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Math. Ann. DOI 10.1007/s00208-006-0057-z

Mathematische Annalen

Ultracontractivity and embedding into L∞ A. Bendikov · T. Coulhon · L. Saloff-Coste

Received: 2 June 2006 © Springer-Verlag 2006

Abstract Given a self-adjoint semigroup e−tA satisfying an ultracontractivity bound of the type e−tA 2→∞ ≤ em(t) , we find conditions on the sequence 1/n An f 2 that imply that f is a bounded function. Sobolev’s classical embedding theorem says that, when A is the Laplace operator on Rd , Ak f 2 < ∞ for some k > d/4 suffices to imply that f is bounded. In the cases we are interested 1/n in, the desired condition involves the whole sequence An f 2 and depends on the behavior of the ultracontractivity function m. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Ultracontractivity functions and their transforms 2.1 The functions F and . . . . . . . . . . . . 2.2 The functions N and Q . . . . . . . . . . . . 3 The eventual ultracontractivity of e−sF(A) . . . .

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Research of A. Bendikov was supported by the Polish Goverment Scientific Research Fund, Grant 1 PO3 A 03129. Research of T. Coulhon was partially supported by the European Commission (IHP Network “Harmonic Analysis and Related Problems” 2002–2006, Contract HPRN-CT-2001-00273-HARP). Research of L. Saloff-Coste was partially supported by NSF grant DMS-0102126. A. Bendikov Mathematical Institute, Wrocław University, Wrocław, Poland T. Coulhon Département de Mathématiques, Université de Cergy-Pontoise, Cergy-Pontoise Cedex, France L. Saloff-Coste (B) Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853, USA e-mail: [email protected]

A. Bendikov et al. 4 Embeddings into L∞ . . . . . . . . . . . . . 5 Relations with Nash type inequalities . . . . 6 Discussion of specific behaviors . . . . . . . 6.1 The classical case M(t) = c + d4 log(1 + 6.2 6.3

. . . . . . . . . . . . . . . . . . . . . 1 ), c, d > 0 t

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The case M(t) = c1 + c2 [log(1 + 1t )]λ , c1 , c2 > 0, λ > 1 . . . . . . . . . . . . . . . . . The case M(t) = c1 + t−λ , c1 , λ > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

λ

6.4 The case M(t) = c1 ec2 [log(1+ t )] , c1 , c2 , λ > 0 . . . . . . . . . 6.5 Regular variation . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 The case M(t) = exp(g(1/t)), g ∈ SRλ , λ > 0 . . . . . . . . . . 7 Further results concerning tame behaviors . . . . . . . . . . . . . . 7.1 Probability spaces with polynomially bounded eigenfunctions 7.2 Convolution semigroups on abelian groups . . . . . . . . . . . 7.3 The case M(t) = g(log(1 + 1/t)), g ∈ SRλ , λ > 1 . . . . . . . . 8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The infinite dimensional torus . . . . . . . . . . . . . . . . . . 8.2 Symmetric Lévy generators on R . . . . . . . . . . . . . . . .

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1 Introduction One of the classical uses of Sobolev embedding theorem is to show that an L2 function on Rd having k derivatives in L2 with k > d/2 is a bounded function. This has been generalized as follows. Let e−tA be a semigroup of self-adjoint operators on L2 (X, µ), where (X, µ) is a σ -finite measure space. Assume that, for all t ∈ (0, 1), e−tA 2→∞ = sup e−tA g∞ ≤ Ct−ν/4 . g2 ≤1

Then any function g ∈ L2 (X, µ) such that Ak g ∈ L2 (X, µ) for some k > ν/4 (roughly speaking, this corresponds to 2k derivatives in L2 , with 2k > ν/2) must be a bounded function. See, e.g., [14, Théorème 1] and the references therein. The aim of the present paper is to obtain results in this spirit when the semigroup e−tA satisfies an ultracontractivity bound of the type e−tA 2→∞ ≤ em(t) ,

t>0

(1.1)

with a function m which tends to infinity at least as fast as log 1/t as t tends to 0. We call such a function m an ultracontractivity function for e−tA . More precisely, we would like to obtain equivalences between (1.1) and properties such as g∈

∞ 0

1/n

n

Dom(A )

and

An g2 lim sup φ(n) n→∞

≤ 1 ⇒ g ∈ L∞ (X, µ),

(1.2)

where the function m in (1.1) and the function φ in (1.2) are related in some explicit way. We call any function φ such that (1.2) holds an embedding

Ultracontractivity and embedding into L∞

function for A. Following the terminology of [14], we call (1.2) a generalized Gagliardo-Nirenberg inequality (although it is rather an embedding property than an inequality). Similar questions were discussed in [16] which focussed on problems related to the long time behavior of the semigroup. In this paper, the focus is on the short time behavior. We will also relate these properties to Nash type inequalities and characterize those functions f on the real line such that e−f (A) 2→∞ < ∞. In fact, the connection between Nash inequalities and ultracontractivity bounds is sharp when the explosion of m at 0 is not too fast, whereas embedding properties of the form (1.2) give tight results when the explosion of m at 0 is fast enough. There is a common zone where both techniques give excellent results (see Theorem 5.3), and two exclusive zones where apparently only one of them stays sharp. See the examples in Sect. 6. In Sect. 7, we consider the case where X has finite measure and A has discrete spectrum (with L∞ bounds on the eigenfunctions), and the case of invariant operators on abelian locally compact metric groups. In these two cases, we prove sharp generalized Gagliardo–Nirenberg inequalities even when m does not belong to the favorable zone. In the final Sect. 8, we exhibit families of concrete examples, namely invariant diffusions on infinite dimensional tori and symmetric Lévy semigroups on the real line, which display the whole variety of behaviors considered in earlier sections.

2 Ultracontractivity functions and their transforms This section is of a technical nature, but introduces definitions that are crucial throughout this paper. We start with a non-increasing non-negative function M, which will later coincide with the (short time) ultracontractivity function m for e−tA , and define four transforms of M. Given two non-negative functions f , g, we write f ≈ g to signify that there exist finite positive constants a, b such that af ≤ g ≤ bf on the relevant domain. We write f ∼ g (say, at infinity) if f ≤ (1 + o(1))g and g ≤ f (1 + o(1)).

2.1 The functions F and In this section, we set up the machinery which is necessary to connect ultracontractivity estimates with embedding properties in L∞ . Definition 2.1 Let M be a non-increasing non-negative function defined on (0, +∞) and such that M(0+ ) = ∞. For non-negative x, set F(x) = FM (x) = inf {tx + M(t)}, t>0

(2.3)

A. Bendikov et al.

and (x) = M (x) = sup

x

t>0

t

e−M(t)/x .

(2.4)

The function F is non-negative, non-decreasing, concave, equals M(∞) at 0 and tends to ∞ at ∞. It satisfies F(x) = o(x) at infinity and ∀ a ∈ [1, ∞),

∀ x ∈ (0, ∞),

F(ax) ≤ aF(x).

(2.5)

The function is directly related to F by F(y) (x) = sup ye1− x ,

x > 0.

(2.6)

y>0

Indeed, we have inf t>0 {ty+M(t)} F(y) ty M(t) x = sup sup ye1− x − x sup ye1− x = sup ye1− y>0

y>0

y>0 t>0

M(t) ty = sup sup ye1− x e− x t>0

= sup

y>0

t>0

x t

e−

M(t) x

= (x).

Lemma 2.2 The function at (2.4) has the following properties: 1. 2.

is non-negative, non-decreasing, convex and satisfies limx→∞ [(x)/x] = ∞; For x ≥ M(∞), F ◦ (x) ≥ x.

Proof of Lemma 2.2 It is clear that is non-negative and non-decreasing. It is convex because x → xe−a/x is convex for any a ≥ 0. By (2.6), for any A > 0, (x) ≥ Axe−F(Ax)/x . Since F(x) = o(x) at infinity, it follows that (x) ≥ Ax/2 for x large enough. Hence (x)/x tends to infinity. To prove the second assertion, note that (x) ≥ ye1−(F(y)/x) for any y > 0 and pick y such that F(y) = x. Such a real y exists if x ≥ M(∞). Remarks

If M(t) ≤ C1 + C2 log 1 + 1t , then (x) = ∞ for all x > C2 . If M(t) ≥ c1 + c2 log 1 + 1t , then (x) < ∞ for all x ≤ c2 . 2. When M is convex, one can recover M from F by the Legendre inversion formula (see, e.g., [20]): M(t) = supx≥0 {−tx + F(x)}. 3. To see how one can retrieve M from , introduce the increasing func

(x) = x log[x−1 (x)], we have tion M(t) = M(e−t ), t ∈ R. Setting

(x) = supτ ∈R {τ x − M(τ )}. When M is convex, the Legendre inversion 1.


(τ )} and one can recover M from formula yields M(t) = supτ ∈R {τ t −

, its Legendre transform [

]∗ , and using the formula by computing ∗

M(t) = [] (log 1/t). 4. If we are given two non-negative non-increasing functions M1 ≤ M2 then, with obvious notation, we have F1 ≤ F2 , 1 ≥ 2 . If M1 , M2 are assumed to be convex, then F1 ≤ F2 implies M1 ≤ M2 by Remark 2. If t → M1 (e−t ), M2 (e−t ) are assumed to be convex, then 1 ≥ 2 implies M1 ≤ M2 by Remark 3. 2.2 The functions N and Q In this section, we set up the machinery which is necessary to connect ultracontractivity estimates with Nash inequalities. Definition 2.3 Let M be a non-increasing non-negative function defined on (0, +∞) and such that M(0+ ) = ∞. For any real x, set N(x) = sup{xt − tM(1/t)}.

(2.7)

t>0

When M ∈ C 1 , set also Q(x) =

−M ◦ M−1 (x) 0

if x > M(∞) otherwise.

