friedrich-schiller-universität jena · 2009. 7. 15. · issn 1068-3623, journal of contemporary...

ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2009, Vol. 44, No. 2, pp. 117–145. c© Allerton Press, Inc., 2009.Original Russian Text c© M. Zahle, 2009, published in Izvestiya NAN Armenii. Matematika, 2009, No. 2, pp. 65–96.

Markov Processes

Potential Spaces and Traces of Levy Processes on h-sets

M. Zahle1*

1University of Jena, Mathematical InstituteReceived January 20, 2009

Abstract—Potential spaces and Dirichlet forms associated with Levy processes subordinate toBrownian motion in R

n with generator f(−Δ) 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 with0 < α < 2 corresponding to α-stable Levy processes. For general (tame) Bernstein functions fpotential representations of the trace spaces, the trace Dirichlet forms, and the trace processes onfractal h-sets are derived. Here we suppose the trace condition

∫ 1

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

MSC2000 numbers : 60J45, 28A80, 60G51DOI: 10.3103/S1068362309020071

Key words: Subordinate Levy processes; potential spaces; traces on fractals; time change.

Dedicated to the 80th birthday of Klaus Krickeberg

1. INTRODUCTION

Let h be a continuous non-decreasing positive function on (0, 1] satisfying the doubling conditionh(2r) ≤ const h(r). μ is called an h-measure on R

n if it is a Borel measure with compact support Fsuch 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 (fractal)

set F . Such tracing procedures have been studied in the literature from the analytical as well as from theprobabilistic point of view. Here we continue the papers [23], [24], [9], [14] combining both the analyticaland probabilistic side.

Our starting point is a Levy process Xf in Rn subordinate to Brownian motion according to

a Bernstein function f , i.e., the pseudodifferential operator f(−Δ) with Laplace operator Δ is itsinfinitesimal generator. The domain of the corresponding Dirichlet form is a Bessel-type potential spacein R

n in the sense of [4], [10]. For the special case of symmetric α-stable Levy processes these are

the classical Bessel potential spaces which agree with the Besov spaces Bα/22,2 (Rn). In [14] we have

interpreted the above Bessel-type potential spaces as Besov spaces of generalized smoothness andproved some properties well-known for the classical Besov spaces Bs

p,q(Rn). In particular, a quarkonialrepresentation 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 above

*E-mail: [email protected]

117

118 ZAHLE

Euclidean function spaces. Moreover, under some additional conditions, this is equivalent to an exten-sion of the trace procedure given in [12] for classical Besov spaces. The analytical approach was appliedto determining the domain of the Dirichlet form of the trace of the Levy process Xf on F via time changein the sense of [7], [6].

The aim of the present paper is to complete some results in [14] and to work out the correspondingpotential theory. The related notions and results in R

n are presented in section 1.In 1.1 the subordinate process Xf and some potential properties of independent interest are studied.

In particular, we show that the associated potential operator Uf has a Riesz-type kernel Kf satisfyingKf (x) ≤ const |x|−n f(|x|−2)−1, x �= 0 (under an additional growth condition on f for n ≤ 2). If fsatisfies some lower scaling condition then the opposite estimate for a different constant holds true.(See Theorem 1.1.2). In 1.2 similar estimates for Riesz-type potentials of arbitrary order associated withcomplete Bernstein 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 kernel isconstructed which is more appropriate for our purposes. The relationship between the Dirichlet form ofXf 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 condition∫ 1

0h(r)f(r−2) r−(n+1)dr < ∞.

In 2.1 the trace Hilbert spaces are determined. In particular, we show that the trace operator trμ

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 the f-Riesz-Besselpotential 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 Uf

μ (x) =∫

Kf (x − y)μ(dy) is bounded and uniformly

continuous on the whole Rn. This leads to a pointwise representation of the trace potential operator Uf

μ .The main results are formulated in Theorem 2.2.5. In particular, the scalar product in Hf,1(μ) satisfies

〈u, v〉Hf,1(μ) =⟨√

Dfμu,

√Df

μv

⟩

L2(F,μ)

=: Efμ(u, v),

for Dfμ := (Uf

μ )−1. This is the fractal counterpart to the Euclidean version for the f-Bessel potentialspace Hf,1(Rn).

Relationships to Besov spaces of generalized smoothness and auxiliary tools for sections 1 and 2 arepresented in the Appendix. In particular, the compactness of the operators trμ and Uf

μ on appropriatefunction spaces is shown. Furthermore, we prove global versions of Tauberian-type theorems for theLaplace transform and estimates for the Fourier transform of Riesz-Bessel-type kernels.

In section 3 we infer that Ufμ may be interpreted as the potential operator of the trace of the Levy

process Xf on the h-set F : A trace process Xfμ is obtained from Xf via time change according to

the positive continuous additive functional with Revuz measure μ (in the sense of [7]). For the specialcase of symmetric α-stable Levy processes and d-sets F such a time change was considered in [15],[8], [9]. Here we extend the latter approach. Our main result is that the above quadratic form Ef

μ is theDirichlet 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 Xf

μ and the Dirichlet form satisfies thePoincare inequality. (This also refines the corresponding results from the 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 in terms

JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 44 No. 2 2009

POTENTIAL SPACES AND TRACES OF LEVY 119

of the Levy system and some Feller measures of X. Applied to our situation this leads to a representationof the Dirichlet form Ef

μ by means of those Feller measures of Xf .

2. LEVY PROCESSES SUBORDINATE TO BROWIAN MOTION AND ASSOCIATEDPOTENTIAL SPACES

2.1. 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 (2.4) and (2.6) below considered as definitions for the potential measureVf and the potential kernel Kf associated with the Bernstein function f . The classical analyticalcounterpart concerning Riesz and Bessel potentials may be found in many textbooks on potentialanalysis.

