Sobolev–Orlicz inequalities, ultracontractivity and spectra of time changed Dirichlet forms

Math. Z. (2007) 255:627–647DOI 10.1007/s00209-006-0037-8 Mathematische Zeitschrift

Sobolev–Orlicz inequalities, ultracontractivityand spectra of time changed Dirichlet forms

Ali Ben Amor

Received: 5 April 2005 / Accepted: 3 March 2006 /Published online: 5 September 2006© Springer-Verlag 2006

Abstract Let E be a regular Dirichlet form on L2(X, m), μ a positive Radon measurecharging no sets of zero capacity and � an N-function. We prove that the Sobolev-Orlicz inequality(SOI) ‖f 2‖L�(X,μ) ≤ CE1[f ] for every f ∈ D(E) is equivalent to acapacitary-type inequality. Further we show that if D(E) is continuously embeddedinto L2(X, μ), the latter one implies some integrability condition, which is nothingelse but the classical uniform integrability condition if μ is finite. We also prove thata SOI for E yields a Nash-type inequality and if further μ = m and � is admissible,it yields the ultracontractivity of the corresponding semigroup. After, in the spirit ofSOIs, we derive criteria for D(E) to be compactly embedded into L2(μ), provided μ

is finite. As an illustration of the theory, we shall relate the compactness of the latterembedding to the discreteness of the spectrum of the time changed Dirichlet formand shall derive lower bounds for its eigenvalues in term of �.

Keywords Sobolev–Orlicz inequality · Ultracontractivity · Measure · Spectrum

1 Introduction

In this paper we continue our investigations which we began in [3,4] devoted to givenecessary and sufficient conditions for the validity of Sobolev-type inequalities (ormore generally trace inequalities) for regular Dirichlet forms and give criteria fortheir domain to be compactly embedded into some Lebesgue space.

Here is a short description of our program: Let X be a locally compact separablemetric space and m a positive Radon measure on X whose support is X. Let E be aregular Dirichlet form on L2 := L2(X, m) and μ a Borel measure on X charging nosubsets of zero capacity. It is well known that the Sobolev’s inequality (SI for short)

This work has been supported by the Deutsche Forschungsgemeinschaft.

A. Ben Amor (B)FB Informatik/Mathematik, Universität Osnabrück, 49069 Osnabrück, Germanye-mail: [email protected]

628 A. Ben Amor

(SI) :

⎛⎝

∫

X

|f | 2κ dμ

⎞⎠

κ

≤ CE1(f , f ), for every f ∈ D(E), 0 < κ ≤ 1,

may be extended to a much more general inequality, namely Sobolev–Orlicz inequal-ity(SOI). On the other hand the SI is equivalent to (See [1,3,4,13,20]) the capacitaryinequality (CI’):

(CI′) :(μ(K)

)κ ≤ CCap(K), for every compact K.

Now substituting (SI) by the SOI:

(SOI) : ‖f 2‖L�(μ) ≤ CE1(f , f ) for every f ∈ D(E),

suggests to replace (CI’) by an appropriate CI involving the conjugate function of �,�. This was done, first, by Mazja [19, Satz 4.2] for the gradient energy form on R

d andby Carron for the energy form on a nice Riemannian manifold [7] with μ = m.

Also Kaimanovich [18] extends the result to the case where the transition kernelof the Dirichlet norm is a probability measure and μ = m.

One of our aims in this work is to establish an equivalence between (SOI) anda capacitary-type inequality for general Dirichlet forms. This step is an extension of[3, Theorem 3.1],[13, Theorem 3.1] on one hand and of [19, Satz 4.2] and [5, Theorem10.4] on the other hand (where the case μ = m is considered). Using this equivalencewe show that if Iμ is bounded (defined below), then both (SOI) and a CI imply someuniform integrability condition (see Theorem 3.2) which reduces to the usual one ifμ(X) < ∞.

Further we prove that a (SOI) with an N-function � leads to a Nash-type inequalityand if moreover μ = m and � is admissible, the latter one leads to the ultracontrac-tivity of the semigroup e−tH for every t > 0, i.e. e−tH maps continuously L1 into L∞for every t > 0. This result will play a central role for deriving lower bounds for theeigenvalues in case where H has a discrete spectrum.

Then we derive necessary and sufficient conditions for the domain of the Dirichletform, D(E), to be compactly embedded into some L2(μ)-space for finite measuresμ. Precisely, relying on establishing a relationship between the compactness of Iμ

and the compactness of the trace of E1 on the support of μ, we shall prove that thecompactness of the embedding

Iμ := (D(E), E1) → L2(μ), f �→ f , (1.1)

implies a (SOI) with some N-function �, provided μ is finite. The converse is alsotrue, provided � is admissible.

The scope of (SOI) in the special case μ = m was extensively studied and itsimportance is well known. However, we stress that the general case is still of greatimportance, especially for the study of the perturbed Dirichlet form E + μ+ − μ−[2,6,22] (localization of the essential spectrum, estimate of the number of the neg-ative eigenvalues). It is of great interest as well, for the study of the trace of theDirichlet form E on the support of the measure μ as done by Fukushima–Uemura[13] and as we shall illustrate in the fourth and last paragraphs.

Time changed Dirichlet forms 629

2 Preliminaries

For a positive Radon measure μ on X and 1 ≤ p ≤ ∞, we shall denote by Lp(μ) theusual (real) Lebesgue space of Borel measurable (equivalence classes) of functionson X equipped with the usual norm ‖ · ‖Lp(μ). If μ = m, the space Lp(m) will bedenoted simply by L2 further the notation a.e. means m-a.e.. The space of continuousfunctions with compact support in X will be denoted by Cc(X).

Let E be a regular Dirichlet form defined in L2. We denote by D(E) the domainof E . By the regularity we mean that D(E) ∩ Cc(X) is dense in D(E) with respect tothe norm Eα := E + α(·, ·)L2 for some α > 0 and dense in Cc(X) with respect to theuniform norm.