(2.8)

Remarks 1. 2. 3.

The function N is convex and satisfies N(x) ≤ 0 for all x ≤ M(∞). The function M is convex if and only if t → tM(1/t) is convex. Thus, if M is convex, the inverse Legendre transform gives M(t) = supx>0 {x − tN(x)}. When M is C 1 and convex (hence Q monotone increasing), the function Q grows faster than x at infinity. Indeed, b ≤ 2Q(b)

b b/2

ds =− Q(s)

b

d [M−1 ](s)ds = M−1 (b/2) − M−1 (b), ds

b/2

and M−1 tends to 0 at infinity. 4. If M1 ≤ M2 then N1 ≥ N2 and, by Remark 2, the converse holds if M1 , M2 are assumed to be convex. Note that Q does not enjoy such properties because of the use of M in its definition. Lemma 2.4 Let M be a non-increasing non-negative C 1 function defined on (0, +∞) and such that M(0+ ) = ∞. Let F, , N, Q be as in (2.3), (2.4), (2.7) and (2.8).

A. Bendikov et al.

1.

= bM(at). The associated functions

Q

are

, N, Fix a, b > 0 and set M(t) F, given by

F = aF(x/ab),

2.

= ab(x/b),

N(x) = abN(x/b),

Q(x) = abQ(x/b)

(x) = aF ◦ (x/b). and we have

F ◦ Assume that M is convex. Then Q ≥ N and: (2a) If there exists b ∈ (0, ∞) such that −tM (t) ≤ bM(t)

(2b)

(2.9)

then, for any ε > 0 we have, Q(x) ≤ (b/ε)N((1 + ε)x) for all x large enough. If there exists a > 0 such that −tM (t) ≥ aM(t)

(2c)

for all t small enough,

for all t small enough

(2.10)

then F ◦ (x) ≤ (1 + (1/a)) x for x large enough. If both (2.9) and (2.10) holds then N ≈ Q ≈ at infinity.

Remark Condition (2.9) requires that M does not grow too fast at zero, whereas condition (2.10) requires that M does not grow too slowly. Proof Part 1 is proved by inspection. We now prove that Q ≥ N when M is convex. Given x, the point t0 where the maximum of xt − tM(1/t) is attained is such that x = M(1/t0 ) − (1/t0 )M (1/t0 ). Thus N(x) = −M (1/t0 ). As M is decreasing we have x ≥ M(1/t0 ), i.e. M−1 (x) ≤ 1/t0 . Since M is convex, −M is non-increasing and −M ◦ M−1 (x) ≥ −M (1/t0 ) = N(x). Together with Remark 1, this proves that Q ≥ N. To prove (2a), fix x large enough and ε > 0, and define t1 by x = (1 + ε)M(1/t1 ). Then N(x) ≥ εt1 M(1/t1 ) ≥ −(ε/b)M (1/t1 ) and thus N(x) ≥ −(ε/b)M ◦ M−1 (x/(1 + ε)). This gives the desired inequality. To prove (2b), for any x large enough, define τ = τ (x) by x = −M (τ ) (note that τ goes to 0 when x goes to infinity). Then F(x) = xτ + M(τ ) ≤ xτ − (τ/a)M (τ ) = (1 + 1/a)xτ . To have an estimate for , for x large enough, define σ = σ (x) to be such that (x/σ )e−M(σ )/x = sup(x/s)e−M(s)/x = (x). s>0

Such a σ ∈ (0, ∞) exists because M is C 1 and (2.10) implies that M grows faster than a positive power of 1/t when t tends to zero. We have x = −σ M (σ ) and (x) ≤ x/σ = −M (σ ). Since F is non-decreasing, this gives F ◦ (x) ≤ F(−M (σ (x))) ≤ (1 + (1/a)) [−M (σ (x))]τ (−M (σ (x))).


But τ (−M (σ (x))) = σ (x). Hence F ◦ (x) ≤ (1 + (1/a)) [−M (σ (x))σ (x)] = (1 + (1/a)) x. Finally, we prove (2c). By hypothesis, we have aM(t) ≤ −tM (t) ≤ bM(t)

forall t small enough.

(2.11)

As above, (x) = (x/σ )e−(1/x)M(σ ) where x = −σ M (σ ). By (2.11), this gives (x) ≈ −M (σ ) and x ≈ M(σ ). As (2.11) implies that the two positive decreasing functions M, −M satisfy M(2t) ≥ cM(t) and −M (2t) ≥ −cM (t) for some c > 0 and all t small enough (i.e. M and −M are doubling near 0), we conclude that (x) ≈ −M ◦ M−1 (x) = Q(x) at infinity. 3 The eventual ultracontractivity of e−sF(A) Let (X, µ) be a σ -finite measure space. Recall that there is a one-to-one correspondence between semigroups (Ht )t>0 of self-adjoint contractions on L2 (X, µ) and non-negative, possibly unbounded, self-adjoint operators A on L2 (X, µ) via the relation Ht = e−tA , that is, −A is the infinitesimal generator of Ht . Let A be a non-negative self-adjoint operator on L2 (X, µ). Let Eλ , λ ∈ [0, ∞), be the spectral resolution of A so that ∞ λdEλ .

A= 0

For any function f bounded from below we can consider the bounded selfadjoint operator −sf (A)

e

∞ =

e−sf (λ) dEλ .

0

The following result relates an ultracontractivity estimate of e−tA with function M to the eventual ultracontractivity of e−sF(A) , that is, the ultracontractivity of e−sF(A) for s large enough. Theorem 3.1 Let Ht = e−tA , t > 0, be a self-adjoint semigroup of contractions on L2 (X, µ). Let M and F be as in Definition 2.1 with M convex. (i) Assume that Ht 2→∞ ≤ eM(t) for all t > 0. Then, for any function f defined on the positive semi-axis and such that f ≥ F, the operator e−sf (A) defined on L2 (X, µ) is bounded from L2 (X, µ) to L∞ (X, µ) for all s > 1, uniformly on any interval (s0 , ∞), s0 > 1. (ii) Assume that for some non-negative function f ≤ F the operator e−sf (A) is bounded from L2 (X, µ) to L∞ (X, µ) for all s ≥ s1 and set C(s) =

A. Bendikov et al.

e−sf (A) 2→∞ . Then we have e−tA 2→∞ ≤ C(s)esM(t/s) for all t > 0 and all s ≥ s1 . For the proof, we need the following Lemma. Lemma 3.2 Let M and F be as in Definition 2.1 and set ∞ ψ(x) =

e−xs de−M(s) .

0

For all positive x, we have e−F(x) ≤ ψ(x) ≤ (1 + F(x))e−F(x) . In particular, at infinity, − log (ψ) ∼ F. Proof For fixed x > 0, consider the convex positive function mx (t) = xt + M(t) on (0, ∞). The function mx tends to ∞ at 0 and at ∞. Let tx be the smallest t at which mx attains it minimum so that F(x) = mx (tx ). We have ∞

−xt

e

de

−M(t)

∞ =x

0

−(xt+M(t))

e

∞ dt ≥ x

e−(xt+M(t)) dt

tx

0

≥ xe−M(tx )

∞

e−xt dt = e−(xtx +M(tx )) = e−F(x) .

tx

This proves the desired lower bound. For the upper bound, write ∞

−xt

e

de

−M(t)

∞ =x

0

e−(xt+M(t)) dt

0

⎛ ⎜ ≤ x⎝

F(x)/x

e−(xt+M(t)) dt +

0

⎞ ⎟ e−xt dt⎠

F(x)/x

F(x)/x

≤x

∞

−F(x)

e

∞ dt +

0

e−u du

F(x) −F(x)

= F(x)e

−F(x)

+e

.

That − log ψ ∼ F at infinity follows easily from these two bounds.


Proof of Theorem 3.1 We use some ideas from [16, Proposition 1.3]. To prove the first statement, set ∞ ψt (x) =

e−xs de−tM(s/t) .

0

Applying Lemmas 3.2 and 2.4 with Mt (s) = tM(s/t), t > 0 fixed, shows that ∀ t > 0,

∀x > 0,

ψt (x) ≥ e−tF(x) .

As a consequence, e−tf (A) ψt (A)−1 is a bounded operator on L2 (X, µ) with norm less than one . Hence, for any f ≥ F and any t > 1, we have e−tf (A) 2→∞ = ψt (A)e−tf (A) ψt (A)−1 2→∞ ≤ ψt (A)2→∞ e−tf (A) ψt (A)−1 2→2 ∞ ∞ −Mt (s) −sA ≤ e 2→∞ de ≤ eM(s) de−Mt (s) 0

∞ ≤

0

eMt (s)/t de−Mt (s) =

0

e−tM(∞)

τ −1/t dτ =

0

te−M(∞)(t−1) , (t − 1)

which proves the claim. To prove the second statement, let f ≤ F and note that etx e−sf (x) = etx−sf (x) ≥ etx−sF(x) ≥ etx−s(τ x+M(τ )) = ex(t−sτ )−sM(τ ) , for all s, t, τ , x > 0. In particular, for τ = t/s, we get e−tx esf (x) ≤ esM(t/s) . It follows that (e−sf (A) )−1 e−tA 2→2 ≤ esM(t/s) . Hence, for all s ≥ s1 , e−tA 2→∞ ≤ e−sf (A) 2→∞ (e−sf (A) )−1 e−tA 2→2 ≤ C(s)esM(t/s) . This ends the proof of Theorem 3.1. 4 Embeddings into L∞ Our first result involving the function is the following theorem.

A. Bendikov et al.

Theorem 4.1 Let M, F, be as in Definition 2.1 with M convex. Assume that only takes finite values and that there exists T > 0 such that g∈

∞

1/n

Dom(An ) and

lim sup n→∞

0

An g2 ≤ 1 ⇒ g ∈ L∞ (X, µ). e−1 (n/T)

Then e−sF(A) 2→∞ < ∞.

