ON A CLASS OF PSEUDO-DIFFERENTIAL OPERATORS ON R

ON A CLASS OF PSEUDO-DIFFERENTIAL OPERATORS ON R

× Z

GENERATING FELLER SEMIGROUPS

KRISTIAN P. EVANS, NIELS JACOB and OWEN C. MORRIS

We study a class of pseudo-dierential operatorsq(x, Dx, Dk)acting on functions dened on Rⁿ×Z^m. Sucient conditions on the symbol q(x, ξ, ω)are given to imply that −q(x, Dx, Dk) extends to a generator of a Feller semigroup, hence, of a Feller process with state space Rⁿ×Z^m. Some geometric interpretations of the transition density are given.

AMS 2010 Subject Classication: 47D07, 35S99, 60J35.

Key words: Feller semigroups, pseudo-dierential operators, Q-matrices, analysis onRⁿ×Z^m.

INTRODUCTION

In recent years, the second named author initiated a programme to study Levy-type processes with state space Rⁿ in terms of intrinsic geometries as- sociated with these processes by using their symbols. The idea was and is to establish a theory analogous to elliptic as well as sub-elliptic diusions where transition functions are best understood in terms of volume growth and expo- nential decay where the volume as well as the decay is related to certain metrics associated with the process.

In [8] we worked out our programme for some classes of Levy-processes (X_t^ψ)t≥0 with a real-valued characteristic exponent ψ. The most important observation was that we need two quite dierent metrics. Let the transition density of (X_t^ψ)t≥0 be given by

(0.1) p_t(x−y) =

Z

Rⁿ

e^i(x−y)·ξe^−tψ(ξ)dξ.

We seek ptto satisfy

(0.2) pt(x−y)λ⁽ⁿ⁾(B^dψ,t(0,1))e^{−δ2ψ,t(x,y)},

whereabmeans that the quotient ^a_b is bounded away from zero and bounded, and

REV. ROUMAINE MATH. PURES APPL. 59 (2014), 1, 5575

(0.3) dψ,t(ξ, η) =p

tψ(ξ−η), as well asδ_ψ,t is determined by

e^{−δ2ψ,t(x,y)}= Z

Rⁿ

e^i(x−y)·ξe^−tψ(ξ)

p_t(0) dξ= pt(x−y) p_t(0) .

Thus, dψ,t is a metric in the space where the characteristic exponent is dened while δ_ψ,t lives on the state space. Note that in the case of Brownian motion we have d^BMψ,t (ξ, η) =√

t|ξ−η|andδ^BM_ψ,t (x, y) = ^|x−y|√

2t . For the Cauchy process (n= 1)we have

(0.4) d^cψ,t(ξ, η) =p

t|ξ−η|

and

(0.5) δ^c_ψ,t(x, y) = s

ln

|x−y|²+t² t²

, the symmetric Meixner process on Ryields

(0.6) d^Mψ,t(ξ, η) =p

tln cosh(ξ−η) and

(0.7) δ^M_ψ,t(x, y) =





∞

X

j=1

ln

1 + |x−y|² (t+ 2j)²





1 2

,

compare with [8]. The conjecture is that for a Levy process where ψ¹² gives a metric generating the Euclidean topology and which has the volume doubling property (with respect to the Lebesgue measure) the estimate (0.2) always holds. In [8] some rst results are proved and reasons are given why to conjec- ture (0.2). We also have indicated how the case of variable coecients might behave.

One of the main problems is that p

ψ(ξ−η) is not a metric leading to a geodesic or length space. The Fourier transform switching from the x−space to the ξ−space is dicult to handle when metric properties are looked at, and in particular we lack an understanding to which extentp

q(x, ξ−η)for certain q :Rⁿ×Rⁿ → R, q(x,·) being for each x ∈Rⁿ a characteristic exponent of a Levy process, can be used to construct on Rⁿ, seen as a manifold, a type of Riemann metric.

Harmonic analysis is a powerful tool on locally compact abelian groups.

In [3] we worked out that under some conditions Q−matrices as generators of

Markov chains onZ^m can be viewed as pseudo-dierential operators with sym- bols dened onZ^m×T^m and that at least for diagonal estimates for transition densities a volume growth interpretation is possible, just as in (0.2) forx =y. Thus, there are quite a few interesting indications that this geometric or metric approach to jump-type processes has much more to reveal. However, we are short of suitable examples to get further related insights.

The main purpose of this paper is to construct a toy class of Feller semigroups, hence, Feller processes, with state space Rⁿ×Z^m using pseudo- dierential operators not being translation invariant (with respect tox∈Rⁿ).

Having control on the symbol should help to investigate further our geomet- ric programme. As it turns out, some machinery is needed to construct these semigroups and in this paper we restrict ourselves to this construction only.