The positive selfadjoint operator associated to E via the representation theorem isdenoted by H and is defined as follows:

D(H12 ) = D(E), E[f ] := E(f , f ) = (H

12 f , H

12 f )L2 · (2.1)

For every open subset � ⊂ X and every α > 0 we define [12, p. 61]

L� := {f ∈ D(E), f ≥ 1 a.e. on �} (2.2)

and

Capα(�) :={

inf f∈L�Eα[f ] if L� �= ∅

+∞ if L� = ∅For an arbitrary subset � ⊂ X we define

Capα(�) = inf�⊂ω, ω open

Capα(ω)· (2.3)

It is known that Capα defines a capacity on X. For α = 1 we shall denote the corre-sponding capacity by Cap.

A property is said to be satisfied quasi-everywhere (q.e. for short) if it holds trueup to a set of zero capacity. A function f is said to be quasi-continuous (q.c.) if forevery ε > 0 there is an open subset O such that Cap(O) < ε and the restriction of fto X \ O is continuous.

We recall (cf. [12, p. 65]) that the regularity property of Dirichlet forms impliesthat every element of D(E) can be corrected so as to become q.c.. In the sequel weshall implicitly assume that the elements of D(E) has been corrected in this way. Letus observe that together with Urysohn’s lemma the regularity property implies thatevery compact subset has finite capacity.

From now on we assume that all measures under consideration are Radon measuresand they do not charge subsets of zero capacity, we also denote by Iμ the mapping:

Iμ := (D(E), E1) −→ L2(μ), f �→ f . (2.4)

For the convenience of the reader we also recall some facts about Orlicz spaces. Afunction � : [0, ∞) → [0, ∞) is called a Young’s function if it is convex, �(0) = 0 andlimx→∞ �(x) = ∞. A Young’s function is called an N-function if moreover �(x) = 0if and only if x = 0, limx→0 �(x)/x = 0 and limx→∞ �(x)/x = ∞.

To every Young’s function �, one can associate an other Young’s function �, whichis defined by

�(x) = sup{yx − �(y) : y ≥ 0

}, x ≥ 0.

630 A. Ben Amor

The function � is called the complementary function of �. From the very definitionthe following Young’s inequality holds true

xy ≤ �(x) + �(y), x, y ≥ 0.

We also note that if � is an N-function then � also is.The following important inequality, valid for N-functions, (see [21, p.14]) will be

used later:

x < �−1(x)�−1(x) ≤ 2x, x > 0. (2.5)

We shall, occasionally, make use of a special class of N-functions: An N-function �

is said to satisfy the ∇2-condition (� ∈ ∇2 for short), if there is l > 1 and t0 > 0 suchthat

�(t) ≤ 12l

�(lt), ∀ t ≥ t0.

By [21, Corollary 5, p. 26], if � ∈ ∇2, then � satisfies the following growth conditionat infinity: There is C > 0, ε > 0 and t0 > 0 such that

�(t) ≥ Ct1+ε , ∀ t ≥ t0. (2.6)

This means that � has at least polynomial growth at infinity.For a positive Radon measure μ and a Young’s function � the Orlicz space

L�(μ) := L�(X, μ) is the space of μ-measurable (equivalence classes) of functions fon X such that

‖f‖L�(μ) := sup

⎧⎨⎩

∣∣∫

X

fg dμ∣∣ :

∫

X

�(|g|) dμ ≤ 1

⎫⎬⎭ < ∞ (2.7)

It is well known that(L�(μ), ‖ · ‖L�(μ)

)is a Banach space whose norm is equivalent

to the Luxemburg norm:

‖f‖(�) := inf

⎧⎨⎩λ > 0 :

∫

X

�(|f |/λ) dμ ≤ 1

⎫⎬⎭ . (2.8)

Precisely we have

‖f‖(�) ≤ ‖f‖L�(μ) ≤ 2‖f‖(�), ∀ f ∈ L�(μ). (2.9)

We also recall that if � is an N-function and B is a Borel subset then (see [21, p. 79])

‖1B‖L�(μ) = μ(B)�−1(1/μ(B)). (2.10)

For Orlicz spaces the Hölder’s inequality reads as follows (see [21, Inequality(4),p. 48]):

∫

X

|fg| dμ ≤ 2‖f‖L�(μ)‖g‖L�(μ). (2.11)

Let us emphasize that in some places (and only in some!) we would pay no atten-tion to the constants appearing in the inequalities. So that we shall denote them allby C.


3 SOIs and their consequences

In this paragraph we shall be concerned with two questions: First, show how a SOIand an isocapacitary-type inequality are equivalent to each other and that they bothimply some uniform integrability condition.

Second, which consequences can one get from the (SOI)? Namely we shall provethat a (SOI) with an N-function yields a Nash-type inequality which, in turns, undersome supplementary condition, relative to � (admissible), implies the ultracontrac-tivity of the related semigroup.

3.1 SOIs: necessary and sufficient conditions

The following generalizes both [19, Satz 3.1], [18, Theorem 3.1] and [4, Theorem 3.1].

Theorem 3.1 Let � be an N-function. Then the following assertions are equivalent:

(i) There is a constant C1 such that

‖f 2‖L�(μ) ≤ C1E1[f ], ∀ f ∈ D(E). (3.1)

(ii) There is a constant C2 such that

(CI) : μ(K)�−1(1/μ(K)) ≤ C2Cap(K), ∀ K compact.

Moreover the constants may be chosen so that C2 ≤ C1 ≤ 4C2.

Let us stress that if the Dirichlet form E is transient then the same result holds trueif on changes E1 by E and the 1-capacity by the corresponding one.

