/2sm−1}

s−n/2m exp{−

∣∣e− m−1∑1

xi∣∣2/2sm}

dxm−1 . . . dx1 .

Therefore the above expression is equal to

r2m−n∫

Rn

. . .

∫Rn

∞∫0

s−n/21 exp{−|x1|2/2s1}V α(r2s1) ds1

. . .

∞∫0

s−n/2m−1 exp{−|xm−1|2/2sm−1}V α(r2sm−1) dsm−1

∞∫0

s−n/2m exp{−

∣∣e− m−1∑1

xi∣∣2/2sm}

V α(r2sm) dsm dxm−1 . . . dx1 .

10

From the left side of (17) we infer that

∞∫0

∞∫0

s−n/2 exp{−|x|2/2s}V α(r2s) ds

≥ const r−2f(r−2)−α1∫

0

s−n/2 exp{−|x|2/2s} ds

=: const r−2f(r−2)−α1∫

0

F (s, |x|) ds .

Applying this to each of the inner integrals above we get the lower estimate

r2m−n(r−2)mf(r−2)−mα const = const r−nf(r−2)−mα ,

since the inner integral

1∫0

. . .

1∫0

∫Rn

. . .

∫Rn

F (s1, |x1|) . . . F (sm−1, |xm−1|)F(sm,

∣∣e− m−1∑1

xi∣∣)

dxm−1 . . . dx1 dsm . . . ds1

=

1∫0

. . .

1∫0

(s1 + . . . sm)−n/2 exp{−1/2(s1 + . . . sm)} ds1 . . . dsm

is finite.In order to get the upper estimate we use the right side of (17) for

const

∞∫0

s−n/2 e−|x|2/2s V α(r2s) ds

≤∞∫0

s−n/2 e−|x|2/2s V α(r2s) r−2s−1f(r−2s−1)−α ds

=

1∫0

. . . ds +

∞∫1

. . . ds .

If s ≥ 1 we have f(r−2s−1)−α ≤ sαf(r−2)−α according (8). For s ≤ 1 by the lower scalingassumption (7) on f , f(r−2s−1)−α ≤ const sαδf(r−2)−α. Therefore the above integral does notexceed

const r−2f(r−2)−α∞∫0

max(sαδ−1, sα−1) s−n/2 e−|x|2/2s ds

=: r−2f(r−2)−α∞∫0

G(s, |x|) ds .

Applying this inequality to each of the inner integrals in the above expression for kf,2mαn we getin the same way as for its lower estimate the upper estimate

const r−nf(r−1)−mα .

11

Here we use that∞∫0

. . .

∞∫0

∫Rn

. . .

∫Rn

G(s1, |x1|) . . . G(sm−1, |xm−1|)G(sm,

∣∣e− m−1∑1

xi∣∣)

dxm−1 . . . dx1 dsm . . . ds1

= const

∞∫0

. . .

∞∫0

m∏1

max(sαδ−1i , sα−1

i )(m∑1

si)−n/2 exp{−1/2m∑1

si} ds1 . . . dsm <∞ ,

since α ≤ 1 , δ ≤ 1, and mα < n/2.

The last estimates also show that that when differentiating the equation

kf,2mαn (r) = π−n/2∞∫0

s−n/2 e−r2/2s Vα,m(ds)

on the right side we can take the derivative under the integral which leads to

kf,σn (r) = −2πr kf,σn+2(r) .

1.3 Resolvents and Bessel-type potentials

Recall that the resolvent Rfλ of the operator f(−∆) (or of the Lévy process Xf ) is the inverse ofthe operator λ id+f(−∆). For λ = 1 and the classical case f(r) = rα/2 , 0 < α ≤ 2, the operatorRf1 has a kernel whose Fourier transform (1 + |ξ|α)−1 is equivalent to (1 + |ξ|2)−α/2. The lattercoincides up to a constant with the Fourier transform of the Bessel potential kernel Gα/2(x) inRn. It is known that Gα/2(x) rapidly decreases as x → ∞ and is equivalent to the Riesz kernelKα(x) if x→ 0. (These operator properties hold for all 0 < α < n.)For a general (complete) Bernstein function f , 1 + f is also a (complete) Bernstein function.Moreover, for fixed ρ ∈ (0,∞] we have 1 + f(r) ≤ const f(r) , r > ρ. Then for 0 < s < n, we callthe function

Gf,s(x) := K1+f,s(x) , x ∈ Rn\{0} ,Bessel-type kernel of order s associated with f , or briefly (f, s)-Bessel kernel. Since Gf,s is radialwe denote

gf,sn (|x|) := Gf,s(x) = k1+f,sn (|x|) .

For s = 2 we use again the superscript f instead of (f, 2).Then Theorem 1.1.2 and the local version of Theorem 1.2.1 imply the equivalence of the Riesz-typeand the Bessel-type kernels in a neighborhood of 0:

1.3.1 Corollary. Let f be a Bernstein function. If n ≤ 2 we suppose additionally thatlimr→∞ r−nf(r−2)−1 = 0. Then we have:

(i)gfn(r) ≤ const r−nf(r−2)−1 , r > 0 .

(ii) If the potential measure of f has a monotone density and

f(λr) ≥ constλδf(r) , λ ≥ 1 , r > 1/r0 ,

for some 0 < δ ≤ 1 and r0 ∈ (0,∞] then the opposite estimate is valid, i.e.,

gfn(r) ≥ const r−nf(r−2)−1 , r < r0 .

12

(iii) If f is a complete Bernstein function with the lower scaling condition from (ii) then theestimates hold for arbitrary 0 < s < n:

(const)−1r−nf(r−2)−s/2 ≤ gf,sn (r) ≤ const r−nf(r−2)−s/2 , r < r0 .

For application to the above Lévy processes we need the following modification of the (f, 2)-Besselkernel Gf with Fourier transform

FGf (ξ) = (1 + f(|ξ|2))−1 .

