Subordination in the Sense of S. Bochner-An Approach ... · For continuous negative definite...

Math. Nachr. 178 (1996), 199-231

Subordination in the Sense of S. Bochner-An Approach through Pseudo Differential Operators

By NIELS JACOB of Munich and RENE L. SCHILLING of Erlangen

(Received December 2, 1994)

1. Introduction

In his monograph Harmonic Analysis and The Theory of Probability [3] BOCHNER showed that for the investigation of LQvy processes the Fourier transform is one of the most important tools. Let { X t } t > ~ - be a LQvy process taking values in R" and starting at 0 E Rn. Then we have

EO e-i(ttxt) = e-t$(E) (1.1) 0 with a continuous negative definite function $J : IR" -+ Q: . It is useful to rewrite (1.1) in terms of the underlying semigroup {pt}t>o - of sub- probability measures

thus establishing a one-one correspondence between $J and {pt}'>o. - By

the convolution semigroup {pt}t20 always defines a (strongly continuous) contraction semigroup of linear operators { T t } t > ~ - whose generator A has - in suitable function spaces - the representation

1991 Mathematics Subject Classification. Primary 47D07; Secondary 35805, 60535. Keywords and phmses. Subordination, pseudo differential operators, Markov semigroups

200 Math. Nachr. 178 (1996)

Conversely, starting with the generator A it is possible to construct { T t } t 2 ~ as well as {Xt}t20. These considerations show that the whole information on { X t } t > ~ , {Tt}t>o, or A is already contained in the function $J : R" -+ C. Now, the prominent rsle played by the Fourier transform becomes clear.

In his book BOCHNER also handles the question how to get new LBvy processes from a given one. An important possibility to do this is a procedure called subordination. This concept is most simply explained either on the level of the convolution semigroup {pt}t?o or on the level of the process {Xt)t>o itself. Suppose that { P ~ } ~ > o is another convolution semigroup of sub-probability measures on [0, 00). Then the vague integral

(1.5)

defines an new convolution semigroup of sub-probabilities on R" and, hence, both a

new operator semigroup { T i } and a new LBvy process { X f ) , 2 0 . The superscript t2o

f refers to the subordinator { p t } t > o - which is completely characterized by one (and only one) function,

where f : (0,oa) + IR is a Bernstein function. The semigroup (Tif} calculated as a Bochner integral

can be t>o

(1.7)

its generator being given by

A f u = - (27r)-"I2 ea('?c)f($(t)) G(() d( . (1.8) lRII The corresponding process { X [ } allows the following interpretation: the subordi-

nator {pt}t?o defines a LBvy process {Y,}t>o taking values in [0, oa), starting almost surely in 0 E R, and having almost surelyincreasing sample paths. Without loss of generality, we may assume that the processes { X t } t ~ o and {Y,}t>o - are stochastically independent. Rewriting (1.5)

(1.9) Po{w : X[(w) E B } = Po{w : XY,(~) (W) E B}, for all Bore1 sets B c R" ,

we may identify the processes X , f ( u ) = XY,(~ , (W) . The subordination of LBvy processes and the related potential theory is well studied.

Next to the book by BOCHNER [3] and the references given there we refer to [2, 7, 8). It is, for example, possible to introduce a symbolic calculus for A and Af and related operators, e. g., the resolvent Rx of A, see [2, 151 and other authors.

The concept of subordination is, however, not only restricted to semigroups arising from convolution semigroups, or - on the level of Markov processes - to (trans- lation invariant) Levy processes. Suppose that L(z , D) is a symmetric second order

t>o

Jacob/Schilling, Subordination and pseudo differential operators 201

elliptic differential operator generating a Feller (or a sub - Markovian) semigroup, that is, a strongly continuous semigroup of contraction operators Tt : Coo(lRn) + Coo(IRn), t 2 0, such that 0 5 u(z) 5 1 implies 0 5 Ttu(z) 5 1. (In case of a sub-Markovian semigroup one replaces Coo(lRn) by L2(Rn) with the inequalities holding almost everywhere only.) Again, we can define the semigroup {T!} by (1.7) and the subor- dinated process by

t2o

(1.10) P"(w : Xf(w) E B } = P"{w : Xy,(,,(w) E B } , for all Bore1 sets B c IR".

In this setting, it is, however, much more difficult to find the generator of Tif . Moreover, the corresponding process is, clearly, no L6vy process and its properties are difficult to study. Since the symbol L(z , () of L(x, D) and the Bernstein function f are the only known objects it should be possible to trace all interesting properties of

{ L o

{ T i ) , Af, and { X i } back to L(z,<) and f . t2o t>o

In this paper we want to examine Bochner's theory of subordination by using meth- ods from the theory of pseudo differential operators. Roughly speaking, our idea is to work on the level of the generator and to use a type of symbolic calculus in order to

get approximations for Af and { T t ) which will also lead to an approximation on the level of the process.

In some sense, the paper is introductory for we only consider one rather simple type of generators, a strongly elliptic symmetric differential operator

t >o

(1.11)

imposing rather restrictive assumptions on the coefficients akl and c. Although it would be sometimes possible to use some existing symbolic calculus, we strive to avoid using any of these calculi. This is for two reasons: firstly, we will work on IR" and not on a compact manifold without boundary, thus, estimates in Sobolev spaces H 8 ( R n ) , s 2 0, can only be obtained under restrictions on the coefficients, see, f.ex., [13], and functional calculi as were developed by SEELEY [27] and STRICHARTZ [28] cannot be adapted easily. Secondly, we will later extend our results to much more general operators, pseudo differential operators with non -classical symbols as considered in [19, 171. Therefore, in order to estimate the resolvent {Rx}x>o of A, we also avoid using a calculus for parameter dependent symbols as was developed by GRUBB [ll, 121 and others.