Proof (i)⇒(ii): Follows directly from (2.10) and the definition of the capacity.(ii)⇒(i): The main input in the proof of this implication is the ‘strong CI’ (see [4,13]):

∞∫

0

Cap{|f | ≥ t} d(t2) ≤ 4E1[f ], ∀ f ∈ D(E). (3.2)

Observe that by the regularity property of E , it suffices to prove inequality (3.1) onD(E) ∩ Cc(X). Further by Markov property for E i.e. E[|f |] ≤ E[f ], ∀f ∈ D(E), itsuffices to prove it for positive elements from D(E) ∩ Cc(X).

Let f be such a function. Then by definition and use of Fubini’s theorem we get

‖f 2‖L�(μ) = sup

⎧⎨⎩

∣∣∫

X

f 2g dμ∣∣ :

∫

X

�(|g|) dμ ≤ 1

⎫⎬⎭

= sup

⎧⎪⎨⎪⎩

∣∣∞∫

0

⎛⎜⎝

∫

{f≥t}g dμ

⎞⎟⎠ d(t2)

∣∣ :∫

X

�(|g|) dμ ≤ 1

⎫⎪⎬⎪⎭

.

632 A. Ben Amor

For t ≥ 0, set Et := {f ≥ t} . From the latter inequality we derive

‖f 2‖L�(μ) ≤∞∫

0

⎛⎝sup

⎧⎨⎩

∣∣∣∣∣∣

∫

X

1Et g dμ

∣∣∣∣∣∣:∫

X

�(|g|) dμ ≤ 1

⎫⎬⎭

⎞⎠ d(t2) =

∞∫

0

‖1Et‖L�(μ) d(t2)

=∞∫

0

μ(Et)�−1(1/μ(Et)) d(t2) ≤ C2

∞∫

0

Cap(Et) d(t2)

≤ 4C2E1[f ],where the latter inequality is obtained from inequality (3.2).

Next we shall use Theorem 3.1 to prove that for measures μ such that Iμ is bounded,the (SOI) and (CI) imply the fact that the unit ball of D(E) is nearly L2(μ)-uniformlyintegrable. Here we say that a family F of Borel measurable functions is Lp(μ)-nearlyuniformly integrable (1 ≤ p < ∞) if

(NUI) : limλ→∞ sup

f∈F

∫

{|f |≥λ}|f |p dμ = 0. (3.3)

If the measure μ is finite, the latter definition reduces to the classical definition ofuniform integrability [11].

From now on we denote by B the unit ball of (D(E), E1).

Theorem 3.2 Assume that Iμ is bounded. If (SOI) holds true with some N-functionthen B is L2(μ)-nearly uniformly integrable.

Remark 3.1 At the beginning of the proof of Theorem 4.1, we shall prove that If(SOI) holds true with an N-function then Iμ is automatically bounded.

Proof Assume that (SOI) holds true with some N-function �. Since Iμ is boundedby [3, Theorem 3.1] there is a constant C such that

μ(K) ≤ CCap(K), for all K compact. (3.4)

Now let f ∈ Cc(X) ∩ D(E) be positive such that E1[f ] ≤ 1 and λ > 0. Then by Hölderinequality we have

∫

{f≥λ}f 2 dμ ≤ 2‖f 2‖L�(μ)‖1{f≥λ}‖L�(μ)

≤ 2CE1[f ]μ{f ≥ λ}�−1(

1μ{f ≥ λ}

)

≤ 4C(

�−1(

1μ{f ≥ λ}

))−1

, (3.5)

where the latter inequality follows from (2.5). By the ‘weak CI’

Cap{f ≥ λ} ≤ λ−2E1[f ],


and inequality (3.4) we obtain μ{f ≥ λ} ≤ C/λ2. Since by the properties of �,limλ→∞ �(t) = ∞, we achieve

∫

{f≥λ}f 2 ≤ C

(�−1(λ2/C)

)−1 → 0 as λ → ∞, (3.6)

uniformly in f , implying that B is L2(μ)-nearly uniformly integrable.

3.2 Some consequences of SOIs

For �(t) = tp/p, p > 1, inequality (3.1) yields the SI. Further, SI leads to Nashinequality which in turns leads to the ultracontractivity of the related semi group (see[25]). Our next task is to prove that the first implication still holds true in our generalsetting: Namely a (SOI) with an N-function leads to a Nash-type inequality.

First we shall fix some notations. For an N-function � we set

(s) := 1s�−1(1/s)

, s > 0 and κμ := sup{((μ(K))Cap(K))−1, K compact

},

(3.7)

where the ratio is understood to be zero if Cap(K) = 0.We observe that satisfies the following property:

0 < s1 ≤ s2 ⇒ (s2) ≤ 2(s1). (3.8)

Indeed, from inequality (2.5) we infer

(s2) < �−1(1/s2) ≤ �−1(1/s1) ≤ 2(s1).

Theorem 3.3 Assume that (SOI) holds true with an N-function �. Then for everyf ∈ D(E) \ {0} and every ε ∈ (0, 1) we have

E1[f ] ≥ (4κμ)−1(1 − ε)

⎛⎝

∫

X

f 2 dμ

⎞⎠

(2‖f‖2

L1(μ)

ε∫

X f 2 dμ

). (3.9)

For �(t) = d−2d td/d−2, d > 2 then (s) = cs−2/d. Whence if moreover μ = m, then

(SOI) is the classical (SI) and (3.9) is the usual Nash inequality:⎛⎝

∫

X

f 2 dm

⎞⎠

(1+2/d)

≤ C‖f‖4/dL1 E1[f ], ∀f ∈ D(E).

For the gradient energy form on ‘nice’ manifolds, Grigor’yan proved in [15, Lemma6.3] a similar inequality.

Proof We follow an idea of Grigor’yan with some modifications.By Markov property we learn that it suffices to make the proof for positive ele-

ments from D(E) \ {0}. Let f be such an element and ε ∈ (0, 1). We first suppose thatμ(X) < ∞.

