Pseudo-Differential Operators and Markov Processes

{x) < 1, such that supp 4> C G and 4>\K = 1. B. Let G C M" be an open set and let {Gj \ j G / } be a family of open sets Gj C R n such that G = U , G J Gj- Then there exists

18

Chapter 2 Essentials from Analysis

a family of functions {<j>j \j £ I}, called a partition of unity subordinated to the covering {Gj | j £ I}, such that j < 1, and for j £ I we have supp j C Gj. Moreover, the family {supp <j>j \ j £ I} is locally finite and Yjj£i4>j{x) = 1 for all x £ G. In addition, for any e > 0 there exists a function ip £ Cg°(R n ) such that 0 < ip < 1, diam(supp ip) < (1 + e)nll2 and 2 n 2 (^DkeZ", V'jbfa) = i> (x — k), is a partition of unity, i.e. J2keZ tp (x — k) = 1 for all x £ R™. It has the property that for m £ No given, there exists a constant M = M(n,m,e) such that \daipk(x)\ < e~^M for all \a\ < m, and there exists a number N = N(e, n) such that the intersection of N + 1 of the supports of the functions ipk is empty. C. For any £ > 0 there exists 4>e £ C°°(R) such that -e < (f>E(x) < 1 + e for all x £ R, 4>e\[o,i] = 1 and whenever x < y it follows that 0 < (j)e{y) — 4>e(x) < y — x. A proof of part A and B is given in [237], Corollary 1.3.2 and Theorem 1.5.2, respectively. A proof of part C is indicated in [102], Problem 1.2.1. In some situations we have to work with lower semicontinuous or upper semicontinuous functions. A function f : G —>• (—00,00], G C R n open, is said to be lower semicontinuous at XQ £ G if f(a;o) < 00, then for every e > 0 there exists a neighbourhood U{XQ) C G such t h a t i(x) > / ( z o ) — £ for all x £ U{XQ), and if f(a;o) = 00, then for every M > 0 there exists a neighbourhood U{XQ) C G such t h a t i(x) > M for all x £ U(x0). If f is lower semicontinuous at all points x £ G, then it is called lower semicontinuous on G. T h e set of all lower semicontinuous functions on G is denoted by %i.s.c{G). We call a function g : G —>• [—00, 00) upper semicontinuous if —g : G —> (—00, 00] is lower semicontinuous. The set of all upper semicontinuous functions on G is denoted by rHu.s.c.{G). P r o p o s i t i o n 2 . 1 . 4 A. Leti,g £ 'Hi.s.c.{G) and a > 0. Thena-i, f+g andiAg are lower semicontinuous too. B. For any non-empty set % c 'Hi.s.c. {G) the function defined by u(x) := sup {f(a;) | / £ V.} is also lower semicontinuous. C. Let u : G —¥ [0, 00) be a non-negative lower semicontinuous function. Then we have u(x) := sup {i(x)

I f £ C 0 (R") andO < i(x) < u(x) } .

D. Let u £ T-li.s.c.{G). Then there exists a sequence ({V)V£H of functions f„ : G —> R such that f„ < f„+i for all u G N and VL{X) = lim iu{x) — supf„(a:).

continuous

2.1

19

Calculus Results

E. Let u £ V.i.s.c.{G) and K C G a compact set. such that u(x0) = inf u(x). xeK

Then there exists XQ £ K

A proof of Proposition 2.1.4 is given (partly) in [126], p.89, or in [8], § 3. D e f i n i t i o n 2 . 1 . 5 A family {ij : G -» [—00,00] \j £ 1} is called filtering increasing if for j \ , J2 £ I there exists J3 £ I such that ij1 < f,3 and f,-2 < f,-3. T h e o r e m 2.1.6 Let H C 'Hi.s.c.{G) be a filtering increasing family such that f(a;) ;= sup{g(a;) | g £ T-L} is upper semicontinuous. Then f is continuous and f is approximated uniformly on compact sets K C G by the family H, i.e. for every e > 0 and every compact set K C G there exists g£ £ % such that 0 < f(ir) - ge(a;) < s for all x £ K. A proof of Theorem 2.1.6, which is often called Dini's theorem, is given in [8], § 4. Let G C R n be an open set. We call u : G —> C Holder continuous exponent A £ (0,1] if sup K

g

) - y i

<

C

< oo

for all x,y £ G. For A = 1, Holder continuous functions are Lipschitz ous, i.e. \u(x) - u(y)\ < c\x - y\

with

(2.39)

continu-

(2.40)

holds for all x,y £ G. T h e space C m ' A ( G ) , m £ N 0 and 0 < A < 1, consists of all u £ Cm(G) such t h a t da(u) is Holder continuous with exponent A for all a, \a\ = m. Finally, we have a look at the divergence theorem. Let G C R n be an open set such t h a t dG is a smooth oriented manifold. Further, for u £ Cm(G) denote by C m ( G ) the subspace of all functions with the property t h a t for | a | < m every function da(u) : G —> C has an unique continuous extension onto G. Let F : G ->• R n , F = ( F i , . . . , F n ) and Fj £ C ^ G ) , be a vector field on G c R". T h e n the divergence theorem says I divFdx= JG

I JdG

F-ncr(dz),

(2.41)

20

Chapter 2 Essentials from Analysis

where ft = ( n i , . . . , nn) is the outer normal to dG, and a is the surface measure on dG. In particular, for u, v G C 1 (G) we have a formula for integration by parts: / u - — d a ; = / u • v • n 7 cr(da;) — / v-—da;. 9XJ J9G JG dxj

(2.42)

JG

Suppose that G C 1 " is an arbitrary open set and let u, v G C0X}(G). We may extend both functions to R n by defining them to be zero in Gc. Then u, v £ Cg°(E") and supp u, supp v are compact sets contained in G. Let Bfl(O) C R" be a ball such that supp u, supp v C B fl (0). We find u,v G C°°(B fl (0)) and u| 9 B f l ( 0 ) = v| aBn (o) = 0. Thus (2.42) yields

r

dv

r

JG

9XJ

JBR(0)

/ u^— da; = /

2.2

dv

u - — dx = dxj

r JBR{0)

du

v - — dx = dxj

r £>u JG

v—— da;. dxj

Some Topology

We collect some more or less well known results from topology, mainly first results on topological vector spaces. As our major reference for these topics we mention the monograph [255] of W. Rudin, where those parts of topology needed by a working analysist is presented in a clear and straightforward way. We assume the reader to be familiar with basic concepts such as the definition of a topology, the notion of open, closed and compact sets and that of a base for a topology, as well as the notion of a neighbourhood etc. Moreover, we take for granted that the reader is familiar with notions of convergence (sequences and nets), separating axioms, weaker and stronger topologies, etc. By definition a topological vector space (X, O) is a vector space X (over M. or C) such that the vector space addition and the scalar multiplication are continuous with respect to the topology O and the product topologies, respectively. If B0 is a base (of neighbourhoods) of 0 G X, then Bx := {x + B0 \ Bo £ Bo} is a base (of neighbourhoods) of x £ X, and the topology of a topological vector space is completely determined by a base Bo of 0 G X. A set G C X is called convex if for all A £ [0,1] and all x, y G G it follows that Aa; + (1 — X)y G G. We call G an absorbing set, if for any x G X there exists A > 0 such that x G AG := {y G X \y = Xzandz G G}. The set G is a balanced set whenever x G G and |A| < 1 implies Aa; G G. A bounded set G C X

2.2 Some Topology

21

in a topological vector space is a set where to every neighbourhood V of 0 £ X exists A > 0 such that G C XV for some A > 0. Finally, we call (X, O) a locally convex topological vector space, if there exists a base of 0 £ X consisting of convex, balanced and absorbing sets. Locally convex topological vector spaces are characterised by families of separating seminorms. By definition a family P of seminorms on a vector space X is called separating, if for any x £ X \ {0} there exists p £ P such that p(x) ^ 0. Proposition 2.2.1 Let P be a separating family of seminorms on the vector space X and B'0 the family of subsets Up>£ C X defined for s > 0 and p £ P by UPt£ := {x £ X \p(x) < £ } . Then the set Bo of all finite intersections of elements ofB'0 form a base ofO £ X which turns X into a locally convex topological (Hausdorff) space. Conversely, let X be a locally convex topological vector space and for a convex, balanced and absorbing set U C X define the Minkowski functional Pu(x) : = i n f { A > 0 \x€ AC/}. Then P\j is a seminorm and the family of all these seminorms generates the topology on X in the sense described in the first part of this proposition. A linear operator A : X —» Y mapping a topological vector space X into another topological vector space Y is continuous if and only if it is continuous in 0 £ X. Moreover, when the topologies in X and Y are generated by families Px and Py of separating seminorms, then A is continuous if and only if for any q £ Py there exist finitely many seminorms pj £ Px, 1 < j < I = l(q), such that q(Ax) < c max Pj(x)

(2-43)

holds. In particular, a linear functional u : X —¥ C is continuous if and only if there exist finitely many seminorms pj £ Px, 1 < j < I = 2(u), such that |u(a;)| < c max Pj(x)

(2.44)

holds. Let q be a seminorm on X. We call q continuous with respect to the family Px if (2.43) holds with A = id^- Suppose that Pi and P 2 are

22

Chapter 2 Essentials from Analysis

two separating families of seminomas each turning the vector space X into a locally convex topological vector space. These topologies on X are equivalent if and only if every seminorm in Pi is continuous with respect to Pi and every seminorm of P\ is continuous with respect to Pi. In this situation we call Pi equivalent to P2. For a locally convex topological vector space we denote by X* the space of all continuous linear functionals u : X —¥ C, i.e. X* is the dual space of X. Let X be a locally convex topological vector space. The weak topology on X is the weakest topology on X which makes every u. £ X* continuous. Note that every weak neighbourhood of 0 € X contains a neighbourhood of the type {x e X I \UJ(X)\ < £j for 1 < j < m}

(2.45)

where u, € X* and £j > 0. Next we want to introduce a suitable topology on X*. For this let x € X be given. We can define a linear functional lx on X* by lx(u) := u(x) for all u<EX*. Denote the set of all these linear functionals on X* by X. The weakest topology on X* which makes all elements of X continuous is called the weak-*-topology on X*. Every weak-*-neighbourhood of 0 S X* contains a neighbourhood of the form {u € X* I | u ( ^ ) | < £j, for 1 < j < m}

(2.46)

where Xj G X and £j > 0. For the weak-*-topology we have the following compactness result. Theorem 2.2.2 (Banach—Alaoglu) Let X be a topological vector space and U(0) C X a neighbourhood ofOeX. Then the set K:={ueX*

I [u(sc)| < I for allx £ U(0) }

(2.47)

is weak-*-compact. For a separable space X, i.e. there is a countable dense set in X, we have more results. Theorem 2.2.3 A. Let X be a separable topological vector space and let K C X* be a weak-*-compact set. Then there exists a metric on K turning K into

2.3

23

Measure Theory and Integration

a metric space with metric topology equivalent to the weak-*-topology (on K). B. Let U(0) C X be a neighbourhood ofO£X,X being a separable topological vector space. Further let (u„)„gN be a sequence in X* such that |u„(:r)| < 1 for all x £ U(0) and i / £ N . Then there exists a subsequence (u„fc)fc6N of (u„)i,gN and u £ X* such that for all x £ X we have lim ul/k(x) = u(x). k—>oo

R e m a r k 2 . 2 . 4 In general we call a topology metrizable, if there exists a metric generating an equivalent topology. A locally convex topological vector space which is metrizable such that the metric space is complete is called a Frechet space. W h e n working with mappings f : X —>• Y from a topological vector space into another, it is often sufficient to restrict the attention to sequentially continuous mappings, i.e. mappings with the property t h a t xv —>• x in X implies t h a t f(a;„) ->• f(a:) in Y.

2.3

Measure Theory and Integration

We assume the reader to be familiar with the basic notions and results of measure theory and the theory of integration including the notion of properties which do hold almost everywhere (a.e. as abbreviation), the completion of a afield, central theorems like the Caratheodory extension theorem, the dominated and monotone convergence theorems, the theorems of Tonelli and Fubini, the Radon-Nikodym theorem, etc. For these topics and for the following results which we state without proofs and without reference, our standard references are the book [18] of H. Bauer, the beautiful work [217] of P. Malliavin, and the monograph [256] of W. Rudin. Let Q be a set and S C V(£l). We call S a n-system in fl, whenever A,B € S implies A D B e S. A 7r-system is called a d-system in fi, if ft G S, A,B £ S and A C B then B\A £ S, and for any sequence (j4„)„ e N, Au £ S such t h a t A„ C Av+\ it follows t h a t (Ji/gN ^ 6 &. T h e o r e m 2.3.1 ( M o n o t o n e class t h e o r e m ) Suppose that S is Then it follows that d ( 5 ) = a(S).

air-system.