1.3.2 Lemma. For any tame Bernstein function f in the sense of Definition 1.1.4 and R > 0there exists a positive continuous radial function GfR on Rn\{0} rapidly decreasing at infinity suchthat for |x| < R, GfR(x) coincides with the f-Riesz kernel Kf (x) and

(const)−1(1 + f(|ξ|2))−1 ≤ FGfR(ξ) ≤ const(1 + f(|ξ|2))−1 . (27)

The proof is given in Appendix C. We conjecture that this result may be extended to Gf,s andKf,s for arbitrary 0 < s < n. For the classical case f(r) = r this is true: Recall that theclassical Riesz kernel of order s is given by K id,s(x) = Kf (x) = const |x|−(n−s) with Fouriertransform FKs(ξ) = |ξ|−s. The corresponding Bessel kernel Gs(x) = gsn(|x|) of order s satisfiesFGs(ξ) = (1 + |ξ|2)−s/2, where 0 < s < n.

1.3.3 Corollary. For any 0 < s < n and R > 0 there exists a positive continuous radial functionGsR on Rn \ {0} rapidly decreasing at infinity such that for |x| < R, GsR coincides with the Rieszkernel Ks(x) and

(const)−1(1 + |ξ|2)−s/2 ≤ FGsR(ξ) ≤ const(1 + |ξ|2)−s/2 ,

i.e., the Fourier transform of GsR is equivalent to that of the Bessel kernel of order s.

The proof in Appendix C corrects that from [24, (3.2)]. Note that for 0 < s < 2 this is a specialcase of Lemma 1.3.2.

1.4 Dirichlet forms and Bessel-type potential spaces

The Dirichlet form of Xf in L2(Rn) can be represented as

Ef (u, v) =∫

Rn

f(|ξ|2)Fu(ξ)Fv(ξ) dξ

=∫

Rn

√f(−∆)u(x)

√f(−∆)v(x) dx.

(28)

Its domain is the (f, 1)-Bessel-type-potential space Hf,1(Rn). For general s > 0, the Hilbert spaceHf,s(Rn) is given by the scalar product

〈u, v〉Hf,s(Rn) =∫

Rn

(1 + f(|ξ|2))s Fu(ξ)Fv(ξ)dξ . (29)

(For our purposes we only consider real valued functions.)In the case s = 1 we obtain

〈u, v〉Hf,1(Rn) = Ef1 (u, v)

in terms of the Dirichlet forms.(For ψ(ξ) = f(|ξ|2), Hf,s(Rn) agrees with the space Hψ,s considered in Jacob [10], [11].)

13

By Lemma 1.3.2 for 0 < s ≤ 2 and tame Bernstein functions f the norm in Hf,s(Rn) is equivalentto

||u|Hf,s(Rn)|| ∼( ∫

Rn

(FGf,sR (ξ))2 |Fu(ξ)|2 dξ,)1/2

(30)

i.e., u ∈ Hf,s(Rn) if and only if

u(x) = Gf,sR ∗ w(x) =∫

Rn

Gf,sR (x− y)w(y) dy (31)

for some w ∈ L2(Rn) and the (f, s)-Riesz-Bessel kernel Gf,sR . The space Hf,1(Rn) provided withthe corresponding scalar product ∫

Rn

(FGf,sR (ξ))2 Fu(ξ)Fv(ξ) dξ (32)

will be denoted by Hf,sR (Rn).

Below we will need only the space Hf,1(Rn), or equivalently, Hf,1R (Rn) with the associated kernel

Gf,1R together with the kernel Gf,2R . Recall that these kernels coincide on B(0, R) with the Riesz-type kernels Kf,1, resp. Kf,2 = Kf (in the notations of section 1.1). Moreover, Kf,1 = K

√f .

2 Traces on h-sets

2.1 Traces of potential spaces

Let now F ⊂ Rn be a compact h-set with h-measure µ as in the introduction. (Note that all suchµ are equivalent to the Hausdorff measure Hh|F .) Throughout this section we assume that f is atame Bernstein function in the sense of Definition 1.1.4.Choose an arbitrary R > 0 such that F is contained in the ball B(0, R). We are interested intraces on F of the Bessel-type potential spaces Hf,1

R (Rn) equivalent to Hf,1(Rn) (cf. (32)).In [14] under the trace condition

1∫0

h(r)rn+1 f(r−2)

dr <∞ (33)

and some growth condition on the function f the existence of the trace space on F is proved in thesense of embedding in L1(F, µ). (For general approaches to the trace problem see Jonsson/Wallin[13], Jonsson [12] and Triebel [20], [21], [22].) Condition (33) is the integral version of condition(29) in [14] for p = 2. There f needs not be a Bernstein function.

In the present paper we will use the tracing procedure in the sense of Triebel in the L2(F, µ)-setting:

For Schwartz functions u puttrµ u := u|F .

In view of Theorem A2 in the Appendix we have

|| trµ u|L2(F, µ)|| ≤ const ||u|Hf,1R (Rn)||.

Since the Schwartz functions are dense in Hf,1R (Rn) the trace operator trµ admits a continuous

extensiontrµ : Hf,1

R (Rn) → L2(F, µ) =: L2(µ).

14

(Actually, as in [22, Theorem 7.16] one obtains a more explicit representation of trµ in terms ofwavelet expansions.) From this we infer the following.

2.1.1 Theorem. If the tame Bernstein function f and the gauge function h for the measure µsatisfy the trace condition (33) then the trace space

Hf,1R (F, µ) = Hf,1

R (µ) := trµ(Hf,1R (Rn)) (34)

is determined. Moreover,

|| trµ v|L2(Fµ)|| ≤ const ||v|Hf,1R (Rn)|| . (35)

The norm||u|Hf,1

R (µ)|| := inf{||v|Hf,1

R (Rn)|| : v ∈ Hf,1R (Rn) , trµ v = u

}(36)

induces a Hilbert space structure in Hf,1R (µ) and satisfies

||u|L2(µ)|| ≤ const ||u|Hf,1R (µ)|| . (37)

2.1.2 Remark. (i) If we replace in the tracing procedure the Hilbert space Hf,1R (Rn) by its

classical variant Hf,1(Rn) then the Bernstein function needs not to be tame. In this versionthe theorem can be proved as in [22, Theorem 7.16] by means of wavelet expansions with amore explicit representation of the operator trµ.