Clearly

f 2 ≤ (f − s)2+ + 2sf , ∀s ≥ 0. (3.10)

634 A. Ben Amor

Setting �s : {f > s}, A := ∫X f dμ, B := ∫

X f 2 dμ and using Hölder inequality, we get

B ≤∫

�s

(f − s)2 dμ + 2sA ≤∫

�s

f 2 dμ + 2sA

≤ 2‖1�s‖L�(μ)‖f 2‖L�(μ) + 2sA = 2(μ(�s))

‖f 2‖L�(μ) + 2sA

≤ 2(μ(�s))

κμE1[f ] + 2sA. (3.11)

On the other hand we have μ(�s) ≤ s−1A. So that using (3.8) and choosing s = εB/2Awe get the desired result for finite μ. Now for arbitrary μ, set Bk the open ball of Xcentered at x0 with radius k and μk := 1Bkμ. Then

E1[f ] ≥ (4κμk)−1(1 − ε)

⎛⎝

∫

X

f 2 dμk

⎞⎠

⎛⎝ 2‖f‖2

L1(μk)

ε∫

X f 2 dμk

⎞⎠

≥ (4κμ)−1(1 − ε)

⎛⎝

∫

X

f 2 dμk

⎞⎠

⎛⎝ 2‖f‖2

L1(μk)

ε∫

X f 2 dμk

⎞⎠

→ (4κμ)−1(1 − ε)

⎛⎝

∫

X

f 2 dμ

⎞⎠

(2‖f‖2

L1(μ)

ε∫

X f 2 dμ

)as k → ∞, (3.12)

which was to be proved.

The latter result is the main input to show that (SOI), in some cases, implies theultracontractivity property of the related semigroup Tt := e−tH , for t > 0.

Let � be an N-function and defined as before. We say that � is admissible iffurther

(s(s)

)−1 is integrable at zero, i.e.,

α∫

0

(s(s)

)−1 ds < ∞ for some α > 0. (3.13)

Remark 3.2 A sufficient condition for � to be admissible is that �−1 being integrableat infinity. In particular, we derive from the property (2.6) that N-functions belongingto ∇2 are admissible. Indeed, for every � ∈ ∇2, there is C, t0 > 0 such that

�−1(t) ≤ Ct1

1+ε , ∀ t ≥ t0.

Let � be an admissible N-function. Set κ := κm. Following Grigor’yan, we set γ

the function defined via

t := 8κ

γ (t)∫

0

1s(s)

ds,

and for every ε ∈ (0, 1)

βε(t) := 2εγ (2(1 − ε)t)

.


Let us notice that Coulhon made an analogue assumption in [9, Proposition II.1].

Theorem 3.4 Assume (SOI) with an admissible N-function � and μ = m. Then forevery t > 0, Tt is ultracontractive and

‖Tt‖L1,L∞ ≤ βε(t/2)et, ∀ε ∈ (0, 1). (3.14)

It follows that for every t > 0, Tt is an integral operator, with kernel pt(x, y) definedeverywhere and such that

ess supx,y∈X

pt(x, y) ≤ βε(t/2)et, ∀t > 0. (3.15)

We note that, in an other context, Coulhon [9, Proposition II.1] established a similarresult with a bound which is different from ours.

Proof We are going to prove the boundedness of Ht := e−t(H+1) from L1 into L2 withthe bound β

1/2ε , which is by the symmetry of Tt and duality equivalent to the claim of

the theorem.For this purpose, thank to the Markov property for Ht:

|Htf | ≤ Ht|f |, a.e. ∀ f ∈ L2,

and to the fact that L2 ∩ L1 is dense in L1, it suffices to prove that for every t > 0 andevery f ∈ L2 ∩ L1, f ≥ 0 a.e. and ‖f‖L1 = 1, we have

‖Htf‖2L2 ≤ βε(t).

Fix f ∈ L2 ∩ L1 such that f ≥ 0 a.e. and ‖f‖L1 = 1. Set

ft(x) := Htf (x), J(t) :=∫

X

ft(x)2 dm(x) = ‖Htf‖2L2 . (3.16)

Observing that for every t > 0, ft ∈ D(H) \ {0}, differentiating with respect to t,applying Nash inequality (3.9) and using inequality (3.8) , we achieve

J′(t) = −2((H + 1)Htf , Htf

) = −2E1[ft]

≤ −12(1 − ε)κ−1

⎛⎝

∫

X

f 2t dμ

⎞⎠

(2‖ft‖2

L1

ε∫

X f 2t dm

)

≤ −14(1 − ε)κ−1

⎛⎝

∫

X

f 2t dm

⎞⎠

(2

ε∫

X f 2t dm

)

= −14(1 − ε)κ−1J(t)

( 2εJ(t)

), ∀t > 0. (3.17)

The latter inequality leads to

J′(t)J(t)

( 2εJ(t)

) ≤ −14(1 − ε)κ−1. (3.18)

636 A. Ben Amor

Integrating between 0 and t and making the change of variable s = 2ε−1J(t)−1 weobtain

κ

2ε−1J(t)−1∫

0

dss(s)

≥ κ

2ε−1J(t)−1∫

2ε−1J(0)−1

dss(s)

≥ 14(1 − ε)t, (3.19)

which by the definition of βε yields

J(t) ≤ βε(t), (3.20)

which gives the first part of the theorem.The rest of the proof follows from Dunford-Pettis theorem (see [23, Theorem 46.1

p. 471]).

Let us observe that for μ = m, inequality (3.1), with E1 replaced by E , leads to thefollowing Faber-Krahn-type inequality

(m(B)�−1(1/m(B))

)−1 ≤ λ1(B), ∀ B open bounded. (3.21)

Here λ1(B) := inf{E[f ] :∫

X f 2 dm = 1, f ∈ D(E), f = 0 q.e. on X \ B} is a sort of a‘Dirichlet eigenvalue’.

It is well known that in some special cases inequality (3.1) and (3.21) are equivalentto each other. For example if �(t) = tp

p , p > 1 (see [16]) or if X is a nice Riemannianmanifold and E is the gradient energy form (see [7]).

However it is still an open question whether the both inequalities are equivalentin general or not!