Here d(S) denotes the d-system generated by S and cr(S) is the cr-field generated by S.

24

Chapter 2 Essentials from Analysis

C o r o l l a r y 2.3.2 Let ft be a set and S a •n-system in Q. Further let rl be a vector space of real-valued functions f : fi —»• R such that 1 £ %, XA € H for all A € S, and for any sequence (fv)v€^, f„ G H, such that 0 < iv < f„+i and f = supf„ < oo it follows that f € T-L. Then "H contains all real-valued bounded functions

which are measurable with respect to o~(S).

For a locally compact space G its B orel-o-field is denoted by B(G), but for G = R™ we write B^n\ Note t h a t B^ is already generated by the bounded open sets in R™. T h e Lebesgue measure on R n is denoted by \(n\dx). By definition measures are non-negative, and measures on B(G), G being a locally compact space, are called Borel measures. Let fi be a Borel measure. We call H locally finite if fJ-(K) < oo for any compact set K C G. A Borel measure is inner regular if for any B € B{G) we have / i ( 5 ) = sup {n(K)

| K C B and K c o m p a c t } ,

and \i is outer regular if for any B € B{G) it follows t h a t H(B) = inf {fi(U)

| B C Uand Uopen} .

A Borel measure fj, is called a Radon measure if it is locally finite and inner regular. On fi(") Radon and Borel measures coincide. T h e support supp fi of a Radon measure /x is the complement of the largest open set G such t h a t /i(G) - 0. Let (fi, .4) be an arbitrary measurable space. T h e total mass of a measure H on (Q,,A) is denoted by ||/x|| := //(fi), and .MjJ~(fi) is the set of all bounded measures on (fi,-4), i.e. fi € A4jj"(fi) implies ||/x|| < oo. By Ml(fl) we denote the set of all probability measures, i.e. measures fi with total mass ||/z|| = 1. A sub-probability measure is an element of .Mjj"(fi) such t h a t ||/x|| < 1. For a G 0 the Z>irac measure at a is denoted by e a , but in case fi = R n and a = 0 we often write £ instead of £Q. Now let G be a compact space. A signed measure on (G, B(G)) is a mapping /i : B(G) —> R such t h a t we can write y,{A) = Hi(A) — fJ>2(A) for two measures /xi,/X2 G A^j"(G). Whenever p is a signed measure there exist two measures Hi and y% in M£(G) such t h a t fi = (1° — (i% where /i° and /i° are mutually singular, i.e. there exists a Borel set A0 C G such t h a t

A(A°) = /i?(G) and /^(^°) = 0.

(2.48)

2.3

25

Measure Theory and Integration

Let (fi, A, //) be a measure space. We call AG A a fi-atom if fi(A) > 0 and for any B C A, B e A, either //(£?) — 0 or //(A \ 5 ) = 0 holds. In particular we have / / ( J B ) = 0 or n(B)

=

p(A).

T h e total mass of // is given by ||//|| := //°(G) + //°(G). T h e set of all signed measures is denoted by M.(G). Let G be a locally compact space. A signed Radon measure // is given by two mutually singular Borel measures //J and / J ° . W i t h |//| := /i° + ^2 w e consider the completion of 23(G) with respect to |//| and denote this cr-field by B^G). For A G B^{G) such t h a t |//|(A) < oo we define //(A) = //?(A) — n\(A). T h e bounded signed Radon measures on G are denoted by M.b(G). It follows t h a t (Mb, ||-||) is a Banach space. T h e o r e m 2.3.3 The Banach space (Mb(Rn), ||.||) is the dual space of n Coo(]R ;]R). For /J, G ^Vl6(]Rn) a linear continuous functional on Coo(]R™;R) is defined by UH

/" u(a:)/x(da;) = /" u(a:) ^°(da;) 7R" 7R"

/

u(x) (4(dx).

(2.49)

JU"

This theorem is also related to the various variants of Riesz' representation theorem. A linear functional I on C ( G ; R ) or any of its subspaces, G being a locally compact space, is called a positive functional if l(u) > 0 for all u > 0, u € C(G; R) (or in case where I is defined only on a subspace of C(G; R), u > 0 should be taken only from this subspace). T h e o r e m 2 . 3 . 4 A. Let G be a metrizable locally compact space which is countable at infinity. Then I is a positive linear functional on Co (G; R) if and only if there exists a unique locally finite Borel measure fj, such that Z(u) = / u{x) //(da;)

(2.50)

JG

for all u € Co(G;R). B. In case that G is compact, I is a positive linear functional on C ( G ; R ) if and only if (2.50) holds with a unique measure // £ Ai^(G) and all u e C(G; R ) . C. A linear functional I on Coo(R n ; R) is positive if and only if there exists a unique Borel measure // on R™ such that (2.50) holds for all u G Coo(R n ;R). Moreover, for all u G C 0 0 ( R " ; R ) it follows that JRn u(x) //(da;) < oo. We may also extend the considerations to complex-valued measures. This is possible in two equivalent ways. Either we define a complex-valued measure

26

Chapter 2 Essentials from Analysis

with the help of two pairs (M?, M°) and (^°, fi°) of mutually singular measures by (i(A) = fi{A) - &{A) + i(£(A) -

fi(A))

(2.51)

whenever (2.51) makes sense, or we introduce complex-valued measures as the dual space of Co(G), now we consider of course complex-valued functions. T h e total mass of M given by (2.51) is now defined by ||/i|| := Y^=i M?(^)> a n d IMI is the measure $^,- =1 M?- We denote the space of all complex-valued measures by MC(G); M.f(G) is the set of all bounded complex-valued measures. Let ( f i i , - 4 i , / / i ) and (0,2, .4.2, M2) be two measure spaces. T h e product measure of MI and M2 is denoted by Mi ® M2- We introduce the mapping Ak : Rnk ->• R n , (xi,... ,Xk) i-+ xi + ... + Xk and give D e f i n i t i o n 2.3.5 Let Mj G At+(R™), 1 < j < k, be measures. The image of Mi ® • • • ® Mfc under Ak is called the convolution of these measures and is denoted by Mi * . ..*/zjb := Ak(fJ*i <8>...® Mfc)-

(2.52)

Obviously we have Mi * . . . *Mfc G A^j"(R n ) and ||MI * • • • *Mfc|| = ||MI|| • . . . • ||/ifc||. T h e convolution is associative and commutative, i.e. (Mi * fj,2) * M3 = Mi * (A*2 * M3) and Mi * M2 = M2 * Mi

(2.53)

for all Mj ^ A^j"(R"). For any non-negative measurable function on Rn we have /

i(x) (Mi *M2)(dz) =

JR"

/

/

f(a; + y)Mi(da;)M2(dy)

,/R" ./R"

= 1 1

i(x + y)^(dy)tx1(dx)

(2.54)

which yields in particular (Mi*M2)(5)=/

Mi(5-y)M2(d2/)= /

M2(S-z)Mi(dz)

(2.55)

n

for all i? G B( \ Using (2.51) we may extend the convolution to .Mf,(R n ) as well as to M%(Rn), and (2.53) remains valid for this extension. Moreover we have Mi * (M2+M3) = Mi * M 2 + M i *M3

(2.56)

2.3

27

Measure Theory and Integration

and «(Mi * M2) = {afJ-i) * M2 = Mi * (<*M2), a > 0, € R or € C depending on whether Hj € M^(Rn), respectively. In each case we have fj,*e

(2-57) Mb{Rn)

= £*H = H

or

Mf(Rn),

(2.58)

Thus it follows t h a t _Mb(R n ) and Mf(Rn) are algebras with respect to the vector space operations and the convolution as product. Moreover, £ is a unit in these algebras. For integrable functions f, g : R n ->• C or R, iX^ and gA^n) are elements in M%(M.n) or .Mi,(R n ), respectively. Thus we can consider the convolution iX^ * gA< n \ and we find f A ^ * gA
i(x-y)g(y)dy

(2.59)

denotes the convolution of the two functions f and g. Next we want t o introduce some topologies on M.£(G) etc. Let G b e a locally compact space. Further let (fJ.l/)l/^ be a sequence in A1b(G) and /io €: Aib(G). We say t h a t (/x„)i,gN converges in norm to /io if and only if lim ^ - / i o H = 0 .

(2.60)

is—too

T h e sequence (/x^)j/gN converges weakly to / J 0 if for all u 6 Cb(G;R) we have lim

/ u(a:) /i„(da:) = / u(x) fio(dx).

j/->oo JG

(2-61)

JG

Whenever (2.61) holds only for all u € Coo(G;R), the sequence (nu)ueN is said to converge with respect to Coo to /xo- But if (2.61) holds only for all u G Co(G;M), we say t h a t (/x„)„gN converges vaguely to /J,Q. Obviously, we have the implications norm convergence => weak convergence => convergence with respect to C^o => vague convergence. Moreover, the notion of vague convergence is also well defined on M(G). O n M.b{G) we have

28

Chapter 2 Essentials from Analysis

T h e o r e m 2.3.6 A sequence (/xI/)^eN» Vv G M-biG), converges weakly if and only if there exists a constant c > 0 and a dense set D C C ^ (G, R) such that ||/z„|| < c and JGu(x) \iv(da;) converges for all u € D . Moreover, (iit/)i/€N converges weakly if and only if it converges with respect to C ^ and for every £ > 0 there exists a compact set K C G such that \^iv\{Kc) < e for all i / £ N , v sufficiently large. In addition we have T h e o r e m 2 . 3 . 7 Suppose that (/i„)„eN, Vv G M^(G), converges vaguely to H € M^(G) and that lim ^(G) = fi(G). Then (£t„)„gN converges weakly to v —>oo /X.

C o r o l l a r y 2.3.8 A sequence of probability measures on G converges weakly if and only if it converges vaguely. R e m a r k 2 . 3 . 9 The topology referring to the weak convergence of measures often called the Bernoulli topology.

is

A subset M C A4~£(G) is said to be vaguely bounded if sup

[

u(x) (i(dx)

< oo

(2.62)

JG

for a l l u e C 0 ( G ; ] T h e o r e m 2 . 3 . 1 0 A subset of M~£{G) is relative vaguely compact if and only if it is vaguely bounded. Furthermore we have T h e o r e m 2 . 3 . 1 1 For any /i € A ^ R " ) there exists a sequence of finite linear combinations of Dirac measures converging weakly, hence also with respect to CQO and vaguely, to (i. Since we may consider the space M%{G) as the complexification of M.b{G), we may define the notions of convergence introduced above also on Mf(G). In particular, for families {r]t)t>o, Vt S M^(G), we may define ^lim-fo+,-77,) dt

S_K)

S

(2.63)

2.3

29

Measure Theory and Integration

in these topologies. For example for the vague convergence (2.63) does mean t h a t for a l l u G C 0 (G) lim - ( / u(x) rjt+s(dx) «->o s \JG

-

/ u(x) JG

r]t{dx)

exists and leads to a measure fi, defined by / u(x) fi(dx) = lim - ( / JG

u ( i ) Vt+s{dx) -

s->0 S \JG

[ u(x) r,t(dx)) Jo

(2.64) J

for all u G C 0 ( G ) . Let (Cl, A) be a measurable space and f, g : Cl —)• R be measurable functions. We write (f A g)(x) := min(i(x),g(x))

and (f V g)(x) := max(f(x),

g(x)).

(2.65)

Both functions f Ag and f Vg are measurable too. Thus we may define f+ := f VO as the positive part of f and f ~ := — (f A 0) as the negative part of f. We always have t h a t f+, f_ > 0 and f + (a:) • i~(x) = 0 for x € 0 . Moreover it follows t h a t f=f+-f"

and | f | = f + + r .