(ii) Below we will show that the norm ||u|Hf,1R (µ)|| does not depend on the choice of the radius

R as above (cf. (42)). Therefore we will also write

Hf,1R (µ) = Hf,1(µ)

for the trace space provided with this Hilbert norm. Note this norm is equivalent, but doesnot coincide with that of Hf,1(µ) arising from the above tracing procedure starting with theHilbert space structure Hf,1(Rn) instead of Hf,1

R (Rn).

(iii) We have the Hilbert space decompositions

Hf,1R (Rn) =

{u ∈ Hf,1

R (Rn) : trµ u = 0}⊕Hf,1(µ)

Hf,1(Rn) ={u ∈ Hf,1(Rn) : trµ u = 0

}⊕ trµ(Hf,1(Rn))

with the usual identifications. In particular, the corresponding extension operators

Extfµ,R : Hf,1(µ) → Hf,1R (Rn) , for tame f ,

Extfµ : trµ(Hf,1(Rn)) → Hf,1(Rn) , for general f ,(38)

are determined by these decompositions.

(iv) In the case of tame f for different µ with given gauge function h the norms in the trace spaceare all equivalent.

2.2 Potential representation of trace spaces and quadratic forms

Recall that Hf,1(Rn) is the space of (f, 1)-Bessel potentials and agrees for tame f with the spaceHf,1R (Rn) of (f, 1)-Riesz-Bessel potentials in the sense of norm equivalence (cf. (29)-(32)). In [23]

for the classical case f(r) = rα/2 and d-sets F a similar potential representation is derived for thetrace spaces Hf,1(µ). We now will extend this potential approach to arbitrary (tame) Bernsteinfunctions f and h-measures µ.

15

Definition. The (f, µ)-potential (function) of µ is given by

Ufµ (x) :=∫Kf (x− y)µ(dy), x ∈ Rn .

For the next property we need only the upper h-regularity of the measure µ: µ(B(x, r)) ≤consth(r).2.2.1 Proposition. Under the growth condition (10) on f , if n ≤ 2, and the trace condition (33)on f and h, the function Ufµ is bounded and uniformly continuous.

Proof. The boundedness easily follows from Theorem 1.1.2: Denoting mx(r) := µ(B(x, r)) we getfor rmax := diam(F ),

∫Kf (x− y)µ(dy) =

rmax∫0

kfn(r) dmx(r)

= mx(r) kfn(r)|rmax0 −

rmax∫0

(kfn)′ (r)mx (r) dr

= mx(r) kfn(r)|rmax0 +

rmax∫0

const r kfn+2 (r)mx (r) dr

according to (12). Since mx(r) < ch(r) and kfn(r) < const r−nf(r−2)−1 by (11), the trace condi-tion

1∫0

h(r) r−(n+1) f(r−2)−1 dr <∞

implies the boundedness of the second summand. Moreover, the convergence of this integral leadsto lim

r→0mx(r) kfn(r) = 0.

In order to prove the uniform continuity we estimate as follows:

|Ufµ (x)− Ufµ (z)| ≤∫|Kf (x− y)−Kf (z − y)|µ(dy)

≤∫

|x−y|≥2−k

|z−y|≥2−k

∣∣Kf (x− y)−Kf (z − y)∣∣µ(dy)

+∫

|x−y|<2−k

(Kf (x− y) +Kf (z − y))µ(dy)

+∫

|z−y|<2−k

(Kf (x− y) +Kf (z − y))µ(dy).

Because of Theorem 1.1.2 and the continuity of Kf (x) outside x = 0, for fixed k ∈ N the firstsummand tends to 0 as |x− z| → 0. Furthermore, by the same theorem and the monotonicity of

16

f and h, ∫|x−y|<2−k

Kf (x− y)µ(dy) =∫

r<2−k

kfn(r) dmx(r)

≤ const∫

r<2−k

r−n f(r−2)−1 dmx(r)

= const∞∑j=k

∫2−(j+1)≤r<2−j

r−n f(r−2)−1 dmx(r)

≤ const∞∑j=k

2jn f(22j)−1 h(2−j).

The last expression tends to 0 as k → ∞, since the convergence of the series is equivalent to thetrace condition (33).Finally, we show that

limk→∞

∫|x−y|<2−k

Kf (z − y)µ(dy) = 0

uniformly in x and z. Then, by symmetry arguments, the third summand vanishes as k → ∞uniformly in x, z. By Theorem 1.1.2,∫

|x−y|<2−k

Kf (z − y)µ(dy) ≤ const∫

<x−y|<2−k

|z − y|−n f(|z − y|−2)−1 µ(dy).

From the property (8) of the Bernstein function f one easily deduces that the function rn f(r−2)is nondecreasing in r. Therefore the last integral may be estimated as follows. It is equal to∫

|x−y|<2−k

|z−y|<|x−y|

|z − y|−n f(|z − y|−2)−1 µ(dy)

+∫

|x−y|<2−k

|z−y|≥|x−y|

|z − y|−n f(|z − y|−2)−1 µ(dy)

≤∫

|z−y|<2−k

|z − y|−n f(|z − y|−2)−1 µ(dy)

+∫

|x−y|<2−k

|x− y|−n f(|x− y|−2)−1 µ(dy).

Now we can apply the first part of the above estimates in order to conclude that the last summandstend to 0 as k →∞ uniformly in z and x, resp.

Next we will introduce the analogue of the f -Riesz potential operator Ufw = Kf ∗w , w ∈ L2(Rn),from section 1.1 on the trace space. Note that Uf may also be considered as an operator actingon distributions w in Rn.For any u ∈ L2(µ) the distribution idµ u := uµ is given by uµ(ϕ) :=

∫ϕ(x)u(x)µ(dx) for Schwartz

17

functions ϕ. In the proof of Corollary A.3 in the Appendix it is shown that for the corresponding f -Riesz-Bessel potential operator UfR in the case of tame Bernstein functions f we have the embedding

UfR ◦ idµ : L2(µ) → Hf,1R (Rn).