4 SOIs, compactness of Iµ and relationship to time changed Dirichlet forms

In this section we shall discuss to which extend is a (SOI) equivalent to the compact-ness of Iμ, provided μ is finite. Further, to illustrate the importance of such results weshall relate them to the compactness of a new Dirichlet form, namely the time changedDirichlet form or the trace of the Dirichlet form E1 on the support of μ. For μ = m, adirect use of Theorem 3.4 yields the following

Proposition 4.1 Assume that m is finite. Then a (SOI) with an admissible N-functionimplies the compactness of Im.

We mention that the latter result was obtained by Cipriani [8, Theorem 4.2], by adifferent method, for more general quadratic forms under the additional assumptionthat H is a Persson’s operator. Further by [8, Theorem 4.6] every operator H associ-ated to a local Dirichlet form is a Persson operator. Therefore Proposition 4.1 is newonly for Dirichlet forms of jump type.

4.1 Relationship to time changed Dirichlet forms

To illustrate the relevance of the compactness property of Iμ (and hence the existenceof a (SOI)), we shall relate it to the compactness of the resolvent of a new Dirichletform.


We start with the definition of the trace of the Dirichlet form E on some subsets.To this end we denote by Fe the extended Dirichlet space of E (see [12, p. 35]).

Let μ be a positive Radon measure charging no set of zero capacity and A thepositive continuous functional whose Revuz measure is μ. We denote by F the sup-port of μ and by F the support of A. It is known [12, p. 265] that F is a quasi-supportfor μ, μ(F \ F) = 0 and that by [12, theorem 4.6.2] if two elements from Fe coincideμ-a.e. then they coincide q.e. as well.

We introduce the subspaces

FX−F := {f ∈ Fe : f = 0 q.e. on F}, (4.1)

and HF the E-orthogonal complement of FX−F in the space Fe so that the followingdecomposition holds true [12, p. 265]:

Fe = FX−F ⊕ HF . (4.2)

Let P be the orthogonal projection onto HF . We define the trace of E on L2(F, μ)

as follows (see [12, p. 266 and Theorem 4.6.5]):

D(E) := {f ∈ L2(F, μ) : f = u − μ a.e. on F, for some u ∈ Fe},E[f ] = E[Pu], f = u − μ a.e. on F

= inf{E[u] : u ∈ D(Fe), u = f − μ a.e. on F}.It is known that E is also a Dirichlet form in L2(F, μ) [12, p. 266] and the Dirichletspace (E , D(E)) is called the time changed Dirichlet space or the trace of the space(E , D(E)) on F relative to μ.

We denote by H the operator associated to E via the representation theorem. Togive an explicit formula for the resolvent of H we give some auxiliary notations: Set Aμ

tthe PCAF associated to μ and for α, p ≥ 0 the operators R, Uα,μ defined respectivelyon the spaces L2(F, μ) and L2(μ) by

Rf (x) := Ex

∞∫

0

e−Aμt f (Xt) dAμ

t and Uα,μp f (x) := Ex

∞∫

0

e−αt−pAμt f (Xt) dAμ

t .

Here Xt designate the process associated to E . Clearly for every α, p > 0 and everyf ∈ L2(μ), Uα,μ

p f is the α-potential of the signed measure fμ with respect to theDirichlet form E + pμ. Further by [3, Theorem 3.1], for every α, p > 0, Uα,μ

p isbounded on L2(μ) and by [12, p. 274]

(H + 1)−1 = R. (4.3)

We further introduce the auxiliary operators

Kμα := L2(μ) → L2(μ), f �→ Kμ

α f := Uα,μ1 f μ − a.e. ∀α ≥ 0 (4.4)

Then by [12, p. 266], R = Kμ0 on L2(F, μ).

Now set T the trace of E1 on the support of μ and T the positive selfadjoint operatorrelated to T via the representation theorem.

Assume further that Iμ is bounded. Then for every f ∈ L2(μ) the signed measurefμ has finite energy integral. Set K1 its 1-potential w.r.t. to E . From the latter discus-sion (change E by E1) we get that if Iμ is bounded, then the resolvent of T is given

638 A. Ben Amor

by

T−1 = K1|L2(F,μ)= (

IμI∗μ)|L2(F,μ)

. (4.5)

Indeed: Let f ∈ L2 and g ∈ D(T ). Then there is g ∈ D(E) such that g = g μ-a.e. on F.From the definition of K1 we get

T (K1|L2(F,μ)f , g) = E1(K1f , g) =

∫

X

f g dμ

=∫

F

fg dμ, (4.6)

which gives the latter identity, whose use yields the following

Lemma 4.1 Assume that Iμ is bounded. Then The operator Iμ is compact if and onlyif the time changed Dirichlet form T has discrete spectrum.

4.2 SOIs and compactness of Iμ

Theorem 4.1 Assume that the measure μ under consideration is finite. If (SOI) holdstrue with an admissible N-function �, then Iμ is compact.

Proof First, we claim that if a (SOI) inequality holds for E with an N-function � thenIμ is bounded. Indeed, by [3, Theorem 3.1] showing this claim amounts to show thefollowing capacitary inequality

μ(K) ≤ CCap(K), ∀K ⊂ X compact, (4.7)

where C is a positive constant.Now by Theorem 3.1, for every compact subset K ⊂ X we have

μ(K) ≤ C(�−1(1/μ(K))

)−1Cap(K)

Further since μ is finite, there is ρ > 0 such that μ(X \ Bρ) ≤ 1. So that by the latterinequality and the fact that �−1 is increasing, we get

μ(K) ≤ C(�−1(1/μ(Bρ))

)−1Cap(K), ∀K ⊂ Bρ

and

μ(K) ≤ C(�−1(1))

)−1Cap(K), ∀K ⊂ X \ Bρ ,

which yields (4.7) and therefore the boundedness of Iμ.On the other hand, if a (SOI) inequality holds for E with an admissible N-function,

then it also holds true for T with the same function. Indeed from the very definitionof T , we get

‖f 2‖L�(F,μ) ≤ CT [f ], ∀ f ∈ D(T ). (4.8)

Now we get the result from Proposition 4.1 and Lemma 4.1.

