The domain of the Ornstein-Uhlenbeck operator on an <span class="mathjax-formula">$L^p$</span>-space with invariant measure

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The Domain of the Ornstein-Uhlenbeck Operator on an L^p-Space with Invariant Measure

GIORGIO METAFUNE – JAN PR ¨USS – ABDELAZIZ RHANDI – ROLAND SCHNAUBELT

Abstract.We show that the domain of the Ornstein-Uhlenbeck operator onL^p (R^N, µd x)equals the weighted Sobolev spaceW²^,^p(R^N, µd x), whereµd xis the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative Dore-Venni theorems.

Mathematics Subject Classification (2000):35J15 (primary), 35K10, 47A55, 47D06 (secondary).

1. – Introduction

In recent years the Ornstein-Uhlenbeck operator

Lu(x)= 1 2

N i,j=1

qi jDi ju(x)+ N i,j=1

ai jxiDju(x)

= 1

2 trQ D²u(x)+ Ax,Du(x), x∈R^N, and its associated semigroup T(·) on, say, Cb(R^N) given by

(1.1) (T(t)ϕ)(x)=(2π)⁻^N² (detQt)⁻¹²

RNϕ(e^{t A}x−y)e⁻¹²^<^Q⁻¹^t ^y^,^y^>d y, x ∈R^N, t >0, This work was done while A. Rhandi visited the University of Halle, Germany. He wishes to express his gratitude to the Deutsche Forschungsgemeinschaft (DFG) for the financial support.

He further thanks D. Pallara and the University of Lecce for their kind hospitality while the work was initiated. He gratefully acknowledges the support by the INDAM.

Pervenuto alla Redazione il 5 ottobre 2001.

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have attracted a lot of interest. These activities are in particular motivated by the fact that T(·) is the transition semigroup of the Ornstein-Uhlenbeck process (see [11])

X(t,x)=e^{t A}x+ _t

0

e⁽^t⁻^s⁾^Ad W(s)

on_R^N, where W is an N-dimensional Brownian motion with covariance matrix Q, i.e.,

(T(t)ϕ)(x)=E[ϕ(X(t,x))].

The main purpose of this paper is to determine the domain of the realization Lp of L in a certain weighted Lebesgue space L_µ^p = L^p(R^N, µd x) assuming that Q = (qi j) is a real, symmetric, positive definite N × N-matrix and that A=(ai j) is a real N ×N-matrix whose eigenvalues are contained in the open left half plane. These hypotheses, kept throughout Sections 1-3, ensure that the matrices

(1.2) Qt =

_t

0

e^{s A}Qe^{s A}^∗ds, t ∈(0,∞],

are well defined, symmetric, and positive definite. (If t ∈(0,∞), then one can allow for an arbitrary real A.) Moreover, the Gaussian measure µd x given by the weight

µ(x)=(2π)⁻^N² (detQ_∞)⁻¹²e⁻¹²^Q⁻^∞¹^x^,^x, x ∈R^N,

is the unique invariant measure for the semigroup T(·), i.e., µ is the only probability measure for which

(1.3)

RN(T(t)ϕ)(x)µ(x)d x =

RNϕ(x)µ(x)d x, ϕ∈C_b(R^N), t ≥0, see [11, Theorems 11.7, 11.11]. As a result, T(·) extends to a semigroup of positive contractions on L_µ^p for 1 ≤ p≤ ∞ and it is not difficult to see that T(t)ϕ(x)is still defined by (1.1) forϕ∈L_µ^p,x∈R^N, andt>0. The semigroup T(·) is strongly continuous on L_µ^p if 1 ≤ p < ∞, analytic for 1 < p < ∞ (but not for p = 1), and its generator Lp is the closure of L defined on the Schwartz class_S(R^N). We refer, e.g., to [16] for the proofs of these properties.