Let (Xf (t))t≥0 be a Levy process subordinate to Brownian motion (B(t))t≥0 in Rn. This means that

the characteristic function is given by

E exp(i〈Xf (t), ξ〉) = exp(−tψ(ξ)) (2.1)

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 f

t )t≥0 of Xf is given by f(−) (for the Laplace operator )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 Laplace

transform is given by∫

e−rsηft (ds) = e−tf(r) . (2.2)

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

T ft u(x) =

∫ ∞

0

∫

Rn

u(y) ps(x − y) dy ηft (ds) (2.3)

where

pt(x) := π−n/2 t−n/2 e−|x|2/2t

denotes the density of the distribution of B(t).

Let Rfλ, Rηf

λ and Uf := Rf0 , Vf := Rηf

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

Rfλu(x) =

∫ ∞

0e−λt T f

t u(x) dt

Rηf

λ u(r) =∫ ∞

0e−λt

∫u(r − s) ηf

t (ds) dt, λ ≥ 0,

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

∫ ∞

0e−rs Vf (ds) =

1f(r)

. (2.4)

The distribution function of Vf will be denoted by

V f (s) :=∫

1[0,s] dVf , s ≥ 0 .


120 ZAHLE

Furthermore, the potential operator of Xf is representable as

Ufu(x) =∫

Rn

u(y)Kf (x − y) dy

with density kernel

Kf (x) =∫ ∞

0ps(x) dV f (s) . (2.5)

Convergence of the integral (under an additional condition on f if n ≤ 2) follows from Theorem 1 below.

The analogue of equation (2.4) then reads:

FKf (ξ) = f(|ξ|2)−1 (2.6)

where F denotes the Fourier transform, in general, in the distributional sense. Therefore Kf has a radialrepresentation:

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 results andreferences cf. [10, 4.9]). In particular, f(r) is then a Bernstein function itself. Moreover, a function 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 (twoversions):

f(λr) ≥ const λδf(r), λ ≥ 1, r > 1/r0 (2.7)

(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. (2.8)

Remark 1. (i) Complete Bernstein functions are characterized by the representation

f(r) = a + br +∫ ∞

0(1 − e−rs)

∫ ∞

0+e−stτ(dt) ds (2.9)

for some a, b > 0 and a measure τ with∫ 10+ t−1τ(dt) +

∫ ∞1 t−2τ(dt) < ∞ (cf. Schilling [18]

or [10, Theorem 4.9.29]).

Then a sufficient condition for (2.7) is that τ has a density τ satisfyingτ(λt) ≥ const λδ τ(t), t > 0, λ ≥ 1, for some 0 < δ ≤ 1.

(ii) Condition (2.7) in terms of f(r) := rf(r)−1 reads as follows:

f(λr) ≤ const λ1−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 ).

Theorem 1. (i) For any Bernstein function f and n ≥ 3 we have

limr→∞

r−nf(r−2)−1 = 0. (2.10)



(ii) Under the condition (2.10) (which is restrictive only for n ≤ 2) the potential density kfn of

the associated subordinate Levy process Xf in Rn, i.e. of the potential operator Kf given by

(2.6) possesses the following properties:

kfn(r) ≤ const r−nf(r−2)−1, r > 0. (2.11)

kfn is differentiable and

(kfn)(r) = −2πr kf

n+2(r). (2.12)

(iii) If the Bernstein function f satisfies the lower scaling condition (2.7) for r > 1/r0 (or r < r0)and the associated potential measure Vf determined by (2.4) has a monotone density V f

then the opposite estimate is valid:

kfn(r) ≥ const r−nf(r−2)−1, r < r0, (2.13)

(resp. r > 1/r0).In particular, if (2.7) is given on (0,∞), i.e., r0 = ∞, then (2.13) 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 monotone

decreasing.

Proof. (i): We use the representation (2.5),

πn/2kfn(r) =

∫ ∞

0s−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. (2.14)

Furthermore, substituting in (2.8) 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. (2.15)

In particular,

lim sups→∞

s−1f(s−1)−1 < ∞ (2.16)

which proves (i).(ii): (2.14) and (2.15) 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/2ds = const r−n f(r−2)−1.

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

∫ ∞

0s−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)


122 ZAHLE

= 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 (2.14), (2.15), (2.10) 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

0ϕr(s) dV f (s) = V f (s)ϕr(s)

⏐⏐⏐⏐

r2

0

−∫ r2

0V f (s) ϕr(s) ds

In view of (2.14) the first summand does not exceed const r−n f(r−2)−1. Moreover,∣∣∣

∫ r2

0V f (s) ϕr(s) ds

∣∣∣ ≤ V f (r2)

∫ r2

0|ϕr(s)| ds

≤ const f(r2)−1 r2

∫ r2

0s−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 (2.12) 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 yields theestimates

(const)−1r−1f(r−1)−1 ≤ V f (r) ≤ const r−1f(r−1)−1 (2.17)

for r < r0 (resp. r > 1/r0) on the left side and r > 0 on the right side. The assertion (2.13) is now aconsequence of (2.17) and (2.5):

const rn−2kfn(r) = rn−2

∫ ∞

0s−n/2 e−r2/2s dV f (s)

≥ rn−2

∫ r0

0s−n/2 e−r2/2s V f (s) ds

=∫ r0/r2

0s−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 (2.17) for r < r0 we infer for the case r2 < r0,

const rn−2kfn(r) ≥ const

∫ 1

0s−n/2e−1/2s V f (r2s) ds

≥∫ 1

0s−n/2 e−1/2sr−2s−1f(r−2s−1)−1 ds

≥ r−2f(r−2)−1

∫ 1

0s−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∫ r0

0 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 this casewe have rf(r)−1 = r2L(g)(r) for some Bernstein function g, i.e.,

f(r)−1 = r L(g)(r) = r

∫ ∞

0e−rsg(s) ds.