We proceed now to prove that some converse to Theorem 4.1 holds. This wasalready observed by Cipriani, for μ = m [8, Theorem 5.1], but for more generaloperators of Persson’s type.


Theorem 4.2 Assume that the measure μ is finite and that Iμ is compact. Then there isan N-function � and a constant C such that

‖f 2‖L�(μ) ≤ CE1[f ], ∀ f ∈ D(E). (4.9)

Proof The proof is inspired from Cipriani [8].The key idea is to show that the family

F ′ := {f 2 : f ∈ D(E), E1[f ] ≤ 1},is uniformly integrable.

From the assumption, F ′ is relatively compact in L1(μ). Hence it is σ(L1(μ),L∞(μ))-sequentially compact. It follows, by Dunford–Pettis criterion (see[11, Corollary 11 p. 294]), that

limμ(B)→0

∫

B

f 2 dμ = 0, (4.10)

uniformly in f ∈ F ′. Let λ > 0 and f ∈ Cc(X) ∩ D(E) such that f ≥ 0 and E1[f ] ≤ 1.Since Iμ is bounded, making use of [3, Theorem 3.1] together with the ‘weak CI’, weachieve

μ{f ≥ λ} ≤ CCap{f ≥ λ} ≤ Cλ2 , (4.11)

which implies limλ→∞ μ{|f | ≥ λ} = 0 uniformly in f ∈ B, by the regularity of E . Henceby (4.10), we get

limλ→∞

∫

{|f |≥λ}f 2 dμ = 0, (4.12)

uniformly in f ∈ B. Whence B is L2(μ)-uniformly integrable. By la Vallée Poussintheorem [21, Theorem 2, p. 3] or [10, p. 38] there is a Young function � such thatlimt→∞ �(t)

t = ∞ and

supf∈B

∫

X

�(f 2) dμ < ∞.

Hence by the very definition of ‖ · ‖(�) we get

‖f‖(�) ≤ CE1[f ], ∀ f ∈ D(E). (4.13)

Followimg an idea of Cipriani, we can assume that � is an N-function. For, if not,we modify � on a suitable interval [0, t0], t0 > 0 to get an N-function without affecting(4.13), because by [21, Theorem 3, p. 155], for finite measures, changing � on a finiteinterval gives the same Orlicz space with an equivalent norm, and the proof is finished.

Remark 4.1 Along the lines of the proof we have in fact proved that, for finite mea-sures, the L2(μ)-uniform integrability of B implies (SOI) with an N-function. Further,in the beginning of Theorem 4.1, we proved that for finite measures the occurrence ofa (SOI) with an N-functions yields the boundedness of Iμ. Therefore, with the help ofTheorem 3.2, we conclude that if μ is finite then the following equivalence holds true:(SOI) with an N-function ⇔ the L2(μ)-uniform integrability of the unit ball of D(E).

640 A. Ben Amor

5 Lower bounds for eigenvalues of time changed Dirichlet forms

Let T be as in the latter section and assume that a (SOI) inequality holds true forD(E) with an admissible N-function. Then it holds also true for D(T ) (the trace ofE1!). Hence if moreover μ is finite, from Theorem 4.1 together with Lemma 4.1, welearn that T has discrete spectrum. Further the first eigenvalue satisfies the followingFaber–Krahn-type inequality

λ1 ≥ C(μ(F)�−1(1/μ(F)))−1 = C(μ(F)). (5.1)

We are going to prove that the same type of isoperimetric inequality for the highereigenvalues also holds true.

We denote by (λk)k≥1 the eigenvalues of T , ordered in an increasing way and byH the operator associated to T .

Theorem 5.1 Under the above assumptions we have

λk ≥ 1 − ε

16κμ

(γ

(2γ −1

(2μ(F)

k

))), ∀ 0 < ε < 1. (5.2)

Proof By Theorem 3.4 (changing E by E1), we conclude that for every t > 0 theoperator Ht := e−tH has a kernel pμ

t which is bounded by βε(t/2), μ a.e. Whence Htis a Hilbert–Schmidt operator and

∞∑k=1

e−tλk = ‖Ht/2‖2HS =

∫

F

∫

F

(pμ

t/2(x, y))2 dμ(x) dμ(y)

≤ βε(t/2)

∫

F

∫

F

pμt/2(x, y) dμ(y) dμ(x) = βε(t/2)

∫

F

(Ht/21)(x) dμ(x)

≤ μ(F)βε(t/2). (5.3)

Thereby

ke−tλk ≤ μ(F)βε(t/2), (5.4)

which implies

λk ≥ 1t

log

(k

μ(F)βε(t/2)

). (5.5)

Whence changing t by t(1−ε)

we get

λk ≥ 1 − ε

tlog

(kεγ (t)2μ(F)

)∀ t > 0. (5.6)

Choosing t such that 2μ(F) = kεγ (t/2), we conclude that there is θ ∈ (t/2, t) such that

λk ≥ (1 − ε)

16κμ

(γ (θ)). (5.7)

Now we get the result by observing that γ increases and using property (3.8).


6 Two applications

6.1 Ultracontractivity and spectra for traces of s-stable processeson semi (d, �)-sets

As an illustration, we shall use the above obtained results to derive some spectralproperties of the generator of the s-stable process over d-sets. The novelty here (com-pared to Triebel [24, Theorem 23.2 p. 349]) is that we do no more assume that thed-measure related to the d-set under consideration has compact support. We shallonly assume that it is finite.