∀ s ≥ T,

In particular, for all t > 0 and all s ≥ T, there exists a constant C(s) such that e−tA 2→∞ ≤ C(s)esM(t/s) . Proof Let g be in L2 (X, µ) with g2 = 1. Then, spectral theory and the definition of give n −tF(A)

A e

1/n g2

1/n n −tF(x) ≤ sup x e x>0

= sup xe−(t/n)F(x) = e−1 (n/t). x>0

n n 1/n ≤ e−1 (n/t) Thus, the function h = e−tF(A) g is in ∞ 0 dom(A ) and A h2 for all n. By hypothesis, this implies that h = e−tF(A) g belongs to L∞ (X, µ) if t ≥ T. Hence, by the closed graph theorem, ∀ t ≥ T,

C(t) = es−tF(A) 2→∞ < ∞.

By Theorem 3.1, this gives that e−tA 2→∞ ≤ C(s)esM(t/s) for all t > 0 and all s ≥ T. Our next result provides a partial converse for Theorem 4.1. Theorem 4.2 Let M, F be as in Definition 2.1 with M convex. Assume that φ is a non-negative function such that F ◦ φ(x) ≤ κx.

(4.1)

for some κ > 0 and all x large enough. Assume further that there exists t0 > 0 such that e−t0 F(A) 2→∞ < ∞.


Then, for any T > 2et0 κ, the implication g∈

∞

1/n

Dom(An )

and

lim sup n→∞

0

An g2 ≤ 1 ⇒ φ(n/T)

g ∈ L∞ (X, µ)

holds true. The proof of this theorem requires two lemmas. For any n ≥ 1, consider the following functions. Mn (t) = [M(t1/n )]n ,

t > 0;

Fn (x) = inf {xt + Mn (t)} = FMn (x), t>0

F(n) (x) = [F(x1/n )]n ,

(4.2) x > 0;

x > 0.

(4.3) (4.4)

Lemma 4.3 Let M, F be as in Definition 2.1. For each integer n ≥ 1, the function Mn is non-increasing and convex. The function Fn is non-decreasing, concave and satisfies Fn (x) = o(x) at infinity. Moreover Fn ≤ F(n) ≤ 2n−1 Fn .

(4.5)

Proof Computing derivative, we have Mn (t) = −t−(1−1/n) [M(t1/n )]n−1 [−M (t1/n )]. Since t−(1−1/n) and the two other factors are non-increasing, Mn is non-decreasing, i.e., Mn is convex. The stated properties of Fn = FMn easily follow (see Definition 2.1). The double inequality (4.5) follows easily from an + bn ≤ (a + b)n ≤ 2n−1 (an + bn ), a, b > 0. The usefulness of this lemma is apparent in Lemma 4.4 below and comes from the fact that F(n) is not, in general, a concave function on (0, ∞) (it is concave on a neighborhood of infinity, the neighborhood depending on n). Lemma 4.4 Let M, F be as in Definition 2.1. Let A be a non-negative self-adjoint operator on a Hilbert space H. For any u ∈ H with u = 1, and any integer n, we have [F(A)]n u1/n ≤ 2F(An u1/n ). Proof By spectral decomposition, the definition of F2n and the right-hand side inequality in (4.5), we have ∞ [F(A)] u = [F(λ)]2n dEλ u, u n

2

0

A. Bendikov et al.

∞ = [F(2n) (λ2n )]dEλ u, u 0 2n

≤2

∞ [F2n (λ2n )]dEλ u, u, 0

where ·, · denotes the scalar product in H. Since F2n is concave, Jensen’s inequality gives ⎛ [F(A)]n u2 ≤ 22n F2n ⎝

∞

⎞ λ2n dEλ u, u⎠

0

= 2 F2n An u2 ≤ 22n F(2n) An u2 2n . = 22n F An u1/n 2n

Here we have used (4.5) to obtain the second inequality. This proves Lemma 4.4. Remark Recall that a function F : [0, ∞) → [0, ∞) is a Bernstein function if F is smooth and satisfies (−1)n+1 F (n) ≥ 0 for n ≥ 1 (here, F (n) is the n-th derivative of F). Then F(n) (x) = [F(x1/n )]n is also a Bernstein function (see [10]). In particular, F(n) is concave for all n. Hence, for any Bernstein function F, the conclusion of the lemma can be improved to [F(A)]n u1/n ≤ F(An u1/n ), and, in that case, the proof follows immediately from spectral theory and Jensen’s inequality. In particular, for F(x) = xα , α ∈ (0, 1), we have Kolmogorov’s inequality Aαn u ≤ An uα u1−α . Proof of Theorem 4.2 Set T = 2et0 κS with S > 1. Fix g ∈ that g2 = 1 (this is no loss of generality) and

∞ 0

dom(An ) such

for n large enough.

(4.6)

1/n

An g2 lim sup ≤ 1. n→∞ φ(n/T) Then we have 1/n

An g2

≤

1+S φ(n/T) 2


To show that g ∈ L∞ (X, µ), write et0 F(A) g2 ≤

∞ n ∞ n t0 t0 1/n n 2F An g2 [F(A)]n g2 ≤ n! n! 0

0

∞ n t0 [(1 + S)F ◦ φ(n/T)]n ≤ C1 + n! 0

≤ C1 + C2 + = C1 + C2 +

∞ ((1 + S)κt0 n/T)n 0 ∞ 0

n! (n/e)n 1 + S n < ∞. n! 2S

Here we have used Lemma 4.4 to obtain the second inequality, (2.5) and (4.6) to obtain the third inequality and the hypothesis (4.1) to obtain the fourth inequality. The finite constants C1 , C2 appear here to account for the finite number of terms to which both (4.6) and (4.1) do not apply because n is not large enough. Thus et0 F(A) g ∈ L2 (X, µ). By hypothesis, e−t0 F(A) is bounded from L2 (X, µ) to L∞ (X, µ). Hence, g = e−t0 F(A) [et0 F(A) g] is in L∞ (X, µ). This ends the proof of Theorem 4.2. We can now state one of the main theorems of this paper. Theorem 4.5 Let M, F, be as in Definition 2.1 with M convex. Assume that there exists a finite positive constant κ such that F ◦ (x) ≤ κx,

for all x large enough.

(4.7)

Let −A be the infinitesimal generator of a self-adjoint semigroup of contractions on L2 (X, µ). Then each of the properties listed below implies the one following it. 1. 2. 3.

∀ t > 0, e−tA 2→∞ ≤ eM(t) . ∀ t > 1, e−tF(A) 2→∞ < ∞. There exists T > 0 such that g∈

∞ 0

4. 5.

1/n

Dom(An )

and

lim sup n→∞

An g2 ≤ 1 ⇒ g ∈ L∞ (X, µ). (n/T)

∀ t ≥ T, e−tF(A) 2→∞ < ∞. For each s ≥ T there exists a finite positive constant C(s) such that, for all positive t, e−tA 2→∞ ≤ C(s)esM(t/s) .

This statement follows from Theorems 3.1, 4.1 and 4.2. Roughly speaking, this theorem says that properties 1, 2, 3 above are equivalent under the hypotheses of the theorem. Lemma 2.4 shows that Theorem 4.5 applies if M is convex, C1 , and satisfies (2.10).

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5 Relations with Nash type inequalities For the results in this section, we need to assume more than the mere fact that e−tA , t > 0, is a semigroup of self-adjoint contractions. We will assume that (e−tA )t>0 is sub-Markovian, i.e., e−tA is self-adjoint on L2 (X, µ) and satisfies 0 ≤ f ≤ 1 ⇒ 0 ≤ e−tA f ≤ 1, for all t > 0. In fact, what is really needed is simply that e−tA acts on L1 (X, µ) and L∞ (X, µ) with norm at most 1. Recall the following result from [15] (the functions m, θ ,

θ of [15] √are related

x), θ (x) = to our functions M, N, Q, by log m(t) = 2M(t), θ (x) = xN(log √ 2xQ(log x)). Here ·, · denotes the scalar product in L2 (X, µ). Proposition 5.1 Let −A be the infinitesimal generator of a sub-Markovian semigroup on L2 (X, µ). Let M, N and Q be as in Definition 2.3. 1. [15, Proposition II.2] Assume that, for all t > 0, e−tA 2→∞ ≤ eM(t) . Then ∀ f ∈ Dom(A) 2.

with f 1 ≤ 1, f 22 N(log f 2 ) ≤ Af , f

(5.8)

holds true. [15, Proposition II.1] Assume that the Nash type inequality ∀ f ∈ Dom(A)

with f 1 ≤ 1, f 22 Q(log f 2 ) ≤ Af , f

(5.9)

holds true. Then, for all t > 0, e−tA 2→∞ ≤ eM(t) . The next theorem follows from Proposition 5.1 and Lemma 2.4. Theorem 5.2 Let −A be the infinitesimal generator of a sub-Markovian semigroup on L2 (X, µ). Let M be a C1 convex non-increasing function defined on (0, +∞) and such that M(0+ ) = ∞. Let N be defined by (2.7). Assume that M satisfies (2.9), that is, there exists b such that −tM (t) ≤ bM(t) for all t small enough. Then the following properties are equivalent. 1.

There exist c1 , c2 ∈ (0, ∞) such that ∀ t > 0,

2.

e−tA 2→∞ ≤ ec1 M(c2 t) .

There exist c3 , c4 ∈ (0, ∞) such that ∀ f ∈ Dom(A)

with f 1 ≤ 1,

c3 f 22 N(c4 log f 2 ) ≤ Af , f .