Therefore we introduce the operator

Ufµ := trµ ◦UfR ◦ idµ (39)

from L2(µ) into Hf,1R (µ). Ufµ does not depend on the choice of the radius R of the ball containing

F , since the f -Riesz kernel Kf and the f -Riesz-Bessel kernel GfR coincide on F . Proposition 2.2.1implies that in the definition (39) of Ufµu for bounded u the trace is pointwise determined:

2.2.2 Corollary. Under the trace condition (33) for any bounded Borel function u on F we havethe following:

(i)∫Kf (x− y)u(y)µ(dy) , x ∈ Rn, is a continuous version of Uf ◦ idµ u .

(ii)∫Kf (x− y)u(y)µ(dy) , x ∈ F , is a continuous version of Ufµ u determined in this way for

general Bernstein functions f (with the growth condition (10) if n ≤ 2).

In the sequel Uf ◦ idµ u and Ufµu will be understood in this sense. In section 3 below we will provethat Ufµ is the potential operator of the trace of the Lévy process Xf on the h-set F . To this aimwe now introduce the associated quadratic form using the following auxiliary results which are ofindependent interest.Throughout the rest of this section we assume the trace condition (33)and f to be tame.Using Corollary A.3 from the Appendix one derives as in [23, Theorem 3.1] the following.

2.2.3 Lemma.√Ufµ maps L2(µ) onto Hf,1

R (µ) and is an isometry, i.e.,

⟨√Ufµ u,

√Ufµ v

⟩Hf,1

R (µ)= 〈u, v〉L2(µ), (40)

u, v ∈ L2(µ).

Furthermore, Theorem 3.2 in [24] extends to our situation using similar Fourier transform argu-ments:

2.2.4 Lemma.〈Ufµ u, v〉L2(µ) =

⟨√Uf (uµ),

√Uf (vµ)

⟩L2(Rn)

, (41)

u, v ∈ L2(µ).

(In the reference papers the special case of d-sets F and f(r) = rα/2 is treated.)

Equation (40) implies that the operators√Ufµ and Ufµ are invertible. Denote

Dfµ := (Ufµ )−1 ,

hence, √Dfµ =

(√Ufµ

)−1.

18

Moreover, from (40) we conclude that the domain of√Dfµ is Hf,1

R (µ) and the scalar product inthis space does not depend on the choice of R:

〈u, v〉Hf,1R (µ) =

⟨√Dfµu,

√Dfµv

⟩L2(µ)

, (42)

u, v ∈ Hf,1R (µ) and we can denote Hf,1

R (µ) =: Hf,1(µ). (Recall that Hf,1R (µ) = trµ(H

f,1R (Rn)). For

the scalar product in the classical version trµ (Hf,1(Rn)) formula (42) does not hold as equality.Here we have only equivalence.)As in the special case of d-sets and f(r) = rα/2, Df

µ may be interpreted as fractal pseudodifferentialoperator. The Euclidean counterpart is

Df :=(Uf )−1 = f(−4),√Df =

√f(−4).

In these terms formula (42) may be rewritten as⟨√Dfµu,

√Dfµv

⟩L2(µ)

=⟨√

Df u,√Df v

⟩L2(Rn)

if

u = Ufµ u′, v = Ufµ v

′ ,(43)

u = Uf (u′µ), v = Uf (v′µ)

for some u′, v′ ∈ L2(µ).As in the proof of Theorem 4.1 (ii) in [24] one shows that the correspondence u 7→ u in (43)generalizes to a linear Riesz-type extension operator

extfµ : Hf,1(µ) → Lf,12 (Rn) (44)

into the space of (f, 1)-Riesz potentials

Lf,12 (Rn) :={w =

√Ufw′ : w′ ∈ L2(Rn)

}.

Note that extfµ is different from the Bessel-type extension operators Extfµ and Extfµ,R from section2.1.Finally, we consider the quadratic forms

Ef (u, v) =⟨√

Df u,√Df v

⟩L2(Rn)

in L2(Rn) with domain Hf,1(Rn) (cf. (28)) and

Efµ (u, v) :=⟨√

Dfµu,

√Dfµv

⟩L2(µ)

(45)

in L2(µ) with domain Hf,1(µ). Recall that

||u|Hf,1(Rn)||2 = Ef (u, u) + ||u|L2(Rn)||2

In view of (42) the analogue on the trace space reads

||u|Hf,1(µ)|| = Efµ (u, u) ,

i.e., here we do not need additionally the L2(µ)-norm.

Because of the norm equivalences both Ef and Efµ are closed and regular quadratic forms on thecorresponding L2-spaces, since the Schwartz functions are dense in Hf,1(Rn) and Hf,1(µ).Summarizing the main results of this section we obtain the following.

19

2.2.5 Theorem. For any tame Bernstein function f in the sense of 1.1.4 and any gauge functionh satisfying the trace condition (33) we have:

(i)Efµ (u, v) = 〈u, v〉Hf

R(µ) = 〈Extfµ,R u,Extfµ,R v〉Hf,1R (Rn)

for the Riesz-Bessel-type extension operator Extfµ,R from (38). In particular, the scalarproduct in Hf,1

R (µ) does not depend on R > 0 with F ⊂ B(0, R).

(ii) Efµ is a closed any regular quadratic form in L2(µ) with domain Hf,1(µ) := Hf,1R (µ).

(iii)Efµ (u, v) = Ef (extfµ u, extfµ v),

u, v ∈ Hf,1(µ), for the Riesz-type extension operator extfµ : Hf,1(µ) → Lf,12 (Rn) determinedby

extfµ u(x) =∫Kf (x− y)u′(y)µ(dy), x ∈ Rn,

if u(x) =∫Kf (x − y)u′(y)µ(dy), x ∈ F , for some bounded Borel function u′ on F , and by

continuation to all u according to the above quadratic form.

(iv)||u|L2(µ)||2 ≤ const Efµ (u, u) , u ∈ Dom Efµ (46)

(Poincaré inequality).

(Note that (iv) is a consequence of the embedding of Hf,1(µ) into L2(µ).)

