Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator

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Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator

ROBERTHALLER-DINTELMANN ANDJULIANWIEDL

Abstract. Replacing the Gaussian semigroup in the heat kernel estimates by the Ornstein-Uhlenbeck semigroup onR^d, we define the notion of Kolmogorov kernel estimates. This allows us to show that under Dirichlet boundary conditions Ornstein-Uhlenbeck operators are generators of consistent, positive, (quasi-) contractiveC₀-semigroups onL^p()for all 1≤ p< ∞and for every domain ⊆R^d. For exterior domains with sufficiently smooth boundary a result on the location of the spectrum of these operators is also given.

Mathematics Subject Classification (2000):47D06 (primary); 35K20 (secon- dary).

1.

Introduction

Heat kernel estimates have proved to be a powerful tool for the analysis of elliptic differential operators. Besides many other things, they allow the extension of a given semigroup on L^p⁰for some p₀to the whole scale ofL^p-spaces for 1≤ p<

∞in a consistent way, transferring certain nice properties to all these semigroups, such as analyticity, and yielding p-invariance of the spectrum.

Dealing with Ornstein-Uhlenbeck operators onL^p()for unbounded domains we evidently cannot expect to get heat kernel estimates, as the spectrum of these operators in L^p(

R

^d)already depends heavily on p. Nevertheless, the well-known representation for the Ornstein-Uhlenbeck semigroup on L^p(

R

^d) that is due to A. N. Kolmogorov and given by

T(t)f(x)=

R^dk_t(e^{t B}x−y)f(y)dy=(k_t∗ f)(e^{t B}x), where the Kolmogorov kernel is

k_t(x)= 1

(4π)^d²(detQ_t)¹² exp

−1

4Q⁻_t ¹x·x

with Q_t = _t

0

e^{s B}Qe^{s B} ds, Pervenuto alla Redazione il 4 luglio 2005 e in forma definitiva il 7 novembre 2005.

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looks very similar to the heat semigroup. So it is a natural idea to consider Kol- mogorov kernel estimates replacing the Gaussian semigroup by the Kolmogorov semigroup (T(t))t≥0, thus getting a majorising semigroup that is well adapted to operators of the Ornstein-Uhlenbeck type. These estimates then allow us to extend semigroups onL^p⁰to the whole scale of L^p-spaces for 1 ≤ p < ∞, analogously to the case of heat kernel bounds.

Having established this idea, in the sequel we apply it to Ornstein-Uhlenbeck operators with Dirichlet boundary conditions inL^p(), where 1≤ p<∞andis a domain in

R

^d. Ornstein-Uhlenbeck operators are differential operators, formally given by

(

A

^u)(x)= d i,j=1

q_{i j}D_iD_ju(x)+Bx· ∇u(x), (1.1)

where Q = (q_{i j})^d_i_,_j₌₁ ∈

R

^d^×^d is a symmetric and positive definite matrix and B=(b_{i j})^d_i_,_j₌₁∈

R

^d^×^d\ {0}. They first appeared in stochastic analysis, describing a Brownian motion with an additional drift. In this context one usually works in spaces of continuous functions on

R

^d or in the spacesL^p(

R

^d, µ), whereµis the invariant measure of the underlying process.

Recently, it became clear that an analytic treatment of these operators is of great interest. For instance, looking at the Stokes equation in the exterior of a rotating obstacle leads to operators of the Ornstein-Uhlenbeck type, see [14], [13]

and [9]. Thus one is interested in their behaviour on L^p() with respect to the Lebesgue measure for domains and especially exterior domains.

Passing from the invariant measure to the Lebesgue measure changes the properties of the operator completely. The spectrum is no longer contained in the nega- tive real axis, instead it contains a vertical line (cf. [15]), so the semigroup(T(t))t≥0

on L^p(

R

^d)is not analytic, as it is in L^p(

R

^d, µ) (cf. [16]), and it even fails to be eventually norm-continuous.

Whereas Ornstein-Uhlenbeck operators are well understood in L^p(

R

^d) (cf.