The equality D(L2)= W_µ²^,² has been first proved by A. Lunardi in [14]

making heavy use of interpolation theory. A simpler proof of the same result can be found in [7]. Recently, this result was extended to p∈ (1,∞) in the symmetric case by A. Chojnowska-Michalik and B. Goldys in [3, Theorem 3.3], who studied (as in [7]) the infinite dimensional version of L where _R^N is replaced by a separable real Hilbert space, and by G. Da Prato and V. Vespri in [10, Theorem 2.2], who allowed for more general drift terms of gradient type on _R^N. Both approaches are based on maximal regularity results from [2]

and [1], respectively. We also refer to the previous papers [6], [9], [12], [20].

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In our main Theorem 3.4 we establish the equality D(L_p) = W_µ²^,^p for 1< p<∞. In Section 2 we first diagonalize Q and Q_∞ simultaneously and describe the resulting drift matrix A₁. This allows to decompose L= L⁰+B, where L⁰is a symmetric, diagonal Ornstein-Uhlenbeck operator and B generates an isometric group on L_µ^p. Then we determine D(Lp) in three steps. The one- dimensional case is first settled in Lemma 3.1 by rather elementary calculations.

In Proposition 3.2 we then establish that D(L⁰_p)=W_µ²^,^p using the Dore-Venni theorem [13], Lemma 3.1, and elliptic regularity in L^p(R^N). In a final step we deduce that D(Lp)= D(L⁰_p)=W_µ²^,^p employing a perturbation argument based on a non commutative Dore-Venni type theorem, see [17].

In the last section of this paper we characterize the domain of the Ornstein- Uhlenbeck operator L in L^p(R^N), 1< p<∞, for an arbitrary real drift matrix Aapplying again the results in [17]. As a byproduct, we can prove L^p estimates for L. We remark that Schauder estimates for L have been already obtained in [8].

Notation. The space of continuous functions f having continuous (resp.

continuous and bounded) partial derivatives D_αf up to order k is denoted by C^k(R^N) (resp. C_b^k(R^N)) and the corresponding weighted Sobolev space by W_µ^k^,^p = W^k^,^p(R^N, µd x) = {f ∈ L_µ^p : D_αf ∈ L_µ^p, |α| ≤ k}, where k ∈ N0, 1 ≤ p < ∞, C_b⁰(R^N) = Cb(R^N), W_µ⁰^,^p = L_µ^p = L^p(R^N, µd x), and α is a multi index. The Schwartz class is designated by _S(R^N) and the space of test functions by C₀^∞(R^N). We write L^p(R^N) if the underlying measure is the Lebesgue measure. By a slight abuse of notation we write x f or xj f for the functions x → x f(x) or x → xjf(x), where x = (x1,· · · ,xN) ∈ R^N. The symbol c denotes a generic constant.

2. – Preparations

In this section we collect some results needed in the next section. For reader’s convenience we give complete proofs of known facts.

The Ornstein-Uhlenbeck operator L is called symmetric if the semigroup T(·) is symmetric in L²_µ. The next lemma is a slight modification of [3, Theorem 2.2].

Lemma 2.1.The equality A Q_∞+Q_∞A^∗= −Q holds. Moreover, the following properties are equivalent.

(a) L is symmetric.

(b) Q_∞A^∗= A Q_∞.

(c) QtA^∗= A Qt for all t ∈(0,∞). (d) Q A^∗= A Q.

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Proof.Observe that (1.2) yields the formula (2.1) Q_t+e^{t A}Q_∞e^{t A}^∗ = Q_∞.

The first assertion can be verified by taking in (2.1) the derivative at t = 0.

The equivalence of (a) and (d) was shown in [3, Theorem 2.2]. The implication (d)⇒(b) is an immediate consequence of (1.2). Assertion (c) follows from (b), due to (2.1). Finally, (c) implies (d) by differentiation.

Given an invertible real N × N-matrix M, we introduce the similarity transformation

M :C(R^N)→C(R^N); (Mu)(y)=u(M⁻¹y) .