3 Dirichlet forms and trace processes

For the notions from the theory of Markov processes and Dirichlet forms used in this section cf.Fukushima, Oshima and Takeda [7] and Fukushima [6]. First note that the operator Df = (Uf )−1

is the infinitesimal generator of the primary Lévy process Xf and Ef is its Dirichlet form. Wenow will show that the quadratic form Efµ from (45) is the Dirichlet form of the trace of Xf onthe fractal h-set F . As in the special case of d-sets and f(r) = rα/2 (see Kumagai [15], Fukushimaand Uemura [8], Hansen and Zähle [9]) we use the method of time change for Markov processes.

Under the trace condition (33) the measure µ is of finite energy integral:

||v|L1(µ)|| ≤ const√Ef1 (v, v)

for any continuous v ∈ Dom Ef . This follows from ||v|L1(µ)|| ≤ ||v|L2(µ)||, the equality Ef1 (v, v) =||v|Hf,1(Rn)||2 and Theorem A.2. (A more direct proof can be given as in Triebel [22, Remark7.15] for the special case f(r) = rα/2 and more general µ.) Therefore the h-measure µ can beinterpreted as the Revuz measure of a finite positive continuous additive (random) functional(briefly PCAF) Lfµ w.r.t the primary Lévy process Xf : According to the Theorem 5.1.1 and5.1.2. and Lemma 5.1.3 in [7], Lfµ is determined (up to equivalence) by the property that for anynonnegative bounded Borel function w and λ > 0,

Ex( ∞∫

0

e−λt w(Xf (t)) dLfµ(t))

is a quasi continuous version of Rfλ(wµ) (for the resolvent Rfλ = (idλ+Df )−1 of the process Xf ).

20

Definition. The PCAF Lfµ(·) is called the µ-local time of the process Xf on the fractal supportF of µ.

Letting λ→ 0 and using Rf0 = Uf we infer that

E(·)

( ∞∫0

w(Xf (t)) dLfµ(t))

= Uf (wµ)

Ef -quasi everywhere. According to Proposition 2.2.1 the function

Uf (wµ)(x) =∫Kf (x− y)w(y)µ(dy)

is continuous on Rn. Interpreting it as a version of the above conditional expectation we get forany nonnegative bounded Borel function w and any x ∈ F ,

Ex( ∞∫

0

w(Xf (t)) dLfµ(t))

=∫Kf (x− y)w(y)µ(dy) = Ufµw(x) .

Therefore Ufµ may be interpreted as the potential operator of the time changed process Xfµ obtained

as follows: Setτfµ (t) := inf{s > 0 : Lfµ(s) > t} and Xf

µ(t) := Xf (τfµ (t))

According to [7, Theorem 6.2.1], Xfµ is a normal right continuous strong Markov process on the

support F of the PCAF Lfµ, where µ(F4F ) = 0. Its transition function determines a stronglycontinuous Markov semigroup (Tt)t≥0 on L2(µ). By construction,

∞∫0

Ttw(x) dt = Ufµw(x) , x ∈ F ,

i.e., Ufµ is the potential operator of the semigroup(Tt

)t≥0

.Recall that in the case of tame f the inverse operator of Ufµ was denoted by Df

µ. Therefore in thiscas the the corresponding Dirichlet form is given by

Efµ (u, v) =⟨√

Dfµ u,

√Dfµ v

⟩L2(µ)

.

The domain of Efµ agrees with the trace space Hf,1(µ) and Efµ is regular. In view of [7, Theorem7.2.1], Efµ determines a µ-symmetric Hunt process Xf

µ on F . We call Xfµ the µ-trace of the Lévy

process Xf on the fractal h-set F . By construction, Xfµ is a version of the time changed process

Xfµ on F .

Summarizing the above results we obtain:3.1 Theorem. Let µ be an h-measure with compact support F and Xf be a Lévy process in Rnsubordinate to Brownian motion according to the tame Bernstein function f . Under the tracecondition (33) the following holds true.

(i) There is a uniquely determined Hunt process Xfµ on F with potential operator Ufµ , where

Ufµw(x) =∫Kf (x− y)w(y)µ(dy) , x ∈ F ,

for any bounded Borel function w.

21

(ii) Its Dirichlet form is given by the trace of the Dirichlet form Ef of Xf on F :

Efµ (u, v) =⟨√

Dfµ u,

√Dfµv

⟩L2(µ)

with Dom Efµ = Hf,1(µ), the trace of the f-Bessel potential space Hf,1(Rn) on F .Efµ satisfies the Poincaré inequality (46).

(iii) Xfµ is a version of the time changed process Xf

µ given by the µ-local time Lfµ of Xf on F .

The time changed process Xfµ exists for general Bernstein functions f satisfying the trace condition

(33). Ufµ defined as in (i) is its potential operator (provided the growth condition (10) is fulfilledif n ≤ 2).

Remark. A probabilistic interpretation of Efµ as the time changed Dirichlet form and of the tracespace Hf,1(µ) as its domain follows from the general approach in [7, 6.2]. In the above approachwe have used the method of time change in order to obtain the Markov property of the regularquadratic form Efµ introduced in Section 2. For the special case of d-measure and f(r) = rα/2 in[23] this Markov property was proved in a purely analytical way. The general case needs someadditional tools.

A Associated Besov spaces of generalized smoothness

In [14] an interpretation of the potential spacesHf,1(Rn) as Besov spaces of generalized smoothnessBσ,N2,2 (Rn) with σj = 2j , Nj =

√f−1(22j), j ∈ N0, was used. (For a survey on spaces of generalized

smoothness, the corresponding notions and related results we refer to Farkas and Leopold [5] andto Bricchi [2].) The choice of the admissible sequence σ and the strongly increasing sequence Nis not unique. For our purposes it is more convenient to work with the classical version Nj = 2j .Then the corresponding σ is determined by

σj :=√f(22j) . (47)

(The norm equivalence follows from comparing the Fourier analytical definitions of Hf,1(Rn) andBσ,N2,2 (Rn).) Moreover, this sequence is admissible, i.e.,

c0σj ≤ σj+1 ≤ c1σj , j ∈ N,

since for a Bernstein function we have

f(22j) ≤ f(22(j+1)) ≤ 4f(22j)

in view of (8).As in the literature we write Bσ2,2(Rn) for the space Bσ,N2,2 (Rn) with Nj = 22j and choose σaccording to (47).For a gauge function h as in Section 1 we define the admissible sequence σ∗ by

σ∗j :=σ√

2j√h(2−j)

. (48)

Then (σ∗j )2 = f(22j)

2j h(2−j) , and therefore

∞∑j=0

(σ∗j )−2 <∞ (49)

22

is a discrete version of the trace condition (33). In [14, Section 2] for σ∗ and N as above (andmore general N) the Besov spaces of generalized smoothness Bσ

∗,N2,2 (F ) =: Bσ

∗

2,2(F ) on the frac-tal support F of µ are introduced by means of quarkonial decompositions. Moreover, the traceoperator as a continuous extension from Schwartz functions to Bσ2,2(Rn) and in the more explicitquarkonial decomposition exists: [14, Theorem 15] implies the following.

