CAPACITIES ASSOCIATED WITH KERNELS.
THE GAUSSIAN CASE
DENIS FEYEL and ARNAUD DE LA PRADELLE
We study some capacities on a Lusin space, which are dened by a kernel. A particular interest is brought to the Hermite transform on the abstract Wiener space in innite dimension.
AMS 2010 Subject Classication: 60J45, 60J57, 60J40, 60J80, 31D05.
Key words: capacities, functional spaces, Gaussian measure, convolution, innite dimension.
Introduction. The capacity is a natural notion issued from the classical potential theory and Sobolev spaces in nite or innite dimension. After that, functional capacities arised [47, 9, 11]. The goal of this note is to introduce a Hermite capacity, in the white noise theory [7, 10]. In this framework, it appears that the points of the Cameron-Martin space have non zero Hermite capacity and is dense in the quasi-topology associated to this capacity.
In the rst part, we recall some generalities on functional capacities associ- ated with Feller kernels. Next, we prove some results in the case of convolution kernels. In the last part, we look at the convolution with a Gaussian measure, which is the Hermite transform in nite or innite dimension.
Some functional capacities. In all the paper, we call Lusin space every topological space homeomorphic with a Borel subset of a compact metrizable space. For many other denitions, the reader can read [3, p. 72 to 80]. If E is Lusin, denoteCb(E)the space of bounded continuous functions onE. Consider a Markov Feller kernel on E, that is a Borel kernel U such that U1 = 1 and U f ∈ Cb(E) for everyf ∈ Cb(E). If µis a probability measure, dene for every g≥0 l.s.c.
c(g) = Inf
Np(f)
U f ≥g ,
whereNp is the norm in Lp(µ). Observe thatc(1)≤1. For h∈IRE, we put
REV. ROUMAINE MATH. PURES APPL. 59 (2014), 1, 7785
c(h) = Inf c(g)
g≥ |h|, g l.s.c. ,
As usual, we say that a property holds quasi-everywhere if it holds but on a set P withc(P) = 0. Such a set is called a polar set.
Proposition 1. a) Forhn≥0, andh=P
nhnwe havec(h)≤P
nc(hn). b) c(U f)≤Np(f).
Proof. a) Suppose P
nc(hn) < λ, let gn ≥ hn l.s.c. be such that P
nc(gn)< λ, then there exist somefn≥0such thatU fn≥gnandP
nNp(fn)<
λ. Putf =P
nfn we have U f ≥g and c(h)≤c(g)≤Np(f)≤λ.
b) The inequality is obvious for f ≥ 0 l.s.c. In the general case, take λ > Np(f), take g ≥f and g l.s.c. with Np(g)< λ, we get c(U f) ≤c(U g)≤ Np(g)< λ.
Lemma 2. For g≥0 l.s.c. we have c(g) = Inf
Np(f)
U f ≥g quasi-everywhere .
Proof. Denotec0(g)the right hand member. The inequalityc0(g)≤c(g)is obvious. Next suppose g≤U f quasi-everywhere, then c(g) ≤c(U f)≤Np(f), hence, c(g)≤c0(g).
Theorem 3. Let h≥0 be a function. Then c(h) = Inf
Np(f)
U f ≥h quasi-everywhere
and the inmum is achieved. Moreover, for p > 1 it is achieved at a unique point.
Proof. Let c0(h) be the right hand member. Ifh≤U f quasi-everywhere, we havec(h)≤c(U f)≤Np(f)so thatc(h)≤c0(h). Next, suppose thatc(h)<
λ, there exist g ≥ h, g l.s.c. with c(g) < λ. It then follows c0(h) ≤ c0(g) = c(g) < λ. Next, let fn ≥ 0 be a sequence with U fn ≥ h quasi-everywhere and LimnNp(fn) = c(h). Let f be a weak cluster point of the sequence fn. There exists a sequence fn0 such that fn0 ∈Conv(fn, fn+1, . . .) which strongly converges to f. We have U fn0 ≥ h quasi-everywhere, so that c(h) ≤ Np(fn0).
Moreover,c(U f−U fn0)converges to 0. By extraction of a subsequence we obtain U f ≥hquasi-everywhere so thatc(h)≤Np(f)≤Lim InfnNp(fn) =c(h). The unicity of f follows from the strict convexity of the norm Np for p >1.
Daniell property. Assume the Daniell property, that is for every se- quence fn ∈ Cb(E) which decreases to 0, then c(fn) tends to 0. Recall that L1(c) is the closure of Cb(E) under the seminorm c(f), that is f ∈ L1(c) if there exists a sequence fn ∈ Cb(E) such that c(f −fn) converges to 0 [5, 9].
The seminorm cis called a functional capacity.