(2.66)

Note however t h a t f+ and f - do not have disjoint supports since they coincide on the set {x £ fi | f(a;) = 0 } . In case t h a t fl = G is a topological space and A = B{G) is the Borel cr-field, the set of all measurable functions f: G —>• C is denoted by B(G), and Bf,{G) is the set of all bounded functions f S B(G). When we want to emphasise t h a t we have to work with real-valued functions we write f G B(G;W) or f G Bb(G;R), respectively. For any measure space (CI, A,n) the spaces Lp(Cl,n), 1 < p < oo, are the usual Lebesgue spaces (of equivalence classes) of measurable functions f : £~i —>C with finite norm

Hf|| L P :=^JfW| P Md^))

P

,l
(2.67)

These spaces are Banach spaces and L2(Cl,[i) is a Hilbert space. Often we write ||.||o for | | . | | L 2 and (., .)o for the corresponding scalar product. T h e norm ||f|| L oc : = e s s . s u p | u ( a ; ) |

(2.68)

30

Chapter 2 Essentials from Analysis

is often denoted by ||.||oo- F ° r a finite measure space, i.e. ^i(fi) < oo, it follows t h a t L 9 (fi, n) C L p (fi, fi) provided q > p, and we have the estimate

||u||Lp
(2.69)

for all u G L 9 (fi,/i). For an unbounded measure space, i.e. /i(fi) = oo, there is in general no relation between L 9 (fi,/x) and L p (fi,/i) for p ^ q in the sense t h a t there are elements in L p (fi,/x) not belonging to L 9 (fi,/x) and there are also elements in L 9 (fi,/z) not belonging to L p (fi,/z). For (SI,A,(J.) = (Rn,B(n\\W) we write L p ( R n ) , 1 < p < oo, for p n L (R ,A(™)) and for a Borel subset G C R " we write L P (G) instead of L p ( G , A ( " ) | G ) . As usual we write L p (ft,/x;R) or L P ( G ; R ) if we want to point out t h a t we are working with real-valued functions. L e m m a 2 . 3 . 1 2 For 1 < qi < qi < oo it follows L9l(Rn)nL92(Rn) c

p|

that

Lr(R").

(2.70)

qi
T h e space Ljoc(G), G C R™ open, consists of all measurable functions u : G —> C which are locally integrable, i.e. for any VL G L X (G), or equivalently for any compact set K C G we have xKn € L1(G). P 1 For any p, 1 < p < oo, we have L (G) C L, oc (G). Holder's inequality for the spaces Lp(fl,fJ,) says

/

u(x)v(x)

fi(dx) < | | u | | L p | | v | | L p ( , K p < o o ,

(2.71)

for u G L p (fl, /x), v G L P (Q, fi) and - + h = 1- Furthermore we have L e m m a 2 . 3 . 1 3 Assume that JQU(X)Y(X) /j,(dx) > 0 for some u G Lp(Q,fi) p and all v G L (fi, fi), - + ^r = 1, and v > 0 a.e. Then it follows that u > 0 a.e. Another important inequality is Minkowski's

integral

inequality:

L e m m a 2 . 3 . 1 4 Let f : R n x R™ —> C be measurable and 1 < p < oo. we have

f \i(.,y)\dy

< f

\\f(.,y)\\hPdy.

Then

(2.72)

2.3

31

Measure Theory and Integration

A first application of (2.72) yields part A of the next lemma, which is often called Young's inequality. L e m m a 2 . 3 . 1 5 A. For u G L p ( R n ) , 1 < p < oo, and v G L ^ R " ) the convolution (u*v)(x) =

u(x-y)v(y)dy Jm.™

(2.73)

defines an element in L p ( R n ) such that l|u*v||LP<||u||LP||v||L1.

(2.74)

B. For u G L p ( R n ) and v G L P ' ( R " ) , ± + ± = 1 and I < p < oo, it follows that u * v G C&(R n ) and

l|u*v|| 0 o <|H| L l ) ||v|| L P /.

(2.75)

R e m a r k 2.3.16 The proof of part B of Lemma 2.3.15 uses the following tinuity result lim||f(. + y ) - f ( . ) | | L P = 0

con-

(2.76)

y-»0

for all f G L p ( R n ) and 1 < p < oo. For 1 < p < oo the space Cg°(R") is dense in L p ( R n ) . Moreover, there is an important technique to smooth out elements in L P ( R " ) , namely Friedrichs mollifier. Denote by j : R n —> R the function

j(*)=={^ ( ( N a - i ) _ 1 ) '!:!;;. where c^ 1 = Jlxl
e

- l ) " 1 ) da:. For e > 0 set j E (a;) : = e - " j ( f ) . It

G Cg°(R"), je(x) > 0, supp j

J e ( u ) : = (j e * u ) ( i ) = / is called the Friedrichs

^ e

= Be(0) and / R „ j e ( i ) da; = 1.

j e ( i - y)u(y) dy

mollifier.

(2.78)

32

Chapter 2 Essentials from Analysis

P r o p o s i t i o n 2 . 3 . 1 7 For any u G L P ( R " ) , 1 < p < oo, we have J e ( u ) G C°°(R") n L p ( R " ) anrf J £ : L p ( R n ) -> L P (R") is a Ziraear operator. Moreover we have IIWHLP

< IHIL?

(2.79)

and lim || J e ( u ) - u|| L P = 0.

(2.80)

£->0

In case p = 2 we also have (JE(u),v)0 = (u,Js(v))0 for all u, v G L 2 ( R n ) , i.e. J £ is a bounded selfadjoint Moreover, for u G Co(R n ) we have lim/

jv(x - y)u(y) dy = u(x) =

*-e- (JTJ)TJ>O converges vaguely to EQ and (^(x

operator on L 2 ( R n ) .

u(y) ex(dy),

(2.81)

— .))v>o converges vaguely to ex.

Note t h a t the approximation result holds also true for u € Coo(R n ). Moreover, the special type of j e is not necessary if only the convergence is of interest. For any <j> € L 1 ( R n ) such t h a t / R „ <j>(x) dx = 1 let us define <j>e(x) = e~n ( | ) . Then it follows t h a t ||u * <j>E - u|| L P -> 0 as e ->• 0 for all u G L p ( R n ) , 1 < p < oo. Once again the result remains true for u G C 0 0 ( R n ) . A proof for this is given in [289]. Now let u, v G L P ( R " ) , 1 < p < oo, be functions such t h a t u > 0 a.e. and 0 < v < 1 a.e., respectively. It follows t h a t J £ (u) > 0 and 0 < J £ (v) < 1 hold. Furthermore, since |((0 V u{x)) A 1) - ((0 V v(x)) A 1)| < |u(x) - v(a;)| a.e.

(2.82)

holds for all u,v G L p ( R n ) , we find t h a t a sequence ( u „ ) „ e N , uv G Cg°(R n ), converging in L p ( R n ) to u G L P ( R " ) , has the property t h a t ((0 V u„) A 1 ) „ 6 N converges in L p ( R n ) to (0 V u) A 1 and ( 0 V u „ ) M £ C 0 ( R n ) . T h e next theorem characterises the conditionally compact subsets in L P (G). T h e o r e m 2.3.18 A subset K C L P ( G ) , G c l " a Borel set, 1 < p < oo, is conditionally compact if and only if the following three conditions hold

33

2.3 Measure Theory and Integration 1. sup ||u||LP < c; 2. lim sup ||u(. + h) - u ( . ) | | L P = 0; |h|->0u€if

3. lim sup

G\BR(0)U

= 0.

X

LP

iVoie rAai for a bounded Borel set G C R™ t/ie Zas£ condition is empty. Let g : G —• R be a strictly positive measurable function and consider the measure fx := gA(")|G on G. The space L P (G, /x) consists of all measurable functions u such that u • g G L P (G), or equivalently, for any v G L P (G) the function ^ • v belongs to L P (G, fi). Moreover, we have ||u|| LP , G •. = ||ug|| LP , G j. Thus a set K C L P (G, fj,) is conditionally compact if and only if the set Kg := {v = <7-u | u £ i Y } i s conditionally compact in L P (G). Let fi be a, bounded measure on B^n\ For any u G L 1 (E n ) we may define the integral Ju(x — y) fi(dy) and it follows that /

u(x-

y) (j,(dy) dx <

/

|u(a; - y)\ dx fi{dy),

thus we find that x (->• Ju(x — y) n{dy) defines an element in L 1 (R n ). More generally, we may introduce the notion of a kernel. Definition 2.3.19 Let (SI, A) and (SI', A') be two measurable spaces. We call K : SI x A' ->• [0, oo] a kernel from (SI, A) to (SV, A') if x i-> K(x, A') is for all A' € A' a measurable function and A' H-> K(X, A') is for all x G SI a measure. A kernel K is called a Markovian kernel ifK(x,Sl') — 1 for all x G SI, and it is called a sub-Markovian kernel ifK(x, SI') < 1 for all x G SI. Given a kernel K from (SI, A) to (SI', A'). We may define an operator K op on all non-negative measurable functions u : SI' —> K by (K op u)(x) := J u(x')K(x,dx').

(2.83)

It follows that K op u is a non-negative measurable function on SI. Suppose that Kop is an operator from all non-negative measurable functions u : ST.' —• R to the space of all measurable functions v : SI —> R having the representation (2.83). Then the right hand side in (2.83) is called the kernel representation of K op .

34

Chapter 2 Essentials from Analysis

T h e o r e m 2 . 3 . 2 0 Suppose thatK.op is an operator from all non-negative measurable functions u : Cl' —>• R to all measurable functions v : Cl —> R. Assume further that Kop has the property that u > 0 implies K o p u > 0, and that for every sequence (u„)j,gN of non-negative measurable functions \xv : f2' —> R such that 0 < u„ < u „ + i it follows that supKopU^ = K o p ( s u p u t / ) . Then there exists a kernel K /rom (fi,»4) io (Cl',A') Kopu(z) = /

such that Kop has the kernel

representation

u{x')K(x,dx')

for all measurable functions

(2.84)

u : $7' —>• R, u > 0.

Clearly, we may extend K o p in the usual way to suitable integrable functions. In particular, if all the measures (K(a;, .)) x en are bounded, K o p is well defined on the space of all bounded, real-valued measurable functions u : Q' —¥ R and K o p u is a bounded measurable function on fi. Suppose t h a t N : G" x B(G) -> [0,oo], G,G' C R n Borel sets, has the property t h a t for every x € G' the mapping N(x,.) : 13(G) —• [0, oo] is a measure. Still it is possible to define the operator N op u(a;) := /

u(y)N(x,dy).

JG

P r o p o s i t i o n 2 . 3 . 2 1 Suppose that for all u € C Q ° ( G ) the

mapping

a; M- / u(y) N(z, dy) is continuous. Then x \-t N(a;, A) is measurable for every A S B(G), N(:r, dy) is a kernel.

i.e.

This proposition is proved by approximating the functions x A , ^ € B(G), pointwise by suitable sequences of functions belonging to Co°(G). Suppose t h a t Ki is a kernel from (£li,Ai) to ( ^ 2 , ^ 2 ) and K2 is a kernel from (^2,^.2) to (^3,^.3). Suppose further t h a t the measures (Ki(a;i, . ) ) I i e n i as well as (K 2 (a;2, -))x 2 en2 a r e bounded. Then the operators K„ p , K„ p and K ^ o K ^ are defined by K^p := ( K i ) o p and K ^ := ( K 2 ) o p , respectively, acting on bounded measurable functions, and K^ p o K ^ is just the usual composition of operators. For a bounded, measurable function u : $^3 —>• R we find Kopu(x1):=(K2opoKfp)u(x1)=

[ [ Jn2 Jn3

u(x3)K2(x2,dx3)K1(x1,dx2).

2.3

35

Measure Theory and Integration

On the other hand, by Theorem 2.3.20 the operator Kop has a kernel representation with a kernel K3i(a:i, dx^) which yields K31(Xl,A3)=

f

K2(x2,A3)K1(x,dx2)

(2.85)

for all x\ S fli and A3 e .43. In particular, when a family (Kt)t>o of kernels from (R™, Bn) into itself is given, we may consider the kernels K,it{x,A)=

[

Ks(y,A)Kt(x,dy).

(2.86)

Often we have to use the following two results. L e m m a 2 . 3 . 2 2 Let E be a metric space and (£l,A,n) be a measure space. Further suppose that u : ExQ, —» R is a function with the following properties: u> i-t \i(x,u>) is for all x G E integrable with respect to fi, x i-> U(X,OJ) is continuous at XQ £ E for all ui £ ft, and there exists a function h £ L 1 (r2,/n) such that |f(a;,o;)| < h(u>) for all x £ E and w £ fi. Then the function x i->JQ{(x,u>) fi(dui) is continuous at XQ. L e m m a 2 . 3 . 2 3 Let I c R be a non-empty interval and (Cl,A,fJ,) a measure space. Assume that u : I x fi —>• R has the following properties: u(x,.) £ L 1 (fl,/i) for all x € 1, for all ui £ Q, the function x H> U(X, UI) is differentiate on I, and there exists a function h £ L 1 (f2,//) such that \-^u(x,u>)\ < h(w) for all x £ I and u> £ fi. Then the function x i-» Jn u(x,u>) /i(du>) is differentiate, the function to i-> -^n{x,ijj) is an element in L 1 (fi!,/i) for all x £ 1, and — I u{x,u)

fi{du) = I -^(x,u>)

Ai(dw).