A.1 Proposition. Under the trace condition (49) we get for σj =√f(22j) and σ∗j given by (48),

trµ(Bσ2,2(Rn)) = Bσ∗

2,2(F ) .

Remark. Since Bσ2,2(Rn) agrees with the potential space Hf,1(Rn), for tame Bernstein functionsf the space Bσ

∗

2,2(F ) provides a quarkonial interpretation of the trace potential space Hf,1(µ)introduced in Section 2.

We now consider trµ as a mapping from Bσ2,2(Rn) into L2(µ). Recall that for u ∈ L2(µ) themeasure uµ may be considered as a tempered distribution idµ u defined by

idµ u(ϕ) =∫ϕ u dµ =

∫trµ ϕ u dµ

for Schwartz function ϕ. In this way the identification operator idµ is the dual to trµ. It is well-known that L2(µ)′ = L2(µ) and Bσ2,2(Rn)′ = Bσ

−1

2,2 (Rn) (see, e.g., [5, Theorem 3.1.10] in a moregeneral context).With the method of wavelet expansions used in Triebel [22, Theorem 7.16] (for the classical caseσj = 2js) one proves the following:

A.2 Theorem. Under the condition (48) and (49) on an admissible sequence σ the operators

trµ : Bσ2,2(Rn) → L2(µ) (50)

idµ : L2(µ) → Bσ−1

2,2 (Rn) (51)

are compact.

Next we consider the potential operator UfR given by

UfR u(x) :=∫

GfR(x− y)u(y) dy (52)

for the f -Riesz-Bessel kernel GfR from 1.3.2. Similarly as for the classical spaces B−s2,2(Rn) and

Bs2,2(Rn) (cf. Triebel [19]) one shows for the sequence σ from (47) and tame Bernstein functionsf the lifting property

UfR : Bσ−1

2,2 (Rn) → Bσ2,2(Rn)

in terms of Fourier representations. From this and Theorem A.2 we obtain the following.

A.3 Corollary. For any tame Bernstein function f the operator

Ufµ = trµ ◦ UfR ◦ idµ

is a compact self-adjoint operator on the Hilbert space Hf,1R (µ) satisfying

〈Ufµ u, v〉Hf,1R (µ) = 〈u, v〉L2(µ) ,

u, v ∈ Hf,1R (µ).

23

Proof. The compactness of Ufµ follows from:

id : Hf,1R (µ) → L2(µ) bounded,

idµ : L2(µ) → Bσ−1

2,2 (Rn) compact,

UfR : Bσ−1

2,2 (Rn) → Bσ2,2(Rn) bounded,

trµ : Hf,1R (Rn) → Hf,1

R (µ) bounded

and the norm equivalence for Bσ2,2(Rn) and Hf,1R (Rn). Furthermore, for

B := UfR ◦ idµ ◦ trµ

and u, v ∈ Hf,1R (Rn) we get as in Triebel [20, 28.6],

〈trµ u, trµ v〉L2(µ) = 〈B u, v〉Hf,1R (Rn),

since the scalar product in Hf,1R (Rn) is given by 〈(UfR)−1/2(·), (UfR)−1/2(·)〉L2(Rn). (Use that

〈uµ, v〉L2(Rn) = 〈u, v〉L2(µ) in the sense of dual pairing.) In particular,

||√Bu|Hf,1

R (Rn)|| = || trµ u|L2(µ||,

u ∈ Hf,1R (Rn). Therefore we have the Hilbert space decomposition

Hf,1R (Rn) = N(B)⊕Hf,1

R (µ)

andN(B) = {u ∈ Hf,1

R (Rn) : trµ u = 0},

where N(B) is the null space of the operator B.Since B = UfR ◦ idµ ◦ trµ and Ufµ = trµ ◦UfR ◦ idµ, this yields

〈B u, v〉Hf,1R (Rn) = 〈Ufµ u, v〉Hf,1

R (µ),

hence,〈Ufµ u, v〉Hf,1

R (µ) = 〈u, v〉L2(µ)

for u, v ∈ Hf,1R (µ) identified with their extensions Extfµ,R in the sense of the above Hilbert space

decomposition (see (38)).

B A global Tauberian-type theorem for the Laplace trans-form

A main tool for the estimates in Sections 1 and 2 is the following relationship for the Laplacetransform of monotone functions.

B.1 Theorem. Suppose that w is a positive continuous function on (0,∞) with existing Laplacetransform

w(x) =

∞∫e−xyv(y)dy, x ≥ 0.

Let x0 ∈ (0,∞]

24

(i) If v is monotone then we have

v(x) ≤ constx−1w(x−1) , x > 0 .

(ii) If(a) v is monotone increasing and

w(2x) ≥ constw(x) , x > 1/x0 ,

or(b) v is monotone decreasing and

w(λx) ≤ constλ−δw(x) , λ ≥ 1 , x > 1/x0 , (53)

then we havev(x) ≥ constx−1w(x−1) , x ≤ x0 .

(iii) Suppose that the condition of (ii),(b) is satisfied, v is continuously differentiable with mono-tone decreasing |v|, and

|w(x)| ≤ θx−1w(x) , x > 1/x0 ,

for some θ < 1. Then we have

|v(x)| ≥ constx−2w(x) , x ≤ x0 .

The opposite estimate is true for all x > 0.