Let us briefly outline the plan of the paper. In Section 2 we recall some results on negative definite functions, Bernstein functions and Bochner's theory of subordination. In particular, we give a formula for the generator of a subordinate semigroup in terms of the corresponding Bernstein function, the generator of the original semigroup, and its resolvent. The third section is devoted to introducing the operator A = - L ( z , D ) (as generator of a semigroup) and a suitable class of function spaces to handle Af . We determine the domain of Af in terms of these function spaces. The pseudo differential operator p ( z , 0) with symbol f ( L ( x , I)) is introduced in Section 4 in order

202 Math. Nachr. 178 (1996)

to approximate Af. Since the symbol f ( L ( z , ( ) ) does, in general, not belong t o a classical symbol class, p ( z , 0) is already a non -classical pseudo differential operator. Using results from [20, 191 we prove that p ( z , D ) itself extends both to a generator of a Feller semigroup and to a generator of a sub-Markovian semigroup. In order to control Af - p ( z , D) we need estimates for the resolvent Rx of A. Again, we construct an approximation qx(z, 0) of Rx. Here, the operator qx(z, 0) is a pseudo differential operator with symbol ( L ( z , ( ) + A)-’. Our assumptions on a k l and c guarantee that qx(z,D) is continuous as an operator from H2(RIRn) onto L2(RIRn), and, for suitable values of s 2 0, even from HS+2(R”) into HS(lR”) . Furthermore, we prove that the operator Kx(z, D) := Lx(z , 0) o qx(z, D) -id satisfies certain estimates similar to the resolvent estimates, cf. Theorem 5.9 and Corollary 5.10 below. This leads to estimates for qx(z, D) - Rx; in particular, this operator is of order - 3.

In Section 6 we return to studying the operators Af and p(x, D). Using results from the previous sections, we show for a large class of Bernstein functions the following estimate (1.12) Ilf(Lb1 D))u - P(Z1 D)uIIo I cllullo

which, in turn, gives estimates for T,f-Tj2) and ?/-Tim). Here, { F / } sub- Markovian and { F / } sub - Markovian, resp., Fellerian semigroup generated by L(z , D), while

and { T,(m)}

- p ( z , D ) . In fact, we have for all u 6 L2(Rn)

denotes the

the Fellerian semigroup obtained by subordinating the t>o

t>o

{ T i 2 ) ) t t o are the sub - Markovian, resp., Fellerian semigroups generated by

t>o

(1.13)

and for u E H k ~ a 2 ( R n ) , k > n/2, k 6 IN,

(1.14) ~ ~ F / ~ - T ~ ~ ) ~ ~ ~ 00 5 c’tllullk,aa.

Of course, these inequalities can be interpreted as first (and rather crude) estimates for the transition probabilities for the corresponding Markov and Feller processes.

As already mentioned, the operator L ( z , D ) should be looked upon as a model operator. In forthcoming papers we will treat more general generators of Fellerian and sub - Markovian semigroups. There, the probabilistic consequences of our results and the relations to the work of BOULEAU [4] and HIRSCH [16] will be examined more carefully. Moreover, the case of an elliptic differential operator on a manifold without boundary will be handeled separately using the classical symbolic calculus.

2. Negative definite functions, Bernstein functions, and subordination in the sense of Bochner

In this section we discuss those parts of Bochner’s theory of subordination which will be used later on. Our main references are the books [3] by BOCHNER and [2] by BERG and FORST.


Definition 2.1. A function 11, : R" -+ G is called negative definite if for all m E IN and < I , . . . , Ern E R" the matrix (ll,(E~)+~-ll,(~-Ej))~~=l is positive Hermitian.

For continuous negative definite functions we have the LBvy - Khinchine representation

with a positive constant co 2 0, a vector C E R", a positive semi-definite quadratic form q( a ) on R", and a finite Borel measure v not charging the origin.

In this paper we will always consider real - valued continuous negative definite functions. Such a function, 11, : Rn -+ R say, satisfies

(2.2) 0 I 11,(0 I .$(1+ IEI2) and 11,(0 = 11,(- E ) and, for this reason, we denote a real-valued continuous negative definite function by a2( a ). In this case, the LBvy-Khinchine formula (2.1) formula becomes

a2(E) = c o + q ( O + (l-cos(x,~))--- v(dx) . I.I2 The next class of functions we need is the class of Bernstein functions.

Definition 2.2. An arbitrarily often differentiable function f : (0, GO) -+ IR is called Bernstein function if (2.4) f 2 0 and (-l)j-'f(j) 2 0

hold for all j E IN. By 03 we denote the set of all Bernstein functions and we set

The following analogue of the Lkvy -Khinchine formula holds for Bernstein functions

f(s) = c1 + c2s + lo (1 - e-")p(dr),

with constants c1, c2 2 0 and a Borel measure p on the open interval ( 0 , ~ ) satisfying Jd" 5 P(dX) < 0.

Remark 2.3. Using (2.5) it is easy to see that f has a holomorphic extension onto the right half-plane {z E C : Rez 3 0) which is continuous up to the boundary. When appropriate, we will use this extension without change of notation. The function +:IR-bR

+(s) := f ( 2 s ) , s E IR, is continuous and - comparing (2.5) with (2.1) - also negative definite. This shows the close relationship between continuous negative definite and Bernstein functions.

204 Math. Nachr. 178 (1996)

A crucial point in our considerations is the fact that the composition of a continuous negative definite function a2 : IR" IR with a Bernstein function f : [0, 00) + IR,

f o a 2 : IR" --t IR

is again a continuous negative definite function. Let a2 : IRn -+ IR be a continuous negative definite function and f : [O,m) -+ R

be a Bernstein function. With each of these functions we can associate convolution semigroups of sub-probability measures. In the first case, we have a convolution semigroup {pt}t>o - of symmetric measures on IR" related with a2 via

while f gives rise to a convolution semigroup { p t } t > o - of sub - probability measures on the open interval (0, m),

(2.7)

Combining (2.3), (2.5), (2.6), and (2.7) we find for the function f o a2

= c1 + COC2 + c2q(<) + (1 - cos (2, E ) ) - i- ''I2 uf ( d x ) , 1Xl2

where vf is given by the vague integral

The convolution semigroup referring to f o a2 can be explicitly calculated.