More precisely, we prove that for certain symbols q:Rⁿ×Rⁿ×T^m→R leading to pseudo-dierential operators

(0.8) q(x, D_x, D_k)u(x, k) = (2π)^−m Z

Rⁿ

Z

T^m

e^ixξe^ikωq(x;ξ, ω)˜u(ξ, ω)dωdξ the operator−q(x, D_x, Dk)with a suitable domain extends to a generator of a Feller semigroup onC∞(Rⁿ×Z^m). The main conditions onq are that(ξ, ω)7→

q(x;ξ, ω) is a continuous negative denite function having the decomposition (0.9) q(x;ξ, ω) =q₁(ξ, ω) +q₂(x;ξ, ω),

whereq₁ is a continuous negative denite function comparable uniformly with respect to ω∈T^m to a xed continuous negative denite functionψ:Rⁿ→R having a minimal growth at innity, and q2(x, ξ, ω) satises with respect to x certain smoothness and smallness conditions, compare with A.0, A.1 and A.2.M.

This is a way to construct examples proposed in [6] for the case of state space Rⁿ and we can adopt some ideas here. Note that q₂(x, D_x, D_k) can be viewed as a small pertubation ofq(Dx, Dk)and in [4], see also [5], it was worked out for the state space Rⁿ how this can be used to get much larger classes by using martingale problem techniques. Thus, the restriction to this class is for our purposes not too severe.

In Section 1, we collect auxiliary results, Section 2 gives the core estimates, and in Section 3 the semigroup is constructed, i.e. the conditions of the Hille- Yosida-Ray theorem are veried. In Section 4, we provide some examples and a short observation to our geometric programme.

By dedicating this paper to Professor Boboc, the second named author would like to express his appreciation to a colleague who made signicant con- tributions to potential theory and to a long lasting friendship.

1. AUXILIARY RESULTS

We need some well-known results from harmonic analysis on locally com- pact abelian groups, in particular for Rⁿ, Z^m and the torus T^m as well as on their products and dual groups. Recall that the dual group of Rⁿ can be iden- tied byRⁿand that ofZ^m isT^m. Pontryagin duality yields that(T^m)^∗=Z^m. For a locally compact abelian groupGthe notions of positive denite and neg- ative denite functions are dened, compare with [1]. Note that if ψ:Rⁿ→R is a continuous negative denite function, i.e. ψ(0) ≥0 and for all t > 0 the function ξ 7→ e^−tψ(ξ) is a continuous positive denite function in the sense of Bochner, compare [1], then

(1.1) ψ(ξ)≥0,

(1.2) ψ(ξ)≤c_ψ(1 +|ξ|²),

and

(1.3) 1 +ψ(ξ)

1 +ψ(η) ≤2(1 +ψ(ξ−η)) (Peetre's inequality).

The spaceS(Rⁿ×Z^m)consists of allu:Rⁿ×Z^m →Cwhich are continuous and for every α ∈Nⁿ₀ and all j, l∈N0 =N∪ {0} there existsc =cu,α,j,l such that

(1.4) |∂_x^αu(x, k)| ≤c_u,α,j,l(1 +|x|²)^−j2(1 +|k|²)^−2l holds. The Fourier transform on Rⁿ is dened by

(1.5) u(ξ) = (Fˆ x7→ξu) (ξ) := (2π)⁻ⁿ² Z

Rⁿ

e^−ixξu(x)dx, with inverse Fourier transform

(1.6)

F_ξ7→x⁻¹ v

(x) := (2π)⁻ⁿ² Z

Rⁿ

e^ixξv(ξ)dξ.

On Z^m the Fourier transform is given by

(1.7) (Fk7→ωu) (ω) := X

k∈Z^m

u(k)e^−ikω,

with inverse (acting on functions dened on T^m) (1.8) F_ω7→k⁻¹ v

(k) := 1 (2π)^m

Z

T^m

e^ikωv(ω)dω.

The rationale behind the normalisation of the Fourier transform is to have a probability measure as a Haar measure on a compact group, implying a counting measure on a discrete group. OnRⁿ we use the normalisation as in

[6] which gives the constant 1 in Plancherel's theorem. This allows easy referral to [6].

For u : Rⁿ×Z^m → C we denote the Fourier transform by u˜ and the following holds foru∈S(Rⁿ×Z^m)

˜

u(ξ, ω) = (2π)⁻ⁿ² Z

Rⁿ

X

k∈Z^m

u(x, k)e^−ikωe^−ixξdξ (1.9)

= (2π)⁻ⁿ² X

k∈Z^m

Z

Rⁿ

u(x, k)e^ixξdξe^ikω, which has the inverse

(1.10)

F_ξ7→x⁻¹ F_ω7→k⁻¹ v

(x, k) = (2π)^−n2−m Z

Rⁿ

Z

T^m

v(ξ, ω)e^ikωe^iξxdωdξ.

Moreover we write

(1.11) u(ξ, k) := (Fˆ x7→ξu(·, k)) (ξ, k).