[15, 18, 17]) and for bounded domains, where they can be viewed as a perturbation of lower order of the elliptic diffusion part, there are very few results for unbounded domains. M. Geissert, H. Heck, M. Hieber and I. Wood showed in [10] that in the case of an exterior domainwith sufficiently smooth boundary, a realisation of

A

onL^p()generates aC₀-semigroup and G. da Prato and A. Lunardi treat the case of L²-spaces of convex sets with respect to Neumann boundary conditions and in- finitesimally invariant measures in [4]. The Dirichlet problem in spaces of bounded continuous functions on smooth domains is treated by S. Fornaro, G. Metafune and E. Priola in [8]. An overview may be found in [3].

In this paper we show that for arbitrary domains ⊆

R

^d a realisation of

A

in L²()generates a (quasi-)contractive, positive C₀-semigroup, that has a Kol- mogorov kernel estimate, see Theorem 3.2. In the sequel this allows us to define consistent Ornstein-Uhlenbeck semigroups onL^p()for 1≤ p<∞that have the same contractivity, positivity and domination properties. This immediately gives

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an upper bound on the growth bound of the semigroups and implies that Ornstein- Uhlenbeck operators admit a boundedH^∞-calculus onL^p()(Theorem 4.5).

In the special case of an exterior domain with sufficiently smooth boundary, it turns out that the domain is

W₀¹^,^p()∩W²^,^p()∩ {f ∈L^p(): Bx· ∇f ∈L^p()}

and we even deduce in Theorem 5.2 that the same vertical line as in the case of the whole space is contained in the spectrum of the operator, so the spectral behaviour is the same as for the case=

R

^d. This means that also in this case the semigroup is not eventually norm-continuous. Nevertheless we can show that its growth bound and the spectral bound of its generator coincide, which is no longer clear by standard spectral theory for semigroups.

The paper is organised as follows. In section 2 we introduce the notion of Kolmogorov kernel estimates and prove their main implications. The generation result for

A

ⁱⁿ^L²()is contained in section 3 and in section 4 we show that this semigroup is positive and admits a Kolmogorov kernel estimate. Section 5 finally contains the results for exterior domains.

Notation. Throughout this paper we use the following notation.

For a closed operator(A,D(A))on some Banach spaceXwe denote byσ (A) the spectrum, by(A)the resolvent set and byR(λ,A)=(λ−A)⁻¹, λ∈(A), the resolvent of A. Furthermore, the space of all bounded linear operators on X is denoted by

L

(X).

As usual, for⊆

R

^d ^open, · pstands for the norm of the Lebesgue spaces L^p()whenever the setis clear from the context. We writeW^k^,^p(), orH^k() in the case p = 2, for the Sobolev spaces, C_c^∞() for the space of all smooth functions having compact support inandW₀¹^,^p(), orH₀¹(), is the closure of C_c^∞()in the norm of W¹^,^p() or H¹(), respectively. Furthermore, if X is a function space,X₊stands for the cone of all positive functions inX.

Finally, B_r(x₀)is the open ball of radiusr with centrex₀and, given a matrix B∈

R

^d^×^d, we write tr(B)=_d

j=1b_{j j}for its trace.

2.

Kolmogorov kernel estimates

Given a matrixB∈

R

^d^×^d\ {0}and a positive definite matrixQ∈

R

^d^×^d, we define the Kolmogorov semigroup(K_p(t))t≥0, onL^p(

R

^d)by

K_p(t)f (x)=

R^dk_t(e^{t B}x−y)f(y)dy, f ∈L^p(

R

^d), where the Kolmogorov kernelk_t is given by

k_t(x)= 1

(4π)^d²(detQ_t)¹² exp

−1

4Q⁻_t ¹x·x

with Q_t = _t

0

e^{s B}Qe^{s B} ds.