Proposition 2.4. ([2], p. 70.) Let a2 : IR" + IR and f : [O,m) -+ IR be as above. associated with f oa2 is given by the vague integral The convolution semigroup { p f )

t2o

(2.10) Ptf(d4 = Idrn Ps(dX) pt(ds)

where {p,},20 and {pt}t>o - are the semigroups attached to a2 and f , respectively.

Definition 2.5. The semigroup { p c } of sub-probability measures on IR" is t 2 o

called subordinate to { p t } t > ~ - with respect to f (or {pt}t>o). -

further results on a subclass of Bernstein functions. Later on, we will extend this notion of a subordinate semigroup. We need some


Definition 2.6. A Bernstein function f is said to be a complete Bernstein function if the representing measure p of (2.5) is absolutely continuous with respect to Lebesgue measure on [0, m), the density being given by

m(s) = re-sTp(dr), s 2 0 , I” (2.11)

where p is a measure on the open interval (0,oo) satisfying set of all complete Bernstein functions is denoted by C B 3 and we set

& p ( d r ) < 00. The

ca30 := { f E CBF: f(0) = 0 } .

Remark 2.7. Some authors call the functions in C B 3 also operator monotone for they are the only functions preserving positivity of the quadratic form associated with a Hilbert space operator, see LOWNER [22] and HEINZ [14].

For complete Bernstein functions f E CBF we have the following characterization:

Lemma 2.8. A function f belongs to CBF if and only if

is a Stieltjes transform, that is, i f

(2.12)

holds with c1,cz us in (2.5) and p as in (2.11).

A proof of this lemma and further characterizations of complete Bernstein functions f E CBF are given in [25], Theorem 5.6, see also [26] and the related paper [l].

Example 2.9. The following functions are in CBFo:

(2.13) sa =

(2.14) - -

(2.15) l o g ( l + s ) =

- S

S + X

X arctanz = (2.16)

Let us prove some auxiliary results for complete Bernstein functions.

206 Math. Nachr. 178 (1996)

Lemma 2.10. For all f E B 3 we have

(2.17)

I f f E C B 3 ,

(2.18)

holds for all k E IN and, in particular,

1 (2.19) I f ( " % ) ( I (k+l)!-&if(S), s > 0,

is valid for all k E NO.

Proof . By the inequality 1 + s 5 eTs or, equivalently, s e-" 5 1 - e-T8, we obtain by differentiating (2.5) under the integral sign

I" 00

sf'(s) = czs +

which proves (2.17).

s e - T S p ( d ~ ) 5 c1 + czs + (1 - e - T s ) p ( d T ) = f ( s )

If f E CBF, we obtain by (k + 1)-times differentiating (2.12)

which yields

and the lemma follows. 0

Remark 2.11. The function s H 1 - e-ps, p > 0, is a Bernstein function that is not completely Bernstein. A direct computation shows that (2.18) does not hold for this function.

Bernstein functions are known to be subadditive. In fact, we even have the following stronger property:

Lemma 2.12. I f f E B 3 , then


Proof. Since f 2 0 and f ' 2 0, the first inequality in (2.20) is obvious. Using f" 5 0 and (2.17) we find

f '(T) L f'(3) L 9 f ( s ) , o < s 5 r .

Integrating this, we get

lc3 f ' ( r )d r I 1'' f o d r S = (c- l ) f ( s ) ,

which proves the second inequality in (2.20). 0

Lemma 2.13. Let f E CBT and denote by p and p the representing measures from (2.5) and (2.11), respectively. Then we have for u > -1

(2.21)

if and only if

(2.22)

/ o l s U p ( d s ) < 0O

-p (dr ) < 00 . Ji" rk

Proof. Using (2.5) and (2.11) we find

= I" 1' s*r e-" ds p(dr)

s-'"-l ds. sin cm ra-l a

In example (2.13), we have p(dr) = - Hence, Lemma 2.13 yields for 0 5 u 5 1 that

dr and p ( d s ) = ?r r p a )

(2.23)

208 Math. Nachr. 178 (1996)

is finite if and only if u > a. Next, we will turn our attention to contraction semigroups on a Hilbert space. Let

{Tt}tlo be a semigroup of strongly continuous symmetric contractions on the Hilbert space 'H and denote by A its generator and by {Rx}x>o the corresponding resolvent. Clearly, A is a self adjoint nonpositive operator with dense domain D ( A ) , and Rx is given by Rx = (A - A)-1. As above, pick f E BFo with corresponding convolution semigroup {p t } t>o , - see (2.7). Since for all u E 3.1

111" T, .Pt (dS) l l I 1" IlTtll Pt (ds) 1141 I 1141

the following definition makes sense.

Definition 2.14. The semigroup Ttf defined on 'H by the Bochner integral { ) t>o

(2.24)

is called subordinate to {Tt}tlo with respect to f (or { p t } t > o ) .

It is well-known, see [7] or [24], that { Tf } is again a strongly continuous con-

traction semigroup of symmetric operators with self adjoint generator Af. In the sequel, we will often need the following (Bochner) integral representation of A f :

t>o

Theorem 2.15. Let f E CBF. For all u E D(A) we have u E D ( A f ) and

A f u = C ~ U + C ~ A U + RxAup(dA) 1" (2.25)

where c1, cg are the constants from (2.5) and p is the measure given by (2.11).