(2.87)

We need various results on the integration of functions with values in vector o

spaces. Let I C R be a closed interval such t h a t 1 ^ 0 . Further let (X, \\.\\x) be a Banach space. We denote by C(I; X) the space of all continuous functions u : I —¥ X, and C 1 (I; X) is the space of all these functions which are continuously o

differentiable in I and have onesided derivatives at the endpoints of I. For any compact interval [a, b], a ^ b, we may define the Riemann integral f u(t) dt for u £ C([a, b]; X), and then, as usual, we may define the improper Riemann integrals /»oo

/ Ja

pb

u(t)dt,

/ J— 00

/-oo

u{t)dt

and

/ J — 00

u(t) dt

36

Chapter 2 Essentials from Analysis

for suitable functions u : I —• X, I = [a, oo), I = (—oo, 6] or (—oo, oo), respectively. The following result is taken from the book [88], p.9, of St.Ethier and Th. Kurtz. Lemma 2.3.24 A. Let u G C(l;X) such that Jj ||u(t)|| x dt < oo. Then the integral J. u(£) di exists and we have the estimate

fu(t)dt

< [\\u(t)\\xdt.

(2.88)

In particular, for every compact interval I, every u G C(I;X) is integrable. B. For u G C1([a, b]; X) we have

jT (j|U(i))di = u(6)-u(a).

(2.89)

C. Let (A, D(A)) be a closed operator on X and u G C(I;X) such that u(£) G D(A) for all t G I, Au G C(I; X) and u as well as Au are integrable (over I). It follows that J, u(t) dt G D(A) and we have k(f

u(t) dt j = f Au(t) dt.

(2.90)

Another way for denning an integral for functions u : I —> X is due to S. Bochner [37]. We follow the book [315] of K. Yosida. Let {£l,A,n) be a measure space and u : S l - ^ I a finitely valued function such that ' k

u(u,) = ^ x n »

(2.91)

for Xj G X and Clj C 0, //($!,•) < oo. We call functions of type (2.91) fi-Xelementary functions. The //-integral of u is now denned by

J

A;

u(w) /i(dw) := J2 xolJL<Sli) G X1

( 2 - 92 )

3= 1

Definition 2.3.25 A function u : Cl —• X is said to be Bochner-/z-integrable if there exists a sequence (u„)„gN of Q-X -elementary functions converging li-almost everywhere, such that lim f ||u„(w)-ii(w)|| x /i(da;) = 0.

(2.93)

2.4 Convexity

37

The Bochner integral has many properties analogous to the Lebesgue integral. For elements u G C ( I ; l ) , l c l a c o m p a c t interval, the Riemann integral coincides with the Bochner integral, thus in Lemma 2.3.24 we could have also worked with the Bochner integral. Finally, we mention how to define an integral for functions u : Q —> X where (Q, A, (J,) is a measure space and X is a topological vector space. For details we refer to W. Rudin's monograph [255]. We suppose that X* separates points in X, i.e. for x, y G X, x =£ y, there exists u G X* such that u(a;) ^ u(y). Now let f: 0 —• X be a function and for u G X* define the function u(f) : Q, —> C (or R) by u(f)(w) — u(f(w)). Suppose that u(f) is /Li-integrable. If there is some x G X such that u(x) = f u(f)(w) /z(du;)

(2.94)

Jn for all u G X*, we define the /x-integral of f by x=

I fd/i.

(2.95)

Jn A criterion for the existence of the /x-integral (2.95) is Theorem 2.3.26 Let X and X* be as above and for a compact Hausdorff space G let (G,B(G),fi) be a probability space, i.e. {G,B{G),JJL) is a measure space and fJ-(G) = 1. If i : G —>• X is continuous and if the convex hull of f(G) C X has a compact closure in X, then the ^-integral x = JQidfi exists.

2.4

Convexity

We need only a few notions and results from convexity theory, but also some concrete applications. Our standard references are the lecture notes [17] of H. Bauer, G. Choquet's lectures on analysis [57]-[59], once again W. Rudin's monograph [255], and for Choquet theory we refer to the book of R. Phelps [238]. Let X be a vector space and K C X. The set K is said to be convex if for all x,y G K and A G [0,1] we have Aa; + (1 - X)y £ K. The intersection of arbitrarily many convex sets is convex again, hence for any set H C X we may define its convex hull conv(H) as the smallest convex set containing H, i.e. conv(tf) = p | {K D H I K convex}.

(2.96)

38

Chapter 2 Essentials from Analysis

Let K C X be a non-empty convex set. A point XQ € K is called an extreme point oi K ii K \ {x0} is convex too. Equivalently, xo is an extreme point whenever xo = Xx + (1 — A)y for some x,y & K and A € (0,1) it follows t h a t XQ = x = y. T h e set of all extreme points of K is denoted by ext(K). A subset C c X i s called a cone with vertex at 0 G X , if for all A > 0 it follows t h a t AC C C. We call C a peafced cone if C n ( - C ) = {0}. Further, C is said to be a cone with base if there exists a hyperplane H in X such t h a t 0 g H and for every x G C \ {0} the intersection of {Az | A > 0 } and i7 is non-empty. T h e set B := C C\ H is called a 6ase of C. A cone is said to be convex if it is convex as a subset of X. It follows t h a t a cone C is convex if and only i f C + C c C . A central question is whether we may describe a convex set by using only its extreme points. An important first result is the theorem of M. Krein and P. Milman. T h e o r e m 2.4.1 (Krein—Milman) Every compact convex set K ^ 0 in a locally convex topological vector space is the closure of the convex hull of its extreme points, i.e. K = conv(ext(K)).

(2.97)

Suppose t h a t X is a locally convex topological vector space with dual space X*. Further, let K C X be a non-empty compact set and [i a probability measure on K, i.e. a Radon measure on the Borel sets of K such t h a t fx(K) = 1. We call x € X a barycentre of /U if u(x) = /

u(y) /i(dy)

(2.98)

JK

for all u e l ' . T h e o r e m 2.4.2 ( C h o q u e t ) Let K C X be a non-empty compact convex set and suppose that the induced topology on K is metrizable. Then for every x G K there exists a probability measure ^ on K with barycentre x such that s u p p / i C ext(K). Choquet's theorem may be used to obtain concrete representation results. Let i f be a compact Hausdorff space. Consider a linear subspace H C C(K; 1R) such t h a t 1 G H and denote by H* the dual of H (with the weak-*-topology).

39

2.4 Convexity The set A(H):={leH*

\l(l) = l=\\l\\}

(2.99)

is a compact convex set in H* and every functional lx(i) :— i(x), x G K and i £ H, belongs to A(H). We call H point separating if for all £1,2:2 G K, xi ^ X2, there exists f G H such that i(x\) ^ f(x2). The set dHK := {x G K I lx G ext{k{H)) }

(2.100)

is called the Choquet boundary of iJ. Theorem 2.4.3 Suppose that H is point separating. A point x belongs to djjK if and only if /x = ex is the only probability measure with i(x) = f f(y) M(dy)

(2.101)

JK

for all { € H. Theorem 2.4.4 Let H be a point separating subspace of C(K;W) such that 1 G H. For any I G H* exists a measure fi on K such that l{() = f t(x) n(dx)

(2.102)

JK

for all f G H. Moreover, fJ.(G) = 0 for any Baire set G such that GtldtfK

= 0.

Recall that the Baire sets form the smallest cr-field in K such that all elements f G C(K;W) are measurable. In presenting Theorem 2.4.2-Theorem 2.4.4 we used the monograph [276], but note that there are no proofs for these results in [276]. Proofs were given in [238]. Finally in this section we want to discuss shortly Hausdorff's moment problem. Let (ci/)„gN0 be a sequence of real numbers and put Jc„ := c„ + i — cv. We call the sequence (C 1/ )^ 6 N 0 completely monotone if and only if {-l)k6kcv

>0

for all k,v G N 0 .

(2.103)

40

Chapter 2 Essentials from Analysis

Theorem 2.4.5 (Hausdorff) A sequence {cv)v&io *s completely monotone if and only if there exists a measure on [0,1] such that cu = f tv n(dt), v e No, Jo

(2.104)

holds. A proof of Theorem 2.4.5 is given in [313], pp. 148-154.

2.5

Analytic Functions

The reader is of course assumed to have some knowledge in the theory of complex-valued functions of one complex variable. As a standard reference we mention the monograph [2] of L. Ahlfors which is to our opinion still one of the best books in the field. Furthermore, we would like to mention R. Burckel's book [51] which is ideal in both precision and historical scholarship. From the theory of complex-valued functions of one variable we quote only the theorem of G. Herglotz, see [122], pp.508-511, and Hadarmard's threeline-theorem, see [51], p.147. Theorem 2.5.1 A. (Herglotz) Denote the unit disk in C by D and let f : D —> C be a function. This function is analytic with Re f > 0 if and only if it has the representation

f(z) =ic+ T ^±^

M(dC),

z € D,

(2.105)

where /i is a finite measure on (—n, w] andc = lmf(0) € R. The representation is unique. B. (Hadamard) Let Cl :— {x + iy | 0 < x < 1, y G R} and Cl its closure. Further let f be a bounded continuous function on fi which is analytic in Cl. Then the function M 7 := sup {|f(7 + ij/)| | y e R } satisfies M1 < M^1'Ml for 0 < 7 < 1. Here we want to discuss shortly some basic results from the theory of complex-valued functions of several complex variables and the theory of functions of one variable with values in some topological vector spaces. As standard references for several complex variable theory we mention L. Hormander's book

2.5

Analytic Functions

41

[151] or the book of St. Krantz [189]. A good reference for functions of one complex variable with values in a topological vector space is W. Rudin's book [255], we refer also to the monograph [179] of T. Kato. Let G C C™ be an open set and f : G - > C a continuous function. We call f analytic (or holomorphic) in G if f is analytic in each of its variables, i.e. for any point a = (a\,..., an) £ G the functions Zj H> gj(zj)

:= f(ai,...,

a,j + zj:...,

an), l<j
(2.106)

are analytic in a neighbourhood of 0 S C. An equivalent condition is t h a t for 1 < 3' < n the Cauchy-Riemann equation dZj f = 0 holds in G. Note t h a t for z = x + iy, x,y £ R™, we have d

3

•=-d-+i dxj

— dyj

(2.107)

Further helpful notations are R e z = x, Imz = y and |z| — (\zi\2 + .. .+\zn\2)1^2, a 1 as well as z = z" • ... • z%" for z e C a n d a e Nft. As in the theory of functions of one complex variable we have several uniqueness results. For example we have for an entire function f, i.e. analytic functions f : C™ -> C, the following result. L e m m a 2.5.2 Suppose that the entire analytic function on M.n, i.e. f|mn = 0. Then f is identically zero in C™.

f: C™ —> C

vanishes

From Lemma 2.5.2 it follows in particular t h a t for a function f : M" —> C which can be extended analytically to a function f: C n —• C this extension is unique. Suppose t h a t f : G -> C, G C C open, is analytic and let gj be defined as in (2.106). In this case we may of course use the theory of complex curve integrals for g^ in any open domain where g^ is defined. For example for a simply closed J o r d a n curve 7j lying in the intersection of G with a complex plane Cjta := {z £ C " \z = (ai,...,a,j

+Zj,...,

an)

with a £ G}

we find *

f(z) dzj = 0

(2.108)

42

Chapter 2 Essentials froin Analysis

In fact it is possible to characterise the analyticity of f by conditions on curve integrals as (2.108). Namely one has to require that for all 1 < j < n and all simply closed Jordan curves 7 j , constructed as above, (2.108) holds. Of course, we may restrict our considerations to closed paths which are by definition piecewise continuously differentiable curves in the plane which are closed. As usual we define the index of z £ C with respect to a closed p a t h 7, z & 7, by . . If dw ind 7 (z) . v ; := — - / 2-KiLw-z

v(2.109)

'

Now let us t u r n to functions of one complex variable into a topological vector space. D e f i n i t i o n 2.5.3 Let G C C he an open set and X a topological vector space over the field C. A. We call f : G —• X weakly analytic in G if for all u € X* the function u(f) : G —• C, u(f)(z) = u(f(z)), is analytic. B. The function f is said to he strongly analytic in G if lim

f(w)

w^z

~ f(*} w —z

(2.110)

exists in the topology of X for every z G G. Clearly, every strongly analytic function is also weakly analytic. Moreover we have T h e o r e m 2.5.4 Let G C C he an open set and X a Frechet space over the field C. Further let f : G —> X be a weakly analytic function. Then the following assertions hold: 1. f is a strongly analytic

function;

2. if j is a closed path in G, such that ind 7 (w) — 0 for every w 0 G,

then

we have f[(z)dz

= 0,

(2.111)

and if z £ G and ind 7 (z) = 1

f(z) = J_ I l^L dw. v ;

2niJ7w-

z

(2.112)

2.6

43

Functions and Distributions

Obviously, formula (2.111) is Cauchy's theorem and formula (2.112) is Cauchy's integral formula in this new situation. From (2.111) it follows t h a t if 71 and 72 are two closed paths in G such t h a t for all w £ G we have i n d 7 l ( w ) = ind 7 2 (w), it follows t h a t f f(z) dz=

f

f(z) dz.