Recall that the dual space M(c) of L1(c) consists of signed measures ξ whose the total variation|ξ|is controlled byc since (|ξ|(f)≤ kξkc(f)) [5].
Theorem 4. c is a Choquet capacity [1, 2].
Proof. Let hn ≥0 be an increasing sequence of functions. We obviously have Supnc(hn) ≤ c(h). Suppose that c(hn) < λ for every n. Let fn ≥ 0 withhn≤U fn quasi-everywhere and Np(fn) =c(hn)< λ. As above, letf be a weak cluster point of the sequence fn. By the same way as above, we get c(h)≤λ.
Theorem 5. Let σ ≥0 be a measure on E. Thenσ ∈ M(c) if and only if σ integrates H=U(Lp(µ)). In this case, the norm |σ|c of σ in M(c) is the same as its norm |σ|H in the dual space of H.
Proof. First we have
|σ|H= Sup
f≥0
σ(U f)
Np(f) ≤ |σ|cc(U f)
Np(f) ≤ |σ|c.
Let ϕ∈ L1(c), ϕ≥0, let Rϕ=U f withNp(f) =c(ϕ). We have σ(ϕ)≤σ(Rϕ) =σ(U f)≤ |σ|HNp(f)≤ |σ|Hc(ϕ) hence, |σ|c≤ |σ|H.
Convolution. Consider a locally convex space which is Lusin. As E is metrizable, it follows that the completed space Eb is a separable Frechet space, andEis Borel inEbsince it is Lusin. Letµbe a symmetric probability measure onE. Put Φf =f∗µfor every Borelf ≥0. ThenΦis a Feller Markov kernel on E. Hence, we can dene as above forg≥0 l.s.c.
c(g) = Inf
Np(f)
Φf ≥g , where the norm Np is taken with respect to µ.
Proposition 6. c(1) = 1.
Proof. As Φ1 = 1 we rst get c(1) ≤ 1. Next if Φf ≥ 1, we have the inequalities 1≤f∗µ(0) =R
f dµ≤Np(f) so that1≤c(1).
Proposition 7. Suppose there exists a symmetric compact convex set K whose the Minkowski functional q belongs toLp(µ). Thenc is tight on compact sets.
Proof. We haveq(x)≤q(x+y)+q(y). Thenq ≤Φ(q0)whereq0 =q+µ(q). Put Gn=E−nK, we have Φ(q0)≥q ≥n1Gn, so thatc(Gn)≤Np(q0)/n. We suppose that Φ is an isomorphism from Lp(µ) onto a subspace H ⊂ L1(c) with the seminormkΦfkH=Np(f).
Now, observe the relation for every measure σ≥0 Z Z
f(x+y)dσ(x)dµ(y) =hf, σ∗µi=hσ,Φfi.
Theorem 8. Ifσ ≥0belongs to the dualM(c) thenσ∗µhas a densityσe with respect toµ. Moreover,σe∈Lp0(µ)withp0=p/(p−1)andNp0(eσ) =kσkM.
Proof. For f ≥0 we have
hf, σ∗µi=hσ,Φfi ≤ kσkMc(Φf)≤ kσkMNp(f).
Hence, there exists a function eσ ∈Lp0(µ) such that hf, σ∗µi =R
feσdµ. We have
kσkH0 = Sup
f6=0
hσ,Φfi/kΦfkH= Sup
f6=0
heσ, fi/Np(f) =Np0(eσ), so thatNp0(eσ) =kσkH0 =kσkM.
Corollary 9. If a bounded measure σ ≥ 0 vanishes on polar sets then σ∗µis absolutely continuous with respect to µ.
Proof. If σ vanishes on polar sets, then σ = P
nσn where σn ≥ 0 is majorized by c [6]. Hence, σ∗µ = P
nσn∗µ has a density P
neσn ∈ L1(µ).
The Gaussian case. Recall that (E, H, µ) is an abstract Wiener space if E is a separable Banach space, µ a centered Gaussian measure on E with support E, and H is the Cameron-Martin space, that is H is the dual space of E0 equipped with the topology of L2(µ). As well knownH identies with a subspace of E and the canonical inclusion H⊂E is compact. Observe thatE needs not be Banach, it suces it is Frechet (exampleIR∞with the product of normal measures, in this case H=`2).
Now, let E be a locally convex space which is Lusin. As seen above it is Borel in Eb which is a Frechet separable space. Let µ be a centered Gaussian measure onE whose support isE. Then(E, H, µ)b is an abstract Wiener space, and it can be proved [6] that H ⊂E, so that we shall say that (E, H, µ) is a Lusin abstract Wiener space. Hence, suppose that(E, H, µ)is a Lusin abstract Wiener space. We dene the Hermite transform Φ(f) = f ∗µ [7]. If E is strongly convex, Proposition 7 applies, so that the capacity c associated toΦ is tight on compact sets.