On the other hand,

f(r)−1 =∫ ∞

0e−rsdV f (s) = r

∫ ∞

0e−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.

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

Kα(x) =const|x|n−α

. (2.18)

Its generator is equal to (−)α/2. In analogy to this case we call for a Bernstein function f thefunction Kf the f-Riesz kernel (with generator f(−)). 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−2kf

n(r) (2.19)

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

f(r) ≤ θ r−1f(r), r > 1/r0, (2.20)

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 (2.20) is fulfilled for θ = α/2. More

generally, let f be a complete Bernstein function with representation (2.9). Then a sufficient condition for(2.20) 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 Bernsteinfunction. 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.

Definition 1. We say that a Bernstein function f is tame if it satisfies the lower scaling condition(2.7), the upper scaling condition (2.20), and −V f is nonnegative and monotone decreasing. Forn ≤ 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 provides aclass of tame Bernstein functions.

Lemma 1. If f is a tame Bernstein function, then the derivative of the auxiliary function hfn(r) =

rn−2kfn(r) satisfies

(const)−1r−3f(r−2)−1 ≤ −hfn(r) ≤ const r−3f(r−2)−1 (2.21)

for r > 0 on the right side and for r < r0 on the left side, where r0 is from condition (2.20).Moreover, −hf

n is monotone decreasing.


124 ZAHLE

Proof. Recall that

f(r)−1 =∫ ∞

0e−rs V f (s) ds.

Therefore we choose in Theorem B1,(iii) in the Appendix w := 1/f and v := V f . The condition that−V f is nonnegative and monotone decreasing implies the same properties for −v. Furthermore, thelower scaling condition (2.7) on f yields that on w, and w = −f/f2 together with the upper scalingproperty (2.20) 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 (2.22)

for r > 0 on the right side and r < r0 on the left side.

Now we use the representation

hfn(r) = rn−2kf

n(r) = const rn−2

∫ ∞

0s−n72 e−r2/2s dV f (s)

= const∫ ∞

0s−n/2 e−1/2s V f (r2s) ds.

Then the derivative of hfn equals

hfn(r) = const r

∫ ∞

0s−n/2e−1/2sV f (r2s) ds. (2.23)

Consequently, −hfn is positive and monotone decreasing. Moreover, completely analogous arguments

as in the proof of Theorem 1.1.2, (ii) and (iii), show that (2.22) and (2.23) lead to the desired estimates.

2.2. Riesz-type potentials of arbitrary order

We now turn to Riesz-type potentials where the Fourier transform of the associated kernel Kf,σ(x) =kf,σ

n (|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 the lowerscaling property (2.7). Note that in the case 0 < σ ≤ 2, f(r)σ/2 is also a complete Bernstein function(see [10, Theorem 4.9.29,3]). Obviously, this function fulfills (2.7) if f does so. Therefore Theorem 1.1.2is applicable and we obtain upper and lower estimates for the kernel kf,σ

n . We will show now that underthe condition (2.7) on f these estimates remain true for arbitrary 0 < σ < n.

Theorem 2. Let f be a complete Bernstein function satisfying

f(λr) ≥ const λδf(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 for any0 < σ < n the function f(|ξ|2)−σ/2 is the Fourier transform of a radial function Kf,σ(x) = kf,σ

n (|x|)such that

(const)−1r−nf(r−2)−σ/2 ≤ kf,σn (r) ≤ const r−nf(r−2)−σ/2. (2.24)

kf,σn is differentiable and kf,σ

n (r) = −2πr kf,σn+2(r).

Remark 3. (i) An analysis of the following proof shows that we can replace the global lowerscaling condition on the Bernstein function f by one of its local versions from (2.7) andobtain the estimate (2.23) 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 implies the assertion. Thus, for the first part ofthe theorem it suffices to consider the case σ = 2mα for 0 < α ≤ 1, m ∈ N, 2mα < n, and to prove theexistence of kf,2mα

n and the estimate

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

Consider the kernel kf,2αn = kfα

n associated with fα. According to Theorem 1 it exists and satisfies(2.24). Furthermore,

FKf,2mα(ξ) = f(|ξ|2)−αm =(∫ ∞

0e−|ξ|2s Vα(ds)

)m

where for brevity Vα here denotes the potential measure Vfαof fα. Let Vα,m be the convolution of order

m of the measure Vα. Then we obtain

f(r)−αm =∫ ∞

0e−rsVα,m(ds) (2.25)

and

Kf,2mα(x) = π−n/2

∫ ∞

0s−n/2e−|x|2/2s Vα,m(ds). (2.26)

(The finiteness of the integral will be shown below. Calculating its distributional Fourier transform leadsto 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

∫ ∞

0s−n/21 exp{−|x1|2/2s1}V α(r2s1) ds1

. . .

∫ ∞

0s−n/2m−1 exp{−|xm−1|2/2sm−1}V α(r2sm−1) dsm−1

∫ ∞

0s−n/2m exp

{−

∣∣e −

m−1∑

1

xi

∣∣2/2sm

}V α(r2sm) dsm dxm−1 . . . dx1.

From the left side of (2.17) we infer that∫ ∞

0

∫ ∞

0s−n/2 exp{−|x|2/2s}V α(r2s) ds


126 ZAHLE

≥ const r−2f(r−2)−α

∫ 1

0s−n/2 exp{−|x|2/2s} ds

=: const r−2f(r−2)−α

∫ 1

0F (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 (2.17) for

const∫ ∞

0s−n/2 e−|x|2/2s V α(r2s) ds

≤∫ ∞

0s−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 (2.8). For s ≤ 1 by the lower scaling assump-tion (2.7) on f , f(r−2s−1)−α ≤ const sαδf(r−2)−α. Therefore the above integral does not exceed

const r−2f(r−2)−α

∫ ∞

0max(sαδ−1, sα−1) s−n/2 e−|x|2/2s ds

=: r−2f(r−2)−α

∫ ∞

0G(s, |x|) ds.