A proof of Theorem 2.15 is given in [25] Satz 5.10, Korollar 5.11, see also [26] and [l]. Later, we will use (2.25) only in the case c1 = cg = 0.

3. The operators L(x,D) and f(L(x,D))

Let L ( z , D) be the differential operator

where akl : IRn -+ IR, 1 I I c , 1 I n, are continuously differentiable functions such that

n akl (z) = alk (x) and


hold for dl x E IRn and < E Rn with suitable constants 0 < ic1 5 6 2 . Without loss of generality we may assume that i ~ 1 5 1. Moreover, we assume that

n

(3.3) @&=O k = l

holds for 1 5 1 5 n and that c : Rn --+ IR is a continuous and bounded function satisfying

(3.4) c(x) 2 vo > 0 .

For u, w E Coo0 (Rn) we find

implying that L(z ,D) extends to a self adjoint operator on L 2 ( R n ) with domain H 2 ( R n ) . Moreover, L(s , D) is positive, i. e.,

From (3.5) we deduce that for any A > - vo the operator

(3.6) Lx(z ,D) := L ( x , D ) +Aid

has a bounded inverse Rx, its resolvent at A. The resolvent maps L2(Rn) continuously and bijectively onto H 2 ( R n ) and (3.5) implies the resolvent estimate

(3.7) 1

IlRxll I vo+x where 11 is the operator norm of Rx (considered as a bounded operator on L 2 ( R " ) ) . Thus, we find that - L(x, D) generates a strongly continuous contraction semigroup

{ T i } t2o

(3.8) IlTtll I e-"Ot.

Since - L(z , D) is a Dirichlet operator in the sense of [5], the semigroup {Tt}tlo is sub-Markovian (in the sense of [9]) and the quadratic form

of operators on L2(IRn) which satisfy

defined on H ' ( R n ) is a Dirichlet form. Let f : [O,oo) + R be a complete Bernstein function in the sense of Definition 2.6

and assume that in its representation (2.5) c1 = c2 = 0 hold. By { T i } we denote

the semigroup obtained from {Tt}t?o by subordination with respect to f E CBFo. t>o

210 Math. Nachr. 178 (1996)

Clearly, {T!} is again a symmetric sub-Markovian semigroup on L2(Rn) . In

order to determine the domain of the generator Af of Tf we need some function

spaces. The function a2 : lR" + R,

t>o

{ Itbd

(3.10) a 2 ( 0 := f(lEI2) 1 5 E R",

is a continuous negative definite function and for s 2 0 the norm

(3.11)

turns the space (3.12) H s 1 a 2 ( R n ) := { u E L 2 ( R n ) : 1 1 ~ 1 1 ~ , ~ z < 00)

into a Hilbert space with inner product ( . , . ) s , a ~ and C,oO(R") as a dense subspace. Introducing the negative norms in the sense of Lax,

(3.13)

it follows by standard arguments that H'? (R")] * can be identified with the space [ (3.14) H - S , a2 (R") := { u E S'(R") : Ilull-s,aZ < m} .

In particular, for u E L2(lRn) we find

(3.15) I I u I I - s ,a2 2 = k,, (1 + a2( t ) ) -2S lWI2 4

with L2(Rn) being a dense subspace of H-s1a2(R"), s 2 0. In order to get some embedding theorems, we assume

(3.16) f ( A ) 2

to hold for some T > 0 and all A 2 p > 0. This implies

(3.17) 1 +a2(<) = 1 + f ( l E I 2 ) 2 c'(1 + 1512)T/2, 5 E Eln,

which leads to the continuous embeddings

(3.18) H s y a 2 (R") ~t H s r ( R n )

and - by Sobolev's embedding theorem - for ST > f

(3.19) H s 9 a 2 ( R " ) - Cm(Rn)

where Cm(lRn) denotes the space of all continuous functions vanishing at infinity. Further results on the anisotropic Sobolev spaces H s ~ " 2 ( R n ) , s E lR, can be found in [18] and [19].


The following result is taken from [25] Section 5.3, see also [26].

Theorem 3.1. By - f(L(z, D ) ) we denote the generator of the subordinate semi-

Y O U P {T i}& Then

(3.20) D(- f ( L ( z , D ) ) ) = HlQR")

is valid.

Remark 3.2. The notation f ( L ( z , D ) ) is formal. However, it is possible t o shaw (compare Theorem 2.15) that for u E D ( L ( z , D)) = H2(IRn) we have

(3.21) f(L(z, D))u(z ) =

where we used the representation (2.12) of f E CB.3. For a proof of (3.21) we refer to [l, 25, 261.

It should be noted that - L(z , D) also extends to a generator of a Feller semigroup on C,(IR").

Although we possess a handsome representation formula for f ( L ( z , D)), it is desir- able to have a more concrete formula for this operator. Consider, therefore, L(z , D) as pseudo differential operator. Its symbol is then determined by

cy) 00

L(z , D)Rxu(z) p(dX) = 1 RxL(z, D)u(z) p(dX) 0

4 L ( z , D ) ) ( z , t ) = L(z , t ) (3.22)

thus (3.23)

We are looking for a representation of f ( L ( z , D ) ) as a pseudo differential operator, i. e., we want to find a function : IR" x IR" --+ R such that

(3.24)

holds. Clearly, one would expect

(3.25) m,o = ei("'€)f(qz, D)) e - 4 " 7 € )

P ( Z , t ) := f(L(z75))

but this expression is not well-defined. Instead of (3.25) we will use the symbol

and the associated operator

212 Math. Nachr. 178 (1996)

In some sense, p(x, D ) turns out to be a good approximation of f ( L ( z , D ) ) . We can, however, prove a bit more, namely, that -p (x ,D) itself extends to a generator of a sub-Markovian and even Fellerian semigroup. This will be discussed in the next section.