For u ∈ S(R^N) and v = Mu ∈ S(R^N), one easily calculates that Lu(x) = L˜v(M x), x ∈R^N, where

L˜v= ¹₂ trQ D˜ ²v+ ˜Ay,Dv, Q˜ =M Q M^∗, A˜= M AM⁻¹.

This means that L = ⁻M¹L˜M on _S(R^N) and Q˜_∞ = M Q_∞M^∗. The corresponding Gaussian measure for L˜ is given by

˜

µ(x)=(2π)⁻^N² (detQ˜_∞)⁻¹²e⁻¹²^˜^Q⁻^∞¹^x^,^x= 1

|detM|µ(M⁻¹x), x ∈R^N . As a result, M induces an isometry from L_µ^p onto L_µ^p_˜ and an isomorphism fromW_µ^k^,^ponto W_µ^k_˜^,^p, 1≤ p≤ ∞, k∈N. Recalling that the induced generators Lp and L˜p are the closures of L and L˜ defined on S(R^N), respectively, we arrive at

L_p=⁻M¹L˜_pM with D(L_p)=⁻M¹D(L˜_p) .

There is an invertible real matrix M1 such that M1Q M₁^∗= I and an orthogonal real matrix M2 such that M2(M1Q_∞M₁^∗)M₂^∗ = diag(λ1,· · ·, λN) =: D_λ for certain λj > 0. Taking M = M₂M₁, we have transformed L into the more convenient form described in the next lemma.

Lemma 2.2. (a) There exists a real invertible N × N -matrix M such that Lp = ⁻M¹L˜pM and D(Lp) = ⁻M¹D(L˜p), whereLu˜ = ¹₂u+ ˜Ax,Duand A˜ =M AM⁻¹.Moreover,Q˜_∞= D_λfor certainλj >0,

(2.2) µ(˜ x)=(2π)⁻^N² (λ1· · ·λN)⁻¹² exp



−

j

x²_j 2λj



,

andM :W_µ^k^,^p→W_µ^k_˜^,^p,1≤ p≤ ∞, k∈N0, is an isomorphism.

(b)SettingL˜⁰u= ¹₂u−¹₂D1

λx,Du, Bu= A₁x,Du, and A₁= ˜A+¹₂D1 λ, we can write L˜ = ˜L⁰+ B. Moreover, A₁D_λ = −D_λA^∗₁ and hence the diagonal elements of A₁equal zero. Finally,µ (˜ defined in(2.2))is the invariant measure of the Ornstein-Uhlenbeck semigroup generated byL˜⁰, andL˜⁰is symmetric.

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Proof. (a) holds in view of the discussion above. As regards (b), by Lemma 2.1 we have A D˜ _λ+D_λA˜^∗= −I, and hence A1D_λ = −I−D_λA˜^∗+¹₂I =

−D_λA^∗₁. Finally,µ˜ is the invariant measure for L˜⁰ by the explicit computation of the integral in (1.2) and L˜⁰ is symmetric (in L²_µ_˜) by Lemma 2.1.

In order to determine D(Lp), we may thus assume that L is given by

L=L⁰+B, L⁰u=1 2u−1

2D1

λx,Du, Bu=A1x,Du, where (2.3)

A1D_λ=−D_λA^∗₁, Q_∞=D_λ, µ(x)=(2π)⁻^N²(λ1· · ·λN)⁻¹² exp



−

j

x²_j 2λj

 (2.4) 

for x ∈R^N and certain λj > 0. We recall that µ is the invariant measure for L⁰ and that L⁰ is symmetric.

We further need the following property of the space W_µ¹^,^p which was essentially proved in [16, Lemma 2.3], see also [14, Lemma 2.1] for the case

p=2.

Lemma 2.3. Let1< p< ∞. Ifϕ,D_jϕ∈ L_µ^p, then the function x_jϕbelongs to L_µ^p andx_jϕ_L_µ^p ≤C_p(ϕ_L_µ^p + D_jϕ_L_µ^p)for a constant C_p > 0depending only on p andλj.