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It is well known (cf. [15]) that(K_p(t))t≥0is a positiveC₀-semigroup onL^p(

R

^d)for every 1 ≤ p<∞and it is straightforward by substitution and Young’s inequality that for every f ∈L^p(

R

^d)

K_p(t)f _p =

R^d

(k_t∗ f)(e^{t B}x)^pdx ¹_p

= e⁻^tr(B)^p ^t k_t ∗ f _p≤e⁻^tr(B)^p ^t f _p, (2.1) since k_t ₁ =1. The generator A_Rd,pof this semigroup is the Ornstein-Uhlenbeck operator

A

given in (1.1) and G. Metafune, J. Pr¨uss, A. Rhandi, and R. Schnaubelt showed in [18] that its domain is

D(A_Rd,p)=W²^,^p(

R

^d)∩ {f ∈L^p(

R

^d): Bx· ∇f ∈L^p(

R

^d)} (2.2) for 1< p<∞.

Now, let ⊆

R

^d be an open set, 1 ≤ p < ∞, and let(T(t))t≥0 be a C₀- semigroup onL^p()with generator A.

Definition 2.1. We say that the semigroup (T(t))t≥0 on L^p() satisfies a Kol- mogorov kernel estimateif there exist a matrix B ∈

R

^d^×^d \ {0}, a positive definite matrix Q ∈

R

^d^×^d^, ^M ≥0 andω ∈

R

, such that for all f ∈ L^p()and allt ≥ 0 we have the pointwise estimate

|T(t)f| ≤Me^ω^tK_p(t)| ˜f|, where f˜denotes the trivial extension of f to

R

^d^.

For such semigroups we have the following result.

Proposition 2.2. Let ⊆

R

^d ^{be open,}¹ ≤ p < ∞, and let(T(t))t≥0be a C₀- semigroup on L^p()that satisfies a Kolmogorov kernel estimate for some matrix B, a positive definite matrix Q and constants M andω. Then for1≤q<∞there exist consistent C₀-semigroups(T_q(t))t≥0on L^q(), such that T_p =T and

T_q(t) _L(L^q())≤Me^(ω−^tr(B)^q ⁾^t

for every t ≥ 0. Furthermore, T_q satisfies the same Kolmogorov kernel estimate and if T is a positive semigroup, T_qis also positive.

Proof. Letg∈L^p()∩L^q()andt ≥0. Then, by consistency of the semigroups K_qfor 1≤q<∞and (2.1), we have

T(t)g _q =

|T(t)g|^q ¹_q

≤Me^ω^t

R^d(K_p(t)| ˜g|)^q _q¹

= Me^ω^t K_q(t)| ˜g| q ≤Me^(ω−^tr(B)^q ⁾^t g _q.

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Thus we can extend the operatorT(t)continuously to an operatorT_q(t)onL^q(), obtainingC₀-semigroups that are consistent by construction and that obey the stated norm estimate by the above calculation.

For f ∈L^q()∩L^p()the inequality

|T_q(t)f| ≤Me^ω^tK_q(t)| ˜f|

is immediate by consistency. ThusMe^ω^tK_q(t)| ˜f|−|T_q(t)f|is positive for all these functions and the Kolmogorov kernel estimates follow for arbitrary f ∈ L^q(), sinceL^q()+is closed inL^q().

Finally, the same closedness argument yields the positivity ofT_q, wheneverT is positive.

For every 1 ≤ q < ∞we denote the generator of (T_q(t))t≥0 by A_q. The Kolmogorov kernel estimates also provide consistency results for these operators and their resolvents. We collect them in the next proposition.

Proposition 2.3. Let1≤q <∞. Then

1. R(λ,A_q)f = R(λ,A_r)f for all f ∈ L^q()∩L^r(), all1≤r <∞and all λ∈

C

^with^Re(λ) > λ0:=max ω−^tr⁽_q^B⁾, ω− ^tr⁽_r^B⁾

.

2. The set{f ∈D(A)∩L^q(): A f ∈L^q()}is contained in D(A_q)and A_qf = A f for all such f .

3. |R(λ,A_q)f| ≤M R(λ, ω+A_Rd,q)| ˜f|for all f ∈L^q()and allλ > ω−^tr⁽_q^B⁾. Proof.