4. The operator p(x,D)

Let L(z , D) and f E CBFo be as in Section 3. We define the symbol

p : l R n x I R n - lR

Lemma 4.1. For all < E IR" and x E R" the estimate

(4.3) IP2(Z,Ol I 4o(.)(f(lEl2) + 1)

holds where 90 is given by

Proof . Using the representation (2.12) for f we find by (3.2) and (3.4)

IP2(X, 01


M

- = 40(.)f(K.1I<l2 + vo) ,

where

1 n - (4.5) 4o(.) = -14.) - c(.o)l; ; l g ~ n l a k r ( 4 - akd.O>l * { i0 Since Bernstein functions are subadditive we find by Lemma 2.12, using ~1 5 1,

(4.6) f(.11tI2 + vo) I f(1tI2) + vo I m={l,vo)(l + f ( l < l 2 ) ) which gives (4.3). 0

In the sequel, we will always assume that the function $0 E L1(R") n Lm(IRn). Thus, we have

(4.7) IP2(.,t)l I 40(.) (f(lt12) + 1)

214 Math. Nachr. 178 (1996)

with do E L1(IRn) n L'(IRn). We need estimates like (4.7) for certain partial derivatives of pz(x,<).

Lemma 4.2. Let L(x, D ) and f be as above and suppose that a g a k c and agc belong to L1(IR") n Z'(lRn) for (I: E IN,", 0 < la1 5 q. Then there ezist functions da from L1(lRn) n LM(Rn) such that

(4.8) la:P2(Z,E>I I da(4(f(l<l2) + 1) 7 0 < I 4 5 4 ,

holds.

Proof. Since a # 0, we find

Using the formula for the (I:- th derivative of a composite function, P(f o g ) , where f : IR ---f IR and g : R" --+ R,

with the second sum running over all pairwise different multiindices 0 # /3, y, . . . , w in IN," and all 60, S,, . . . , 6, E IN such that 6,B+S,y+. . .+S,w = (Y and 6p+6,+. . .+S, = j hold (compare [lo] 0.430 for the one-dimensional case), the estimate (2.19), the ellipticity estimate (3.2), and (3.4), we find for 0 < (a1 5 q the desired inequalities

0 once we know that dgakl, LJgc E L1(IRn) n L'(IRn).

The next result will us enable to apply most of the estimates done in [19] to the operator p ( z , D). To do this, we need the following notations:

(4.10)

and for N E IN pick TN such that

p ^ 2 ( ~ , <) = (27r)-"I2 kn ei(z 'v)p2(x, [) dx

(4.11) ( I + lq12)N'2 5 T N 1$1. b l IN

Lemma 4.3. Let q E IN0 and dgakl, agc E L1(lRn) n Loo(IRn) for 0 4 1.1 5 q and suppose that (4.7), (4.8) hold. Then we have for all N E INo, N 5 q,

(4.12) p^2(V,<) 5 TJV IldallLl(1 + 1912)-N'2(f(1[12) 1) * bl<N


Proof . For Q E lNt, (a( 5 N , we have

This implies

Let p ( z , E ) be as above and suppose that &a,lq lI#aJIL1 is small compared with tcl for sufficiently large q E IN. Then it is shown in [19] that - p ( z , 0) extends to a generator of a Feller semigroup. For we will need further estimates of p ( z , D) and p ~ ( z , D) := p ( q D) + Aid we recall the results of [19] in greater detail. The crucial point to transfer the results in [19] is the validity of Lemma 4.3.

We assume that q > %+$+lo, that dguk~, 3gc E L1(IRn)nLW(IRn) for 0 < 1.1 5 q, and that $0 E L1(Rn) where $0 is given by (4.4). The number r 5 2 has already been introduced in (3.16). Since rq > f we get by Sobolev's embedding theorem the continuous inclusions

(4.13) Hqf', (Rn) c+ Hq*aZ (En) - CW(nt">

with u2(<) = f ( 1 < I 2 ) . By "/b we denote the constant

(4.14)

where 40 is given by (4.4) whereas the #a, 0 < la1 5 q, are taken from Lemma 4.2. Note that yb is finite since q > n + 1. The above mentioned smallness condition imposed on IJc#JaJ(Li can now be made explicit

(4.15)

It should be noted that all considerations remain valid if we define

(4.16)

216 Math. Nachr. 178 (1996)

and require 1 5 (1-c)E for some c e (

(1) (4.17) l-JZ;T

instead of (4.15). On CF(Rn) we consider the operator

(4.18) p(z, D)u(z) = (2r)-"l2

and the associated bilinear form B( , a ) given by

As usual, we set Bx(u, v) := B(u, w) + X(u, W)O for A E R. The following estimates are taken from [19]; the proofs rely mainly on Lemma 4.3.

Theorem 4.4. Suppose that q > n + for all u,w E C?(R") and 0 5 t 5 5 + 1

+ 10 and that (4.15) holds. Then we have

(4.20)

(4.21)

(4.22) llP(Zc, D)..Ilt,d 5 c3 Il4l t+l ,a2 ; (4.23) Il4lt+l,aZ L C4(IlP(.,D)4,.2 + Il.llo>.

IB(u7u)I L clIl4~,,2 II4l+,a2 ; B(%'IL) 2 EKl llulli,a* - (72 lluI120 ;

Remark 4.5. (i) We can enlarge the range of t in (4.22) and (4.23) if we enlarge q.

(ii) The conditions 1 - -& < E < 1 and q > n + + 10 are by no means sharp, see HOH [17]. Since we want to give a precise citation, we took the weaker, but exact, conditions from [19].

Theorem 4.4 allows us to extend B as a continuous bilinear form onto H * f (a") and to extend p ( z , D) as a continuous linear operator from Ift+'* (Rn) into Hi* (R") provided that t 5 $+1. The estimates (4.20) - (4.23) remain valid for these extensions and, in what follows, we will not distinguish between the original operators, B and p(s, D), and their extensions.