Proof.It suffices to prove the lemma forϕ∈C₀^∞(R^N)and we may assume that µ is in the diagonal form (2.4). Integrating by parts, we obtain

(2.5)

RN|xjϕ(x)|^pµ(x)d x = −λj

RN|xj|^p⁻¹|ϕ(x)|^psignxj(Djµ)(x)d x

=λj

RN

((p−1)|xj|^p⁻²|ϕ(x)|^p +p|xj|^p⁻¹(Djϕ)(x)|ϕ(x)|^p⁻¹signxj

)µ(x)d x. We set I1 =_R^N|xj|^p⁻²|ϕ(x)|^pµ(x)d x and I2=_R^N |xj|^p⁻¹|Djϕ(x)||ϕ(x)|^p⁻¹ µ(x)d x. H¨older’s inequality yields

(2.6) I₂≤^&

RN|x_jϕ(x)|^pµ(x)d x

'^p⁻_p¹ &

RN|D_jϕ(x)|^pµ(x)d x '¹_p

. In order to deal with I₁, we first consider p≥2. Then, for each ε >0 there is M_ε >0 such that |x_j|^p⁻²≤ε|x_j|^p+M_ε, and hence

(2.7) I1≤M_εϕ_L^p^p

µ+ε

RN|xjϕ(x)|^pµ(x)d x.

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Combining (2.5), (2.6), (2.7) with Young’s inequality, we arrive at xjϕ_L^p^p

µ ≤λj(p−1)M_εϕ^p_L^p

µ+λj(p−1)εxjϕ^p_L^p

µ+λjpxjϕ_L^p⁻^p¹

µ Djϕ_L_µ^p

≤λj(p−1)M_εϕ^p_L^p

µ+λj(p−1)εxjϕ^p_L^p

µ+ε^p⁻^p¹λj(p−1)xjϕ_L^p^p

µ

+ε⁻^pλjD_jϕ_L^p^p

µ,

and hence

(2.8) [1−λj(p−1)(ε+ε^p⁻^p¹)]xjϕ_L^p^p

µ ≤λj(p−1)M_εϕ_L^p^p

µ+λjε⁻^pDjϕ_L^p^p

µ. Therefore the lemma is proved for p≥2 taking a sufficiently small ε >0. If 1< p<2, we write x=(x_j,xˆ) and estimate

(2.9) I1≤

RN−1

& ₁

−1|xj|^p⁻²|ϕ(x)|^pµ(x)d xj +

R\[−1,1]|ϕ(x)|^pµ(x)d xj

' dxˆ

≤ 1

−1|x_j|^p⁻²d x_j

RN−1

= sup

x_j∈[−1,1]|ϕ(x_j,xˆ)|

>_p

µ(0,xˆ)dxˆ+ ϕ_L_µ^p

≤c(D_jϕ_L^p^p

µ+ ϕ^p_L^p

µ)

using the embedding W¹^,^p(−1,1) → L^∞(−1,1). As in (2.8), the assertion follows from (2.5), (2.6), and (2.9).

From the above lemma we deduce thatW_µ²^,^p is a core for Lp if 1< p<∞. Proposition 2.4. For 1 < p < ∞, the operator L_p is the closure of the differential operator L defined on W_µ²^,^p.

Proof. As shown in [16, Lemma 2.1], Lp is the closure of the operator L defined on S(R^N). Let u ∈ W_µ²^,^p and (un)⊂ S(R^N) converge to u in the norm of W_µ²^,^p. Then un∈ D(Lp) and Lpun= Lun converges to Lu in L_µ^p, by Lemma 2.3. Since Lp is closed, u∈D(Lp) and Lpu=Lu.