This should be compared to [15, Theorem II.5, page 516] which is very much in the same spirit. As already mentioned above, our functions M, N, Q are precisely related to the functions m,

θ , θ of [15]. However, the hypothesis made in [15] that m satisfies the condition (D) considered there is (slightly) stronger than the hypothesis that M satisfies (2.9). The conclusion of [15, Theorem II.5, page 516] is stronger than ours because it yields c4 = 1. Putting together Lemma 2.4, Theorem 4.5 and Theorem 5.2, and noting that under the assumptions below M and are doubling functions, we obtain the following result.


Theorem 5.3 Let M, F, be as in Definition 2.1. Assume that M is C1 , convex, and that there are constants a, b ∈ (0, ∞) such that (2.11) holds, that is, aM(t) ≤ −tM (t) ≤ bM(t) for all t small enough. Let −A be the infinitesimal generator of a sub-Markovian semigroup on L2 (X, µ). Then the following properties are equivalent: 1. 2. 3.

There exists c1 ∈ (0, ∞) such that, for all t > 0, e−tA 2→∞ ≤ ec1 M(t) . There exists t0 > 0 such that, for all t > t0 , e−tF(A) 2→∞ < ∞. n There exists C1 ∈ (0, ∞) such that for any function f ∈ ∞ 0 Dom(A ) we have

1/n

An f 2 lim sup (n) n→∞ 4.

≤ C1 ⇒ f ∈ L∞ (X, µ).

There exists C2 ∈ (0, ∞) such that the Nash inequality ∀ f ∈ Dom(A)

with f 1 ≤ 1,

f 22 (log f 2 ) ≤ C2 Af , f + f 22

is satisfied. For another discussion related to Nash inequalities we refer to [5]. 6 Discussion of specific behaviors In this section we compute F, , N, Q for some explicit functions M and spell out how the results of the previous three sections apply. 6.1 The classical case M(t) = c +

d 4

log(1 + 1t ), c, d > 0

Here, the important parameter is d which plays the role of the dimension. A simple computation shows that there are constants c1 , c2 > 0 depending on c, d such that c1 +

d d log(1 + x) ≤ F(x) ≤ c2 + log(1 + x). 4 4

Theorem 3.1(i) says that if, for all t > 0, e−tA 2→∞ ≤ C1 (1 + 1t )d/4 then (I+A)−α is bounded from L2 (X, µ) to L∞ (X, µ) for all α > d/4. Theorem 3.1(ii) says that if (I +A)−d/4 is bounded from L2 (X, µ) to L∞ (X, µ) then there exists a constant C2 such that e−tA 2→∞ ≤ C2 (1 + 1t )d/4 . Compare with the somewhat sharper result in [14, Th.1]. The function satisfies ≡ ∞ on (d/4, ∞) so the results of Sect. 4 (except Theorem 4.2) do not apply. Theorem 4.2 gives a very weak result. Namely, by

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taking φ(x) = ex , it yields that if, for all t > 0, e−tA 2→∞ ≤ C1 (1 + 1t )d/4 then n any function g ∈ ∩∞ 0 Dom(A ) satisfying lim sup e−n/T An g2

1/n

n→∞

≤1

for some T large enough must be bounded. Theorem 3.1 gives a better result. In general, the results of Sect. 4 are not entirely satisfactory when applied to relatively tame ultracontractivity functions M. The functions N and Q satisfy N(x) ≈ Q(x) ≈ exp((4/d)x) for x large enough. Proposition 5.1 (i.e., [15]) yields the equivalence between e−tA 2→∞ ≤ 2(1+2/d) 4/d ≤ C1 (Af , f + f 22 )f 1 . C1 (1 + 1t )d/4 and the Nash inequality f 2 Note however that Theorem 5.2 only yields a weaker result. Namely, it states that a “polynomial” bound e−tA 2→∞ ≤ C1 (1 + 1t )α , for some α > 0, is 2(1+β)

2β

equivalent to a Nash inequality of the type f 2 ≤ C1 (Af , f + f 22 )f 1 for some β > 0. The correspondence between α and β has been lost in this statement (one should have β = 2/α). 6.2 The case M(t) = c1 + c2 [log(1 + 1t )]λ , c1 , c2 > 0, λ > 1 Somewhat tedious computations show that λ

F(x) ≈ 1 + [log(1 + x)] ,

1 x 1/(λ−1) 1− (x) ≈ x exp . λ c2 λ

Theorem 3.1 yields the equivalence between the ultracontractivity property ∃ c1 , c2 > 0,

∀ t > 0,

1

λ

e−tA 2→∞ ≤ ec1 +c2 [log(1+ t )] ,

(6.10)

and ∃ t0 > 0,

∀ t > t0 ,

λ

e−t[log(I+A)] 2→∞ < ∞.

Theorem 4.5 does not apply because, for large x, F ◦ (x) ≈ xλ/(λ−1) and thus condition (4.7) is not satisfied. Nevertheless, we can apply Theorem 4.1 and Theorem 4.2. Theorem 4.1 says that, if there exists T such that g∈

∞ 0

1/n

Dom(An )

and

lim sup n→∞

An g2

1/(λ−1) e(n/T)

≤ 1 ⇒ g ∈ L∞ (X, µ),

then there are constants c1 , c2 > 0 such that (6.10) holds. Theorem 4.2 says that, if there are constants c1 , c2 > 0 such that (6.10) holds, then there exists T


such that g∈

∞

1/n

Dom(An )

and

lim sup n→∞

0

An g2 e(n/T)

1/λ

≤ 1 ⇒ g ∈ L∞ (X, µ).

Thus we have not quite obtained the desired equivalence between ultracontractivity and embedding in this case. Again, this reflects the unsatisfactory nature of the results of Sect. 4 when applied to tame functions M. The functions N, Q satisfy N(x) ≈ Q(x) ≈ x(λ−1)/λ exp (x/c2 )1/λ . Theorem 5.2 gives the equivalence between the ultracontractivity property (6.10) and the Nash type inequality ∀ f ∈ Dom(A)

with f 1 ≤ 1,

c3 f 22 exp(c4 [log(1 + f 2 )]1/λ ) ≤ Af , f + f 22 .

In this statement, no information is given on the relation between the important constants c2 , c4 . The computation of N and Q above and a direct application of Proposition 5.1 yield a more precise result. 6.3 The case M(t) = c1 + t−λ , c1 , λ > 0 This is, in a sense, typical of the cases we want to consider in this work. Section 6.5 treats the more general case when M is regularly varying. Set λ = λ/(1 + λ). One easily computes

F(x) = c1 + (1 + λ)xλ ,

(x) = λ−1/λ e−λ(1+c1 /x) x1/λ ,

and N(x) = λ

x − c1 1+λ

1/λ ,

Q(x) = λ (x − c1 )1/λ , x ≥ c1 .

Theorems 3.1, 4.1, 4.2, 4.5, 5.2 all apply as well as Theorem 5.3 which, for sub-Markovian semigroups, states the equivalence of the following properties: 1. 2. 3.

−λ

There exists c1 > 0 such that, for all t > 0, e−tA 2→∞ ≤ ec1 +t ; λ There exists t0 ∈ (0, ∞) such that, for all t > t0 , e−tA 2→∞< ∞; n There exists a finite positive constant C such that for any f ∈ ∞ 0 Dom(A ), 1/n ≤ C ⇒ f ∈ L∞ (X, µ). lim sup n−1/λ An f 2 n→∞

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4.

There exists C1 ∈ (0, ∞) such that for all f ∈ Dom(A) with f 1 ≤ 1, we have f 22 [log(1 + f 2 )]1/λ ≤ C1 Af , f + f 22 . 1

λ

6.4 The case M(t) = c1 ec2 [log(1+ t )] , c1 , c2 , λ > 0 This is really two different cases depending on whether λ ∈ (0, 1) or λ ∈ (1, ∞). For λ = 1, it reduces to M(t) = (1 + 1t )c2 which is essentially the case treated in Sect. 6.3. For any fixed λ ∈ (0, 1), log F(x) ∼ c2 [log(1 + x)]λ and there are constants c3 , c4 > 0 such that c3 exp c3 (log(1 + x)]λ ≤ F(x) ≤ c4 exp c4 (log(1 + x)]λ . When instead λ ∈ (1, ∞), there exist constants c3 , c4 > 0 such that c3 (1 + x) exp −c4 (log(1 + x)]1/λ ≤ F(x) ≤ c4 (1 + x) exp −c3 (log(1 + x)]1/λ . Theorem 3.1 applies but we will not write the result explicitely. Concerning the function , there are c5 , c6 > 0 such that x exp c5 (log x)1/λ ≤ (x) ≤ x exp c6 (log x)1/λ for x large enough. When λ ∈ (0, 1), tedious computations show that x−1 F ◦(x) tends to infinity at infinity, hence Theorem 4.5 does not apply. When λ ≥ 1, (2.10) holds true, F ◦ (x) ≈ x, and Theorem 4.5 applies. For any λ ∈ (0, ∞), we can still apply Theorems 4.2 and 4.1 which actually give a quite satisfactory result stated below in Theorem 6.1. Computing N and Q, one finds that, for large x, Q(x) ≈ x (log x)

(λ−1)/λ

1/λ −1 , exp c2 log(x/c1 )

and that there are constants c7 , c8 ∈ (0, ∞) such that x exp c7 (log x)1/λ ≤ N(x) ≤ x exp c8 (log x)1/λ . If λ ∈ (0, 1), Lemma 2.4 and Theorem 5.2 apply. If λ > 1, one can check that N is substantially smaller than Q in the sense that, for any fixed A > 1, N(Ax) = o(Q(x)) at infinity. As (2.11) fails if λ = 1, Theorem 5.3 does not apply. However, Theorems 4.1, 4.2 and Proposition 5.1 yield the following result (recall that log(2) (s) = log(1 + log(1 + s))).