Here are some basic notions that we shall use in the sequel (see [24, p. 334]): a closedsubset F ⊂ R

d is called a semi (d, �)-set, where 0 < d ≤ n, if there is a positive measureμ supported by F, a real-valued function � on (0, 1], positive constants C > 0, C1 ≥ 0and 0 < C2 < 1 and a real number b such that:

(i)

μ(Br(x)

) ≤ Crd�(r), ∀ x ∈ F, r ∈ (0, 1]. (6.1)

(ii) The function � is such that

�(r) ≤ C1| log(C2r)|b, ∀ r ∈ (0, 1]. (6.2)

The measure μ is called a semi (d, �)-measure.If b = 0 we say that μ is a semi d-measure and if moreover

μ(Br(x)

) ≥ C−1rd, ∀ x ∈ F, r ∈ (0, 1], (6.3)

we say that μ is a d-measure.By [24, Proposition 22.8], for every pair (d, �) satisfying the above conditions there

is a semi (d, �)-set.

Example 6.1

(1) The restriction of Lebesgue measure on a closed subset of Rn is a semi n-measure.

(2) The d-dimensional Hausdorff measure of a semi d-set is a semi d-measure.

From now on, we fix s such that 0 < s ≤ 1. We designate by E(s) the followingDirichlet form:

D(E(s)) =⎧⎨⎩f ∈ L2(Rn) :

∫

Rn

|f (x)|2|x|2s dx < ∞⎫⎬⎭ , E(s)(f , g) =

∫

Rn

f (x)g(x)|x|2s dx.

It is known that E(s) is a Dirichlet form whose domain the space of Bessel potentialsLs,2(Rn) and is related to the operator (−�)s:

D((−�)s)=L2s,2(Rn), E(s)(f , g)=((−�)sf , g), ∀ f ∈ L2s,2(Rn), g ∈ D(E(s)) = Ls,2(Rn).

From now on we assume that μ is a semi (d, �)-measure such that n − 2s < d ≤ n.For semi d-measures the following result is due to Adams (see [17, p. 214].

642 A. Ben Amor

Lemma 6.1 Let μ be a semi (d, �)-measure. Then

(i) For n > 2s and 2 < p < 2dn−2s (p ≤ 2d

n−2s if b = 0), the following inequality holdstrue

⎛⎝

∫

X

|f |p dμ

⎞⎠

2/p

≤ CE(s)1 [f ], ∀ f ∈ D(E(s)). (6.4)

(ii) For n ≤ 2s and 2 < p < ∞ (p = ∞ if n < 2s) the following inequality holds true

⎛⎝

∫

X

|f |p dμ

⎞⎠

2/p

≤ CE(s)1 [f ], ∀ f ∈ D(E(s)). (6.5)

Proof We give the proof only for b �= 0. For the other case it runs in the same way.For 2 < p < ∞, by Theorem 3.1, we have to show that there is a constant C such

that

(μ(K))1/p ≤ C(Cap(K))1/2, for all K compact. (6.6)

Let G be the kernel of ((−�)s + 1)−1. Then for every compact subset having positivecapacity we have

μ(K)

Cap(K)≤ sup

x∈Rn

∫

K

G(x, y) dμ(y). (6.7)

We shall consider the different cases separately.The case n < 2s.For p = ∞, it is known that the embedding Idx, where dx refers to Lebesgue mea-

sure, is bounded into L∞. Since dx-essentially bounded functions are bounded q.e.and since μ does not charge sets having zero capacity, we conclude that Iμ is boundedinto L∞(μ) as well.

Now assume that p < ∞. In this situation it is known that the kernel G is contin-uous and bounded and that points have positive capacity. Hence there is A > 0 suchthat for every nonempty compact subset K we have Cap(K) ≥ A.

Let K be a compact subset with positive capacity. Thanks to the latter observationsand inequality (6.7), we get

(μ(K))1/p

(Cap(K)1/2≤ A1/p−1/2

(μ(K)

Cap(K)

)1/p

≤ A1/p−1/2

⎛⎝ sup

x∈Rn

∫

K

G(x, y) dμ(y)

⎞⎠

1/p

.

(6.8)

On the other hand, it is not difficult to realize (see [2, p. 14]) that there is a constantC(n), depending only on n, such that

supx∈Rn

∫

Bx(1)c

G(x, y) dμ(y) ≤ C(n) supx∈Rn

∫

Bx(1)

G(x, y) dμ(y). (6.9)


Thus

μ(K)

Cap(K)≤ (1 + C(n)) sup

x∈Rn

∫

Bx(1)

G(x, y) dμ(y)

≤ supx,y

G(x, y)CC1�(1) < ∞,

yielding inequality (6.6).The case n > 2s. The proof runs substantially as the latter case. Recalling that

G(x, y) ≤ A|x − y|2s−n, we have to show an other time that

supx∈Rn

∫

Bx(1)

|x − y|q(2s−n) dμ(y) < ∞,

for 1 < q < dn−2s .

Let q be such a number. Observe that for 0 < t < 1 we have

B(t) := tq(2s−n)μ(Bx(t)) ≤ CC1td−q(n−2)(

log(C2t))b,

so that by assumption on q, limt→0 B(t) = 0. Hence a routine computation yields

∫

Bx(1)

|x − y|q(2s−n) dμ(y) = μ(Bx(1)) + q(n − 2s)

1∫

0

tq(2s−n)−1μ(Bx(t)) dt

≤ C�(1) + CC1

1∫

0

td−1−q(n−2s)| log(C2t)|b dt.

As a final step we have to prove that the latter integral is finite. Clearly it is, if b ≤ 0.On the other hand iterating the integration by parts an other time, we realize that weare lead to prove the claim for 0 < b ≤ 1. For such b, | log C2t|b ≤ | log C2t|. So that itsuffices to make the proof for b = 1.

1∫

0

td−1−q(n−2s)| log(C2t)| dt = C′| log(C2)| + C′′1∫

0

td−1−q(n−2s) dt < ∞, (6.10)

by assumption on q.The case n = 2s. Let q > 1 and q′ its conjugate exponent. Making use of (6.6)

together with Hölder inequality and replacing G by Gq in the inequality (6.9), weobtain

(μ(K))1/2q

(Cap(K))1/2≤ sup

x∈Rn

⎛⎝

∫

K

Gq(x, y) dμ(y)

⎞⎠

1/2q

≤ (1 + C(n))1/2q supx∈Rn

⎛⎜⎝

∫

Bx(1)

Gq(x, y) dμ(y)

⎞⎟⎠

1/2q

.