3. – Main results

We first compute the domain of the one-dimensional Ornstein-Uhlenbeck operator L_p on L_µ^p for 1 < p < ∞. In view of Lemma 2.2 we may assume that Lu= ¹₂u− ₂¹_λxu and µ(x)=(2πλ)⁻¹^/²exp{−x²/2λ}.

Lemma 3.1. If N =1and1< p<∞, then D(Lp)=W_µ²^,^p.

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Proof.Thanks to Proposition 2.4 and Lemma 2.3, it remains to show that

(3.1) u_W²,p

µ ≤c(Lu_L_µ^p + u_L_µ^p)

for u ∈W_µ²^,^p. For a givenu∈W_µ²^,^p, we write f =(I−L)u∈L_µ^p and v=u. Hence,

−1

2v(x)+ 1

2λxv(x)= f(x)−u(x), x∈R. Integrating we obtain

e⁻²¹λξ²v(ξ)−e⁻²¹λx²v(x)=2 _ξ

x

(u(y)− f(y))e⁻²¹λy²d y, ξ ∈R. Since (u− f)e⁻2λ¹ y² ∈L¹(R) and v∈ L_µ^p, the function e⁻2λ¹ξ²v(ξ) tends to 0 as ξ→ ± ∞ and therefore

e⁻2λ¹x²v(x)= −2 _{± ∞}

x (u(y)− f(y))e⁻2λ¹ y²

d y.

Setting ϕ(x)=e⁻^2p¹^λ^x²v(x) and g(x)=2e⁻²¹^p^λ^x²(f(x)−u(x)), we arrive at

ϕ(x)=







 _∞

x

exp 1

2pλ(x²−y²)

g(y)d y, x ≥0,

− _x

−∞exp 1

2pλ(x²−y²)

g(y)d y, x ≤0.

Note that xu_L_µ^p = xϕL^p(R) and 2f −u_L_µ^p = gL^p(R). We thus have to control the L^p(R)-norm of xϕ. It suffices to estimate xϕL^p(R+). H¨older’s inequality and Fubini’s theorem yield

xϕ_L^p^p₍_R₊₎= _∞

0

_∞

x

xe⁻

2p1λ(y+x)(y−x)

g(y)d y

p

d x

≤ _∞

0

_∞

x

xe⁻

p1λx(y−x)

|g(y)|^pd y ^∞

x

xe⁻

p1λx(y−x)

d y p−1

d x

=(pλ)^p⁻¹ _∞

0

_∞

x

xe⁻

p1λx(y−x)

|g(y)|^pd y d x

=(pλ)^p⁻¹ _∞

0

|g(y)|^p _y

0

xe⁻

p1λx(y−x)

d x d y. We further compute

_y

0

xe⁻

p1λx(y−x)

d x = y² ₁

0

te⁻

p1λy²t(1−t)

dt ≤2y² ¹

2 0

e⁻

p1λy²t(1−t)

dt

≤2y² ¹

2 0

e⁻

2p1λy²t

dt ≤4pλ .

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As a result,

xu_L_µ^p = xϕL^p(R) ≤cgL^p(R)≤c(Lu_L_µ^p + u_L_µ^p) .

It follows that u_L_µ^p ≤c(Lu_L_µ^p+u_L_µ^p), and the analogous estimate for u easily follows from those foruandxu. We have therefore established (3.1).

We now treat the N-dimensional, symmetric case. The next result was recently shown in [3, Theorem 3.3] and [10, Theorem 2.2] in more general situations, but with completely different arguments.

Proposition 3.2. If the Ornstein-Uhlenbeck operator L is symmetric, then D(Lp)=W_µ²^,^pfor1< p<∞.

In view of Lemma 2.2 we may assume that L= L⁽¹⁾+ · · · +L⁽^N⁾ with L⁽^j⁾u= 1

2Dj j − xj

2λj

Dj and µ(x)=(2π)⁻^N² (λ1· · ·λN)⁻¹²exp



−

j

x²_j 2λj



.