1. By Proposition 2.2 we know that the growth bounds ofT_qandT_rare at mostλ0. Takingλ∈

C

^{with Re}(λ) > λ0, this allows us to conclude with the help of the Laplace transform

R(λ,A_q)f = _∞

0

e^−λ^tT_q(t)f dt = _∞

0

e^−λ^tT_r(t)f dt = R(λ,A_r)f for every f ∈L^q()∩L^r()by the consistency of the semigroups.

2. Let f ∈ D(A)∩L^q()withA f ∈L^q(). Then, choosingλ > ω+ |tr(B)|, we haveλ∈(A)∩(A_q)and since(λ− A)f ∈L^p()∩L^q(), we get

f = R(λ,A)(λ− A)f = R(λ,A_q)(λ−A)f ∈D(A_q) by part (1) of this proof. The above equality also yields A_qf = A f.

3. Letλ > ω− ^tr⁽_q^B⁾. Then we may again use Laplace transform forR(λ,A_q)and we obtain for f ∈L^q()

|R(λ,A_q)f| = _∞

0

e^−λ^tT_q(t)f dt

≤M _∞

0

e^−λ^te^ω^tK_q(t)| ˜f|dt =M R(λ, ω+A_Rd,q)| ˜f|, sinceT_qalso has a Kolmogorov kernel estimate by Proposition 2.2.

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In the special case that M = 1 in the kernel estimates, the operators A_q − ω+tr(B)/qgenerate contraction semigroups for allq∈(1,∞). Since|tr(B)/q| ≤

|tr(B)|for allq∈(1,∞), the amount of the shift is bounded. ThusAq−ω−|tr(B)|, 1 < q < ∞, is a family of generators of contraction semigroups onL^q(). If in addition the semigroup(T(t))t≥0is positive, by Proposition 2.2 all the semigroups are positive. This immediately yields a bounded H^∞-calculus for their generators, see [12].

Proposition 2.4. The operators A_q −ω− |tr(B)|admit a bounded H^∞-calculus for every 1 < q < ∞, whenever (T(t))t≥0 is a positive semigroup satisfying a Kolmogorov kernel estimate with M =1.

3.

The Ornstein-Uhlenbeck semigroup on

L²()

In the following we want to use Kolmogorov kernel estimates to show that the Ornstein-Uhlenbeck operator

A

is the generator of a positive C₀-semigroup on L^p()for every open and connected subsetof

R

^d. As we already mentioned in the introduction, the key is a generation result for the case p=2 and Kolmogorov kernel estimates. We will prove these two items in this and the following section.

We define the realisation of the Ornstein-Uhlenbeck operator inL²()by D(A_,₂)= H₀¹()∩ {u∈ H_loc² ():

A

^u∈L²()},

(A_,₂u)(x)= d i,j=1

q_{i j}D_iD_ju(x)+Bx· ∇u(x)

= d i,j=1

q_{i j}D_iD_ju(x)+ d i,j=1

b_{i j}x_jD_iu(x), x ∈⊂

R

^d,

where Q = (q_{i j})^d_i_,_j₌₁ ∈

R

^d^×^d is a symmetric and positive definite matrix and B=(bi j)^d_i_,_j₌₁∈

R

^d^×^d\{0}. Note that this notation is consistent with the definition of A_Rd,2 in (2.2). In fact, the domain in (2.2) is clearly contained in the domain given here, so when we have shown in Proposition 3.5 that A_,₂is dissipative, the equality of the two domains follows by the following general observation.

Remark 3.1. LetBbe the generator of aC₀-semigroup on some Banach space X and let A⊇ B be dissipative. Then we already have A= B. In fact, there exists λ >0, such thatλ−Bis surjective andλ−Ais injective by dissipativity. Thus the claim follows by [7, IV.1.21 (5)].

Now we can formulate our result for the case p=2.

Theorem 3.2. Let ⊆

R

^d be a domain. Then the operator A_,₂ generates a positive C₀-semigroup(T_,₂(t))t≥0 on L²()with T_,₂(t) _L(L²()) ≤ e⁻^tr(B)² ^t

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for all t ≥ 0. Moreover for every λ > −tr(B)/2and every t ≥ 0we have the domination properties

|R(λ,A_,₂)f| ≤ R(λ,A_Rd,2)| ˜f|, f ∈L²(),

|T_,₂(t)f| ≤T_Rd,2(t)| ˜f| = K₂(t)| ˜f|, f ∈L²(),

where f denotes the extension of f by˜ 0. In particular, T_,₂fulfills a Kolmogorov kernel estimate with M =1andω=0.