644 A. Ben Amor

So that we are lead to show the finiteness of the latter integral. We recall that there isa constant A such that

G(x, y) ≤ A log(2/|x − y|), for |x − y| ≤ 1. (6.11)

Now using Fubini theorem, integrating by parts and taking the the properties of themeasure μ into account we achieve

∫

Bx(1)

(log(2/|x − y|))q dμ(y) = log(2)qμ(Bx(1)) + qC′1∫

0

td−1| log t|q−1| log(C2t)|b dt.

The rest of the proof runs now in a similar way as in final step of the latter case, gettingthe sought result, which finishes the proof.

Theorem 6.1 Let μ be a semi (d, �)-measure with support F such that n − 2s < d ≤ n.Set T the trace of E(s)

1 on L2(F, μ) and pt the kernel of the related semigroup. Let p > 2be as given in the latter lemma. Then

pt ≤ C1tq

μ × μ a.e., ∀ t > 0, (6.12)

where q is the conjugate of p/2.If further μ is finite, then T has discrete spectrum. Set (λk)k its eigenvalues, ordered

in an increasing way. Then the following inequalities hold true:

λk ≥ Ckp−2

p , ∀ k ≥ 1. (6.13)

Proof By Lemma 6.1, we have a (SOI) with �(t) = p−1tp, 2 < p < ∞, which is anadmissible N-function. Further by Lemma 4.1, Iμ is compact so that by Lemma 4.1,T has discrete spectrum.

For p < ∞, the rest of the proof is obtained by a straightforward application ofTheorem 3.4 and Theorem 5.1.

For n < 2s and p = ∞ we get the proof by passing to the limit (p → ∞) and byobserving that the resulting constant is finite (which follows from the first step in theproof of Lemma 6.1).

6.2 Spectra of certain Dirichlet forms of jump-type on Besov spaces over d-sets

We pursue to work in the same setting. We assume that n = 2s, and that μ is ad-measure. Set

�(t) := et − 1.

Then, by a result due to Mazja [19, Folgerung 6.2, p. 46], the space Ls,2(Rn) is embed-ded into the Orlicz space L�(μ): There is a constant C such that

‖f 2‖L�(μ) ≤ CE1[f ], ∀f ∈ Ls,2(Rn). (6.14)

Let F be the d-set related to μ and � its diagonal. Define δ by δ := s + d2 = n+d

2 .Let B be the form defined by

B[f ] :=∫

F×F\�

(f (x) − f (y))2

|x − y|d+2δdμ(x) dμ(y), D(B) := {f ∈ L2(F, μ) : B[f ] < ∞}.


Then B is a regular Dirichlet form on L2(F, μ) (see [14]) and its domain is the Besovspace B2,2

δ (F). Combining inequality (6.14) with the Jonsson–Wallin trace theorem[17, Theorem 1, p.141] which asserts that

B2,2δ (F) = Ls,2(Rn)|F

and that both the restriction and extension operators are continuous, we obtain

‖f 2‖L�(μ) ≤ CB1[f ], ∀f ∈ B2,2δ (F), (6.15)

where C is a positive constant and � is an admissible N-function.We would like to mention that Fukushima–Uemura [13] proved that for n > 2s

and δ < d2 then a Sobolev-type inequality holds true for B2,2

δ (F). So that the latterresult completes the small gap in the latter mentioned work.

Whence making use of Theorem 3.4, we conclude that the semigroup of the oper-ator related to B has a kernel pμ

t and by Proposition 4.1 it has a discrete spectrumprovided μ is finite.

The corresponding to � is given by

(s) = 1s log(1/s + 1)

, (6.16)

so that the corresponding γ is given by the identity

t = 8κμγ (t) log

(1 + γ (t)

γ (t)

)+ 8κμ log(1 + γ (t)), t > 0. (6.17)

The lower bounds for the eigenvalues are described in the following

Theorem 6.2 Let μ be a finite d-measure. Then B has discrete spectrum. Set λ1 <

λ2 · · · ≤ λk ≤ · · · the eigenvalues of B1. Then

λk ≥ C(1 − ε)

2μ(F)

k

log( k2μ(F)

+ 1), ∀ 0 < ε < 1,

where C is positive constant.

Proof Making use of Theorem 5.1 we get

λk ≥ C(1 − ε)

(γ

(2γ −1

(2μ(F)

k

))). (6.18)

Hence, owing to an idea of Grigor’yan, we get the result if we show that the functionoγ has at most polynomial decay i.e.,

(γ (2t))(γ (t))

≥ δ > 0, ∀t > 0. (6.19)

We proceed now to prove the latter claim. For every t > 0, set Q(t) := (γ (2t))(γ (t)) . By the

properties of and γ , we have Q(t) > 0 for every t > 0. On the other hand we have

Q(t) = γ (t) log(1 + 1/γ (t))γ (2t) log(1 + 1/γ (2t))

.

646 A. Ben Amor

Since γ −1(t) → ∞ as t → ∞, γ also does. So that limt→∞ Q(t) = 1. Further we have

γ −1(t)t

= 8κμ log(1 + 1/t) + 8κμ

log(1 + t)t

, (6.20)

which implies limt→0 γ (t)/t = 0. Thereby

limt→0

Q(t) = limt→0

t − κμ log(1 + γ (t))2t − κμ log(1 + γ (2t))

= 1/2,

which yields (6.19) and the proof is finished.

Acknowledgments The author would like to thanks the referee for his (or her) subtle observations,which led to improve the paper in some places.

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Sobolev–Orlicz inequalities, ultracontractivity and spectra of time changed Dirichlet forms