(2.113)

Moreover, the function f is strongly continuous.

2.6

Functions and Distributions

T h e purpose of this section is to collect various material for functions and distributions as it is needed in later chapters. In principle it should be possible to find these results in L.Hormander's monograph [152]. As further standard reference we mention F. Friedlander [97], W. Rudin [255], and C. Zuily [317]. In our presentation we often used the book [165], First let us introduce some spaces of functions and their topologies. T h e Schwartz space S ( R " ) consists of all functions u € C°°(R n ) such t h a t for all m i , m2 € No R » l i m j ( u ) := sup ((1 + \x\2)m^2 I 6 R

"

V

\dau(x)\)

< 00

(2.114)

\<*\<m2

T h e family ( p m i , m 2 ) m i , m 2 e N 0 forms a family of separating seminorms. family (pa,/3)Q,/3€Nj given by PcAu)

:

=

SU

P \x0daVL(x)\

The

(2.115)

x€R"

is equivalent to the family (Pm1,m2)m1,m2€No a n d <S(Rn) equipped with the topology generated by one of these families is a Frechet space. Suppose t h a t u , v £ S(Rn), then it follows ueP|Lp(Rn),

u - v e < S ( R n ) and u*v<E<S(R").

(2.116)

P>I

Moreover, for any function £ C°°(R n ) which together with all its partial derivatives is polynomially bounded, i.e. \da(f)(x)\ < pa(x) for all a £ NQ and with suitable polynomial pa, the function 4> • u belongs to <S(R"). For <S(R") we have the following density results.

44

Chapter 2 Essentials from Analysis

C o r o l l a r y 2 . 6 . 1 A. The space Cg°(R n ) is dense in <S(R n ). S(Rn) is dense in L p ( R n ) for p > 1 and in C 0 o ( R " ) .

B. The space

Moreover, using the Friedrichs mollifier we have L e m m a 2.6.2 For any u G L p ( R n ) , u > 0 a.e., there exists a (u„)t,gN, u„ G <S(Rn) and u„ > 0, such that lim ||u„ — u|| L P = 0.

sequence

V—XX)

n

n

Let T : R ->• R be a differentiable mapping and u G <S(R n ). In general u o T does not belong to <S(R"). It is sufficient to take T(a;) = CQ € R n for a fixed value XQ such t h a t 4>(CQ) ^ 0. Then u o T is a non-zero constant which does not belong to <S(R n ). However, if T is a diffeomorphism such t h a t all its derivatives are bounded, then u o T £ <S(R n ). Of course, u o T is the pullback of the function u G <S(R") under the mapping T and often it is denoted by T*u := u o T. In particular, for any bijective linear mapping T : R™ —>• R n we have u o T G <S(R") for all u G <S(R"). We introduced already the space Co°(R"), and more generally the space C Q ° ( G ) , G C K " open. Defining a topology on Co°(G) is more complicated. First we give a topology on C°°(G). For this let (Kj)j&^ be a sequence of compact sets Kj C G such t h a t Kj C Kj+\ and U j l i Kj = ^> i-e- (Kj)j€® is a compact exhaustion of G. On C°°(G) we obtain a separating family of seminorms turning C°°(G) into a Frechet space by P«,K-,(u) := sup l ^ u ^ ) ! ,

a G NJ.

(2.117)

xeKj

It is obvious t h a t a second compact exhaustion (K'j)jew will lead to an equivalent family of seminorms. Let K C R™ be a compact set and m G N 0 U {oo}. T h e space C^(K) is defined by Cf(K)

:= {u : R™ -> C | u G C m ( R " ) and u|*= = 0 } .

(2.118)

(Note t h a t this is a definition slightly different to t h a t given in Section 2.1 for the space CQ(E) and a general topological space, but this will not lead to any confusion later on.) On ^(K), m G No, we have a norm (hence a family of separating seminorms) given by Wc(u):=gm(u):=

£ i|a]<m \s

sup|cTu(z)lx€K

(2.119)

2.6

Functions and Distributions

Moreover, C^(K) (
a

45

equipped with the topology generated by the family

Frechet space. Note t h a t for every open set G such t h a t K C G

we have the inclusion C§°(*0 C Cg°(G). Now a topology on Co°(G), G e l " open, is introduced as follows. D e f i n i t i o n 2 . 6 . 3 A convex set C in C£°(G) is an open neighbourhood of 0 G Cg°(G) whenever C n Cg°(if) is /or aH K C G, K compact, an open neighbourhood for 0 G C^(K) in the topology ofCjf(K). Let us summarise some important properties of this topology. T h e o r e m 2 . 6 . 4 A. A sequence (UJ)J^, u,- G Co°(G), converges to uGCo°(G) if and only if there is a compact set K C G such that Uj,u € C'Q'(K) for all j s N and ( U J ) J £ N converges in CQ>(K) to u. 5 . Lei X be a topological vector space and A : C Q ° ( G ) —• X. The operator A is continuous if and only if for any compact set K C G the operator A|c~(/r) *s continuous from C^(K) to X. C. The mapping id : C Q ° ( G ) —>• C°°(G) is continuous. Now we are in a position to generalise the notion of a function. D e f i n i t i o n 2 . 6 . 5 Let G C R™ be an open set. The topological dual space V(G) o/Cg°(G) is the space of distributions on G. Hence, u G V(G) means t h a t u : C Q ° ( G ) —> C is a linear continuous functional. Since Cg°(G) C C°°(G) and Cg°(R") C <S(R"), in the sense of dense and continuously embedded subspaces, any continuous extension from Co°(G) to C°°(G) or from Cg°(R") to «S(R"), respectively, is uniquely determined. D e f i n i t i o n 2.6.6 A. The topological dual space <S'(R n ) of the Schwartz space <S(R") is called the space of tempered distributions. It consists of all distributions u G £>'(R") having a continuous extension to <S(R n ), i.e. 5 ' ( R n ) C 2?'(R n ). B. The topological dual space S'(G) of C°°(G) is called the space of distributions with compact support. It consists of all distributions u G T>'(G) having a continuous extension to C°°(G), i.e. £'(G) C V(G). In case of G = R n we also have £ ' ( R n ) C <S'(R n ). For a distribution u G W(G) we define its support supp u as follows. For G C G C R n , G, G open sets, we have always C Q ° ( G ) C C O ° ( G ) . Hence, we may consider u| C oo«3\

46

Chapter 2 Essentials from Analysis

which is an element in 2^'(G'). Two distributions u, v G 2?'(G) are said to coincide on G C G if their restrictions to C Q ° ( G ) coincide. T h e support of u G T>'(G) is by definition the complement of the largest open set G C G on which u coincides with 0 £ V(G). In the sense of this definition S'(G) is the space of all elements u € V'(G) having a compact support. Let us collect some examples. For u G Ljoc(G) a distribution is defined by u() := fG(x)u(x) dx, 4> G C Q ° ( G ) . Moreover, for a locally finite measure \i on B(G) we may define fi(<j>) := JG (j>(x) fi(dx) which gives also a distribution. In each of these cases we identify u or jj, with the corresponding distribution. Now we have V(G)

C L/ o c (G) C P ' ( G ) ,

(2.120)

and MC(G)

C V'{G).

(2.121)

Furthermore, it follows t h a t L P ( R " ) , 1 < p < oo, all polynomials and all measurable, polynomially bounded functions belong to <S'(R n ). In addition, if/x G Mc(Rn) such t h a t there exists N G N with the property t h a t / R „ (1 + \x\2)~N n \fj,\(dx) < oo, then fi belongs to S ' ( R ) . In particular, we have L e m m a 2 . 6 . 7 For u G L P ( R " ) , 1 < p < oo, and a polynomially measurable <$> it follows that u G £ ' ( R n ) .

bounded

Clearly, X>'(G), S'{G) and S ' ( R n ) are vector spaces and we may consider t h e m as topological vector spaces when taking the weak-*-topology on these spaces. W i t h these topologies we have the following density results. L e m m a 2.6.8 A. The space C Q ° ( G ) is sequentially space <S(R") is sequentially dense in <S'(R™). There are some natural operations on V(G). and-0GCoo(G)

dense in T>'(G). B. The

We may define for u G T>'(G)

(dau)(<j>) := ( - l ) H u ( d a 0 )

(2.122)

(tf>u)(0:=u(^)

(2.123)

and

2.6

47

Functions and Distributions

for all <j> S Cg 0 (G r ). T h e convolution of functions with distributions can be denned in several situations. T h e o r e m 2 . 6 . 9 For a distribution fined by

u and a function

the convolution

{u*)(x):=u(<j>(x-.)) in each of the following

andGG^(G);

2. u e £ ' ( G )

and4>€C°°(G);

3. u G S'(R ) Moreover,

(2.124)

situations:

1. u e P ' ( G )

n

is de-

and <j> G «S(R n ).

in each case u * is a C°° -function

and we have

da(u*) = (dau)*(f> = u*(da), and for u G <S'(R"), <j> G <S(R n ), i/ie function growth.

(2.125) u * 0 /ias ai mosi

polynomial

Using the convolution we may extend the concept of the Friedrichs mollifier to V(G) by setting J £ (u) := u £ := u * j e . It turns out t h a t u £ converges in V{G) to u. Moreover, when u G £'{G) or S'(G), u£ belongs to these spaces too and t h e convergence takes place in the topology of these spaces. It is also possible to define the convolution for two distributions u i and U2 provided one of them has a compact support: (ui *u2)(c/>) : = u i *(u 2 *<£).

(2.126)

It turns out t h a t (2.126) is a reasonable definition and one can prove t h a t ui * u 2 is a distribution. Moreover, we have ui*u2=u2*ui,

(2.127)

d a ( u i * u 2 ) = ( d a U l ) * u 2 = Ui * ( d Q u 2 ) ,

(2.128)

and for u i , u 2 , U3 such t h a t two of them have a compact support, we have (ui * u 2 ) * u 3 = ui * (u 2 * u 3 ) . For many proofs in the theory of distributions one needs

(2.129)

48

Chapter 2 Essentials from Analysis

Theorem 2.6.10 LetG c R n andG' C R m be open sets andip G C°°(GxG'). Further let K C G be a compact set such that tp(x, y) = 0 for all x G Kc. For any u G V(G) and a e N J the function y i-> u(tp(., y)) belongs to C°°(G') and we have d£(uOK.,y))=u(c£V(-,2/))-

(2.130)

Suppose that T : G\ —>• G^ is a diffeomorphism, G\, G^ C R™ being open sets. Clearly, for <j> G Cg°(G,2) it follows that o T G Cg°(Gi), thus in the notation introduced before, the pullback T*0 belongs to Co°(Gi). For u G V'{G2) we define the pullback T*u G V(G{) by

^^:=U(^W(T_1)V)'

(2J31)

for all <j> G Co°(Gi). Obviously, det(dT) denotes the Jacobi determinant of the differential of T. Whenever T is such that T*4> G <S(R") for all

0 define HA : R™ -> R n by x M- Xx. Definition 2.6.11 A distribution u G X>'(R™ \ {0}) is called homogeneous of degree p £ R if H£(u) = Apu

(2.132)

holds. Note that for u G L}oc(Rn \ {0}) we find with 4> G Cg°(Rn \ {0}) (H^u)(^») = A - " ( u , ( H * / ^ ) ) = A_n /

u(x)

JRn\{0}

^^'

u{\x)4>(x) da;,

JR"\{0}

and (H^u)() = Apu() = \p f

u(x)(x) dx,

JR"\{0}

hence u(Xx) = Apu(:r), i.e. u is a homogeneous function of degree p. Note that for p being not an integer less or equal to —n, any homogeneous distribution on V(Rn \ {0}) has a unique extension to D'(R n ). Moreover, we have

2.6

49

Functions and Distributions

L e m m a 2 . 6 . 1 2 Let u G Z>'(Rn \ {0}) be a homogeneous tension to V'{Rn). Then u belongs to <S'(Kn)

distribution

with ex-

We will use two results on the structure of distributions. T h e o r e m 2 . 6 . 1 3 A. Suppose that u G V'(Rn) there exists a number m G No such that

u=

and that supp u = {0}.