Theorem 6.1 Fix λ ∈ (0, ∞). Let −A be the infinitesimal generator of a sub-Markovian semigroup on L2 (X, µ). The following properties are equivalent: 1. 2.

3.

There exist c1 , c2 ∈ (0, ∞) such that, for all t > 0, e−tA 2→∞ ≤ 1 λ ec1 exp(c2 [log(1+ t )] ) . n There exists c3 ∈ (0, ∞) such that for any function f ∈ ∞ 0 Dom(A ) we have 1/n An f 2 < ∞ ⇒ f ∈ L∞ (X, µ). lim sup 1/λ n→∞ nec3 (log n) There exist c4 , c5 ∈ (0, ∞) such that for all f ∈ Dom(A) with f 1 ≤ 1, we have 1/λ ≤ Af , f + f 22 . c4 f 22 log(1 + f 2 ) exp c5 log(2) f 2

Note that this statement does not give information on the relationships between the important constants c2 , c3 , c5 . This reflects in part the fact that Theorem 5.3 does not apply. 6.5 Regular variation This section generalizes the example M(t) = c1 + t−λ by treating functions of regular variation. Recall the following classic definition (see [13]). A measurable positive function f is of regular variation of index λ (at infinity), λ ∈ R, if, for all ρ > 0, lim f (ρx)/f (x) = ρ λ .

x→∞

The class of these functions is denoted by Rλ . Functions in R0 are called slowly varying functions. A positive function f is of smooth regular variation of order λ (i.e., f ∈ SRλ ) if h(x) = log f (ex ) is smooth in a neighborhood of infinity and lim h (x) = λ,

x→∞

lim h(n) (x) = 0

x→∞

for all n = 2, 3, . . . .

According to [13, (1.8.1 )], a function f is in SRλ if and only if it is smooth in a neighborhood of infinity and lim xn f (n) (x)/f (x) = λ(λ − 1)...(λ − n + 1)

x→∞

for all n = 1, 2, . . . .

By [13, Theorem 1.8.2], for any f ∈ Rρ there exist f1 , f2 ∈ SRρ such that f1 ∼ f2 and f1 ≤ f ≤ f2 in some neighborhood of infinity. For λ, α, β ∈ R, the function f (t) = tλ (log t)α (log log t)β is an example of a function in Rλ .

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Proposition 6.2 Let M, F, be as in Definition 2.1. Assume that t → M(1/t) ∈ SRλ for some λ > 0. Set λ = λ/(1 + λ) and M1 (t) = t−1 M(t). Then 1. 2. 3. 4.

F ∈ SRλ , in fact there exists aλ > 0 such that F ∼ aλ M ◦ M1−1 . ∈ SR1/λ , in fact there exists bλ > 0 such that (x) ∼ bλ x/M−1 (x). There exists cλ > 0 such that F ◦ (x) ∼ cλ x at infinity. N ≈ Q ≈ in a neighborhood of infinity. In fact, there exist dλ , dλ > 0 such that N ∼ dλ , Q ∼ dλ .

Proof The function t → xt+M(t) attains its infimum at τ satisfying x = −M (τ ). As tM (t) ∼ −λM(t) as t tends to 0, we have τ x ∼ λM(τ ) (as x tends to infinity). This implies F(x) = τ x + M(τ ) ∼ (1 + λ)M(τ ) ∼ (1 + λ)M ◦ [−M ]−1 (x). It is well known that M ∈ SR−λ implies −M ∈ SR−1−λ and [−M ]−1 ∈ SR−1/(1+λ) . Hence F ∈ SRλ/(1+λ) . This proves the first statement. To prove the second and third statements, consider the function y → ye1−(1/x)F(y) . It attains its supremum at at point z such that x = zF (z). As zF (z) ∼ λ F(z), we have F(z) ∼ x/λ and

(x) ∼ e1−(1/λ ) z ∼ e1−(1/λ ) F −1 (x/λ ) ∼ e1−(1/λ ) (1/λ )1/λ F −1 (x). Hence, (x) ∈ SR1/λ and F ◦ (x) ∼ cλ x. The last assertion follows from the proof of Lemma 2.4(2c). Theorem 5.3 applies in this setting and yields the following result. Theorem 6.3 Let M, F, be as in Definition 2.1 with M convex. Assume further that t → M(1/t) ∈ SRλ for some λ > 0. Set M1 (t) = t−1 M(t). Let −A be the infinitesimal generator of a sub-Markovian semigroup on L2 (X, µ). Then the following properties are equivalent: 1. 2. 3.

There exists c1 ∈ (0, ∞) such that, for all t > 0, e−tA 2→∞ ≤ ec1 M(t) . −1 2→∞ < ∞. There exists t0 > 0 such that, for all t > t0 , e−tM◦M1 (I+A) n There exists C1 ∈ (0, ∞) such that for any function f ∈ ∞ 0 Dom(A ) we have 1/n An f 2 lim sup ≤ C1 ⇒ f ∈ L∞ (X, µ). n/M−1 (n) n→∞

4.

There exists C2 ∈ (0, ∞) such that the Nash inequality ∀ f ∈ Dom(A) is satisfied.

with f 1 ≤ 1,

f 22

log f 2 2 Af , f + f ≤ C 2 2 M−1 (log f 2 )


In order to apply this theorem in concrete cases, one needs to compute M−1 and M1−1 . This can be done by using Proposition 1.5.15 and Proposition 2.3.5 in [13]. The following examples illustrate such computations. Set log(1) (t) = log(1 + t) and log(k) (t) = log(1 + log(k−1) (t)).

(6.11)

Proposition 6.4 Fix λ > 0 and set λ = λ/(1 + λ). Assume that M(t) = t−λ (1/t) where is a slowly varying function. 1. Assume that (t) = k1 (log(i) t)βi . Then

F(x) ∼ xλ (x)λ /λ , 2.

(x) ∼ x1/λ (x)−1/λ at infinity.

Assume that (t) = exp ε(log t)β with β ∈ (0, 1/2) and ε = ±1. Then F(x) ∼ xλ exp ε(1 + λ)−1−β (log x)β and (x) ∼ x1/λ exp −ελ−1−β (log x)β at infinity.

3.

Assume that (t) = exp ε(log t)β with β ∈ (1/2, 1) and ε = ±1. Then there are positive finite constants ci , 1 ≤ i ≤ 4, depending on λ and β, such that, at infinity, xλ exp εc1 (log x)β ≤ F(x) ≤ xλ exp εc2 (log x)β , and x1/λ exp −εc3 (log x)β ≤ (x) ≤ x1/λ exp −εc4 (log x)β .

Proof These results involve tedious computations. A basic tool is the asymptotic inverse formula of [13, Proposition 1.5.15]. For the first statement, one also uses [13, Proposition 2.3.5] (see [13, Example 1, p. 433]). For the second statement, one uses [13, Example 3, p. 435]. The third statement, which is less precise, is obtained by inspection. 6.6 The case M(t) = exp(g(1/t)), g ∈ SRλ , λ > 0 This is a case where M is a rapidly varying function. The proof of the following proposition is along the same lines as the proof of Proposition 6.2. Proposition 6.5 Let M, F be as in Definition 2.1. Let be as in (2.6). Assume that M(t) = exp(g(1/t)) where g ∈ SRλ for some λ > 0. Then

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1. 2. 3.

F(x) ∼ x/g−1 (log x) at infinity; (x) ∼ xg−1 (log x) at infinity; F ◦ (x) ∼ x at infinity.

Proposition 1.5.15 and Proposition 2.3.5 in [13] show how to compute g−1 . Proposition 6.4 above gives formulas in some special cases. Theorem 4.5 applies nicely in this setting and yields the following. Theorem 6.6 Assume that g ∈ SRλ for some λ > 0. Let −A be the infinitesimal generator of a self-adjoint semigroup of contractions on L2 (X, µ). Then the following properties are equivalent. 1.

There exists c > 0 such that ∀ t > 0, e−tA 2→∞ ≤ eexp(cg(1/t)) .

2. 3.

There exists t0 > 0 such that e g−1 (log(I+A)) 2→∞ < ∞. There exists a constant C > 0 such that

−t0

u∈

∞ 0

(I+A)

1/n

Dom(An )

and

lim sup n→∞

An u2 ≤ C ⇒ ng−1 (log n)

u ∈ L∞ (X, µ).

Note that this is a case when Nash techniques break down because N/Q tends to 0 at infinity (see [17, Example 2.3.5]). Indeed, N(x) ≈ xg−1 (log x) whereas Q(x) ≈ x(log x)g−1 (log x). 7 Further results concerning tame behaviors This section presents a different way of relating the behavior of “ultracontractivity functions” to that of “embedding functions”. If M is as in Definition 2.1 and is defined by (2.4), Theorem 4.1 states that the embedding property (1.2) with φ(x) = (x) implies the ultracontractivity bound (1.1) with m(t) = M(t)+c for some constant c. Theorem 4.5 provides a converse only in the case when (4.7) holds, that is, when there exists κ > 0 such that F ◦ (x) ≤ κx. This condition excludes the case where the behavior of M is “tame”. For instance, (4.7) does not hold when M(t) = c1 + c2 [log(1 + 1/t)]λ , c1 , c2 > 0, λ > 1. See Sect. 6.2. Under additional hypotheses on the infinitesimal generator A, this section gives an equivalence between (1.1) and (1.2) that covers more or less exactly the cases where Theorem 4.5 does not apply. 7.1 Probability spaces with polynomially bounded eigenfunctions Let e−tA , t > 0, be a semigroup of self-adjoint contractions on L2 (X, µ). In this section, we assume that (X, µ) is a probability space and that there exists

Ultracontractivity and embedding into L∞ 2 a countable orthonormal basis (ui )∞ 0 of L (X, µ) such that ui ∈ Dom(A) and Aui = λi ui with 0 ≤ λ0 ≤ λ1 ≤ · · · < ∞ and limi→∞ λi = ∞ (eigenvalues are repeated according to their multiplicity). We set

r(i) = #{j : λj ≤ λi }.