In the following we use the notation (

A

0u)(x)=

d i,j=1

q_{i j}D_iD_ju(x) and (

L

^u)(x)=Bx· ∇u(x) for the diffusion and the drift part of

A

, respectively. We start with a simple lemma, that will be useful for many proofs.

Lemma 3.3. Let G ⊆

R

^d be open and u∈ H₀¹(G)be a real-valued function. For anyϕ∈C_c^∞(

R

^d)we have

G

(

L

^u)uϕ= −tr(B) 2

u²ϕ−1 2

u²

L

ϕ.

Proof. Sinceu ∈ H₀¹(G), there is a sequence(u_n)n∈N ⊆C_c^∞(G)converging tou in H¹(G). Therefore, we have

G(

L

^u)uϕ = lim_n_→∞

G(

L

^un)u_nϕ, since Bx is bounded on the support ofϕ. We get

G

(

L

^un)u_nϕ=

G

Bx· ∇u_nu_nϕ= −

G

div

u_nϕBx u_n

= −

G(

L

^un)u_nϕ−tr(B)

G

u²_nϕ−

G

u²_n

L

ϕ.

Lettingntend to∞, we derive

G(

L

^u)uϕ= −^tr⁽₂^B⁾

Gu²ϕ− ¹₂

Gu²

L

ϕ.

Remark 3.4. To be precise, one has to check that integration by parts is allowed in the proof of Lemma 3.3. This will be used again later on, so it might be useful to note the following generalisation. For any open set G, integration by parts is possible ifu∈C_c^∞(G)andv∈H_loc¹ (G).

In this case, there is a compact set K ⊆ G with supp(u) ⊆ K^◦. Thenv ∈ H¹(K)and by the definition of weak derivatives one gets for all 1≤i ≤d

G

u(D_iv)=

K

u(D_iv)= −

K

(D_iu)v= −

G

(D_iu)v.

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In order to show that A_,₂ is a generator of a C₀-semigroup we will apply the Lumer-Phillips theorem. So we first need dissipativity.

Proposition 3.5. The operator

A:= A_,₂+ tr(B) 2 is dissipative in L²().

Proof. Let f ∈ D(A_,₂) = D(A). Writing f = u+ivfor suitable realu, v ∈ L²(), we get

Re

(A f)f =

(Au)u+

(Av)v, so it suffices to show

(Au)u ≤ 0 for real-valued functionsu. Note that u, v ∈ D(A_,₂), since the coefficients of

A

^{are real.}

We chooseη∈C_c^∞(

R

^d)vanishing outside ofB₂(0)withη|B1(0)=1and define ηm(x)=η(_m^x)form∈

N

^{. Since}(Au)u∈L¹()we derive lim_m_→∞

(Au)uηm =

(Au)uby the pointwise convergence ofηmto1and Lebesgue’s Theorem.

Next we choose a sequence(u_n)n∈NofC_c^∞()-functions converging touwith respect to theH¹-norm. Now, partial integration (as in Remark 3.4) yields

(A_,₂u)(x)u_n(x)ηm(x)dx

=

d i,j=1

q_{i j}D_iD_ju(x)u_n(x)ηm(x)dx+

d i,j=1

b_{i j}x_jD_iu(x)u_n(x)ηm(x)dx

=

d i,j=1

q_{i j}D_iD_ju(x)u_n(x)ηm(x)dx+ 1 2

d i,j=1

b_{i j}x_jD_iu(x)u_n(x)ηm(x)dx +1

2

d i,j=1

b_{i j}x_jD_iu(x)u_n(x)ηm(x)dx

= −

d i,j=1

q_{i j}D_iu(x)D_ju_n(x)ηm(x)dx−

d i,j=1

q_{i j}D_iu(x)u_n(x)D_jηm(x)dx

+1 2

d i,j=1

b_{i j}x_jD_iu(x)u_n(x)ηm(x)dx−1 2

d i,j=1

b_{i j}x_ju(x)D_iu_n(x)ηm(x)dx

−1 2

d i,j=1

b_{i j}x_ju_n(x)u(x)D_iηm(x)dx− tr(B) 2

u(x)u_n(x)ηm(x)dx.