J2 c<*da£°

Then

( 2 - 133 )

|a|<m

with some constants ca G C. B. Let u G V{G), G C R™ open, swc/i t/iai / o r aZ/ <j> G Co°(G), > 0, we have u() > 0. Then there exists a measure \i G M.+ {G) such that u() = f 4>{x) n(dx)

(2.134)

JG

for allege

Cg°(G).

(Recall t h a t measures are always non-negative in our definition). In order to apply Theorem 2.6.13.B, let's say in case G = R n , it is sufficient to show t h a t u(|(£| 2 ) > 0 for all <j> G Cg°(R n ). T h e problem is of course t h a t in general the square root of a function <j> G Co°(R™), (j) > 0, need no longer belong to Co°(]R n ). But one can smooth it out by using the Friedrichs mollifier in an appropriate way. Let Gj C Rnj be two open sets and u^- G C(Gj), j — 1,2. On Gi x G 2 C R™1 x R" 2 we define the function ui <8> u 2 by (ui ®u 2 )(a;i,a;2) : = u i ( a j i ) • u 2 ( x 2 ) , Xj G Gj. We call ui®U2 the tensor product of ui and U2- T h e space C 0 X > ( G I ) ( 2 ) C Q D ( G 2 ) is the set of all finite linear combinations Y^T ,- 2= i Uji ®Uj 2 where Ujk G Co°(Gfc). T h e tensor product generalises to distributions. T h e o r e m 2 . 6 . 1 4 Let Uj G V(Gj), j = 1,2. Then there exists a unique distribution u = u i ® u 2 G £>'(Gi x G 2 ) called the tensor product of u i with u 2 such that u(4>i ® fa) = ui(<^i) • u2(2)

(2.135)

50

Chapter 2 Essentials from Analysis

holds for all j G Cg 0 (G : ,). In addition, for all 4> G C Q ° ( G I X G 2 ) we have u(0) = u i ( u 2 ( $ ) = u 2 ( U l ( 0 ) ) where Uj acts on $ as a function

(2.136)

in Xj G Gj.

C o r o l l a r y 2 . 6 . 1 5 Foruj G £'(Gj), hold for all G C°°(Gi x G 2 ) .

j = 1,2, the assertions

of Theorem 2.6.14

Our aim is to describe linear operators Kop : Co°(G 2 ) —>• £>'(Gi) for open sets Gj c R % j = l , 2 . Suppose t h a t K G C(Gi x G 2 ) . We may define an operator on Co°(G 2 ) by (Kop0)(a:i) := /

R O n , ^ ) ^ ) dz2,

(2.137)

JG2

which yields a linear operator Kop : C Q ° ( G 2 ) —> C ( G i ) . If we consider K o p 0 as a distribution we find (K op <£)(V0=K(V>®<£)

(2.138)

for V e Cg°(Gi) and ^ G Cg°(G 2 ) with K G C(Gi x G 2 ) c X>'(Gi x G 2 ) . T h e function K is called the kernel of K o p . T h e following theorem is due to L. Schwartz, [277]. T h e o r e m 2.6.16 ( K e r n e l t h e o r e m ) Let K G T>'(GX x G 2 ) . Then we may define a sequentially continuous linear mapping Kop, Kop : Co°(G 2 ) —> T>'(Gi), (Kop(j))(ip) = K(ip®(j)) for all ip G Cg°(Gi). Conversely, for every sequentially continuous linear mapping Kop : C Q ° ( G 2 ) —¥ T>'{G{) there exists a unique distribution K G T>'(G\ x G 2 ) called the (distribution) kernel ofKop such that (Kop0)(V>) = K{tP ® ) for all V G C§°(Gi) and G Cg°(G 2 ). E x a m p l e 2 . 6 . 1 7 The kernel of id : Cg°(G) -> Cg°(G) is given by (ldkernel)$=

[ ${x,x)dx

/ o r a// $ G C Q ° ( G X G), and we have supp (idfc erne /) = diag(G). For idfcerne; we write often ^diagLet us have a look on some distinguished classes of operators.

(2.139)

2.6

51

Functions and Distributions

P r o p o s i t i o n 2 . 6 . 1 8 Let G c R n be an open set and ¥{x,D) : Cg°(G) -> V(G) be a differential operator, i.e. P(x,D) = X^ia|<m aa(x)Da. Then P(x,D) is a local operator, i.e. supp P ( z , D)u C supp u

(2.140)

forallu£C™(G). A remarkable result of J. Peetre [236] says t h a t every local operator is already a differential operator. P r o p o s i t i o n 2 . 6 . 1 9 Let A : Cg°(R") -> C°°(R n ) be a linear operator which is sequentially continuous and translation invariant, i.e. Th(A(/>) = A(Th4>) for all <j> G Cg°(R"), where for h€Rn we set (Th(j>)(x) = <j>(x - h). Then there exists a distribution K G V'(Rn) such that for all G Cg°(R") {A(j)){x) = (K 1= (j>){x) holds, i.e. translation

invariant

(2.141) operators are convolution operators.

T h e next result is due to N. Dunford and B. Pettis [75] and describes bounded operators from L P (G) into L°°(G), G G B ' " ' , with the help of kernels. We quote it in a formulation given by D. Robinson [249]. T h e o r e m 2 . 6 . 2 0 Let Kop : V(G) -> L°°(G), 1 < p < oo, be a bounded linear operator. Then there exists a kernel function K : G x G —> C, K G L°°(G) ® L P ' ( G ) , I + i = l, such that Kopu(x)=

[ K(x,y)u(y)dy.

(2.142)

JG

Conversely, every operator defined by (2.142) with a kernel function K : G x G —>• C, K G L°°(G) L p (G), i + -i- = 1, defines a bounded linear operator from L P (G) into L°°(G). Moreover, the operator norm ofKop is given by \\Kop\\ = e s s . s u p | | K ( a : , . ) | | L l ) / .

(2.143)

T h e following characterisation of linear bounded operators from L P (G) into L (G), 1 < p,q < oo, is due to A. Bukhvalov [50], and we take it in the formulation given in [10]. 9

52

Chapter 2 Essentials from Analysis

T h e o r e m 2 . 6 . 2 1 A linear operator Kop : L P (G) ->• Lq(G) tion with a kernel function K : G x G —> C, i.e. K o p u(:r) = / K(x, y)n(y)

has a

representa-

dy

JG

for allu £ LP(G) andK(x,.) G V' (G), ^ + p- = 1, if for any sequence (u„)„ e N; u„ G LP(G), such that u„ —> 0 a.e. anrf |u„(a;)| < u(:r) o.e. / o r some u G L P (G), implies that (Kopul,)(x) —>• 0 o.e.

2.7

Some Functional Analysis

We assume t h a t the reader has basic knowledge in functional analysis. But since introductionary courses in functional analysis formulate the leading principles only for Banach spaces and since we need t h e m often also in Frechet spaces, we quote here also results like the H a h n - B a n a c h theorem etc. Our standard references are W. Rudin [255], K. Yosida [315] and M. R e e d / B . Simon [246]. More encyclopaedic works are the books [76]-[78] of N. Dunford and J.Schwartz, and for results in operator theory we mention the two volumes [108]-[109] written by I. Gohberg with coauthors. Besides Banach spaces, Hilbert spaces and Frechet spaces we need a few times quasi-Banach spaces. By definition a quasinorm q on a vector space X (over R or C) is a mapping q : X —> R such that q(x) > 0 and q(a;) = 0 if and only if x = 0, q(Aa;) = |A|q(a;), and instead of the triangle inequality we have only q(x 1 + x-z) < c(q(a;i) + q(£2)) with some c > 1, the case c = 1 refers, of course, to a norm. A quasi-normed space (X, q) is a quasi-Banach space if it is complete with respect to q. Let X\ and X2 be two locally convex topological vector spaces. We say t h a t X\ is continuously embedded into X% if X\ C X2 and the identity m a p id : X\ —¥ X2, x\ M- X\ G X2, is continuous. Sometimes we write X\ •—» X2 when X\ is continuously embedded into X2. In case of two normed spaces (Xi, \\.\\x ) and (X2, \\-\\x2) this does mean that X\ C X2 and ||w||^ 2 < cH^H^ for all u £ X\. T h e norms ||.||-|_ and ||.|| 2 on a vector space X are said to be comparable if Hu^ < c||u|| 2 or ||u|| 2 < cjlw^ holds with some c > 0. T h e y are said to be equivalent if there are two constants C\ > 0, c2 > 0 such t h a t ciM,

< \\u\\2 < c2\\u\\1

2.7

53

Some Functional Analysis

for all u € X. If ll-Hj is comparable with ||.|| 2 on X, let say Hu^ < c||u|| 2 , then (X, ||.|| 2 ) is continuously embedded into (X, ll-ld). But much more caution is necessary when we look to completion. For a normed linear space (X,||.||_jf) we can always construct a Banach space (X, \\.\\x) such t h a t (X, \\-\\x) i s isomorphic and isometric to a dense subspace of X. Recall t h a t a linear mapping T : Y\ —>• Yi between two normed spaces (Yi, | ] . | | y ) and (Y 2 , ||.||y ) is an isometry if||Ty||ya = | | y | | y i f o r a U y G Y i . Now let ll-Hj and ||.|| 2 be two norms on a vector space X. We call these norms compatible, or a pair of coordinated norms, if every Cauchy sequence in b o t h norms converging in one of them to zero also converges to zero in the second norm. Clearly, equivalent norms are compatible, but in general comparable norms are not. T h e o r e m 2 . 7 . 1 Let 11.||j^ and ||.|| 2 be two compatible norms on a vector space X such that \\x\li < c||x|| 2 for all x G X, c> 0. Denoting by Xj the completion of X with respect to \\.\\ • we have

I ^ I

2

- 4 l i .

If these norms are equivalent,

(2.144) then we have

Xi = X2

(2.145)

A proof of Theorem 2.7.1 is given in [103], pp.13-14. Before discussing linear operators, let us state one result on Hilbert spaces which we will need later. T h e o r e m 2.7.2 ( B a n a c h - S a k s ) Let (H,(.,.)u) be a Hilbert space and (xk)k€N o, sequence in H converging weakly to x £ H. Then there exists a subsequence (ajfc,)z€N, h G N, such that the sequence (yjv)iVeN, VN '•= jf Z)/=i xk,, converges in the norm of H to x. Now let us t u r n to linear operators A on topological vector spaces. Let X, Y be two topological vector spaces. We say t h a t (A, D(A)) is a linear operator from X to Y if D(A) C X is a linear subspace and A : D(A) —>• Y is linear. We call D(A) the domain of A. The range of A is given by R(A) := {y G Y | y = Ax for some x G D ( A ) } .

(2.146)

54

Chapter 2 Essentials from Analysis

T h e set Ker(A) := {x £ D(A) | Ax = 0 }

(2.147)

is called the kernel of A (a notion which should not be mixed up with t h a t of the distributional kernel) and the graph of A is the set T(A) := {(a;, y) 6 X x Y \ y = Ax for some x £ D ( A ) } .

(2.148)

Two operators ( A i , D ( A i ) ) and (A2, D(A2)) from X to Y are equal if a n d only if D(A X ) = D(A 2 ) and AlX = A2x for all x G D(Ai) ( = D(A 2 )). D e f i n i t i o n 2.7.3 Let (A, D(A)) be a linear operator topological vector spaces. A. We call A continuous continuous where D(A) carries the topology induced closed operator, if T(A) is closed in X x Y. C. The it has a closed extension. Recall t h a t (A, D(A)) is an extension D(A) and Arc = Ax for x € D(A).

from X when A by X. operator

to Y, both being : D(A) —¥ Y is B. We call A a A is closable if

of (A,D(A)) if and only if D(A) C

Well known results for continuous linear operators from a normed space (X, H-ll^-) into a normed space (Y, ||.|| y ) are 1. a densely denned linear continuous operator (A, D(A)) from X to Y has a unique continuous extension to X; 2. a linear operator is continuous if and only if it is bounded, i.e. ||Ax||y < c | | x | | x for all x £ D(A). Let (A,D(A)) be a continuous linear operator from (X, \\.\\x) to (Y, ||.||y). The operator norm of A is defined by ||A|| := | | A | | x > y :=

sup

i^in.