We consider the semigroup of positive and self-adjoint contractions T⁽^j⁾(t)ϕ(x)=2πλj(1−e⁻^t^/λ^j)⁻¹²

Rϕ(e⁻

t

2λjxj−y,xˆ)exp

− ^y²

2λj(1−e⁻^t^/λj)

d y

on L_µ^p, where we write x =(xj,xˆ)∈R^N. Its generator in L_µ^p is the operator L⁽_p^j⁾ = L⁽^j⁾ with domain D(L⁽_p^j⁾)= {u ∈ L_µ^p : Dju,Dj ju ∈ L_µ^p}, see the next lemma.

Since T(·) is the product of the commuting semigroups T⁽^j⁾(·), the generator Lp is the closure of the sum L⁽_p¹⁾+ · · · +L⁽_p^N⁾ defined on D(L⁽_p¹⁾)∩ · · · ∩ D(L⁽_p^N⁾). The operator (I −L⁽_p^j⁾) admits bounded imaginary powers on L_µ^p, 1< p<∞, with power angle

θ(L⁽_p^j⁾):= lim

|s|→∞

1

|s|log(I −L⁽^j⁾)^{i s} ≤ π 2

due to the transference principle [5, Section 4], see [4, Theorem 5.8]. The semigroup T⁽^j⁾(·) is symmetric on L²_µ so that θ(L⁽₂^j⁾) = 0 by the functional calculus for selfadjoint operators. Hence, θ(L⁽_p^j⁾) < ^π₂ by the Riesz-Thorin interpolation theorem. Since the resolvents of L⁽_p^j⁾ commute, we can apply the Dore-Venni theorem [13] in the version of [18, Corollary 4]. As a consequence, L⁽_p¹⁾+ · · · +L⁽_p^N⁾ is closed on the intersection of the domains and so

D(L_p)=

;N j=1

D(L⁽_p^j⁾)= {u∈L_µ^p: D_ju,D_{j j}u∈L_µ^p for j =1,· · ·,N}.

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Let u∈D(L_p). In order to check D_{i j}u∈L_µ^p, we set v(x)=u(x)exp

− 1 2pD1

λx,x

, x∈R^N. Notice that v∈L^p(R^N) and

Dj jv(x)=

=

Dj ju(x)−2xj

pλj

Dju(x)− 1 pλj

u(x)+ x²_j (pλj)²u(x)

>

exp

−1 2pD1

λx,x

for j =1,· · · ,N and x ∈R^N. Lemma 2.3 shows that x_jD_ju,x²_ju∈L_µ^p, hence

|x|²u∈ L_µ^p. This implies that D_{j j}v,|x|²v∈L^p(R^N). From standard regularity properties of the Laplacian it follows that D_{i j}v∈L^p(R^N) for i, j =1,· · · ,N. On the other hand,

(3.2)

D_{i j}u(x)=

&

D_{i j}v(x)+ x_i pλi

D_jv(x)+ x_j pλj

D_iv(x)+ x_ix_j p²λiλj

v(x) '

×exp 1

2pD1 λx,x

for i,j =1,· · ·,N and x ∈R^N. For i = j we have, writing x =(xi,xˆ), xjDiv_L^pp(RN)=

RN−1|xj|^p

R|Div(x)|^pd xidxˆ

≤c

RN−1

R|Diiv(x)|^pd xi

¹

2 |xj|^p

R|v(x)|^pd xi

¹

2 dxˆ

≤cD_iiv_L^p²p(RN)x_j²v_L^p²p(RN)

so that x_jD_iv ∈ L^p(R^N). Hence, D_{i j}u ∈ L_µ^p by (3.2). This means that D(L_p)=W_µ²^,^p.

The following lemma has been used in the proof of the preceding result.

Lemma 3.3. The generator of the semigroup T⁽^j⁾(·)in L_µ^p is the operator L⁽_p^j⁾= L⁽^j⁾with domain D(L⁽_p^j⁾)= {u∈ L_µ^p: D_ju,D_{j j}u∈L_µ^p}.