For u ∈ L²(Rⁿ×Z^m) the Fourier transform is dened as an extension using Plancherel's theorem, i.e. using

(1.12) ||u||_L2(Rⁿ×Z^m)=||˜u||_L2(Rⁿ×T^m)

or

(1.13) Z

Rⁿ

X

k∈Z^m

|u(x, k)|²dx= Z

Rⁿ

Z

T^m

|˜u(ξ, ω)|²dξdω, which holds foru∈S(Rⁿ×Z^m) or u∈L¹(Rⁿ×Z^m)∩L²(Rⁿ×Z^m).

Since Rⁿ×Z^m is a locally compact space, the space of all continuous functions vanishing at innity is dened and is denoted byC∞(Rⁿ×Z^m). It is straightforward to see thatS(Rⁿ×Z^m) is dense inC∞(Rⁿ×Z^m)with respect to the supremum norm|| · ||∞which makesC∞(Rⁿ×Z^m)a Banach space. For u ∈S(Rⁿ×Z^m), anys≥0, and a xed continuous negative denite function ψ:Rⁿ→Rwe dene

||u||²_Hψ,s(

Rⁿ×Z^m) :=

Z

Rⁿ

X

k∈Z^m

(1 +ψ(ξ))^s|ˆu(ξ, k)|²dξ (1.14)

= (2π)^−m Z

Rⁿ

Z

T^m

(1 +ψ(ξ))^s|˜u(ξ, ω)|²dωdξ

= (2π)^−m Z

T^m

Z

Rⁿ

(1 +ψ(ξ))^s|˜u(ξ, ω)|²dξdω.

It follows that

(1.15) H^ψ,s(Rⁿ×Z^m) :={u∈L²(Rⁿ×Z^m)| ||u||_ψ,s<∞}

is a Banach space and H^ψ,t(Rⁿ×Z^m) ⊂H^ψ,s(Rⁿ×Z^m) ⊂L²(Rⁿ×Z^m) for t > s in the sense of continuous embeddings. Moreover, since onS(Rⁿ×Z^m)

(1.16) u(x, k) =

F_ξ7→x⁻¹ (F_x7→ξu) (x, k) it follows for s > _2rⁿ

0 that

(1.17) (1 +ψ(ξ))¹² ≥c0,ψ(1 +|ξ|²)^r20, r0 >0, c0,ψ >0, implies

(1.18) ||u||_∞≤c||u||_Hψ,s(Rⁿ×Z^m),

and H^ψ,s(Rⁿ×Z^m) is continuously embedded into C∞(Rⁿ×Z^m), for details we refer to [10]. Note that with

(1.19) ψ(D)u(x, k) =F_ξ7→x⁻¹ (ψ(·)ˆu(ξ, k))(x, k) we have

(1.20) ||u||_Hψ,s(Rⁿ×Z^m) =||(1 +ψ(D_x))^s2u||_L2(Rⁿ×Z^m).

In some of our arguments we will refer to the Levy-Khinchine formula on Rⁿ as well as T^m. For this we refer to [1], in particular§18. Note that onT^m a quadratic form is by denition a mapping q:T^m →Rsatisfying

(1.21) 2q(ξ) + 2q(η) =q(ξ+η) +q(ξ−η).

Further, for a continuous negative denite function q : T^m → C the operator

(1.22) q(D_k)v(k) =F_ω7→k⁻¹ (q(·)F_k7→ωv(·)) (k)

is local if and only if q(ω) = c+il(ω) +q₀(ω) where c ≥ 0, l : T^m → R is a continuous homomorphism, and q0 is a continuous, non-negative quadratic form, compare with [1], Theorem 18.27. In particular, forq(ω) =ω−^ω_2π² dened on S¹=T¹ the corresponding operatorq(D_k)is not local.

Furthermore, we will often use the estimate (1.23)

Z Z

k(x−y)u(x)v(y)dxdy

≤ ||k||_L1||u||_L2||v||_L2

for k∈L¹, u, v∈L².

2. BASIC ESTIMATES

Letq :Rⁿ×Rⁿ×T^m →Rbe a continuous function which we decompose as

(2.1) q(x;ξ, ω) =q₁(ξ, ω) +q₂(x;ξ, ω)

where the reader should have in mind the decomposition by freezing the coef- cients, i.e. for some x₀ ∈Rⁿ

(2.2) q(x, ξ, ω) =q(x0;ξ, ω) + (q(x;ξ, ω)−q(x0;ξ, ω)).

We assume the following for the rest of the paper:

A.0 For all x∈Rⁿthe functions(ξ, ω)7→q(x;ξ, ω) and(ξ, ω)7→q1(ξ, ω) are negative denite.

Note that A.0 implies q₁(ξ, ω)≥0 andq(x;ξ, ω)≥0.