Applying this inequality to each of the inner integrals in the above expression for kf,2mαn we get in the

same way as for its lower estimate the upper estimate

const r−nf(r−1)−mα.

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

∫ ∞

0s−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,σ

n+2(r).



2.3. Resolvents and Bessel-type potentials

Recall that the resolvent Rfλ of the operator f(−Δ) (or of the Levy process Xf ) is the inverse of the

operator λ id +f(−Δ). For λ = 1 and the classical case f(r) = rα/2 , 0 < α ≤ 2, the operator Rf1 has

a kernel whose Fourier transform (1 + |ξ|α)−1 is equivalent to (1 + |ξ|2)−α/2. The latter coincides upto a constant with the Fourier transform of the Bessel potential kernel Gα/2(x) in R

n. It is knownthat Gα/2(x) rapidly decreases as x → ∞ and is equivalent to the Riesz kernel Kα(x) if x → 0. (Theseoperator 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 call thefunction

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,s

n (|x|).For s = 2 we use again the superscript f instead of (f, 2).

Then Theorem 1 and the local version of Theorem 2 imply the equivalence of the Riesz-type and theBessel-type kernels in a neighborhood of 0:

Corollary 1. 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.

(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 Levy processes we need the following modification of the (f, 2)-Besselkernel Gf with Fourier transform

FGf (ξ) = (1 + f(|ξ|2))−1.

Lemma 2. For any tame Bernstein function f in the sense of Definition 1 and R > 0 there existsa positive continuous radial function Gf

R on Rn \ {0} rapidly decreasing at infinity such that 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. (2.27)


128 ZAHLE

The proof is given in Appendix C. We conjecture that this result may be extended to Gf,s and Kf,s forarbitrary 0 < s < n. For the classical case f(r) = r this is true: Recall that the classical Riesz kernelof order s is given by K id,s(x) = Kf (x) = const |x|−(n−s) with Fourier transform FKs(ξ) = |ξ|−s.The corresponding Bessel kernel Gs(x) = gs

n(|x|) of order s satisfies FGs(ξ) = (1 + |ξ|2)−s/2, where0 < s < n.

Corollary 2. For any 0 < s < n and R > 0 there exists a positive continuous radial function GsR

on Rn \ {0} rapidly decreasing at infinity such that for |x| < R, Gs

R coincides with the Riesz kernelKs(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 special caseof Lemma 2.

2.4. 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(−Δ)u(x)

√f(−Δ)v(x) dx.

(2.28)

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. (2.29)

(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]).By Lemma 2 for 0 < s ≤ 2 and tame Bernstein functions f the norm in Hf,s(Rn) is equivalent to

||u|Hf,s(Rn)|| ∼(∫

Rn

(FGf,sR (ξ))2 |Fu(ξ)|2 dξ,

)1/2(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 (2.31)

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

corresponding scalar product∫

Rn

(FGf,sR (ξ))2 Fu(ξ)Fv(ξ) dξ (2.32)

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,1

R

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 .



3. TRACES ON h-SETS

3.1. Traces of potential spaces

Let now F ⊂ Rn be a compact h-set with h-measure μ as in the introduction. (Note that all such

μ are equivalent to the Hausdorff measure Hh|F .) Throughout this section we assume that f is a tameBernstein function in the sense of Definition 1.

Choose an arbitrary R > 0 such that F is contained in the ball B(0, R). We are interested in traces on F

of the Bessel-type potential spaces Hf,1R (Rn) equivalent to Hf,1(Rn) (cf. (2.32)). In [14] under the trace

condition∫ 1

0

h(r)rn+1 f(r−2)

dr < ∞ (3.1)

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 (3.1) is the integral version of condition (2.29) 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 put

trμu := u|F .

In view of Theorem A2 in the Appendix we have

|| trμu|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

extension

trμ : Hf,1R (Rn) → L2(F, μ) =: L2(μ).

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

Theorem 3. If the tame Bernstein function f and the gauge function h for the measure μ satisfythe trace condition (3.1) then the trace space

Hf,1R (F, μ) = Hf,1

R (μ) := trμ(Hf,1R (Rn)) (3.2)

is determined. Moreover,

|| trμv|L2(Fμ)|| ≤ const ||v|Hf,1R (Rn)|| . (3.3)

The norm

||u|Hf,1R (μ)|| := inf

{||v|Hf,1

R (Rn)|| : v ∈ Hf,1R (Rn) , trμv = u

}(3.4)

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

||u|L2(μ)|| ≤ const ||u|Hf,1R (μ)|| . (3.5)

Remark 4. (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 version thetheorem can be proved as in [22, Theorem 7.16] by means of wavelet expansions with amore explicit representation of the operator trμ.


130 ZAHLE

(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. (3.10)). 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) : trμu = 0}⊕ Hf,1(μ)

Hf,1(Rn) ={u ∈ Hf,1(Rn) : trμu = 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,(3.6)

are determined by these decompositions.

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

3.2. 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,1

R (Rn) of (f, 1)-Riesz-Bessel potentials in the sense of norm equivalence (cf. (2.29)-(2.32)). In [23]for the classical case f(r) = rα/2 and d-sets F a similar potential representation is derived for the tracespaces Hf,1(μ). We now will extend this potential approach to arbitrary (tame) Bernstein functions fand h-measures μ.