(7.12)

Thus, if all the eigenvalues λi are distincts, we have r(i) = 1 + i but if the multiplicity of the eigenvalues grows with i, r(i) can be much larger than i. We consider the condition such that ui ∞ ≤ Cr(i)a ,

there exist C, a ≥ 0

(7.13)

which may or may not be satisfied. Theorem 7.1 Let (X, µ) and (e−tA )t>0 be as described above. Let M, be as in Definition 2.1. Assume that M is C 1 , that t → −tM (t) is decreasing on (0, ∞) and tends to ∞ when t tends to 0. Assume that there exists b > 0 such that (2.9) holds, that is, m(t) ≤ bM(t) for all t small enough, and that does not take the value +∞. Assume also that condition (7.13) is satisfied. If, for all t > 0, we have e−tA 2→∞ ≤ eM(t) then, for any T > 2(1 + a)(1 + b), g∈

∞

1/n

Dom(An )

0

and

lim sup n→∞

An g2 ≤ 1 ⇒ (n/T)

g ∈ L∞ (X, µ).

We will need the following lemmas. Lemma 7.2 Let (X, µ) and (e−tA )t>0 be as described above. Let M be as in Definition 2.1 and N be the Nash function associated to M by (2.7). Assume that e−tA 2→∞ ≤ eM(t) . Then, for any integer k, λk ≥ N log r(k) . Proof By the spectral and ultracontractivity hypotheses concerning A, the semigroup e−tA admits a bounded kernel ht (x, y) which is given by ht (x, y) =

∞

e−tλi ui (x)ui (y)

0

and satisfies h2t (x, x) = ht (x, ·)22 ≤ e−tA 22→∞ ≤ e2M(t) . Using the hypothesis that µ(X) = 1, we obtain h2t (x, x)dµ(x) = M

∞ 0

e−2tλi ≤ e2M(t) .

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By Definition (7.12), this implies r(k)e−2tλk ≤ e2M(t) . √ Thus, for any t > 0, λk ≥ t−1 log r(k) − M(t) or, equivalently λk ≥ sup τ log r(k) − τ M(1/τ ) = N log r(k) . τ >0

Lemma 7.3 Assume that M and are as in Theorem 7.1 and N as in (2.7). Then, for any fixed K, y0 , t > 0, lim sup x→∞

2 y K(y) 1 min log ≤ −(1 + b)−1 K−b . x y>y0 y + t N(x)

Proof From the definition of N in terms of M, it follows that, for each x large enough, there exists t0 = t0 (x) ∈ (0, ∞) at which the maximum defining N(x) is attained; this t0 satisfies x = −t0 M (t0 ) + M(t0 ),

N(x) = −M (t0 ),

(7.14)

and tends to 0 when x tends to infinity. In particular, for x large enough, (2.9) implies −t0 M (t0 ) < bM(t0 ). For , if y is such that (y) is finite, then there exists t1 = t1 (y) ∈ (0, ∞) at which the maximum defining is attained and this t1 satisfies y = −t1 M (t1 ),

M(t1 )

(y) = −M (t1 )e t1 M (t1 ) .

(7.15)

Because we assume that t → −tM (t) is decreasing, t1 (y) is uniquely determined by the equation y = −t1 M (t1 ). Now, fix x large enough. Let t0 be as above and set y = −t1 M (t1 ) with t1 = Kt0 . Then M(t1 )

(y) = −M (t1 )e t1 M (t1 ) . As m is decreasing, we have −KM (t1 ) ≤ −M (t0 ) = N(x). Thus y log

K(y) N(x)

≤ −M(t1 ).

By (2.9), for small enough t0 (i.e., large enough x), we have M(t0 ) ≤ Kb M(Kt0 ) and thus x = −t0 M (t0 ) + M(t0 ) ≤ (1 + b)M(t0 ) ≤ (1 + b)Kb M(t1 ). It follows


that y2 K(y) y log ≤− (1 + b)−1 K−b x. y+t N(x) y+t Since y tends to infinity with x, we conclude that lim sup x→∞

2 y (y) 1 min log ≤ −(1 + b)−1 K−b x y>y0 y + t N(x)

as desired. Proof of Theorem 7.1 We want to show that for any T > 2(1 + a)(1 + b), the condition g∈

∞

1/n

Dom(An )

and

lim sup n→∞

0

An g2 ≤1 (n/T)

(7.16)

implies that g∞ < ∞. For any function g ∈ L2 (X, µ), let g = gk uk , gk = g, uk , be the expansion of g along the orthonormal basis (ui )∞ 0 . We have 2 λ2n and thus, if g satisfies (7.16), for any K > 1, there exists |g | An g22 = ∞ k 0 k nK such that for all n > nK we have |gk | ≤ e

1/n n log An g2 /λk

≤ en log(K(n/T)/N (log

√

r(k)))

,

(7.17)

where we have used Lemma 7.2 and the assumption on g to obtain the √ second inequality. Now, let n, z be such that n ≤ z < n+1 and K(n/T)/N(log r(k)) ≤ 1. Then, we have K(z/T) K(n/T) z2 √ ≤ √ . log n log z+1 N log r(k) N log r(k) Hence, for all k large enough, inf

n>nK

K(n/T) n log √ N log r(k)

K(y/T) y2 log ≤ inf √ y>nK y + 1 N log r(k) K(y) y2 log = T inf . √ y>nK /T y + T N log r(k)

By Lemma 7.3, given ε ∈ (0, 1) there exists kε such that, for all k > kε , we have inf

n>nK

K(n/T) √ n log N log r(k)

≤−

T(1 − ε) log r(k). (1 + b)Kb

A. Bendikov et al.

Reporting this in (7.17) yields |gk | ≤ r(k) |uk | ≤ Cr(k)a . Hence

−

T(1−ε) 2(1+b)Kb

, for k > kε . By hypothesis,

⎛ ⎞ ∞ T(1−ε) − +a |g| = gk uk ≤ C ⎝g2 (1 + kε )r(kε )a + r(k) 2(1+b)Kb ⎠. kε +1

k

Since T > 2(1 + a)(1 + b) and ε ∈ (0, 1), K ∈ (1, ∞) are arbitrary, we can pick ε, K such that −

T(1 − ε) + a < −1. 2(1 + b)Kb

− T(1−ε) +a Since r(k) ≥ 1+k, for such a choice of ε, K the series r(k) 2(1+b)Kb converges and it follows that g ∈ L∞ (X, µ). Theorem 7.1 is proved. Let us illustrate the hypotheses made in Theorem 7.1, especially (7.13), by looking at the case when X = G is a compact metrizable group equipped with its normalized Haar measure and −A is the infinitesimal generator of a vaguely continuous convolution semigroup of symmetric probability measures (µt )t>0 . Hence, e−tA : f → e−tA f = f × µt , t > 0, is a self-adjoint semigroup of contractions on L2 (G). By the Peter–Weyl theorem, representation theory provides us with a complete set of orthonormal eigenfunctions for A, formed 1/2 of normalized matrix coefficients dρ ρi,j where ρ runs over of all irreducible representations (modulo equivalence) and dρ = dim Vρ is the dimension of the irreducible unitary representation (ρ, Vρ ). Note that the ρi,j depends on A, d

namely, ρi,j (x) = ρ(x)ei , ej where (ei )1 ρ is an orthonormal basis of Vρ which diagonalizes Aρ : Vρ → Vρ where Aρ v = Aρ(x) v|x=e . Because we assume that G is metrizable, this orthonormal system is countable. Let us first consider the case when G is abelian. In this case, all the irreducible representations are of dimension 1 and the matrix coefficients (one for each irreducible representation class) are the characters γ of G which form the dual group G of G. In particular, viewed as a function on G, any γ has modulus |γ | ≡ 1. Thus, in this case, (7.13) holds with a = 0 and Theorem 7.1 applies (Sect. 7.2 below generalizes this result to the case when G is a metrizable locally compact abelian group). When G is not abelian (7.13) holds with a = 1/2. Indeed, for any eigenvalue λ = λi , there exists a representation ρ and a corresponding normalized matrix 1/2 coefficient u = dρ ρi,j such that u is a normalized eigenfunction for λ. Moreover, λ must have multiplicity at least dρ (because each irreducible representation appears with multiplicity equal to its dimension in the regular representation). As |ρk,l (x)| ≤ 1, it follows that 1/2 u∞ ≤ d1/2 . ρ ≤ r(i)


If we further assume that for each t > 0, the measure µt is central, that is, satisfies µt (xV) = µt (Vx) for any x ∈ G and any Borel set V, then (7.13) holds with a = 1/4. Indeed, in this case, an eigenvalue appearing at an irreducible representaion ρ appears in fact with multiplicity at least d2ρ . Hence we have 1/4 ui ∞ ≤ d1/2 . ρ ≤ r(i)

7.2 Convolution semigroups on abelian groups In what follows G is a locally compact metrizable abelian group and G denotes its dual (which is a locally compact σ -compact abelian group). The reader unfamiliar with this setting can assume that G = G = Rd . The references [11, 19, 23] provide detailed background. Both G and G are equipped with their respective Haar measures dx and dγ which are chosen in such a way that the Fourier transform f → fˆ : fˆ (γ ) = (x, γ )f (x)dx G

is a unitary map from L2 (G, dx) to L2 (G, dγ ). Let (µt )t>0 be a vaguely continuous convolution semigroup of symmetric probability measures on G (a measure ν is symmetric if ν(V) = ν(−V) for any Borel set V). By [11, Theorem 8.3], we have µt (γ ) = e−tψ(γ ) ,

γ ∈ ,

where ψ is a continuous negative definite function. Let −A be the infinitesimal generator of the self-adjoint semigroup of contractions f → µt ×f on L2 (G, dx). Theorem 7.4 Let M, be as in Definition 2.1. Assume does not take the value +∞ and M is C 1 . Assume also that t → −tM (t) is decreasing on (0, ∞), tends to ∞ when t tends to 0, and that there exists b > 0 such that (2.9) holds. If, for all t > 0, e−tA 2→∞ ≤ eM(t) then, for any T > 2(1 + b), we have g∈

∞ 0

1/n

Dom(An ) and lim sup n→∞

An g2 ≤ 1 ⇒ (n/T)

g ∈ L∞ (G, dx).