For u ∈ H we denote ku(x) = ehu,xi−|u|2/2, where hu, xi is the pseudo scalar product on H×E [8].
Theorem 10. Let u∈H be a Cameron-Martin point. We have c({u}) = e−|u|2/2.
Proof. Let h ≥ 0 be an l.s.c. function with h(u) ≥ 1 and f ≥ 0 with Φf ≥ h. We have h(u) ≤ Φ(f)(u) = hf, kui ≤ N2(f) e|u|2/2. It follows that h(u) ≤c(h) e|u|2/2 and c(h)≥e−|u|2/2 so that c({u})≥e−|u|2/2 and {u} is not polar. Next, consider the function f(x) = ehu,xi−3|u|2/2. We have Φ(f)(v) = ehu,vi−|u|2. We then have1{u} ≤Φ(f) quasi-everywhere since {u}is not polar, then
c({u})≤c(Φf)≤N2(f) = e−|u|2/2 and the result.
Letπ:E →IR∞be a linear continuous map such thatπ(µ)is the normal measure onIR∞(cf. [6]). We denoteπn the projection onIRnandπn∗ =π−πn. For a Borel f ≥0
Fnf(x) = Z
f(πnx+π∗ny)dµ(y)
as well known,Fnf is the conditional expectation off with respect toµon the σ-algebra generated byπn.
Lemma 11. For every f ≥0 we haveΦf◦πn= Φ(Fnf). Proof. Obvious, for f = 1. We have
Φf(πnx) = Z
f(πnx+y)dµ(y)
= Z
f(πnx+πny+π∗nz)dµ(y)dµ(z) = Z
Fnf(x+y)dµ(y) = Φ(Fnf)(x).
Lemma 12. Let h≥0. Then c(h◦πn)≤c(h).
Proof. Let c(h) < λ. Let f ≥ 0 with h ≤Φf and N2(f) < λ. We have h◦πn ≤ (Φf)◦πn = Φ(Fnf) so that c(h◦πn) ≤ N2(Fnf) ≤ N2(f) < λ. The result follows.
Lemma 13. Let σ ∈ M+(c). The sequence σn = πn(σ) has kσnkM ≤ kσkM and converges weakly to σ.
Proof. Forϕ∈ L1(c)we haveσn(ϕ) =σ(ϕ◦πn)≤ kσkc(ϕ◦πn)≤ kσkc(ϕ). Then the inequality holds. Forϕ∈ Cb(E) we getσ(ϕ) = Limnσ(ϕ◦πn) by the Lebesgue theorem, and the result follows by equicontinuity.
Theorem 14. H is quasi-dense. In particular we have c(g) = Inf
N2(f)
Φf ≥g on H .
Proof. Let ϕ∈ L1(c) be such that ϕ≥0 on H. For every measureσ ≤c we get σ(ϕ) = Limnσn(ϕ) ≥ 0 since σn is carried on H. Next, suppose that Φf ≥gon H, we getσ(g)≤σ(Φf)for every σ≤cso thatc(g)≤N2(f).
Corollary 15. If G quasi-open andG∩H = Ø, then c(G) = 0.
Proof. Forε >0there exists an open setω withc(ω)< εandω∪Gopen.
We have c(G)≤c(ω∪G) =c((ω∪G)∩H) ≤c(ω)< ε. As ε shrinks to 0 we getc(G) = 0.
Example. Suppose thatA≥0is a symmetric operator and thatB =A2<
I is Hilbert-Schmidt. We do not suppose thatAis Hilbert-Schmidt. Letπnbe the projector on a nite dimensional eigenspace ofA, putAn=Aπn=πnAπn, thenAn is symmetric and An≤An+1 ≤A < I.
The operator A has a µ-measurable linear extension on E [8], we put µA=A(µ)the image measure, which is a Gaussian measure on E.
The measure σn = µAn = An(µ) has a Laplace transform bσn(x) = e|Anx|2/2 andR
σbn(x)dσn≤detS−1<+∞, whereS=√
I−B2. Then the se- quenceσbn=µbAn is bounded inH. It follows thatkσnkM=kσnkH0 ≤detS−1/2 so that there is a weak cluster σ ∈ M. Obviously σ =µA, hence, σn weakly converges to σ=µA.
It follows that σbn weakly converges to µbA in H hence, also in L1(c). As the sequence bσn increases there is convergence quasi-everywhere to σb = bµA. One has
kbσk2H= Z
σbdσ= Sup
n
Z
bσndσ = Sup
n
Z
σbndσn= Sup
n
kbσnk2H, then we see that bσn strongly converges to bσ inHand in L1(c).
Proposition 16. E−H is not polar.
Proof. If Ais not Hilbert-Schmidt, then µA(H) = 0, hence,µA is carried by E−H.