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

Ufμ (x) :=

∫Kf (x − y)μ(dy), x ∈ R

n.

For the next property we need only the upper h-regularity of the measure μ: μ(B(x, r)) ≤ consth(r).

Proposition 1. Under the growth condition (2.10) on f , if n ≤ 2, and the trace condition (3.1) onf and h, the function Uf

μ is bounded and uniformly continuous.

Proof. The boundedness easily follows from Theorem 1: Denoting mx(r) := μ(B(x, r)) we get forrmax := diam(F ),

∫Kf (x − y)μ(dy) =

∫ rmax

0kf

n(r) dmx(r)

= mx(r) kfn(r)|rmax

0 −∫ rmax

0(kf

n)′ (r)mx (r) dr

= mx(r) kfn(r)|rmax

0 +∫ rmax

0const r kf

n+2 (r)mx (r) dr

according to (2.12). Since mx(r) < ch(r) and kfn(r) < const r−nf(r−2)−1 by (2.11), the trace condition

∫ 1

0h(r) r−(n+1) f(r−2)−1 dr < ∞



implies the boundedness of the second summand. Moreover, the convergence of this integral leads to

limr→0

mx(r) kfn(r) = 0.

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

|Ufμ (x) − Uf

μ (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 and the continuity of Kf (x) outside x = 0, for fixed k ∈ N the first summandtends to 0 as |x − z| → 0. Furthermore, by the same theorem and the monotonicity of 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 the tracecondition (3.1). 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 → ∞ uniformlyin x, z. By Theorem 1,

∫

|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 (2.8) of the Bernstein function f one easily deduces that the function rn f(r−2) is


132 ZAHLE

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 summands tendto 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 2.1 on the trace space. Note that Uf may also be considered as an operator actingon distributions w in R

n.For any u ∈ L2(μ) the distribution idμ u := uμ is given by uμ(ϕ) :=

∫ϕ(x)u(x)μ(dx) for Schwartz

functions ϕ. In the proof of Corollary 4 in the Appendix it is shown that for the corresponding f-Riesz-Bessel potential operator Uf

R in the case of tame Bernstein functions f we have the embedding

UfR ◦ idμ : L2(μ) → Hf,1

R (Rn).

Therefore we introduce the operator

Ufμ := trμ ◦ Uf

R ◦ idμ (3.7)

from L2(μ) into Hf,1R (μ). Uf

μ 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 1 implies

that in the definition (3.7) of Ufμ u for bounded u the trace is pointwise determined:

Corollary 3. Under the trace condition (3.1) for any bounded Borel function u on F we have thefollowing:

(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 (2.10) if n ≤ 2).

In the sequel Uf ◦ idμ u and Ufμu will be understood in this sense. In section 3 below we will prove

that Ufμ is the potential operator of the trace of the Levy process Xf on the h-set F . To this aim we now

introduce the associated quadratic form using the following auxiliary results which are of independentinterest.Throughout the rest of this section we assume the trace condition (3.1)and f to be tame.Using Corollary 4 from the Appendix one derives as in [23, Theorem 3.1] the following.

Lemma 3.√

Ufμ maps L2(μ) onto Hf,1

R (μ) and is an isometry, i.e.,

⟨√Uf

μ u,

√Uf

μ v⟩

Hf,1R (μ)

= 〈u, v〉L2(μ), (3.8)

u, v ∈ L2(μ).



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

Lemma 4.

〈Ufμ u, v〉L2(μ) =

⟨√Uf (uμ),

√Uf (vμ)

⟩

L2(Rn), (3.9)

u, v ∈ L2(μ).

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

Equation (3.8) implies that the operators√

Ufμ and Uf

μ are invertible. Denote

Dfμ := (Uf

μ )−1,

hence,√

Dfμ =

(√Uf

μ

)−1.

Moreover, from (3.8) we conclude that the domain of√

Dfμ is Hf,1

R (μ) and the scalar product in thisspace does not depend on the choice of R:

〈u, v〉Hf,1

R (μ)=

⟨√Df

μu,

√Df

μv⟩

L2(μ), (3.10)

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

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

R (Rn)). For thescalar product in the classical version trμ (Hf,1(Rn)) formula (3.10) does not hold as equality. Here wehave only equivalence).

As in the special case of d-sets and f(r) = rα/2, Dfμ may be interpreted as fractal pseudodifferential

operator. The Euclidean counterpart is

Df :=(Uf )−1 = f(−),√

Df =√

f(−).

In these terms formula (3.10) may be rewritten as⟨√

Dfμu,

√Df

μv⟩

L2(μ)=

⟨√Df u,

√Df v

⟩

L2(Rn)

if

u = Ufμ u′, v = Uf

μ v′ ,(3.11)

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 �→ u in (3.11) generalizesto a linear Riesz-type extension operator

extfμ : Hf,1(μ) → Lf,1

2 (Rn) (3.12)

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 section 3.1.

Finally, we consider the quadratic forms

Ef (u, v) =⟨√

Df u,√

Df v⟩

L2(Rn)


134 ZAHLE

in L2(Rn) with domain Hf,1(Rn) (cf. (2.28)) and

Efμ (u, v) :=

⟨√Df

μu,

√Df

μv⟩

L2(μ)(3.13)

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 (3.10) 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 the

corresponding 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.

Theorem 4. For any tame Bernstein function f in the sense of 1 and any gauge function hsatisfying the trace condition (3.1) we have:

(i)

Efμ(u, v) = 〈u, v〉

HfR(μ)

= 〈Extfμ,R u,Extf

μ,R v〉Hf,1

R (Rn)

for the Riesz-Bessel-type extension operator Extfμ,R from (3.6). In particular, the scalar

product in Hf,1R (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,1

R (μ).