In the proof of Theorem 4.4 we use commutator estimates of the type

(4.24)

cf. [19] Theorem 2.1. Moreover, using the Friedrichs mollifier, Theorem 4.4 implies that, if (4.25) BX(%'U) = (f,4)O, 4 E C,-(R"),

Jacob/Schilling, Subordination and pseudo differential operators

holds for some f E Hfi+1*a2(R"), the left-hand side u is already of class HF+2iaa (IR,"). By an approximation argument, see [18] Theorem 9.3 or [21] Proposition 4.1, we deduce that -p(x, D) satisfies the positive maximum principle on H%+2ia2 (IR").

217

Theorem 4.6. Under the above assumptions, - p ( x , D ) , defined on ~ + + 2 , a' (R") C Cm(R") n L 2 ( R n ) , extends to (i) a generator of a Feller semi-

group {Tim'} and (ii) to a generator of a sub - Markovian sernagroup t>o

The domain of the generator of { T j 2 ) } i s given by H1>a2(Rn). t2o

Remark 4.7. (i) The proof that - p ( x , D) extends to a generator of a contractive sub - Markovian semigroup is given in [20].

(ii) The smallness condition (4.15) is due to the method, in particular to the fact that we want to have the estimates (4.20) - (4.23). Using a martingale problem approach HOH [17] succeded in constructing the Feller semigroup without a smallness condition like (4.15). He, however, could not derive (4.20) - (4.23) for the corresponding bilinear form and operator, respectively.

(iii) Note that (4.20) - (4.23) also imply that the sub-Markovian semigroup {St}t>o, obtained by extending { T,(OO) I c,nL2 } onto L2(Rn), coincides with { T,(2)} t to . In

particular, this yields Tt(m)u(x) = T,(2)u(x) almost everywhere for all u E H t ~ a a ( R " ) if Ht.a'(IRn) L--) Cm(Rn). The proof relies, essentialy, on the fact that H ~ + 2 ~ a a ( R n ) is an operator core for both the generator of {St}t?o and p x ( z , D) .

We have now two pairs of semigroups: On the one hand, the Feller semigroups { T,(m)}t>o and {?{},to generated by - p(x, 0) and - f(L(z, D)) (considered as

operators on Cm(R")), respectively. On the other hand, the sub- Markovian semigroups { T,(2)} t to and { generated by -p(x, D) and - f (L(x , 0)) (considered

as operators on L 2 ( R n ) ) , respectively.

t2o

Our aim is to compare {T,(2)}t20 and {T/}t20' {T,(m)}t>o and {F/}t201 and p ( z , 0) and f (L(z , D)). For this purpose we need an approximation of the resolvent {Rx)x2o of L ( x , D ) .

5 . A first -order approximation of Rx

Let L(z , 0) be as in Section 3 and A > - vo. We want to construct an approximation of the resolvent Rx of L(z , D) + X id. Since the symbol of L ( z , D) + X id is given by L(z , r ) + A, a natural candidate of such an approximation is the pseudo differential operator

218 Math. Nachr. 178 (1996)

whose symbol qx(z, <) is given by

(5.2)

Our first aim is to prove that for suitable s 2 0 the operator qA(z,D) maps H"(lRn) continuously into H"+2(lEtn). To this end we need some auxiliary considerations.

Fix a point zo E Rn and decompose qx(z,<)

where TA(Z, <) is given by

By we denote the function

By the assumptions made in Section 4, we have $0 E L1(Rn) n LO"(IRn).

Lemma 5.1. Let rx(z,<) and $o(z) be as above. Then we have

(5.7)

Proof . Without loss of generality we may assume that L(z , <) 5 L(z0, c ) and IEI I 1. By the definition of T X ( Z , <) we find

hence,


On the other hand,

yields

proving (5.7). The case L(z , ( ) 2 L(z0,t) can be handled in a similar fashion. 0

In order to get estimates for ~ $ T A ( ( z , ~ ) , 0 < I q, we observe that for Q # 0

(5.8) azaTX(z, t ) = a,*qx(z, t ) holds.

Lemma 5.2. For 0 < (a( 5 q we assume agakl, agc E L1(IRn) n Loo(IRn). Then there ezist functions E L1(Rn) n LCO(IRn) such that

(5-9)

holds.

Proof . In view of (5.8) it is enough to estimate

a axi Suppose that la\ = 1, i.e., that a,* = - for some 1 5 j 5 n. This gives

220 Math. Nachr. 178 (1996)

where we again used ~1 5 1. To prove the general estimate we can either proceed by induction with respect to la1 or use - as in the proof of Lemma 4.2 - formula

0 (4.9) for the partial derivatives of a composite function.

The next Lemma follows from Lemmas 5.1 and 5.2 by very much the same arguments as did Lemma 4.3 from 4.1 and 4.2.

Lemma 5.3. Let q E IN0 and dgakl, a;c E L1(Rn) n Lw(IRn) for 0 < la\ 5 q and suppose that $0 E L'(IRn), cf. (5.5) and (5.6). Then we have

The continuity of qx(x , D) is now easy to prove.

Theorem 5.4. Suppose that for q > n + 2 the assumptions of Lemma 5.3 are fulfilled. Then the operator qx(z, D), A 2 0, maps L2(Rn) continuously into H2(IRn) and we have the estimate (5.11) Ilqx(z, D)4lZ I c I l 4 l o where the constant c depends only on ~1~ uo and n.