Letψ:Rⁿ→Rbe a xed continuous negative denite function satisfying with some r₀ >0 and c_0,ψ >0the estimate

(2.3) (1 +ψ(ξ))¹² ≥c_0,ψ(1 +|ξ|²)^r20. We assume that q1 satises:

A.1 For all (ξ, ω)∈Rⁿ×T^m we have

(2.4) q1(ξ, ω)≤γ1(1 +ψ(ξ)), and

(2.5) 1 +q1(ξ, ω)≥γ0(1 +ψ(ξ)), whereγ₁ and γ₀ are independent of ξ and ω.

Further, we assume for q₂ the condition A.2.M below with M being de- termined later dependent on r0 and nand maybe further parameters.

A.2.M For M ∈N0 the functionx→q₂(x;ξ, ω) isM-times continuously dierentiable and for all α∈Nⁿ₀,|α| ≤M the estimate

(2.6) |∂_x^αq2(x;ξ, ω)| ≤ϕα(x)(1 +ψ(ξ)) holds with ϕ_α ∈L¹(Rⁿ) independent of ω and ξ. By (2.7) qˆ2(η;ξ, ω) = (Fx7→η(q2(x;ξ, ω)))(η;ξ, ω)

we denote the Fourier transform of x7→q₂(x;ξ, ω) for (ξ, ω)∈Rⁿ×T^m xed.

By standard arguments, compare with [7], Lemma 2.3.2.B, we get Lemma 2.1. Under A.2.M we have

(2.8) |ˆq2(η, ξ, ω)| ≤γ˜M,n,m

X

|α|≤M

kϕ_αk_L1(Rⁿ)(1 +|η|²)^−M2 (1 +ψ(ξ)).

We introduce the corresponding pseudo-dierential operators q(x, D_x, D_k)u(x, k) :=F_ξ7→x⁻¹ F_ω7→k⁻¹ (q(x, ξ, ω)˜u(ξ, ω))(x, k), (2.9)

and

q1(Dx, Dk)u(x, k) :=F_ξ7→x⁻¹ F_ω7→k⁻¹ (q1(ξ, ω)˜u(ξ, ω))(x, k).

(2.10)

Using Plancherel's theorem we immediately derive from A.1 the following estimates for a detailed calculation compare with [10].

Proposition 2.2. Assume A.1 and let s ≥ 0. For u ∈ S(Rⁿ×Z^m) it holds that

(2.11) kq₁(D_x, D_k)uk_Hψ,s(Rⁿ×Z^m)≤γ₁kuk_Hψ,s+2(Rⁿ×Z^m)

and

(2.12) kq₁(D_x, D_k)uk_Hψ,s(Rⁿ×Z^m)≥κ˜₀kuk_Hψ,s+2(Rⁿ×Z^m)−κ˜₁kuk_L2(Rⁿ×Z^m)

with constants ˜κ₀ >0 and ˜κ₁ ≥0.

Remark 2.3. By (2.12) we have for >0 kq₁(Dx, Dk)uk²_Hψ,s(Rⁿ×Z^m)

≥γ0(1−ε)kuk²_Hψ,s+2(Rⁿ×Z^m)−κεkuk²_L2(Rⁿ×Z^m). (2.13)

The Friedrichs mollier Jis the standard one acting only on the variable x, i.e. with

(2.14) j(x) :=

c0exp((|x|²−1)⁻¹), |x|<1

0, |x| ≥1 ,

whereR

Rⁿj(x)dx= 1 determines c₀, we dene for u:Rⁿ×Z^m→C (2.15) J_ε(u)(x, k) :=

Z

Rⁿ

j_ε(x−y)u(y, k) dy, wherej(x) :=⁻ⁿj ^x

As proved in detail in [10] we have.

Proposition 2.4. A. The operator J is a symmetric contraction on L²(Rⁿ×Z^m) and

(2.16) lim

ε→0kJ_ε(u)−uk_L2(Rⁿ×Z^m)= 0.

Moreover, we have

(2.17) J_ε(u)∈ \

t≥0

H^ψ,t(Rⁿ×Z^m), and for u∈H^ψ,s(Rⁿ×Z^m),s≥0, the following hold (2.18) kJ_ε(u)k_Hψ,s(Rⁿ×Z^m)≤ kuk_Hψ,s(Rⁿ×Z^m); as well as

(2.19) lim

ε→0kJ_ε(u)−uk_Hψ,s(Rⁿ×Z^m)= 0.

B. Let s≥0, ε∈(0, ρ) and assume for u∈L²(Rⁿ×Z^m) that (2.20) kJ_ε(u)k_Hψ,s(Rⁿ×Z^m)≤c_u,s

for all ε∈(0, ρ) with c_u,s independent of ε. Then u∈H^ψ,s(Rⁿ×Z^m).

Mimicking the proof of Theorem 2.3.16 and Corollary 2.3.18 in [7] we nd, compare with [10], a commutator estimate, recall [A, B] =AB−BA.