(iii)

Efμ (u, v) = Ef (extf

μ u, extfμ v),

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

2 (Rn) determinedby

extfμ u(x) =

∫Kf (x − y)u′(y)μ(dy), x ∈ R

n,

if u(x) =∫

Kf (x − y)u′(y)μ(dy), x ∈ F , for some bounded Borel function u′ on F , and bycontinuation to all u according to the above quadratic form.

(iv)

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

μ (3.14)

(Poincare inequality).

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



4. 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 theinfinitesimal generator of the primary Levy process Xf and Ef is its Dirichlet form. We now will showthat the quadratic form Ef

μ from (3.13) is the Dirichlet form of the trace of Xf on the fractal h-set F . Asin the special case of d-sets and f(r) = rα/2 (see Kumagai [15], Fukushima and Uemura [8], Hansenand Zahle [9]) we use the method of time change for Markov processes.Under the trace condition (3.1) 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 6. (A more direct proof can be given as in Triebel [22, Remark 7.15] forthe special case f(r) = rα/2 and more general μ.) Therefore the h-measure μ can be interpreted as theRevuz measure of a finite positive continuous additive (random) functional (briefly PCAF) Lf

μ w.r.tthe primary Levy process Xf : According to the Theorem 5.1.1 and 5.1.2. and Lemma 5.1.3 in [7], Lf

μ isdetermined (up to equivalence) by the property that for any nonnegative bounded Borel function w andλ > 0,

Ex

( ∫ ∞

0e−λt w(Xf (t)) dLf

μ(t))

is a quasi continuous version of Rfλ(wμ) (for the resolvent Rf

λ = (id λ + Df )−1 of the process Xf ).

Definition 2. The PCAF Lfμ(·) is called the μ-local time of the process Xf on the fractal support

F of μ.

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

E(·)

(∫ ∞

0w(Xf (t)) dLf

μ(t))

= Uf (wμ)

Ef -quasi everywhere. According to Proposition 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 for any

nonnegative bounded Borel function w and any x ∈ F ,

Ex

( ∫ ∞

0w(Xf (t)) dLf

μ(t))

=∫

Kf (x − y)w(y)μ(dy) = Ufμ w(x).

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

μ obtainedas follows: Set

τ fμ (t) := inf{s > 0 : Lf

μ(s) > t} and Xfμ(t) := Xf (τ f

μ (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(FF ) = 0. Its transition function determines a strongly continuous

Markov semigroup (Tt)t≥0 on L2(μ). By construction,∫ ∞

0Ttw(x) dt = Uf

μw(x), x ∈ F,

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

(Tt

)t≥0

.


136 ZAHLE

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

μ. Therefore in this cas thethe corresponding Dirichlet form is given by

Efμ (u, v) =

⟨√Df

μ u,

√Df

μ v⟩

L2(μ).

The domain of Efμ agrees with the trace space Hf,1(μ) and Ef

μ is regular. In view of [7, Theorem 7.2.1],Ef

μ determines a μ-symmetric Hunt process Xfμ on F . We call Xf

μ the μ-trace of the Levy process Xf

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

μ on F .Summarizing the above results we obtain:

Theorem 5. Let μ be an h-measure with compact support F and Xf be a Levy process in Rn

subordinate to Brownian motion according to the tame Bernstein function f . Under the tracecondition (3.1) the following holds true.

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

μ , where

Ufμ w(x) =

∫Kf (x − y)w(y)μ(dy), x ∈ F,

for any bounded Borel function w.

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

Efμ (u, v) =

⟨√Df

μ u,

√Df

μv⟩

L2(μ)

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

Efμ satisfies the Poincare inequality (3.14).

(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 (3.1). Ufμ defined as in (i) is its potential operator (provided the growth condition

(2.10) is fulfilled if n ≤ 2).

Remark 5. A probabilistic interpretation of Efμ as the time changed Dirichlet form and of the trace

space 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 Ef

μ introduced in Section 3. 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.

AppendixASSOCIATED BESKOV SPACES OF GENERALIZED SMOOTHNESS

In [14] an interpretation of the potential spaces Hf,1(Rn) as Besov spaces of generalized smoothnessBσ,N

2,2 (Rn) with σj = 2j , Nj =√

f−1(22j), j ∈ N0, was used. (For a survey on spaces of generalizedsmoothness, the corresponding notions and related results we refer to Farkas and Leopold [5] and toBricchi [2].) The choice of the admissible sequence σ and the strongly increasing sequence N is notunique. For our purposes it is more convenient to work with the classical version Nj = 2j . Then thecorresponding σ is determined by

σj :=√

f(22j) . (A.1)



(The norm equivalence follows from comparing the Fourier analytical definitions of Hf,1(Rn) andBσ,N

2,2 (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 (2.8).

As in the literature we write Bσ2,2(R

n) for the space Bσ,N2,2 (Rn) with Nj = 22j and choose σ according to

(A.1).

For a gauge function h as in Section 1 we define the admissible sequence σ∗ by

σ∗j :=

σ√2j

√h(2−j)

. (A.2)

Then (σ∗j )

2 = f(22j )2j h(2−j)

, and therefore

∞∑

j=0

(σ∗j )

−2 < ∞ (A.3)

is a discrete version of the trace condition (3.1). In [14, Section 2] for σ∗ and N as above (and moregeneral N ) the Besov spaces of generalized smoothness Bσ∗,N

2,2 (F ) =: Bσ∗2,2(F ) on the fractal support F

of μ are introduced by means of quarkonial decompositions. Moreover, the trace operator as a continuousextension from Schwartz functions to Bσ

2,2(Rn) and in the more explicit quarkonial decomposition exists:

[14, Theorem 15] implies the following.