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Note that, thanks to the bounded supports of the functionsηmandu_n, all integrals in the above calculations are well defined.

The H¹-convergence of the sequence(un)n∈Nyields

nlim→∞

d i,j=1

q_{i j}D_iu(x)D_ju_n(x)ηm(x)dx =

d i,j=1

q_{i j}D_iu(x)D_ju(x)ηm(x)dx

and

nlim→∞

d i,j=1

b_{i j}x_jD_iu_n(x)u(x)ηm(x)dx=

d i,j=1

b_{i j}x_jD_iu(x)u(x)ηm(x)dx, sincex_jηm(x)is bounded. The other summands can be treated analogously, so we derive

(A_,₂u)(x)u(x)ηm(x)dx = lim

n→∞

(A_,₂u)(x)u_n(x)ηm(x)dx

= −

d i,j=1

q_{i j}D_iu(x)D_ju(x)ηm(x)dx−

d i,j=1

q_{i j}D_iu(x)u(x)D_jηm(x)dx

− 1 2

d i,j=1

b_{i j}x_ju(x)u(x)D_iηm(x)dx− tr(B) 2

u(x)u(x)ηm(x)dx. Next, we want to pass to the limit m → ∞, so we have to consider the terms containing derivatives ofηm. The equality(D_iηm)(x)= _m¹(D_iη)(_m^x)implies

x_jD_iηm(x)= x_j

mD_iη x m

=0 for |x| m >2, and therefore all functionsx_jD_iηm(x), 1≤i,j ≤d, are bounded with

|x_jD_iηm(x)| =x_j

m D_iη x m

<2 ∇η _∞ and have support inside{x∈

R

^d ^:^m ≤ |x| ≤2m}.

Now letε >0. Sinceu²∈L¹(), there is a compact subsetK_ε ⊆with

\K_ε

u²≤ ε 2 ∇η _∞.

If we choosem₀large enough, we have K_ε ∩supp(x_jD_iηm)= ∅for allm ≥m₀ and therefore

u²(x)x_jD_iηm(x)dx =

\K_ε

u²(x)x_jD_iηm(x)dx ≤ε.

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This proves

mlim→∞

d i,j=1

b_{i j}x_ju(x)u(x)D_iηm(x)dx =0.

Since D_iηm ∞≤ _m¹ ∇η _∞, we also have

mlim→∞

d i,j=1

q_{i j}D_iu(x)u(x)D_jηm(x)dx =0. Finally we obtain dissipativity of Aby

(Au)u= lim

m→∞

(A_,₂u+tr(B) 2 u)uηm

= −

d i,j=1

q_{i j}D_iu(x)D_ju(x)dx ≤0, asQ is positive definite.

In order to show that A_,₂is a generator, it remains to be proven thatλ−Ais surjective for some fixedλ >0. This will be done by approximating the solution uof the resolvent problem(λ− A)u= f, f ∈ L²(), with solutions of the same problem on bounded and regular subdomains of.

By [5, II.4, Lemma 1], there exists an increasing sequence(n)n∈Nof bounded subdomains of , that have aC²-boundary, such that =

n∈Nn. Note that the specific choice of this sequence is not important, since by dissipativity of Aa solution of the resolvent problem is unique, whenever it exists.

Since the coefficients of

L

are bounded on bounded sets, by standard perturbation theory, the operator

A

+tr(B)/2 generates aC₀-semigroup on L²(n)for every n ∈

N

, when we equip it with the domain D = H₀¹(n)∩ H²(n). As D(A_n_,₂)contains D, we get D(A_n_,₂) = D again by Remark 3.1. This coin- cidence of the domains even gives us some more precious information. Since the generator A_n_,₂+tr(B)/2 is dissipative it even generates a contraction semigroup onL²(n)for everyn∈

N

. Thus the bounds on the resolvent do not depend onn, which will be important in the following.