Proof. First case: Suppose that v is monotone increasing. Then we use arguments from Bertoin[1, III, 1, Prop.1]. For any z > 0 one obtains

w(x) =

∞∫0

e−xy v(y) dy ≥ 1x

∞∫0

e−y v(yx

)dy

=1x

∞∫z

e−y v(yx

)dy ≥ 1

xv

( zx

)e−z

by monotonicity. Hence,v

( zx

)≤ ez x w(x) , x > 0 , z > 0. (54)

This impliesv(x) ≤ e x−1w(x−1) , x > 0 ,

i.e., the first part of (i).

For the lower estimate we proceed as follows. Let x > 1/x0.

x w(x) =

z∫0

e−y v(yx

)dy +

∞∫z

e−y v(yx

)dy

≤ v( zx

)+

∞∫z

e−y e−y/2x

2w

(x2

)dy

≤ v( zx

)+ w

(x2

) x

24e−z/2

≤ v( zx

)+ a w(x) x e−z/2

25

for some a > 0 by monotonicity for the first integral and (54) for the second one and by thedoubling condition on w. Hence, choosing z sufficiently large we get

v( zx

)≥ (1− ae−z/2) x w(x) ,

i.e.,

v(x) ≥ (1− ae−z/2)z

xw

( zx

)≥ const

1xw

(1x

), x < x0 .

In the last inequality we have used that the doubling condition (ii),(a) on w implies

w(zy) > const(z) w(y) , z > 1 , y > 1/x0 .

Thus (ii) with condition (a) is proved.

Second case: Suppose that v is monotone decreasing . Then the upper estimate follows from

w(x) =1x

∞∫0

e−y v(yx

)dy ≥ 1

x

z∫0

e−y v(yx

)dy

≥ 1xv

( zx

)(1− e−z)

which yields

v( zx

)≤ x w(x)

1− e−z, x > 0, z > 0 ,

in particular, the second case of (i) ,

v(x) ≤ (1− e−1)−1x−1w(x−1) , x > 0 ,

andv

(yx

)≤ (1− e−1)−1x

yw

(x

y

), x > 0, y > 0 . (55)

Furthermore,

x w(x) =

z∫0

e−x v(yx

)dy +

∞∫z

e−y v(yx

)dy

≤z∫

0

e−y v(yx

)dy + v

( zx

)e−z

≤ (1− e−1)−1 x

z∫0

e−y1yw

(x

y

)dy + v

( zx

)e−z

because of (55). Under the condition (ii),(b), for 1/y > 1 we have w(x/y) ≤ const yδw(x), x >1/x0, y < z, for sufficiently small z. Then for z < 1 the last expression does not exceed

b x w(x)

z∫0

e−y yδ−1 dy + v( zx

)e−z

26

for some constant b > 0. Choosing z small enough so that b∫ z0e−y yδ−1 dy < 1

2 we conclude

v( zx

)≥ const x w(x) , x > 1/x0 ,

for some z < 1. Hence,

v(x) ≥ constz

xw

( zx

)≥ const

1xw

(1x

), x < zx0,

since w is monotone decreasing. Changing the constant if necessary we infer the inequality for allx < 1/x0. Thus, (i) and (ii) are proved.

The estimates in (iii) can be shown similarly: Denote w(x) := w(x) + xw(x) = w(x) − x|w(x)|.Then we get

(1− θ)w(x) ≤ w(x) ≤ w(x)

using the assumption on the derivative of w on the left side, where x > 1/x0.Furthermore, differentiating the equation

xw(x) =

∞∫0

e−y v(yx

)dy

we obtain

w(x) = w(x) + xw(x) =1x2

∞∫0

e−y y(−v

(yx

))dy .

Since −v = |v|, the above inequalities lead to

(1− θ)w(x) ≤ 1x2

∞∫0

e−y y∣∣∣v (y

x

)∣∣∣ dy ≤ w(x) ,

where x > 1/x0 on the left side.The remaining arguments are completely analogous to the those in (ii),(b) using the monotonicityof |v| and regarding the factor e−yy instead of e−y under the integral.

Remark. In the last theorem we may replace everywhere the role of the intervals (1/x0,∞) and(0, x0) by (0, x0) and (1/x0,∞), respectively. The proofs are similar.

C Estimates for the Fourier transform of Riesz-Bessel-typekernels

Proof of Lemma 1.3.2

Let GfR(x) =: gfn,R(|x|) be an arbitrary radial C1-function on Rn > {0} rapidly decreasing atinfinity such that GfR(x) = kfn(|x|), 0 < |x| < R, for the f -Riesz kernel kfn. If suffices to find sucha version that the desired estimates for the Fourier transform FGfR(ξ) hold for sufficiently small

27

and sufficiently large |ξ|. The case of small ξ is evident because of the rapid decay at infinity.Since GfR is radial, FGfR(ξ) may be evaluated at ξ = (0, . . . , 0, |ξ|):

FGfR(ξ) = const

∞∫−∞

e−i|ξ|xn

∞∫−∞

. . .

∞∫−∞

gfn,R(|x|) dx1 . . . dxn−1

dxn

= const

∞∫−∞

e−i|ξ|y∞∫0

gfn,R(√s2 + y2)sn−2 ds dy

= const

∞∫0

cos(|ξ|y)∞∫y

tgfn,R(t)(t2 − y2)n−3

2 dt dy

= const

∞∫0

sin(|ξ|y) 1|ξ|

∞∫1

hfn,R(yt) ν(dt) dy ,

where ν is the δ-measure at point 1 if n = 3,

ν(dt) := t−(n−3)(t2 − 1)n−3

2 −1dt , if n > 3 ,

hfn,R(r) := rn−2gfn,R(r) , r > 0 ,

and we have used integration-by-parts and variable substitutions.The last expression is equal to

const

∞∫1

1|ξ|2

∞∫0

sin v hfn,R

(tv

|ξ|

)dv ν(dt) .

The inner integral may be rewritten as

IfR(|ξ|, t) =∞∑k=0

π∫0

(hfn,R

(t

|ξ|(w + kπ)

)− hfn,R

(t

|ξ|(w + (k + 1)π)

))sinw dw

=∞∑k=0

π∫0

t|ξ| (w+(k+1)π)∫

t|ξ| (w+kπ)

(−hfn,R(r)

)dr sinw dw .