Proof. Let W_µ²^,^p^,^j = {u ∈ L_µ^p : D_ju,D_{j j}u ∈ L_µ^p}. As in Proposition 2.4 one verifies that L⁽_p^j⁾ is the closure of L⁽^j⁾ defined on W_µ²^,^p^,^j and therefore the equality D(L⁽_p^j⁾)=W_µ²^,^p^,^j will follow from the estimate

(3.3) D_ju_L_µ^p + D_{j j}u_L_µ^p ≤c(L⁽^j⁾u_L_µ^p + u_L_µ^p)

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for u ∈ W_µ²^,^p^,^j. By density, it suffices to prove (3.3) for u ∈ C₀^∞(R^N). By Lemma 3.1 and writing x =(xj,xˆ), µ(x)=µj(xj)µ(ˆ xˆ), we have

R

|Dju(xj,xˆ)|^p+ |Dj ju(xj,xˆ)|^pµj(xj)d xj

≤c

R

|L⁽^j⁾u(xj,xˆ)|^p+ |u(xj,xˆ)|^pµ(xj)d xj.

Now (3.3) follows multiplying by µ(ˆ xˆ) and integrating with respect to xˆ. We now come to the main result of our paper.

Theorem 3.4. Assume that Q is real, symmetric and positive definite, that A is real with eigenvalues in the open left halfplane, and that1 < p < ∞. Then the generator Lpof the Ornstein-Uhlenbeck semigroup T(·)on L_µ^pis the Ornstein- Uhlenbeck operator L defined on W_µ²^,^p.

Proof. Taking into account Proposition 2.4 and Lemma 2.2, it suffices to show that L=L⁰+B is closed on the domain W_µ²^,^p (see (2.3)).

(a) Consider the group S(t)ϕ(x)=ϕ(e^{t A}¹x) for ϕ∈L_µ^p. Due to (2.4) we have −A₁= D_λA^∗₁D1

λ and trA₁=0. This yields S(t)ϕ_L^p^p

µ=(2π)⁻^N²(λ1· · ·λN)⁻¹²

RN|ϕ(e^{t A}¹x)|^pexp

−1 2D1

λx,x

d x

=(2π)⁻^N²(λ1· · ·λN)⁻¹²

RN|ϕ(y)|^pexp

−1 2D1

λe⁻^{t A}¹y,e⁻^{t A}¹y

d y

=(2π)⁻^N²(λ1· · ·λN)⁻¹²

RN|ϕ(y)|^pexp

−1

2e^{t A}^∗¹D1

λy,e⁻^{t A}¹y

d y

=ϕ_L^p^p

µ.

Hence, S(·) is a group of isometries on L_µ^p. It is then easy to see that S(·) is strongly continuous on L_µ^p and that its generator Bp coincides with the operator B on C₀^∞(R^N). Since C₀^∞(R^N) is dense in L_µ^p and S(·)-invariant, it is a core for Bp. Using Lemma 2.3 and the density of C₀^∞(R^N) in W_µ²^,^p, we deduce that the domain D(Bp) contains W_µ²^,^p and that Bpu = Bu for u ∈ W_µ²^,^p. In particular, D(L⁰_p)∩D(Bp)=W_µ²^,^p.

Since B_p generates a positive contraction semigroup on L_µ^p, w− B_p has bounded imaginary powers with power angle θ(B_p) ≤ π/2 on L_µ^p for every w >0 thanks to the transference principle [5, Section 4], see [4, Theorem 5.8].

By the same argument I−L⁰_p has bounded imaginary powers with power angle θ(L⁰_p)≤π/2. Moreover, L⁰₂ is self adjoint on L²_µ and thus has power angle 0 on L²_µ. By interpolation we obtain that θ(Bp)+θ(L⁰_p) < π for 1< p<∞.