Fixing λ > 0 and f ∈ L²() we find a unique solutionu_n ∈ H₀¹(n)∩ H²(n)for the problemλu_n−A_n_,₂u_n− ^tr⁽₂^B⁾u_n = f|_n for everyn∈

N

^with

u_n _L2(n) = R

λ,A_n_,₂+ tr(B) 2

f

n

L²(n) ≤ 1

λ f _L2(), (3.1) independently of the domainn.

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As by [1, Lemma 3.22], the trivial extension ofu_nis an element ofH¹(

R

^d)and hence of H₀¹(), we may regard(u_n)n∈Nas a sequence inH₀¹(). The next lemma will show that even the gradients of the functionsunare bounded independently of the chosen domain, which does not follow by elliptic regularity alone.

Lemma 3.6. Let G⊆

R

^dbe a bounded domain with C²-boundary and u∈H₀¹(G)∩

H²(G)be a solution ofλu− A_G_,₂u−^tr⁽₂^B⁾u=g for someλ >0. Then ∇u ₂≤

2

λ Q⁻¹² g ₂.

Proof. Letu=v+iwwith real-valuedv, w∈H₀¹(G)∩H²(G). Then D^αu ²₂=

G

D^αu D^αu=

G

(D^αv+i D^αw)(D^αv−i D^αw)

= D^αv ²₂+ D^αw ²₂ for any multiindexαwith|α| ≤2.

G

Q¹²∇vQ¹²∇v=

G

Q∇v∇v= −

G

(

A

0v)v

=

G

λv−A_G_,₂v−tr(B) 2 v

v−λ

G

vv+

G

(

L

v)v+tr(B) 2

G

vv

=

G

λv−A_G_,₂v−tr(B) 2 v

v−λ

G

vv

=

G

Re(g)R

λ,A_G_,₂+tr(B) 2

Re(g)−λ R

λ,A_G_,₂+ tr(B) 2

Re(g)

²

2

.

We conclude by (3.1)

∇v ²₂= Q⁻¹²(Q¹²∇v) ²₂≤ Q⁻¹² ²2

λ Re(g) ²₂. Repeating the same calculations forw, we obtain

∇u ²₂ ≤ Q⁻¹² ^{2 2}_λ Re(g) ²₂+ Q⁻¹² ^{2 2}_λ Im(g) ²₂= Q⁻¹² ^{2 2}_λ g ²₂. Now we can prove the main result of this section.

Proposition 3.7. Let ⊆

R

^d be a domain. Then A_,₂is the generator of a C₀- semigroup(T_,₂(t))t≥0on L²()with T_,₂(t) ≤e⁻^tr(B)² ^t.

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Proof. It only remains to show that for a fixedλ >0 and for every f ∈L²() there exists a functionu∈H₀¹()∩ {v∈H_loc² ():

A

v∈L²()}withλu− Au= f.

We consider the sequence(u_n)n∈N of trivial extensions of the solutions onn

mentioned above. In view of Lemma 3.6, it is bounded in H₀¹(), so there exists a weakly convergent subsequence(u_n_k)k∈N. We denote its limit byuand show in the following thatuis the desired solution.

As a first step, we will prove thatu∈ H_loc² (). Fix two compact setsK₁,K₂⊆ with K₁ ⊆ K₂^◦. Then, by construction,K₂ ⊆ n_k for sufficiently largek. The coefficients of

A

are bounded on K₂, so [11, Theorem 9.11] implies that there is a constantC depending onK₁,K₂and the bound of the coefficients onK₂, such that u_n_k _H2(K₁)≤C( u_n_k ₂+ f ₂), so there exists a weakly convergent subsequence (u_n_kl)l∈N in H²(K₁). Letvdenote its limit. Since the sequence(u_n_kl|K1)l∈N also converges weakly tou|K₁ in L²(K₁), we derivev = u|K₁ from the uniqueness of weak limits inL²(K₁). This showsu∈H_loc² ().