Proof . We decompose qx(z, D) similarly to (5.3),

(5.12)

Clearly,

which proves (5.11) for the operator qx(z0,D). v E L 2 ( R n ) C IT2(Rn) we find

We now turn to rx(z ,D) . For

I ( r x ( ~ 7 ~ ) ~ l w ) o l


implying - L2(IRn) is dense in H - 2 ( R " ) - that ~ITx(I ,D)uI(z I cllullo and the theorem follows. 0

Corollary 5.5. Suppose that for q > n + 2 + s the assumptions of Lemma 5.3 are fulfilled. Then the operator qx(x , D ) satisfies

(5.13) l l Q X ( ~ , ~ ) 4 3 + 2 I cs l l ~ l l 3 . Proof . For qx(zo ,D) the above estimate is trivial. In order to estimate T X ( Z , D) we

proceed as in Theorem 5.4. If w E L2(Rn) C H-2-s (R"), we find

which gives (5.13) for T X ( Z , 0) and the corollary is proved. 0

Our next aim is to prove that the operator

222 Math. Nachr. 178 (1996)

is of order -1, that is, for all u E L2(RRn) we have

(5.15) IlK(x(z, m4I1 5 c Ilullo *

To do this, note that Kx(z , D) is a pseudo differential operator whose symbol Kx(s , <) is given by

= K:(& 0 + K 3 z , 0 + K?(z, <I . In order to estimate the Kf(x,D)u's, we need some analogues of Lemma 5.2 and

Lemma 5.3. For later use, we derive estimates of type

for certain values of u and r . By our assumptions we find for IT;($, <)

where a:'$,,,, a:'$k E L1(Rn) n Lm(Rn) for la1 5 q - 1. Clearly, (5.16) implies for some 4; E L1(IRn) n Loo(IRn)

(5.17)

Using the formula for derivatives of composite functions we find

(5.18)

with 4; E L'(Rn) fl Lm(Rn), la1 I q - 1. If 0 5 0 5 3, we may deduce

(5.19)


If q > n + 2, this proves the following proposition.

Proposition 5.6. Under the above assumptions we have for 0 5 u 5 3 C (5.20) llK:(., D)ulll-zu 5 (vo + A)" M o '

i. e., the operator Ki(x, 0) maps L2(Rn) continuously into H 1 - z u ( R n ) .

By the same reasoning as above, we find for K:(z, D)

with 4; E L1(lRn) n L"(Rn) and 0 5 u 5 1. This yields for q > n + 3 the following result:

Proposition 5.7. Under the above assumptions we have for 0 5 u 5 1

c (5.22) IP:h D) U112-Zu 5 + A). Ilullo 1

i. e., the operator K:(z, D ) maps L2(Rn) continuously into H2-2u(Rn).

Finally we get for 0 I u I 1

with 4; E L1(Rn) n L"(Rn). This shows for q > n + 2 the following proposition:

Proposition 5.8. Under the above assumptions we have for 0 5 u 5 1

C (5.24) I I ~ 3 ~ ~ ~ ) U l l Z - Z u 5 (vo + A). 11~110 '

i. e., the operator K i ( x , D) maps L2(Rn) continuously into H2-zu(IR.n).

Let us sum up our results in the following theorem.

Theorem 5.9. Let q > n + 3 and 0 5 u 5 4. Then there exists a constant c, not depending on A, such that

(5.25) llullo CU

holds. I n particular, taking u = 0, the operator Kx(x, D ) is of order -1, i. e., it maps L ~ ( I R ~ ) continuously into H1(R").

224 Math. Nachr. 178 (1996)

Corollary 5.10. For suficientfy large q f IN, 0 5 c 5 a and all s E IR we have

(5.26)

Proof. Combining (5.19), (5.21), and (5.23) we find

with $a E L1(IRn) n Lw(IRn) and Jcrl 5 q - 2. Mimicking the proof of Lemma 4.3 we find for v E L2(IRn)

I J J (1+1t-q0 -q'2(1 + lv12)u-1'2 M V ) l lS(t)I dr)cy (Yo + A)" R" R"

which implies (5.26).

In particular, for u = 5 and s > we get

(5.27)

By our construction, the pseudo differential operator Kx(z , D) with symbol (5.16) satisfies (5.14). Thus, under the above assumptions we can restate our results in the following way:

Theorem 5.11. There is a family of pseudo differential operators Kx(x, D ) , X 2 0, of order -1 satisfying (5.14), i. e., Kx(s ,D) LA(%, D ) o qx(z ,D) - id.

Since Lx(z, D) has for all A > - vo a bounded inverse Rx, the resolvent of L(z , D) at A, we know (5.28) Lx(z,D)Rx = id

while Theorem 5.11 implies

(5.29) Lx(s , D) qx(s , D) = id + KA(z, D) .


Combining (5.28) and (5.29) we find

(5.30) qx(z, D) - Rx = RxKx(z, D) which gives us various estimates for qx(z, 0) - Rx. By the resolvent estimate, we have

x > --Yo, (5.31) IlRxII I > 1

thus, by Theorem 5.9, for 0 I D 5 f II(qx(z,o) - R x ) 4 l o = IIRxKx(x, D b l l o

1 1 I- IlKx(x7 D)UllO I + l l 4 0 . -Yo + x

This implies for 0 5 CT 5 fr 1

(5.32) IIqx(x, 0) - Rxllo = IIRxKx(~1~)llo I + X)"+1 .

Moreover, qx(z ,D) and Rx map L2(R") continuously into H2(IRn), hence,

(5.33) 11(4x(z, D) - Rx)42 L c(-Yo, x)ll4lo. Under our smoothness assumptions on akl and c we know that

Rx : H1(Rn) + H3(Rn)

is also a bounded operator. Using (5.30) this gives

hence C'

(5.34) II(Qx(S, D) - Rx)uIIs I vo+x 11~110 . The last inequality has a nice interpretation: the operator qx(x, 0) approximates Rx with the difference qx(z, 0) - Rx being not only of order - 2 (as would have been expected) but even of order - 3.

We will use these estimates in the next section in order to compare the operators f (L ( z , 0)) and p(z , D) as well as the associated semigroups

and { F/ } tSo and { T , ( ~ ) } ~ ~ ~ , respectively.