(2.149)

For two linear continuous operators (A, D(A)) from (X, \\.\\x) to (Y, ||.|| y ) and (B,D(B)) from (Y, ||.|| y ) to (Z, \\.\\z) such t h a t R(A) C D(B) we have IIBoAII < | | B | | | | A | | .

(2.150)

55

2.7 Some Functional Analysis

The space of all continuous linear operators A : X —• V, (X, \\.\\x) and (Y, \\.\\Y) being two normed spaces, is denoted by B(X, Y), and in case X = Y by B{X). For a Banach space (Y, ||.||y) the space B(X,Y) is a Banach space too. Linear operators (I, D(Z)) from a vector space X to C (or R) are called iinear functionals, and for a topological vector space X the space of all continuous linear functionals, the (topological) dual space of X, is denoted by X*. T h e o r e m 2.7.4 (Hahn—Banach) Let Y be a linear subspace of the vector space X and let p be a seminorm on X. Further let I : Y —> C be a linear functional such that \l(x)\ < p(x) for all x £ Y. Then there exists a linear functional L : X -> C such that L|y = I and |L(a;)| < p(a;) for all x £ X. Another important result for continuous linear operators is the theorem of S. Banach and H. Steinhaus, often called the uniform boundedness principle. T h e o r e m 2.7.5 (Banach—Steinhaus) Let {Aj \j £ 1} be a family of continuous linear operators Aj : X —> Y from a Frechet space X into a topological vector space Y. If for all x £ X the set {AjX \j £ 1} is bounded in Y, then the family {Aj \j £ 1} is equicontinuous, i.e. for every neighbourhood V of Oy £ Y there exists a neighbourhood U of Ox £ X such that Aj(U) C V for all j £ I. Corollary 2.7.6 Let {Aj \j £ 1} be a family of bounded linear operators defined on the Banach space (X,\\.\\x) into a normed linear space (Y, ||-||y)Then the boundedness of the family {\\AJX\\Y \ j £ 1} for each x £ X implies the boundedness o/{||Aj|| \j £ I}. Corollary 2.7.7 Let (Ak)k^N be a sequence of bounded linear operators defined on a Banach space (X, \\-\\x) into a normed vector space (Y, ||.||y). Suppose that for every x £ X there exists Ax £ Y such that lim || AkX — Ax||y — 0 k—*-oo

holds. Then A : X —>• Y is a bounded linear operator such that ||A|| < liminf ||Afc||.

(2.151)

fc—>oo

Next we state some results for closed operators. For this it is convenient to introduce the graph norm for a linear operator (A, D(A)) from a normed space (X, \\.\\x) into the normed space (Y, ||.|| y ) by WXWA;X,Y

••= Mx

+

WAX\\Y-

(2-152)

56

Chapter 2 Essentials from Analysis

In case that X = Y we write only \\x\\A.x:=\\x\\x

+ \\Ax\\x.

(2.153)

Theorem 2.7.8 (closed graph theorem) Let (A,D(A)) be a closed linear operator on a Frechet space X into a Frechet space Y. 7/D(A) = X, then A is continuous. In case of Banach spaces we have Theorem 2.7.9 A linear operator (A, D(A)) from a Banach space (X, \\-\\x) into a Banach space (Y, \\.\\Y) is closed if and only if (D(A), ||-|IAAT V) ^S a Banach space. The following result is due to L. Hormander and taken from [315], p.79. Theorem 2.7.10 Let (Xj,\\.\\x.), j = 1,2, and (X,\\.\\x) be Banach spaces and (Aj,D(Aj)) linear operators from X into Xj. 7/(Ai,D(Ai)) is closed and (A2,D(A2)) is closahle such that D(Ai) C D(A2), then there exists a constant c > 0 such that \\A2x\\Xa < c(\\AlXfXl

+ \\x\\2x)^

(2.154)

holds for all x S D(Ai). Furthermore, we have Theorem 2.7.11 Let (A, D(A)) he a closed linear operator on the Banach space (X, \\-\\x) into itself. Furthermore, let |||.||| be a norm on X which is equivalent to the graph norm. Then it follows that D(A)

= D(A).

For closable operators let us note the following results: Lemma 2.7.12 A linear operator (A,D (A)) from a Banach space (X, \\-\\x) into a Banach space (Y,\\.\\Y) is closable if and only if for every sequence (z,/)„eN, %v S D(A), such that lim xv = 0 and lim Axv = y it follows y = 0. v—>oo

v—>oo

Definition 2.7.13 Let (A, D(A)) he a closahle operator. Its smallest closed extension is called the closure of (A,D(A)) and is denoted by (A,D(A)).

2.7

Some Functional Analysis

T h e o r e m 2 . 7 . 1 4 Let (A, D(A)) be a closable operator on the Banach (X, \\.\\x) i-nt° itself. The domain of its closure is given by D(A) = D ( A ) " " A ; X .

57 space

(2.155)

Let us give some general notions related to the analysis of operators. D e f i n i t i o n 2.7.15 Let (Afc,D(Afc)) be a family of operators from the Banach space (X, \\.\\x) * n *° the Banach space (Y, ||.||y) such that D(Afc) is independent on k, therefore it is denoted just by D. A. We say that (Ak)keN converges strongly to the operator (A,D(A)) i / D ( A ) = D and lim ||A fc u - A u | | y = 0

(2.156)

fc—>oo

for all u £ D. Sometimes we write s-limfc_>ooAfc = A for (2.156). B. We call (Afc)fcgN weakly convergent to (A, D(A)) i / D ( A ) = D and lim l{Aku - Au) = 0

(2.157)

fc—*oo

for all I sY*.

For (2.157) we write also w-lirm-^ooAfc = A.

For a closed operator (A, D(A)) from a Banach space (X, \\.\\x) to a Banach space (Y, \\.\\Y) we call a linear subspace D C D(A) a core of A if A\D = A, i.e. the closure of A|£> is A in the sense of operators. A special class of operators are contractions. A mapping A : X —> X on a Banach space (X, \\.\\x) is called a contraction if ||Aa;|| x < .ftT||a;||x for some 0 < K < 1. It is called a strict contraction UK < 1. Note t h a t we do not assume A to be a linear operator! T h e o r e m 2 . 7 . 1 6 ( B a n a c h ' s fixed p o i n t t h e o r e m ) Let A : X -> X be a strict contraction on the Banach space (X, ||.||^-). Then there exists exactly one fixed point XQ of A, i.e. AXQ = XQ. Moreover, for every xi £ X the sequence (Akx\)k$N converges to XQ. From Theorem 2.7.16 we deduce immediately C o r o l l a r y 2 . 7 . 1 7 Let A : X —> X be a linear operator on the Banach space (X, \\.\\x) such that ||A|| < 1. Then the operator (id — A) has a continuous inverse (id — A ) - 1 : X —> X.

58

Chapter 2 Essentials from Analysis

It turns out t h a t it is convenient to write (u,x) for u(x) if x G X u G X*. Often we have to work with the conjugate of an operator.

and

D e f i n i t i o n 2 . 7 . 1 8 Let (A, D(A)) he a densely defined linear operator from the Banach space (X, \\.\\x) to the Banach space (Y, \\.\\Y)> *-e. D(A) C X is a dense linear subspace. We denote by D(A*) the set D(A*) := {y* G Y*\ 3 z * G X* s. th. = (x*,x)

(y*,Ax)

V:rGD(A)}

(2.158)

and on D(A*) we define A*y* = x*. The operator (A*,D(A*)) is a linear operator from (Y*, | | . | | y . ) to (X*, and we call it the conjugate operator to (A,D(A)).

(2.159) \\.\\x.)

P r o p o s i t i o n 2 . 7 . 1 9 A. Let (A,X) be a continuous operator from the Banach space (X,\\.\\x) to the Banach space (Y, ||.||y). Then (A*,Y*) is also a continuous operator and we have ||A*|| — ||A||. B. For two continuous linear operators (A,X) and (B,X) from X to Y we have ( a A +/3B)* = aA* +/3B*

(2.160)

C. Let (A,X) be a continuous linear operator from the Banach space (X, \\.\\x) to the Banach space (Y, \\.\\Y) and let (B,Y) be a linear operator from the Banach space (Y,\\.\\Y) to the Banach space (Z,\\.\\z). Then we have (BA)* = A*B*, whenever both sides are defined. P r o p o s i t i o n 2 . 7 . 2 0 Let (A, D(A)) be a densely defined linear operator of the Banach space (X,\\.\\x) into itself. The operator (A*,D(A*)) is a weak-*closed operator. If A is closed, then D(A*) is weak-*-dense in X*. In addition, if X is reflexive, then D(A*) is strongly dense in X*. Further we have

2.7

Some Functional Analysis

59

T h e o r e m 2.7.21 (Closed range theorem) Let (A, D(A)) be a densely defined operator from the Banach space (X,\\.\\x) into the Banach space (Y, ||.||y). The following assertions are equivalent 1. R(A) is closed in Y; 2. R(A*) is closed in X*; 3. R(A) = {y G Y | (y*,y) = 0 for ally* G K e r ( A * ) } ; I

R(A*) = {x*eX*

| (a;*, x) = 0 for all x G Ker( A ) } .

Often, when working with operators (A, D(A)) from a Hilbert space (H, (., .)H) into itself, it is reasonable to work with a modified conjugate operator. Let J : H* —»• H be the one-to-one norm-preserving conjugate linear mapping identifying H* with H. Further suppose t h a t (A*,D(A*)) is the conjugate operator to (A, D(A)), D(A) C H a dense subspace, i.e. D(A*) C H* and A* : D(A*) -»• H*. Now we may identify H* with H using J and we obtain the operator J A * J _ 1 : H —>• H. Let us denote for a moment this operator by A'. From the very definition of A* and A' we have for all x G D(A) and y*&H* (y*,Ax)

=

(z*,x)

with z* = A*x. P u t t i n g y = J(y*) and z = J(z*) we find (y*,Ax)

= (J(y*),Ax)H

=

{y,Ax)H

and (z*,x)

= (J(z*),x)H

=

(z,x)H,

which yields (y,Ax)H

=

(z,x)H.

Hence, by the density of D(A) in H we may define an operator A by Ay = z, thus (Ay,x)H

=

(y,Ax)H

60

Chapter 2 Essentials from Analysis

for all x £ D(A) which implies

(j- 1 (Aj/), a :} = <J- 1 z,x), thus J _ 1 ( A 2 / ) = A-*]'1*, A = JA-J-

1

or

= A'.

Let us make the following convention: If (A, D(A)) is a densely defined linear operator on a Hilbert space (H, (., .)H) and if H* is identified with H by J. Then the (Hilbert space) adjoint of (A,D(A)) is the operator J A * J _ 1 with domain J(D(A*)), but we will denote this operator once again by (A*,D(A*)). Thus in case of an operator acting between Banach spaces A* stands for the conjugate operator, but in case of an operator acting between Hilbert spaces, A* stands for the adjoint operator. For Hilbert space adjoints we have T h e o r e m 2 . 7 . 2 2 Let (A,D(A)) be a densely defined linear operator on a Hilbert space (H, (., .)H) and let us identify H* with H. A. The operator (A*,D(A*)) is a closed linear operator on H. B. The operator (A,D(A)) is closable if and only if A** := (A*)* exists and A** is the closure of A, i.e. the smallest closed extension of A. C. The operator A is closed if and only if A = A**. D. If A is bounded, it has an extension to H which we denote once again by A and we have ||A|| = ||A*||. For operators in a Hilbert space we give D e f i n i t i o n 2 . 7 . 2 3 Let (A, D ( A ) ) be a densely defined operator on a Hilbert space (H, ( . , . ) # ) . A. The operator A is called symmetric if A* is an extension of A. B. We call A selfadjoint ifA = A*. Note t h a t for a symmetric operator we have {Ax,y)H-=(x,A.y)H

(2.161)

for all x,y s D(A), whereas for selfadjoint operators we have D(A) = D(A*) and(2.161)forallx,yeD(A). Selfadjoint operators play a central role in spectral theory. Let us recall some of its ba=ic »-esu1ts For-, this let (A. D ( A ) ) be a linear operator on the complex Banach space (X, \\.\\x) m t ° itself.