Proposition 2. Under the trace condition (A.3) we get for σj =√

f(22j) and σ∗j given by (A.2),

trμ(Bσ2,2(R

n)) = Bσ∗2,2(F ) .

Remark 6. Since Bσ2,2(R

n) agrees with the potential space Hf,1(Rn), for tame Bernstein functions

f 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(R

n) into L2(μ). Recall that for u ∈ L2(μ) the measure uμ

may be considered as a tempered distribution idμ u defined by

idμ u(ϕ) =∫

ϕ u dμ =∫

trμϕ 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 Bσ

2,2(Rn)′ = Bσ−1

2,2 (Rn) (see, e.g., [5, Theorem 3.1.10] in a more generalcontext).

With the method of wavelet expansions used in Triebel [22, Theorem 7.16] (for the classical caseσj = 2js) one proves the following:

Theorem 6. Under the condition (A.2) and (A.3) on an admissible sequence σ the operators

trμ : Bσ2,2(R

n) → L2(μ) (A.4)

idμ : L2(μ) → Bσ−1

2,2 (Rn) (A.5)

are compact.


138 ZAHLE

Next we consider the potential operator UfR given by

UfR u(x) :=

∫Gf

R(x − y)u(y) dy (A.6)

for the f-Riesz-Bessel kernel GfR from 2. Similarly as for the classical spaces B−s

2,2(Rn) and Bs

2,2(Rn)

(cf. Triebel [19]) one shows for the sequence σ from (A.1) and tame Bernstein functions f the liftingproperty

UfR : Bσ−1

2,2 (Rn) → Bσ2,2(R

n)

in terms of Fourier representations. From this and Theorem 6 we obtain the following.

Corollary 4. For any tame Bernstein function f the operator

Ufμ = trμ ◦ Uf

R ◦ 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).

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(R

n) bounded,

trμ : Hf,1R (Rn) → Hf,1

R (μ) bounded

and the norm equivalence for Bσ2,2(R

n) and Hf,1R (Rn). Furthermore, for

B := UfR ◦ idμ ◦ trμ

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

〈 trμu, trμv〉L2(μ) = 〈B u, v〉Hf,1

R (Rn),

since the scalar product in Hf,1R (Rn) is given by 〈(Uf

R)−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,1R (Rn)|| = || trμu|L2(μ||,

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

Hf,1R (Rn) = N(B) ⊕ Hf,1

R (μ)

and

N(B) = {u ∈ Hf,1R (Rn) : trμu = 0},

where N(B) is the null space of the operator B.Since B = Uf

R ◦ idμ ◦ trμ and Ufμ = trμ ◦ Uf

R ◦ idμ, this yields

〈B u, v〉Hf,1

R (Rn)= 〈Uf

μ u, v〉Hf,1

R (μ),

hence,

〈Ufμ u, v〉

Hf,1R (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 (3.6)).



AppendixA GLOBAL TAUBERIAN-TYPE THEOREM FOR THE LAPLACE TRANSFORM

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

Theorem 7. Suppose that w is a positive continuous function on (0,∞) with existing Laplacetransform

w(x) =∫ ∞

e−xyv(y)dy, x ≥ 0.

Let x0 ∈ (0,∞]

(i) If v is monotone then we have

v(x) ≤ const x−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) ≤ const λ−δw(x) , λ ≥ 1, x > 1/x0, (B.1)

then we have

v(x) ≥ const x−1w(x−1), x ≤ x0.

(iii) Suppose that the condition of (ii),(b) is satisfied, v is continuously differentiable withmonotone decreasing |v|, and

|w(x)| ≤ θx−1w(x), x > 1/x0,

for some θ < 1. Then we have

|v(x)| ≥ const x−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) =∫ ∞

0e−xy v(y) dy ≥ 1

x

∫ ∞

0e−y v

(y

x

)dy

=1x

∫ ∞

ze−y v

(y

x

)dy ≥ 1

xv

( z

x

)e−z

by monotonicity. Hence,

v( z

x

)≤ ez x w(x), x > 0, z > 0. (B.2)

This implies

v(x) ≤ e x−1w(x−1), x > 0,

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


140 ZAHLE

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

x w(x) =∫ z

0e−y v

(y

x

)dy +

∫ ∞

ze−y v

(y

x

)dy

≤ v( z

x

)+

∫ ∞

ze−y e−y/2 x

2w

(x

2

)dy

≤ v( z

x

)+ w

(x

2

) x

24e−z/2

≤ v( z

x

)+ a w(x) x e−z/2

for some a > 0 by monotonicity for the first integral and (B.2) for the second one and by the doublingcondition on w. Hence, choosing z sufficiently large we get

v( z

x

)≥ (1 − ae−z/2) x w(x),

i.e.,

v(x) ≥ (1 − ae−z/2)z

xw

( z

x

)

≥ const1x

w

(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

∫ ∞

0e−y v

(y

x

)dy ≥ 1

x

∫ z

0e−yv

(y

x

)dy

≥ 1x

v( z

x

)(1 − e−z)

which yields

v( z

x

)≤ 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 ,

and

v(y

x

)≤ (1 − e−1)−1 x

yw

(x

y

)

, x > 0, y > 0. (B.3)

Furthermore,

x w(x) =∫ z

0e−x v

(y

x

)dy +

∫ ∞

ze−y v

(y

x

)dy

≤∫ z

0e−y v

(y

x

)dy + v

( z

x

)e−z

≤ (1 − e−1)−1 x

∫ z

0e−y 1

yw

(x

y

)

dy + v( z

x

)e−z

because of (B.3). 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

0e−y yδ−1 dy + v

( z

x

)e−z



for some constant b > 0. Choosing z small enough so that b∫ z0 e−y yδ−1 dy < 1

2 we conclude

v( z

x

)≥ const x w(x) , x > 1/x0 ,

for some z < 1. Hence,

v(x) ≥ constz

xw

( z

x

)

≥ const1x

w

(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)|. Thenwe 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) =∫ ∞

0e−y v

(y

x

)dy

we obtain

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

∫ ∞

0e−y y

(−v

(y

x

))dy.