6. A comparison of f(L(x,D)) and p(x,D) and of the related semigroups

We suppose that L(z , D) and f E CBFo are as in the above sections. In particular, f has the representation

S

226 Math. Nachr. 178 (1996)

with the measure p satisfying Jt p(dX) + JF continuous negative definite function + : Rn --t IR we have

p(dX) < 00, see (2.12). Thus, for any

In view of (6.3), we thus have for u E H2(Rn) r w


Since f(L(z, D)) and p ( z , D) are bounded operators from Hita2 (Rn) into L2(Rn) , the operator Q,

(6.5) Q4.I := lw XRxKx(z, D)u(z) p(dX) ,

maps H1i a2 (R") continuously into L2(R"). Under some additional assumptions, the operator Q behaves even better.

Theorem 6.1. Let L(x, D ) and f be as above and assume that for 0 5 a I a (6.6) Lrn $ P W < - * Then the operator Q : H2"- ' (Rn) -, L2(IRn) is bounded. In particular, setting a = k , we have

(6.7) Il(f(L(X7 0)) - PbC, D))uJlo I c I I 4 l o * Proof . Choosing s = 2a - 1 in Corollary 5.10 we find for u E L2(RRn)

= cll~l12"-1

and the assertion follows by a standard density argument. 17

Estimate (6.7) shows that the difference of the operators f(L(z, 0)) and p ( z , 0) is of zero order, i.e., the space L2(Rn) is mapped continuously into itself, while each single operator is only continuous from into L2(Rn) . Since the difference has better regularity properties we may speak of a lower order perturbation. In particular, we can improve the estimate

For any 6 > 0 we find

(6.9) Il(f(L(., D)) - P(., D ) ) 4 0 I 6 11~l11,a~ + 4 6 ) llullo Let us now recall PoincarC's inequality. For all u E H1(R") with support in a

compact set K C Rn

holds true with g(R) \ 0 as R \ 0. Thus, for all u E H*(Rn) having compact support with volume vol (suppu) 5 R, Theorem 6.1 implies

(6.10) llullo I dvol ( K ) ) lIuII1

(6.11)

228 Math. Nachr. 178 (1996)

Let us now apply these estimates to compare the semigroups T { t}t>o and {T;2)}t20* In the framework of semigroup theory (6.7) means that the semigroup { T,f } bounded pertubation of { $')}

is a

. Clearly, we can interchange the rdes played by t>n

t2o

_ - - - { T(}tzo and { T,(')} . Therefore, {Ti}

t>o t2o can be obtained from { Tt(2)}tto and

the operator Q (cf. (6.5)) using the Trotter product formula, see [6] p. 93,

(6.12)

where {St}t>O denotes the contraction semigroup { e-t*}t,o generated by the bounded operator - Q.

Furthermore, we can control the norm of T,fu - T,(2)u. Following [23] p. 79 we find for u E L2(IRn)

- -

5 ctllullo.

This proves the following theorem:

Theorem 6.2. Under the assumptions of Theorem 6.1 we have

(6.14)

From Theorem 6.2 (or (6.13)) we obtain estimates for the transition probabilities T t f l ~ ( s ) = ptf(z,B) and Tt(2)lg(z) = pi2'(r ,B) of the corresponding Markov processes

(6.15)

for every bounded Bore1 set B C IR". So far, we have only used (5.25). Using (5.26) and (5.27) we get different estimates.

Suppose that q is so large that (5.26) holds for s = k E 2No. For u E Hk(lR"), k E 2N0, we find because of llL(x, D ) U l l k - 2 - IIUllk - this is due to our smoothness assumptions on the coefficients akl and c-and [ ( L ( z , D ) ) ~ , R A ] = 0 that

1/2 ( Ln Iptf(r, B ) - p j 2 ) ( z , B)I2 d r ) 5 c t (vol (B))'l2

(6.16)

holds with a suitable constant c. By (6.16), (5.26) with D = 4, and choosing k > $ (in order to have Sobolev's embedding theorem at hand) we get


= C'llUllk

This is already the proof of the next theorem.

Theorem 6.3. Suppose that q as suficiently large and 2INo 3 k > 5, i. e., such that Hk(IRn) c Cw(IRn) holds. Then we have for all u E Hk(IRn)

(6.17) Ilf(L(., - P k D).llw I c IlulIk *

By Theorem 4.4 and the commutator estimate (4.24) we know that for any admissible k E IN0 the norms I I u ~ ~ ~ , ~ z and Il(p(z,D) +id)kuIIo are equivalent. Since p ( z , D ) generates { T,(2)} t20, we have ( p ( z , D ) + id), Ti')] = 0 which yields

(6.18)

If k > 2 + 2, we have Hk7a2( lRn) C Hfi+z*a2(IRn) C Coo(Rn), thus, cf. Remark 4.7(iii), for any u E H k i a Z ( R n ) and t 2 0

(6.19) T,(2)u(z) = T,(O0)u(z) almost everywhere.

Theorem 6.4. Suppose that q is large enough and k > 5 + 2. Then we have

(6.20)

for all u E ~ k t a ' ( I R ~ > .

Proof . Pick a u E HklaZ(IRn). Then we find from (6.18) that

T,(') E HkgaZ (IR") c C, (Rn) .

Now, Theorem 6.3 shows that ( f ( L ( z , D ) ) -p(z,D))TS(')u E Hk,a2 (IRn) and the same argument that gave (6.19) also proves !ffv(z) = T{v(z) almost everywhere for any v E H*+2ta2(IR"). Thus, for sufficiently large k,

Ttf(f(L(z, 0)) - P b , D>)T,". = * / ( f W , a) - P(2, D>)T,2'.

almost everywhere. Using the formula

230

the estimate

Math. Nachr. 178 (1996)

finally follows.

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