Proposition 2.5. A. For s ≥ 0 let M > 1 +|s−1|+n be such that A.2.M is fullled. Then it holds

(2.21) k[J_ε, q₂(x, D_x, D_k)]uk_Hψ,s(Rⁿ×Z^m)≤ckuk˜ _Hψ,s+1(Rⁿ×Z^m),

for all ε∈(0,1]and all u∈H^ψ,s+1(Rⁿ×Z^m) with ˜c independent of ε∈(0,1]. B. Under the assumption of A we have for q(x, Dx, D_k) = q1(Dx, D_k) + q₂(x, D_x, D_k) andu∈H^ψ,s+1(Rⁿ×Z^m)

(2.22) k[J_ε, q(x, D_x, D_k)]uk_Hψ,s(Rⁿ×Z^m)≤ckuk˜ _Hψ,s+1(Rⁿ×Z^m), with c˜independent of ε∈(0,1].

In order to handle [(1 +ψ(D_x))]^s2, q₂(x, D_x, D_k)] we rst note ([(1 +ψ(D_x))^s2, q₂(x, D_x, D_k)]u)^∼(ξ, ω)

= (2π)⁻ⁿ² Z

Rⁿ

ˆ

q2(ξ−η, η, ω)((1 +ψ(ξ))^s2 −(1 +ψ(η))^s2)˜u(η, ω) dη and recall, compare with [7], the proof of Theorem 2.3.9,

|(1 +ψ(ξ))^s2 −(1 +ψ(η))^s2| ≤bc_s,ψ(1 +|ξ−η|²)^s2(1 +ψ(η))^s−12 .

Proposition 2.6. Fors >0letM > n+sbe such that A.2.M is fullled.

Then with some constant γ =γ_M,n,m,s,ψ it holds for allu∈H^ψ,s+1(Rⁿ×Z^m) k[(1 +ψ(D_x))^s2, q₂(x, D_x, D_k)]uk_L2(Rⁿ×Z^m)

≤γP

|α|≤Mkϕ_αk_L1(Rⁿ)kuk_Hψ,s+1(Rⁿ×Z^m). (2.23)

Proof. For u∈H^ψ,s+1(Rⁿ×Z^m) and v∈L²(Rⁿ×Z^m) we nd

([(1 +ψ(D_x))^s2, q₂(x, D_x, D_k)]u, v)_L²_(Rⁿ_×Z^m₎

=

(2π)^−n2−m Z

T^m

Z

Rⁿ

Z

Rⁿ

ˆ

q₂(ξ−η, η, ω)((1 +ψ(ξ))^s2 −(1 +ψ(η))^s2) ×

טu(η, ω)˜v(ξ, ω) dηdξdω

≤c_M,n,m,s,ψ X

|α|≤M

kϕ_αk_L1(Rⁿ)(2π)^−m×

× Z

T^m

Z

Rⁿ

Z

Rⁿ

(1 +|ξ−η|²)^−M−s2 (1 +ψ(η))^s+12 |˜u(η, ω)||˜v(ξ, ω)|dηdξdω

≤cM,n,m,s,ψ

X

|α|≤M

kϕ_αk_L1(Rⁿ)k(1 +|·|²)^−M−s2 k_L1(Rⁿ)×

×k(1 +ψ(·))^s+12 u(·,˜ ·)k_L2(Rⁿ×T^m)k˜vk_L2(Rⁿ×T^m)

=γ X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk_Hψ,s+1(Rⁿ×Z^m)kvk_L2(Rⁿ×Z^m), which implies

k[(1 +ψ(D_x))^s2, q₂(x, D_x, D_k)]uk_L2(Rⁿ×Z^m)

= sup

kvk_L2(Rn×Zm)6=0

([(1 +ψ(D_x))^s2, q₂(x, D_x, D_k)]u, v)_L2(Rⁿ×Z^m)

kvk_L2(Rⁿ×Z^m)

≤γ X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk_Hψ,s+1(Rⁿ×Z^m), and the proposition is proved.

Remark 2.7. Note that for s= 0 the proposition is trivial.

With these preparations we can estimate q2(x, Dx, Dk). Since (q₂(x, D_x, D_k)u, v)_L2(Rⁿ×Z^m)

= (2π)^−n2−m Z

Tⁿ

Z

Rⁿ

Z

Rⁿ

ˆ

q₂(ξ−η, η, ω)˜u(η, ω)˜v(ξ, ω) dηdξdω, (2.24)

we nd

|(q₂(x, D_x, D_k)u, v)_L²_(Rⁿ_×Z^m₎|

≤γ˜M

X

|α|≤M

kϕ_αk_L1(Rⁿ)(2π)^−n2−m×

× Z

Tⁿ

Z

Rⁿ

Z

Rⁿ

(1 +|ξ−η|²)^−M2 (1 +ψ(η))|˜u(η, ω)||˜v(ξ, ω)|dηdξdω

≤γ_M⁰ X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk_Hψ,2(Rⁿ×Z^m)kvk_L2(Rⁿ×Z^m), which yields