Computing µeA. 1o. We have
e|Anx|2/2 = 1 +|Anx|2
2 +X
k≥3
|Anx|2k 2kk! .
Every term belongs to L1(c) and converges increasinly in L1(c), so that
|Ax|2 = Limn|Anx|2 exists quasi-everywhere and belongs to L1(c).
2o. The sequence σen strongly converges in L2(µ) to σe = µeA. One has µ∗µAn =µTn withTn=p
I+A2n=√
I+Bn, hence, σen=µeAn = dσn
dµ = det(Tn)−1e|AnTn−1x|2/2 We retrieve N2(eσn)2 =R
bσndσn= detSn−1.
Put fn(x) = exp{[|AnTn−1x|2−Trace(A2nTn−2)]/2} = Λnσen(x). Observe thatΛn= det(Tn) exp[Trace(A2nTn−2)/2]converges to a limitΛ>0of the type
Carleman-Fredholm determinant. One has Λ =Y
i≥1
p1 +βiexp[−βi/2p
1 +βi].
Put hn(x) =hCnx, xi −TraceCn withCn=A2nTn−2. Form≤n one has IE(|hn−hm|2) =Pn
i=m+1c2i withci =βi/(1 +βi) where the βi are the eigen- values of B with their multiplicity order. Hence,hn converges in L2 towards a function Q(x)which belongs to the second Wiener chaos. Finally, we get
eµA(x) = Λ−1eQ(x)/2 ∈L2(µ).
Complexication. Let HC| =H⊕iH be the complexied space of H. A function F dened on HC| is an entire function if it is continuous and its restriction to every nite dimensional subspace is holomorphic. A function F on HC| is special if it is an entire function and F(u)is real for every u∈H.
Lemma 17. Let w=u+ iv∈HC| .Then
1o. The pseudo scalar product onH×E has a unique holomorphic exten- sion on HC| ×E
hw, xi=hu, xi+ ihv, xi.
2o. Dene the entire extension of u→ |u|2 w2 =
Z
hw, xi2dµ(x).
which is a special function on HC| . 3o. The norm on HC| is given by
|w|2= Z
|hw, xi|2dµ(x).
4o. The mapu→ku fromH intoL2(µ) extends in anL2C| (µ)valued map w→kw(x) = exphw, xi −w2/2.
5o. For f ∈ L2(µ), the Hermite transform Φ(f) extends in a special function on HC| given by
Φ(f)(w) = Z
f(x)kw(x)dµ(x) and we have |Φ(f)(w)| ≤N2(f) e|w|2/2.
Proof. Obvious.
The Bergman space. Let ρ be the image measure on EC| = E⊕iE of µ⊗µ by the map z → z/√
2. In nite dimension n, the density of ρ is π−ne−|z|2.
Theorem 18. Suppose (E, H, µ) is nite dimensional. Then the Hermite transform is an isometry of L2(µ)onto the subspaceHof L2C| (ρ) which consists of special functions.
Proof. Let F = Φ(σ)e ∈ H whereσ∈ M(c) is a signed measure. We have F(w) =R
ehw,tidσ(t). Then Z
|F(w)|2dρ(w) =Z
dρ(w)Z Z
ehw,ti+hw,sidσ(t)dσ(s) =Z Z
ehs,tidσ(t)dσ(s), Z
|F(w)|2dρ(w) =N2(σ)e 2.
Hence, Φ is an isometry of L2(µ) onto a closed subspace of L2C| (ρ). It remains to prove that it isH. FirstH is closed. One hasΦ(ku)(v) =R
ku(x+ v)dµ(x) = ehuvi, then
ku(x) =X
n
hu⊗n, hn(x)i
n! ⇒ Φ(ku)(w) =X
n
hu⊗n, w⊗ni n! , wherehnare Hermite polynomials.
Consequently, Φ(hn)(w) = w⊗n and the w⊗n/√
n! are a Hilbert basis of HC| .
It then follows that HC| = Φ(L2C| (µ)) is exactly the Bergman space rel- ative to ρ, and H = Φ(L2E(µ)) is the subspace of special functions. It should be observed that HC| = H ⊕ iH. If (E, H, µ) is innite dimensional, the w⊗n/√
n! always are a Hilbert basis of HC| . As we have |F(w)| ≤ kFke|w|2/2 the series F(w) = P
nhan, w⊗ni converges on HC| . Moreover, it converges in L2C| (E, ρ).
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Received 17 July 2013 Universite d'Evry-Val d'Essonne, Departement de Maths., Boulevard Francois Mitterand,
91025 Evry cedex, France denis.feyel@orange.fr
Universite Paris VI, Laboratoire d'Analyse
Fonctionnelle, Boulevard 4 place Jussieu,
75052 Paris, France adelapradelle@free.fr