Note that hfn,R(r) = hfn(r) for r < R, where as before hfn(r) = rn−2kfn(r) , r > 0.By Lemma 1.1.5, −hfn is positive. Therefore we choose the function gfn,R(r) as above such that0 < −hfn,R(r) ≤ const(−hfn(r)) , r ≥ R. Then we see formally that

0 < IfR(|ξ|, t) ≤ const If (|ξ|, t)

where If is defined as IfR if the function htn,R is replaced by hfn. Formal substitution in the abovedouble integral yields for such GfR(ξ) = gfn,R(|ξ|) ,

0 < FGfR(ξ) ≤ constFKf (ξ) = const f(|ξ|2)−1

.

These arguments are getting strong when calculating the Fourier transform FKf in terms of

28

distributions: For any Schwartz function ϕ we obtain from the above integral estimates

0 <∫FGfR(ξ) ϕ(ξ) dξ = lim

m→∞

∫const

∫|〈ξ,x〉|<2mπ

e−i〈ξ,x〉GfR(x) dx ϕ(ξ) dξ

≤ limm→∞

const∫ ∫|〈ξ,x〉|<2mπ

e−i〈ξ,x〉Kf (x) dx ϕ(ξ) dξ

= const limm→∞

∫Kf (x)

∫|〈ξ,x〉|<2mπ

e−i〈x,ξ〉ϕ(ξ) dξ dx

= const∫Kf (x) Fϕ(x) dx = const

∫f(|ξ|2)

−1ϕ(ξ) dξ .

which leads to the above estimate of the Fourier transform FGfR.Since for large |ξ|,

f(|ξ|2)−1≤ const(1 + f(|ξ|2)

−1

we infer the upper estimateFGfR(ξ) ≤ const(1 + f(|ξ|2)

−1

for these |ξ|.

We next will show the lower estimate for any GfR as chosen before. The above equations and−ffn,R(r) > 0 lead to

FGfR(ξ) = const1|ξ|2

∞∫1

∞∑k=0

π∫0

sinw

t|ξ| (w+(k+1)π)∫

t|ξ| (w+kπ)

∣∣∣hfn,R(z)∣∣∣ dz dw ν(dt)

≥ const1|ξ|2

2∫1

π∫0

sinw

t|ξ| (w+π)∫

t|ξ|w

∣∣∣hfn,R(r)∣∣∣ dr dw ν(dt) .

Here we may replace hfn,R(r) by hfn(r) if |ξ| > 4πR . Since

∣∣∣hfn(r)∣∣∣ is monotone decreasing (seeLemma 1.1.5) we obtain for the last expression with sufficiently large |ξ|, the lower estimate

const1|ξ|3

∣∣∣∣hfn (4π|ξ|

)∣∣∣∣ ≥ const f(|ξ|2

16π2

)−1

≥ const f(|ξ|2)−1 ,

using the left side of (21) and the monotonicity of f .Consequently,

FGfR(ξ) ≥ const(1 + f(|ξ|2)−1

for these ξ and the assertion is proved.

Proof of Corollary 1.3.3

Recall that for 0 < s < 2 the assertion is covered by Lemma 1.3.2.We next consider the case s = 2:

29

Replacing in proof of Lemma 1.3.2 the kernel GfR by G2R and using the corresponding notations

with index 2 instead of f we obtain from those calculations

FG2R|ξ| =

1|ξ|2

∞∫1

I2R(|ξ|, t) ν(dt)

where

I2R(|ξ|, t) =

∞∫0

sin v h2n,R

(tv

|ξ|

)dv .

Note that in this special case we have h2n,R(r) = const for 0 < r < R. Furthermore, for r ≥ R

the function h2n,R can be chosen monotone decreasing. Recall that it rapidly decreases at infinity.

This yields I2R(|ξ|, t) > 0 and

∞∫1

I2R(|ξ|, t) ν(dt) ≤ const

i.e.,

FG2R|ξ| ≤ const

1|ξ|2

.

since the density of the measure ν is less than const t−2.For the lower estimate we use the monotonicity of h and obtain

FG2R(ξ) ≥ 1

|ξ|2

|ξ|∫1

∞∫0

sin v h2n,R

(tv

|ξ|

)dv ν(dt)

≥ 1|ξ|2

|ξ|∫1

∞∫0

sin v h2n,R(v) dv ν(dt)

=1|ξ|2

const ν([1, |ξ|]) ≥ const1|ξ|2

if |ξ| > 2 . The remaining arguments for s = 2 are as in the case 0 < s < 2.

The case 2 < s < n will be reduced to the above situations:Let 2k < s ≤ 2(k + 1). Then we have for 0 < r < R,

hsn,R(r) = rn−2gsn,R(r) = const rs−2

and for the (2k)-th derivative(hsn,R

)(2k) (r) = const rs−2k−2 , if 0 < r < R .

Note that 0 < s− 2k ≤ 2. Therefore we can choose the kernel gsn,R rapidly decreasing at infinity

together with all its derivatives up to order 2k in such a way that the function(hsn,R

)(2k) (r) hassimilar properties as used for the function hs−2k

n,R (r) in the above cases of the proof. Furthermore,

FGsR(ξ) =1|ξ|2

∞∫1

IsR(|ξ|, t) ν(dt)

30

with

IsR(|ξ|, t) =

∞∫0

sin v hsn,R

(tv

|ξ|

)dv .

2k-fold integration-by-parts yields

IsR(|ξ|, t) =t2k

|ξ|2k

∞∫0

sin v(hsn,R

)(2k)(tv

|ξ|

)dv

and for the Fourier transform

FGsR(ξ) =1

|ξ|2k1|ξ|2

∞∫1

∞∫0

sin v(hsn,R

)(2k)(tv

|ξ|

)dv t2k ν(dt) .

From the first part of the proof applied to(hsn,R

)(2k) instead of hs−2kn,R we now infer

const−1

|ξ|2k+(s−2k)≤ FGsR(ξ) ≤ const

|ξ|2k+(s−2k)

for sufficiently large |ξ| . The remaining arguments (for the behavior at small |ξ|) are as before.

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