(2.25) kq₂(x, D_x, D_k)uk_L2(Rⁿ×Z^m)≤γ_M⁰ X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk_Hψ,2(Rⁿ×Z^m),

whereM =n+ 1is sucient. Now, since

kuk_Hψ,s(Rⁿ×Z^m) =k(1 +ψ(Dx))^s2uk_L2(Rⁿ×Z^m), we nd further fors≥0that

kq₂(x, Dx, D_k)uk_Hψ,s(Rⁿ×Z^m)

=k(1 +ψ(D_x))^s2q₂(x, D_x, D_k)uk_L2(Rⁿ×Z^m)

≤ kq₂(x, D_x, D_k)(1 +ψ(D_x))^2suk_L2(Rⁿ×Z^m)

+k[(1 +ψ(D_x))^s2, q₂(x, D_x, D_k)]uk_L2(Rⁿ×Z^m)

≤γ_M⁰ X

|α|≤M

kϕ_αk_L1(Rⁿ)k(1 +ψ(Dx))^2suk_Hψ,2(Rⁿ×Z^m)

+γ X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk_Hψ,s+2(Rⁿ×Z^m). Thus, we have proved

Proposition 2.8. For s≥0 let M > n+ max(s,1)be such that A.2.M holds for the symbol q2 then there exists a constant γ1=γM,n,m,s,ψ such that (2.26) kq₂(x, Dx, D_k)uk_Hψ,s(Rⁿ×Z^m)≤γ1

X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk_Hψ,s+2(Rⁿ×Z^m). Combining these estimates we eventually arrive at

Theorem 2.9. Suppose that A.0, A.1 and A.2.M hold with M > n+ max(1, s). Thenq(x, D_x, D_k)mapsH^ψ,s+2(Rⁿ×Z^m)continuously intoH^ψ,s(Rⁿ× Z^m) and

(2.27) kq(x, D_x, D_k)uk_Hψ,s(Rⁿ×Z^m)≤ckuk_Hψ,s+2(Rⁿ×Z^m)

holds.

Theorem 2.10. Suppose that A.0, A.1 and A.2.M hold withM > n+s+1 and letκ˜₀ as in (2.12). Assume in addition

(2.28) X

|α|≤M

kϕ_αk_L1(Rⁿ)≤ κ˜₀ 2γ1

.

Then there exists a constant δ₀>0 such that for s≥0 we have

(2.29) kq(x, D_x, D_k)uk_Hψ,s(Rⁿ×Z^m)≥δ₀kuk_Hψ,s+2(Rⁿ×Z^m)−κ˜₁kuk_L2(Rⁿ×Z^m). (Note that δ₀ and ˜κ₁ depend on s).

Remark 2.11. Note that since (ξ, ω) 7→ q(x;ξ, ω) is continuous negative denite it follows that foru real-valued (q(x, D_x, D_k)u, u)_L2(Rⁿ×Z^m)≥0which implies forλ≥0that

kq(x, D_x, D_k)u+λuk_L2(Rⁿ×Z^m)

≥ 1

√

2kq(x, D_x, D_k)uk_L2(Rⁿ×Z^m)+ λ

√

2kuk_L2(Rⁿ×Z^m), (2.30)

and for λ≥κ˜1

(2.31) kq(x, D_x, D_k)u+λuk_L2(Rⁿ×Z^m)≥ δ0

√2kuk_Hψ,2(Rⁿ×Z^m). Finally, we can show that elliptic regularity" holds.

Theorem 2.12. Suppose that q(x, Dx, D_k) = q1(Dx, D_k) +q2(x, Dx, D_k) where the symbol q₁ satises A.1 and for M > n+ 2 +s, we assume that the symbol q2 satises A.2.M. If for somef ∈H^ψ,s(Rⁿ×Z^m), s≥0, and λ∈Rⁿ we have a solution u∈H^ψ,s+1(Rⁿ×Z^m) to the equation

q_λ(x, D_x, D_k)u:=q(x, D_x, D_k)u+λu=f, then it follows that under (2.28) u∈H^ψ,s+2(Rⁿ×Z^m).