Proof The proof is similar to that of Theorem 7.1 with some significant technical adjustements. With the notation introduced above, the ultracontractivity of e−tA implies that the measures µt , t > 0, have bounded continuous densities and we write dµt (x) = ft (x)dx. Then e−tA 2→∞ = ft 2 ,

ft = e−tψ .

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Hence, we have ft 22 = e−tψ 22 ≤ e2M(t) . By the second structure theorem for LCA groups (e.g., [23, page 110]), G = Rd × H where H contains a compact subgroup P such that H/P is discrete. Because we assume that G is metrizable, the dual group G is σ -compact ([19, Theorem 4.2.7]) and it follows that H/P is countable. Hence there exists a countable set ⊂ G and a compact U ⊂ G such that !

G=

Uγ

(7.18)

γ ∈

where Uγ = γ + U and Uγ ∩ Uγ has measure 0 if γ = γ . It is convenient to fix the normalization of the Haar measure on so that U has measure 1. Recall that the function √ψ is continuous, satisfies ψ(0) = 0, ψ(ξ ) = ψ(−ξ ) and has the property that ψ is subadditive (see, e.g., [11, Proposition 7.15]). It follows √ that there √ is a constant C such that, for any γ ∈ and any ξ , ζ ∈ Uγ , | ψ(ξ ) − ψ(ζ )| ≤ C. Moreover, by the Riemann-Lebesgue lemma, limξ →∞ ψ(ξ ) = ∞. Hence, for any ε > 0, we have ∀ ξ ∈ Uγ ,

(1 + ε)−1 ψ(γ ) ≤ ψ(ξ ) ≤ (1 + ε)ψ(γ )

(7.19)

for all but finitely many Uγ . Let Zε be the subset of such that (7.19) holds on Uγ when γ ∈ Zε . Let γ1 , γ2 . . . . be an ordering of Zε such ψ(γi ) ≤ ψ(γi+1 ). Let N be related to M by (2.7). We claim that ∀ k = 1, 2, . . . ,

√ ψ(γk ) ≥ (1 + ε)−1 N log k .

(7.20)

Indeed, for any k, ke

−2t(1+ε)ψ(γk )

≤

∞

e

−2t(1+ε)ψ(γi )

≤

1

∞

e−2tψ(ξ ) dξ ≤ e−tψ 22 ≤ e2M(t) ,

1 U γi

and thus (1 + ε)ψ(γk ) ≥ sup t>0

√ √ 1 log k − M(t) = N log k . t

Now, let g be a function on G such that g∈

∞ 0

1/m

Dom(Am )

and

lim sup m→∞

Am g2 ≤ 1. (m/T)


Then, there exists m0 such that for all m > m0 , ((1 + ε)−1 ψ(γk ))2m

|ˆg(ξ )|2 dξ ≤

Uγk

|ˆg(ξ )|2 |ψ(ξ )|2m dξ ≤ ((1 + ε)(m/T))2m . Uγk

Thus Uγk

(1 + ε)3 (m/T) . |ˆg(ξ )|2 dξ ≤ exp 2m log √ N(log k)

By Lemma 7.3, there exists kε such that for all k > kε , we have

−

|ˆg(ξ )|2 dξ ≤ k

T(1−ε) (1+b)(1+ε)3b

.

Uγk

Let Nε be the number of γ contained in \ Zε and write |g| ≤

|ˆg(ξ )|dξ = G

≤

⎛

⎜ ⎝ \Zε

≤

⎜ ⎝ \Zε

γ ∈U

|ˆg(ξ )|dξ

γ

⎞1/2

⎟ |ˆg(ξ )|2 dξ ⎠

⎛

⎞1/2 ⎜ ⎟ 2 + ⎝ |ˆg(ξ )| dξ ⎠ Zε

Uγ

⎛

⎟ |ˆg(ξ )|2 dξ⎠

+

≤ (Nε + kε )

⎜ ⎝ k≤kε

Uγ

⎛

1/2

Uγ

⎛

⎞1/2

⎛

⎞1/2 ⎟ |ˆg(ξ )|2 dξ⎠

+

⎜ ⎝ k>kε

Uγk

⎞1/2

⎞1/2 ⎟ |ˆg(ξ )|2 dξ⎠

Uγk

⎜ ⎟ 2 g2 + ⎝ |ˆg(ξ )| dξ ⎠ k>kε

≤ (Nε + kε )1/2 g2 +

Uγk −

k

T(1−ε) 2(1+b)(1+ε)3b

.

k>kε T(1−ε) As T > 2(1+b) and ε > 0 is arbitrary, we can pick ε > 0 such that 2(1+b)(1+ε) 3b > T(1−ε) − 1. Then the series k 2(1+b)(1+ε)3b converges and it follows that g ∈ L∞ as desired.

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7.3 The case M(t) = g(log(1 + 1/t)), g ∈ SRλ , λ > 1 In this section, we come back to the general setting of Sect. 7.1. Assume that M(t) = g(log(1 + 1/t)), g ∈ SRλ , λ > 1. Straightforward computations show that there are constants c1 , . . . , c4 ∈ (0, ∞) such that, for all x large enough, we have −1 (x)

ec1 (g )

−1 (x)

≤ (x) ≤ ec2 (g )

and ec3 g

−1 (x)

≤ N(x) ≤ ec4 g

−1 (x)

.

One can also show, as in Sect. 6.2, that x−1 F ◦ (x) tends to infinity so that Theorem 4.5 does not apply. However, for any fixed λ > 1 and g ∈ SRλ , under the hypotheses of Theorem 7.1 and assuming that (e−tA )t>0 is sub-Markovian, or under the hypotheses of Theorem 7.4, we obtain the equivalence between the following three properties: 1. 2.

e−tA 2→∞ ≤ ec5 g(log(1+1/t)) ; There exists c5 ∈ (0, ∞) such that, for all t > 0, n There exists c6 ∈ (0, ∞) such that for any u ∈ ∞ 0 Dom(A ) we have 1/n

lim sup n→∞

3.

An u2

−1 (n)

ec6 (g )

< ∞ ⇒ u ∈ L∞ ;

There are constants c7 , c8 ∈ (0, ∞) such that the Nash type inequality ∀ f ∈ Dom(A) with f 1 ≤ 1, c7 f 22 exp(c8 g−1 [log(1 + f 2 )]) ≤ Af , f + f 22 . holds.

Compare with Sect. 6.2 and note that, in the more general setting of Sect. 4, we were not able to prove the equivalence between the embedding property (2) above and the other two properties (1) and (3). 8 Examples In this last section, we present explicit examples of situations where the various type of ultracontractivity behaviors discussed earlier do occur. 8.1 The infinite dimensional torus The results obtain in this paper apply nicely to symmetric Gaussian semigroups (i.e., Brownian motions) on the infinite dimensional torus T∞ = R∞ /Z∞ . The most expeditive way to introduce symmetric Gaussian semigroups on T∞ is to


use Fourier analysis. The dual group of T∞ is Z(∞) , that is, the set of all sequences with finitely many non-zero entries in Z. A convolution semigroup of measures (µt )t>0 is symmetric Gaussian (non-degenerate) if the Fourier transform of µt is of the form θ ∈ Z(∞) ,

µt (θ ) = exp(−tAθ , θ ),

where A = (ai,j ) is symmetric positive definite, that is, satisfies ai,j = aj,i and Aθ , θ =

ai,j θi θj > 0

for all θ = 0.

On smooth cylindrical functions, the infinitesimal generator is given by −Af =

ai,j ∂i f ∂j f .

i,j

The simplest case is when the matrix A is diagonal which we now use to provide very explicit examples. When A is diagonal, we set ai = ai,i and NA (s) = #{i : ai ≤ s}. Obviously, we can obtain functions N = NA with almost any prescribed (non-decreasing) behavior by choosing appropriately the growth of the coefficients ai . For instance, picking ai = log(2) (i) yields log(2) (N(s)) ≈ s, ai = (1 + i)α i gives N(s) ≈ s1/α and ai = ee gives N(s) ≈ log(2) (s). In the following statement, µt (0) denotes the value at 0 ∈ T∞ of the continuous density of the measure µt with respect to Haar measure, if µt admits a continuous density, and µt (0) = ∞ otherwise. With this notation, we have µ2t (0) = e−tA 22→∞ . Theorem 8.1 Let (µt )t>0 be as above. Assume that A is diagonal and NA is regularly varying of index λ ∈ [0, ∞]. Then, we have

log µt (0) ∼

⎧ & 1/t 1 ⎪ NA (x) dx ⎪ x ⎨2 0

cλ NA (1/t) ⎪ ⎪ ⎩ & ∞ −xt 2 0 e dNA (x)

if λ = 0 if λ ∈ (0, ∞) if λ = ∞. i

See [3, Theorem 3.18]. For instance, picking ai = ee yields log µt (0) ∼ 12 log 1t log(2) 1t ; picking ai = (1 + i)α yields log µt (0) ∼ cα t−1/α ; and picking ai = (log(1 + i))α with α ∈ (1, ∞) yields log(2) µt (0) ∼ cα t−1/(α−1) as t tends to 0.