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

(1 − θ)w(x) ≤ 1x2

∫ ∞

0e−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 monotonicity of |v|and regarding the factor e−yy instead of e−y under the integral.

Remark 7. 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.

AppendixESTIMATES FOR THE FOURIER TRANSFORMATION OF RIESZ-BESSEL-TYPE

KERNELS

Proof of Lemma 2Let Gf

R(x) =: gfn,R(|x|) be an arbitrary radial C1-function on R

n > {0} rapidly decreasing at infinity

such that GfR(x) = kf

n(|x|), 0 < |x| < R, for the f-Riesz kernel kfn. If suffices to find such a version that

the desired estimates for the Fourier transform FGfR(ξ) hold for sufficiently small and sufficiently large

|ξ|. The case of small ξ is evident because of the rapid decay at infinity.

Since GfR is radial, FGf

R(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

∫ ∞

0gfn,R(

√s2 + y2)sn−2 ds dy

= const∫ ∞

0cos(|ξ|y)

∫ ∞

ytgf

n,R(t)(t2 − y2)n−3

2 dt dy


142 ZAHLE

= const∫ ∞

0sin(|ξ|y)

1|ξ|

∫ ∞

1hf

n,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−2gf

n,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

∫ ∞

0sin v hf

n,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)))

sin w dw

=∞∑

k=0

∫ π

0

∫ t|ξ| (w+(k+1)π)

t|ξ| (w+kπ)

(−hf

n,R(r))

dr sin w dw .

Note that hfn,R(r) = hf

n(r) for r < R, where as before hfn(r) = rn−2kf

n(r) , r > 0.

By Lemma 1, −hfn is positive. Therefore we choose the function gf

n,R(r) as above such that 0 <

−hfn,R(r) ≤ const(−hf

n(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 ht

n,R is replaced by hfn. Formal substitution in the above double

integral yields for such GfR(ξ) = gf

n,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 distribu-tions: For any Schwartz function ϕ we obtain from the above integral estimates

0 <

∫FGf

R(ξ) ϕ(ξ) dξ = limm→∞

∫const

∫

|〈ξ,x〉|<2mπe−i〈ξ,x〉Gf

R(x) dx ϕ(ξ) dξ

≤ limm→∞

const∫ ∫

|〈ξ,x〉|<2mπe−i〈ξ,x〉Kf (x) dx ϕ(ξ) dξ

= const limm→∞

∫Kf (x)

∫

|〈ξ,x〉|<2mπe−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 estimate

FGfR(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 −f f

n,R(r) >

0 lead to

FGfR(ξ) = const

1|ξ|2

∫ ∞

1

∞∑

k=0

∫ π

0sin w

∫ t|ξ| (w+(k+1)π)

t|ξ| (w+kπ)

∣∣∣hf

n,R(z)∣∣∣ dz dw ν(dt)

≥ const1|ξ|2

∫ 2

1

∫ π

0sin w

∫ t|ξ| (w+π)

t|ξ|w

∣∣∣hf

n,R(r)∣∣∣ dr dw ν(dt).

Here we may replace hfn,R(r) by hf

n(r) if |ξ| > 4πR . Since

∣∣∣hf

n(r)∣∣∣ is monotone decreasing (see Lemma 1)

we obtain for the last expression with sufficiently large |ξ|, the lower estimate

const1|ξ|3

∣∣∣∣h

fn

(4π|ξ|

)∣∣∣∣ ≥ const f

(|ξ|216π2

)−1

≥ const f(|ξ|2)−1 ,

using the left side of (2.21) 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 2Recall that for 0 < s < 2 the assertion is covered by Lemma 2. We next consider the case s = 2:

Replacing in proof of Lemma 2 the kernel GfR by G2

R and using the corresponding notations with index2 instead of f we obtain from those calculations

FG2R|ξ| =

1|ξ|2

∫ ∞

1I2R(|ξ|, t) ν(dt)

where

I2R(|ξ|, t) =

∫ ∞

0sin v h2

n,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

∫ ∞

1I2R(|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

∫ ∞

0sin v h2

n,R

(tv

|ξ|

)

dv ν(dt)

≥ 1|ξ|2

∫ |ξ|

1

∫ ∞

0sin v h2

n,R(v) dv ν(dt)

=1|ξ|2 const ν([1, |ξ|]) ≥ const

1|ξ|2


144 ZAHLE

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−2gs

n,R(r) = const rs−2

and for the (2k)-th derivative(hs

n,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(hs

n,R

)(2k)(r) has similar properties

as used for the function hs−2kn,R (r) in the above cases of the proof. Furthermore,

FGsR(ξ) =

1|ξ|2

∫ ∞

1IsR(|ξ|, t) ν(dt)

with

IsR(|ξ|, t) =

∫ ∞

0sin v hs

n,R

(tv

|ξ|

)

dv.

2k-fold integration-by-parts yields

IsR(|ξ|, t) =

t2k

|ξ|2k

∫ ∞

0sin v

(hs

n,R

)(2k)(

tv

|ξ|

)

dv

and for the Fourier transform

FGsR(ξ) =

1|ξ|2k

1|ξ|2

∫ ∞

1

∫ ∞

0sin v

(hs

n,R

)(2k)(

tv

|ξ|

)

dv t2k ν(dt).

From the first part of the proof applied to(hs

n,R

)(2k)instead of hs−2k

n,R we now infer

const−1

|ξ|2k+(s−2k)≤ FGs

R(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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