In order to finish the proof, it remains to showλu−Au= f. Letg∈C_c^∞() and fix a compactum K ⊆ with supp(g) ⊆ K^◦. Then K ⊆ n_k for k large enough. Let(u_n_kl)l∈N be again a weakly convergent subsequence of(u_n_k)k∈N on

H²(K)and Abe the formal adjoint ofA. Then we conclude

(λu− Au− f)g=

K

(λu−Au− f)g=

K

(λug−u Ag− f g)

= lim

l→∞

K(λu_n_klg−u_n_klAg− f g)= lim

l→∞

K(λu_n_kl −Au_n_kl − f)g=0. Thus the assertion follows by the fundamental theorem of variational calculus.

4.

Domination and positivity of the semigroup

The aim of this section is to prove Kolmogorov kernel estimates for the semigroup (T_,₂(t))t≥0obtained in Proposition 3.7. While doing so, we also obtain positivity of the semigroup. Our method is inspired by the proof of heat kernel estimates for the Dirichlet Laplacian, cf. [2].

Lemma 4.1. Let ⊆

R

^d be a domain, λ > −tr(B)/2 and let u ∈ D(A_,₂), v ∈ H¹()∩ {f ∈ H_loc² () :

A

^f ∈ L²()}be real-valued functions such that v≥0. Then the inequality(λ−

A

)u≤(λ−

A

)va.e. implies u ≤va.e.

Proof. As in the proof of Proposition 3.5, we choose a positiveη ∈C_c^∞(

R

^d)vanishing outside ofB₂(0)withη|B₁(0) =1and defineηm(x)=η(_m^x)form ∈

N

^.

By hypotheses, we have

λ(u−v)−

A

0(u−v)−

L

(u−v)≤0, a.e.

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so we obtain λ

(u−v)ϕηm− d i,j=1

q_{i j}D_iD_j(u−v)ϕηm−

Bx· ∇(u−v)ϕηm ≤0 for allm∈

N

and all positiveϕ∈C_c^∞(). By integration by parts, cf. Remark 3.4, we conclude that

λ

(u−v)ϕηm+ d i,j=1

q_{i j}D_j(u−v)D_iϕηm+ d i,j=1

q_{i j}D_j(u−v)ϕD_iηm

−

Bx· ∇(u−v)ηmϕ≤0.

for allm ∈

N

. Now, this last inequality is even valid for allϕ∈H₀¹()₊by density.

In the following we show that(u−v)⁺ ∈ H₀¹()₊. In order to do so, we choose a sequence(u_n)⊆C_c^∞(), that converges touinH¹(). Then the function (u_n−v)⁺is inH¹()₊for everyn∈

N

^{and since}v≥0, we get supp((u_n−v)⁺)⊆ supp(u_n). Thus(u_n−v)⁺has compact support in, which implies(u_n−v)⁺ ∈ H₀¹()₊. This finally yields(u−v)⁺ ∈ H₀¹()₊, asH₀¹()is a closed subspace of H¹().

Puttingϕ = (u−v)⁺in the above inequality and observing that then all the integrals vanish on the set{u≤v}, we get

λ

(u−v)⁺2

ηm+

∇(u−v)⁺Q∇(u−v)⁺ηm

−

Bx· ∇(u−v)⁺(u−v)⁺ηm+ n i,j=1

q_{i j}D_j(u−v)⁺(u−v)⁺D_iηm≤0. Now Lemma 3.3 yields for the third integral

Bx· ∇(u−v)⁺(u−v)⁺ηm= −tr(B) 2

(u−v)⁺2

ηm

−

Bx· ∇ηm

(u−v)⁺2

.

As all the limits for m → ∞ on the left hand side exist (cf. proof of Proposi- tion 3.5), this implies

−

Bx· ∇(u−v)⁺(u−v)⁺= tr(B) 2

(u−v)⁺₂ . We derive the inequality

λ+tr(B)

2

(u−v)⁺₂ +

∇(u−v)⁺Q∇(u−v)⁺≤0.