Proof. Using (2.29) we nd for ε∈(0,1]that

δ0kJ_ε(u)k_Hψ,s+2(Rⁿ×Z^m)−κ˜1kJ_ε(u)k_L2(Rⁿ×Z^m)− kλJ_ε(u)k_Hψ,s(Rⁿ×Z^m)

≤ kq(x, D_x, Dk)Jε(u)k_Hψ,s(Rⁿ×Z^m)− kλJ_ε(u)k_Hψ,s(Rⁿ×Z^m)

≤ kq(x, D_x, Dk)Jε(u) +λJε(u)k_Hψ,s(Rⁿ×Z^m)

≤ kJ_ε(q(x, D_x, D_k)u+λu)k_Hψ,s(Rⁿ×Z^m)

+k[J_ε, q₂(x, D_x, D_k)]uk_Hψ,s(Rⁿ×Z^m)

=kJ_ε(f)k_Hψ,s(Rⁿ×Z^m)+k[J_ε, q₂(x, D_x, D_k)]uk_Hψ,s(Rⁿ×Z^m). Using Proposition 2.5.Awe get further that

δ0kJ_ε(u)k_Hψ,s+2(Rⁿ×Z^m)

≤ kJ_ε(f)k_Hψ,s(Rⁿ×Z^m)+ ˜κ1kJ_ε(u)k_L2(Rⁿ×Z^m)

+|λ|kJ_ε(u)k_Hψ,s(Rⁿ×Z^m)+k[J_ε, q₂(x, D_x, D_k)]uk_Hψ,s(Rⁿ×Z^m)

≤ kfk_Hψ,s(Rⁿ×Z^m)+ ˜κ₁kuk_Hψ,s(Rⁿ×Z^m)+|λ|kuk_Hψ,s(Rⁿ×Z^m)

+˜ckuk_Hψ,s+1(Rⁿ×Z^m)

=kfk_Hψ,s(Rⁿ×Z^m)+ (˜κ1+|λ|)kuk_Hψ,s(Rⁿ×Z^m)+ ˜ckuk_Hψ,s+1(Rⁿ×Z^m). Now, an application of Proposition 2.4.B yields the theorem.

3. CONSTRUCTING FELLER SEMIGROUPS

We want to apply the Hille-Yosida-Ray theorem to the operator

−q(x, D_x, D_k)in order to construct a Feller semigroup with state spaceRⁿ×Z^m. Recall that according to this theorem a densely dened operator A on the continuous functions vanishing at innity generates a positivity preserving, strongly continuous contraction semigroup, i.e. a Feller semigroup, if and only if it satises the positive maximum principle and λ−A has a dense range for some λ >0, compare with [2].

We can take the dense subset H^ψ,s(Rⁿ×Z^m) for s suciently large to dene −q(x, D_x, D_k). Implementing the positive maximum principle can be achieved by requiring that(ξ, ω)7→q(x;ξ, ω)has for everyx∈Rⁿ a symmetric Levy-Khinchine representation. Note that the case we are interested in will avoid the local parts of generators, however certain non-symmetric cases will work too, compare with [7].

The serious problem is to solve q(x, Dx, Dk)u+λu = f for a dense set of right-hand sides f in C∞(Rⁿ×Z^m). As in [6], see also [7], we suggest to use a variational approach combined with regularity considerations. For this we introduce on S(Rⁿ×Z^m)the sesquilinear form B by

(3.1) B(u, v) := (q(x, D_x, D_k)u, v)_L2(Rⁿ×Z^m). Clearly, B splits according to

(3.2) B(u, v) =B⁽¹⁾(u, v) +B⁽²⁾(u, v), where

(3.3) B⁽¹⁾(u, v) := (q₁(D_x, D_k)u, v)_L2(Rⁿ×Z^m)

and

(3.4) B⁽²⁾(u, v) := (q2(x, Dx, D_k)u, v)_L²_(Rⁿ_×Z^m₎.

A straightforward calculation using Plancherel's theorem yields Proposition 3.1. On S(Rⁿ×Z^m) it holds that

(3.5) |B⁽¹⁾(u, v)| ≤γ1kuk_Hψ,1(Rⁿ×Z^m)kvk_Hψ,1(Rⁿ×Z^m)

and

(3.6) B⁽¹⁾(u, u)≥γ₀kuk²_Hψ,1(Rⁿ×Z^m)− kuk²_L2(Rⁿ×Z^m), where γ0 and γ1 are as in A.1.

This implies

Corollary 3.2. The sesquilinear form B⁽¹⁾ has a continuous extension to H^ψ,1(Rⁿ×Z^m) and (3.5) and (3.6) holds for all u, v∈H^ψ,1(Rⁿ×Z^m). We denote this extension again by B⁽¹⁾.