A. Bendikov et al.

8.2 Symmetric Lévy generators on R On R, consider an operator A of the form 1 Au(x) = − 2

∞

' ( u(x + y) − 2u(x) + u(x − y) d(y),

−∞

where the Lévy measure is a symmetric Radon measure such that

|y|2 d(y) < ∞. 1 + |y|2

This operator, originally defined on smooth compactly supported functions, is extended minimally to a self-adjoint operator in L2 (R). The operator −A is the generator of a translation invariant Markov semigroup Ht = e−tA . Let µt be the measure such that Ht u = µt × u. This semigroup can be described using Fourier transform. Namely, −tψ(ξ ) ) H u(ξ ) t u(ξ ) = e

where ∞ ψ(ξ ) = 2 [1 − cos ξ x]d(x). 0

The function ψ is a continuous negative definite function and µt = e−tψ . See [12, 21]. The asymptotic behaviour of around 0 reflects into the growth of ψ at infinity, and ultimately in the behaviour of µt (0) for small t. See [4] for an explicit connection. Classical examples 1.

Consider = cα |x|−1−2α dx, α ∈ (0, 1). Then ψ(ξ ) = |ξ |2α and d(x) 2 α d and Ht is the symmetric α-stable semigroup. A = − dx

2.

Consider d(x) = |x|−1 e−|x| dx. Then ψ(ξ ) = 2 log(1 + |ξ |2 ), A = 2 log * 2 + d and Ht is the symmetric -semigroup. 1 − dx


By a general result of Berg and Forst [11], the measure µt is absolutely continuous and has a continuous density (with respect to Lebesgue measure) if and only if its Fourier transform µt is in L1 . Moreover, if µt ∈ L1 , e−(t/2)A 22→∞

∞ = µt (0) = 2

e

−tψ(ξ )

∞ dξ = 2

0

e−ts d(s),

(8.21)

0

where (s) = |{t > 0 : ψ(t) ≤ s}|, and x → µt (x) denotes the density of µt . It follows that (Ht )t>0 is ultracontractive if and only if, for all t > 0, e−tψ ∈ L1 . In this case, the density x → µt (x) is continuous. We now describe explicit results relating the behavior of ψ and at infinity to the behavior of µt (0) at 0. We use the notation introduced before Proposition 6.2 concerning regular variation. The following result is a consequence of (8.21) and the Laplace transform techniques of [13, Theorem 1.7.1, Theorem 4.12.2, Theorem 4.12.11(i)], i.e., Karamata’s and Kohlbecker’s theorems. Theorem 8.2 Let α ∈ (0, 2) and ψ0 ∈ Rα . The following properties are equivalent. (a) ψ ∼ ψ0 at infinity. (b) ∼ ψ0−1 at infinity. (c) µt (0) ∼ (1 + 1/α)ψ0−1 (1/t) at 0. 2. Let c > 0, α > 1 and θ ∈ Rα . Set (x) = θ (x)/x. Then the following properties are equivalent. (a) ψ(x) ∼ cα θ (log x) at infinity. (b) log (x) ∼ c−1 θ −1 (x) at infinity. (c) log µt (0) ∼ (α − 1)(αc)α/(1−α) −1 (1/t) at 0.

1.

To discuss further examples, it is convenient to consider the case where there exists an increasing function ω with ω(x) = o(x) at infinity and such that x (x) =

eω(s) ds.

(8.22)

0

In fact, using Polya’s Theorem (e.g., [18, Theorem 4.3.1]), for any ω as above there exists a symmetric convolution semigroup of probability measures (µt )t>0 such that µt (ξ ) = exp(−t −1 (|ξ |)). As −1 (ξ )/ log ξ tends to +∞ at +∞, it follows that µt admits a C∞ density for all t > 0. Using the monotone density theorem [13, Theorem 4.12.10], we derive from Theorem 8.2(2) that, in this

A. Bendikov et al.

context, for any ψ0 ∈ Rα , α ∈ (1, ∞), there is equivalence between ω ∼ ψ0−1 at infinity, and log µt (0) ∼ (α − 1)α α/(1−α) −1 (1/t), where (x) = ψ0 (x)/x. We are now interested in the two limit cases α = ∞ and α = 1. We start with the case α = ∞. To be more precise, we will consider the case where ω is slowly varying. The next theorem follows from [13, Theorem 4.12.12]. Theorem 8.3 Let be defined by (8.22) with ω non-negative increasing with ω(x) = o(x) at infinity. Let x → µt (x) be the smooth density of the associated convolution semigroup. Let ψ0 , ψ0# be a pair of de Bruijn conjugate slowly varying functions (see [13, p. 29 and Appendix 5]). Then the following properties are equivalent. 1. 2.

ω ∼ 1/ψ0 at infinity. log tµt (0) ∼ ψ0# (1/t) at 0.

Example 3 Let α > 0. Referring to (8.22) and the associated Lévy semigroup, the following properties are equivalent. 1. 2.

ω(x) ∼ c(log x)α at infinity log(tµt (0)) ∼ c(log 1/t)α at 0.

Indeed, in this case we have ψ0 (x) = c−1 (log x)−α and, by [13, Theorem A5.2], ψ0# = 1/ψ0 . Finally, we illustrate by an example what happens in the limit case where ω is regularly varying of index α = 1. See [3] for the needed Laplace transform techniques. Example 4 Let α > 0 and recall the notation (6.11) for iterated logarithms. Referring to (8.22) and the associated Lévy semigroup, the following properties are equivalent. 1. 2.

ω(x) ∼ x(log(n) x)−α at infinity log(n+1) (µt (0)) ∼ t−α at 0.

More generally, we have the following result, which proves that the ultracontractivity function of a Lévy semigroup can explode arbitrarily fast at zero. Recall a definition from [21, definition 23.2, p. 147]: a (probability) measure ρ on R is called unimodal with mode a if the functions x → ρ(] − ∞, x]),

x → ρ([x, +∞[)


are convex on ] − ∞, a[ and ]a, +∞[, respectively. That is, ρ = cδa + f (x)dx, where 0 ≤ c < ∞ and f is increasing on x < a and decreasing on x > a. Theorem 8.4 For any function G increasing to infinity at ∞, there exists a symmetric Lévy semigroup with a C∞ unimodal density x → µt (x) such that lim

t→0+

µt (0) = +∞. G(1/t)

˜ be any non-decreasing function from the class R∞ such that Proof Let G ˜ G/G → ∞ at ∞, e.g. one can take ⎛ x ⎞ ˜ G(x) = exp ⎝ G(s)ds⎠ . 0

˜ −1 ∈ R0 and let ω(x) ˜ =G ˜ Define m ˜ = x/m(x) ∈ R1 . By [13, Theorem 1.8.2], there exists ω∗ ∈ SR1 (smoothly varying function, see Sect. 6.5) such that ω∗ is ˜ and ω∗ ∼ ω˜ at ∞. Define f∗ = exp(ω∗ ); then f∗ is increasing, increasing, ω∗ ≥ ω, f∗ ∈ C∞ and 1/f∗2 is eventually convex (at ∞). Indeed, convexity of 1/f∗2 at ∞ follows from the inequality ω∗ − 2(ω∗ )2 < 0 at ∞.

(8.23)

To prove this inequality we note that since ω∗ ∈ SR1 , xω∗ (x) ∼ ω∗ (x) and x2 ω∗ (x)/ω∗ (x) → 0 at ∞ (see Sect. 6.5). In particular, ω∗ = o(ω∗2 ) at ∞ and the result follows. ˜ = o(x) at ∞, ω∗ (x) ∼ ω∗ (x)/x → 0 at ∞, also ω∗ ≥ 0. It Since ω∗ (x) ∼ ω(x) follows that there exists x∗ >> 1 such that ω∗ (x∗ ) < 0 and the inequality (8.23) holds for x ≥ x∗ . Define an increasing smooth function ω such that ω(x) = ω∗ (x) for x ≥ x∗ and ω(x) is concave for 0 < x < x∗ . Then ω satisfies the inequality (8.23) for all x > 0. Hence the function f = exp(ω) is increasing to infinity at infinity and the function 1/f 2 is convexe. Let be defined by (8.22) with ω as above. Let (µt )t>0 be the symmetric convolution semigroup of probability measures such that µt (ξ ) = exp(−t −1 (|ξ |)). Let x → µt (x) be the corresponding smooth density. By Askey’s Theorem [1], the density x → µt (x) is unimodal with mode 0 if its characteristic function t : ξ → exp(−t −1 (|ξ |)) satisfies the following condition: ξ → −t (ξ ) is convex on the interval {ξ > 0}. That this condition holds follows easily from the convexity of the function 1/f 2 . Further

A. Bendikov et al.

we have ∞ µt (0) = 2

e−st f (s)ds ≥

0

∞ = x∗ ∞

=

e−st eω∗ (s) ds ≥

∞ x∗ ∞

e−st f∗ (s)ds

˜ e−st+ω(s) ds

x∗ ˜ −st+s/m(s)

e x∗

∞ ds =

˜ es(1/m(s)−t) ds.

x∗

˜ 0 ) = 1/t. Fot t small enough, define s0 as the unique solution of the equation m(s ˜ increases to infinity at infinity, s0 → ∞ as t → 0. Therefore for 0 < t < T Since m small enough and such that s0 = s0 (t) > x∗ , we obtain s0 µt (0) ≥

˜ es(1/m(s)−t) ds ≥ s0 − x∗

x∗

˜ ˜ −1 (1/t) = G(1/t), ∼m

t → 0.

˜ Hence, because G/G → ∞ at ∞, lim µt (0)/G(1/t) = ∞.

t→0

The proof is finished. Acknowledgments This work was started and finished in Zakopane, Poland. The authors would like to thank Andrzej Hulanicki for his kind invitations.

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