Moreover, for B⁽²⁾ we have

Proposition 3.3. Assume A.2.M with M > n+ 2 then on S(Rⁿ×Z^m) we have the estimate

(3.7) |B⁽²⁾(u, v)| ≤γ˜_M X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk_Hψ,1(Rⁿ×Z^m)kvk_Hψ,1(Rⁿ×Z^m). Proof. We note that

|B⁽²⁾(u, v)|=|(q₂(x, Dx, D_k)u, v)_L²_(Rⁿ_×Z^m₎|

=

(2π)^−n2−m Z

Rⁿ

Z

T^m

Z

Rⁿ

ˆ

q₂(ξ−η, η, ω)˜u(η, ω)˜v(ξ, ω) dηdωdξ

≤γ˜M,n,m

X

|α|≤M

kϕ_αk_L1(Rⁿ)(2π)^−m Z

Rⁿ

Z

T^m

Z

Rⁿ

(1 +|ξ−η|²)^−M2 ×

×(1 +ψ(η))|˜u(η, ω)||˜v(ξ, ω)|dηdωdξ

≤γ_M^∗ X

|α|≤M

kϕ_αk_L1(Rⁿ)(2π)^−m Z

Tⁿ

Z

R^m

Z

Rⁿ

(1 +|ξ−η|²)^−M−12 ×

×(1 +ψ(η))¹²|˜u(η, ω)|(1 +ψ(ξ))¹²|˜v(ξ, ω)|dηdξdω

≤γ˜M

X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk_Hψ,1(Rⁿ×Z^m)kvk_Hψ,1(Rⁿ×Z^m), where we used A.2.M withM ≥n+ 2, and Peetre's inequality.

Corollary 3.4. The sesquilinear form B⁽²⁾ has a continuous extension to H^ψ,1(Rⁿ×Z^m) which we denote again byB⁽²⁾. For this extension estimate (3.7) holds.

Combining Corollary 3.2 and Corollary 3.4 we obtain

Theorem 3.5. Assume A.1 and A.2.M withM > n+2then the sesquilin- ear formB=B⁽¹⁾+B⁽²⁾ is dened onH^ψ,1(Rⁿ×Z^m)and foru, v∈H^ψ,1(Rⁿ× Z^m) we have

(3.8) |B(u, v)| ≤τkuk_Hψ,1(Rⁿ×Z^m)kvk_Hψ,1(Rⁿ×Z^m).

In addition, for γ˜_MP

|α|≤Mkϕ_αk_L1(Rⁿ) ≤ ^γ₂⁰ the following Garding in- equality holds

(3.9) |B(u, u)| ≥ReB(u, u)≥ γ0

2kuk²_Hψ,1(Rⁿ×Z^m)− kuk²_L2(Rⁿ×Z^m), with γ₀ >0 as in A.1.

Proof. From Corollaries 3.2 and 3.4 we deduce that B is dened on H^ψ,1(Rⁿ×Z^m) and it holds

|B(u, v)| ≤ |B⁽¹⁾(u, v)|+|B⁽²⁾(u, v)|

≤



γ₁+ ˜γ_M X

|α|≤M

kϕ_αk_L1(Rⁿ)



kuk_Hψ,1(Rⁿ×Z^m)kvk_Hψ,1(Rⁿ×Z^m). Moreover, we nd

|B(u, u)| ≥ReB(u, u)

≥B⁽¹⁾(u, u)− |B⁽²⁾(u, u)|

≥γ0kuk²_Hψ,1(Rⁿ×Z^m)− kuk²_L2(Rⁿ×Z^m)−γ˜M

X

|α|≤M

kϕ_αk_L1(Rⁿ)kuk²_Hψ,1(Rⁿ×Z^m)

≥γ₀kuk²_Hψ,1(Rⁿ×Z^m)− kuk²_L2(Rⁿ×Z^m)−γ0

2 kuk²_Hψ,1(Rⁿ×Z^m)

= γ₀

2kuk²_Hψ,1(

Rⁿ×Z^m)− kuk²_L2(Rⁿ×Z^m), implying (3.9).

We now introduce the idea of a variational solution to (3.10) q(x, Dx, Dk)u+λu=f.

Denition 3.6. Let λ ≥ 0. We call u ∈ H^ψ,1(Rⁿ×Z^m) a variational solution to the equation (3.10) if for all ϕ∈H^ψ,1(Rⁿ×Z^m) it holds

(3.11) Bλ(u, ϕ) :=B(u, ϕ) +λ(u, ϕ)_L²_(Rⁿ×Z^m)= (f, ϕ)_L²_(Rⁿ×Z^m). Theorem 3.7. Suppose that A.1 and A.2.M, M ≥n+ 2 hold. Then for everyλ≥1equation (3.10) has for every f ∈L²(Rⁿ×Z^m)a unique variational solution u∈H^ψ,1(Rⁿ×Z^m).

Proof. First we note that

|(f, ϕ)₀| ≤ kfk₀kϕk₀≤ kfk₀kϕk_Hψ,1(Rⁿ×Z^m),

wherek·k₀ denotes the norm on L²(Rⁿ×Z^m). Hence, everyf ∈L²(Rⁿ×Z^m) denes a continuous linear functional onH^ψ,1(Rⁿ×Z^m). Further, by Theorem 3.5 the sesquilinear formB_λ is continuous on H^ψ,1(Rⁿ×Z^m) and for λ≥1it also satises the estimate

(3.12) |B_λ(u, u)| ≥ReB_λ(u, u)≥ γ₀

2 kuk²_Hψ,1(Rⁿ×Z^m).