61

2.7 Some Functional Analysis

Definition 2.7.24 The resolvent set p(A) of A consists of all A S C such that Aid — A is surjective and has a continuous inverse (Aid — A ) - 1 defined on R(Aid — A) = X. The set cr(A) := p(A)c is called the spectrum of A. For A £ cr(A) three cases are possible: 1. Aid — A is not invertible, then A belongs to the point spectrum of A; 2. Aid — A is invertible, but its inverse is not continuous, then A belongs to the continuous spectrum of A; 3. Aid — A is injective and its inverse defined on R(Aid — A) is continuous, but R(Aid — A) ^ X, then A is said to belong to the residual spectrum of A. Note that A belongs to the point spectrum of A if and only if there exists x ^ 0, x € D(A), such that Aa; = Xx. We call A an eigenvalue of A and x an eigenvector to the eigenvalue A. The dimension of the space of all eigenvectors corresponding to our fixed eigenvalue A, i.e. dim(Ker(Aid — A)), is called the multiplicity of A. The following results are easy to prove. Theorem 2.7.25 A. The resolvent set p(A) is an open set in C and in each component of p(A) the function R\ defined by A i-» R\ := (Aid — A ) - 1 is analytic. B. For A , / J £ p(A) the resolvent equation holds, i.e. RA - R^ = (ji - A)RARM,

(2.162)

implying RAR^ = R^RA- C. If A is a bounded linear operator on X, i.e. D(A) = X, then the limit TA := lim ||Afc|| exists and is called the spectral k—>oo

radius of A. It follows that rA < ||A|| and a (A) C i? rA (0). A further consequence is Lemma 2.7.26 Let (A, D (A)) be a linear operator such that p(A) ^ 0. Then A is a closed operator. The next result is due to R. Phillips [240]:

62

Chapter 2 Essentials from Analysis

T h e o r e m 2.7.27 Let (A, D (A)) be a densely defined closed linear operator on the complex Banach space (X, ||.||X) into itself. Then we have p(A) = p{A*) and R*x = (Aid - A*)" 1 , A G p(A).

(2.163)

Let A G B(X), (X, \\.\\x) being a complex Banach space. Denote by %(A) the set of all complex-valued functions which are defined and analytic in a neighbourhood of a(A). Let f G H(A) and U C C be an open set such that cr(A) C U, f is defined and analytic on U, and dU consists of a finite number of rectifiable Jordan curves, positively oriented each. Then the following integral exists, is independent of the special choice of U and defines a linear operator on X: f ( A ) z : = - ^ / f(A)RAxdA. 27r* Jdu

(2.164)

The integral (2.164) is often called a Dunford integral. The following theorem summarises Dunford's operational calculus. T h e o r e m 2.7.28 If i £ U(A), then we have f(
and af(A) + /3g(A) = (of + /3g)(A);

2. f • g G H(A) and f(A) • g(A) = (f • g)(A); 3. if the Taylor expansion f(A) = J2T=o ak^k U of a(A), then we have

converges in a neighbourhood

oo

f(A) = VafcA fc (convergence in norm topology); fc=o

(2.165)

4- tf {U)veN is a sequence in H(A) converging to f G H(A) uniformly in a neighbourhood U of a (A), then tv(A) converges in the operator norm topology to f(A); 5. iff G H{A), then f G H(A*) and f(A*) = f(A)*. Later we have to extend this theorem to certain unbounded operators. Here we will state such results only for selfadjoint operators in a Hilbert space. So

63

2.7 Some Functional Analysis

let (H, (., .)H) be a complex Hilbert space. A linear operator P : H —» H is called an orthogonal projection if it is a projection, i.e. P 2 = P, and if H = R(P) © R-(P)"1 where R(P)' L is the orthogonal complement of R(P), i.e. x £ R ( P ) ± if and only if (x, y)u = 0 for all y £ R(P), and © denotes the direct sum. Definition 2.7.29 A family (EA)A£R of orthogonal projections on a Hilbert space (H, (., .)#) is called a resolution of identity on H if the following conditions hold: 1. R(E A ) C R(EM) for A < n; 2. R ( E A ) = f l M > A R ( E M ) ; 3- riA6RR(EA) = {0}; 4- l i n ( U A 6 K R ( E A ) ) = ^ . For a resolution of identity we may define the function X^(Exx,x)H

(2.166)

which is for each x £ H a non-negative increasing function. Hence, we may define for suitable functions f: R —»• C the (improper) Stieltjes integral [f{\)d(Exx,x)H.

(2.167)

JR

Now let (A, D (A)) be a selfadjoint operator on (H, (., .)H). It follows that er(A) C R and for Im A ^ 0, A £ C, we have

IK^-A)-1!!^^-

(2-168)

Theorem 2.7.30 (Spectral theorem for selfadjoint operators) Let (A, D (A)) be a selfadjoint operator on the Hilbert space (H, (., .)#). Then there exists a unique resolution of identity ( E A ) A £ K such that D(A) = lx£H

f X2d{Exx,x)H


(2.169)

and Ax=

lira I / (/:

AdEALr, i e D ( A ) .

(2.170)

64

Chapter 2 Essentials from Analysis

Let us explain the meaning of (2.170). For a bounded interval [a, b] C R and a partition n of [a,b], a = A0 < Ai < . . . < Afc = b, with length K(TT) define the operator k

^M^^ffeXE^.-E^J j=i

where f : K —> C is a continuous function and r = { i i , . . . , i f c } with t,- € [Aj_i, Aj]. T h e operator ST(i,n) is by definition a bounded linear operator on H. We define now the operator

L

b

f(A)dEA:=

lim

ST(f»,

(2.171)

where the limit exists in the sense of the operator norm in

B(H).

We have now the following possibility to define functions of selfadjoint operators. T h e o r e m 2 . 7 . 3 1 Let (A, D ( A ) ) be a selfadjoint operator on H with sponding resolution of identity (EA)AGK- Then we have [ X2d(Exx,x)H=

(2.172)

Jtr(A)

and for any continuous

D(f(A))

X2d(Exx,x)H,

[

JM

corre-

function

:=ix£H

f : cr(A) —>• C we obtain an operator f(A) by

[ |f(A)|2d(EA:r,a;)H
(2.173)

and f(A)a::=

lim ( /

f(A)dEA)a;

AT-s-oo \J[-N,N]n
In particular, for a is selfadjoint and for a bounded. When taking care on possible to use Theorem

for x G D(f(A)).

(2.174)

J

real-valued function f the operator (f(A),D (f(A))) bounded function f the operator (f(A),D (f(A))) is the domains of operators f(A), g(A) etc. it is of course 2.7.31 to establish an operational calculus in the sense

2.7

Some Functional Analysis

65

of Theorem 2.7.28. For example we may find for k € N D(A fc ) = {ue = {ue

D(A) | Au e D ^ " fc 1

D(A - ) | A *

-1

1

) }

u G D(A) } .

Finally we will study sesquiUnear forms on a Hilbert space. T h e main reference is the book [179] of T. Kato, partly we follow our notes [163]. Let (H,(.,.)H) be a complete Hilbert space. Further let D(q) C H be a dense linear subspace. D e f i n i t i o n 2 . 7 . 3 2 A. We call q : D(q) x D(q) —>• C a sesquiUnear form on H with domain D(q) if for all u,v,w £ D(q) and X, (JL € C q(Aw + [iv, w) = Aq(u, w) + /J.q(v, w)

(2.175)

q(u, Xv + fiw) = Aq(u, v) + /2q(u, w)

(2.176)

and

hold. B. A sesquiUnear form q with domain D(q) is said to be sectorially bounded if there exists a real number do and a number 6 £ [0, | ) such that Req(w,u) > d0||«||^

(2.177)

|Imq(u,u)| < (tan0)(Req(u,u)-do||u||^).

(2.178)

and

Obviously every scalar product is a sectorially bounded sesquiUnear form with bound do = 0. Suppose t h a t (q, D ( q ) ) is sectorially bounded. T h e n on D(q) a scalar product is given by ( « . v ) q : = 2 ^ ( u > v ) + q ( u » u ) ) - (do ~ 1 ) ( « , « ) H -

(2-179)

Clearly we have for the corresponding norm ||.||

H I * < ll«llq-

(2-180)

Moreover, we have, see [176], p.311, t h a t |q(u, v) - d0{u, v)H\ < (tan0)||u|| ||v||

(2.181)

66

Chapter 2 Essentials from Analysis

and t h e converse triangle inequality gives |q(u,«)|
(2.182)

for all u,v G D(q). Note t h a t if do > 0, | ( q ( u , i ; ) + q(v,u)) is itself a scalar product on D(q) with a norm equivalent to ||.|| , and (2.180) holds with some C\ > 0 for this norm, i.e. for all u G D(q) we have \\u\\H < ; % ( q ( u , u ) + q ( u , u ) ) 1 / 2 . Further, taking u = v in (2.179) we find R e q ( u , U ) = l ( q ( u , u ) + q ( u , « ) ) = ||«||J - (1 - d0)\\u\\2H.

(2.183)

Now let us consider on the pre-Hilbert space (D(q), (., .) q ) a second sesquilinear form B. Suppose t h a t B is continuous with respect to ||.|| , i.e. |B(u,«)|c2||u||q-c3||u||^

(2.184)

inequality for all u G D(B) = D(q).

(2.185)

In this case ( B , D (B)) is again sectorially bounded since ReB(u,u)

> -c3\\ufH

(2.186)

and | I m B ( u , u ) | < | B ( u , u ) | < CJIIUII^ < — ( R e B ( u , u ) + c 3 ||w||^).

(2.187)

Further it follows t h a t (u,v)q and | ( B ( u , v ) + B(v,u)) + (c 3 + 1)(U,V)H are equivalent scalar products. In particular the completion of D(q) with respect to ||.|| and the completion of D(q) with respect to the norm related to the scalar product defined with the help of B do coincide. D e f i n i t i o n 2 . 7 . 3 3 A. A closed if (D(q), (., .) q ) is a form q is closable if it has its closure, i. e. its smallest

sectorially bounded sesquilinear form (q, D ( q ) ) is Hilbert space. B. A sectorially bounded sesquilinear a closed extension. If q is closable we often denote closed extension, also with q.

2.7

67

Some Functional Analysis

L e m m a 2 . 7 . 3 4 A. A sectorially bounded sesquilinear form q is closed if and only i / R e q is closed, i.e. (ReD(q), Req) is a real Hilbert space. B. A sectorially bounded form q is closable if and only if for any sequence (u„)„ e N, uv G D(q), the relation («„, u „ ) q —• 0 implies that q(uv, uv) —• 0 as u —• oo. W h e n (q, D ( q ) ) is closable we define its closure q as follows. D(q) is the set of all u G H such t h a t there exists a sequence (u^) l /gN, uv G D(q), such t h a t \\uv — u\\ —> 0 as v —» oo and for u,v G D(q) we put q(u,v)

= lim q(uv,vv)

(2.188)

v—>-oo

where u„, w„ G D(q) and \uv — u||

—» 0 as well as 11^ — u|| -4 0 as !/ -> oo.

D e f i n i t i o n 2 . 7 . 3 5 Let q be a closed densely defined sesquilinear form on a Hilbert space H. A linear subspace D C D(q) is called a core of q if the restriction of q to D is closable and has as closure q itself. L e m m a 2 . 7 . 3 6 Let (q, D ( q ) ) be a closed sectorially bounded sesquilinear form. A linear subspace D C D(q) is a core of q if and only if D is dense in D(q) with respect to the norm ||.|| . We want to associate operators with certain sesquilinear forms. For this we note first R e m a r k 2 . 7 . 3 7 Suppose that q is a sectorially bounded sesquilinear form, i.e. (2.177) and (2.178) do hold. This is equivalent to the fact that {q(u, u)\u G D(q), | | u | | i / = l } is contained in the sector Se,d0 := {z G C | |arg(z — cfo)| < 9}. Let (A, D (A)) be an operator on the Hilbert space (H, (., . ) # ) . Its ical range is defined to be the set 6 ( A ) := {(Au,u)H

| u G D(A) und ||u|| H = 1 } .

numer-

(2.189)

D e f i n i t i o n 2 . 7 . 3 8 A. An operator (A, D (A)) on (H, (., . ) # ) into itself is said to be form sectorial (or form sectorially bounded,) if 6(A) C S M o

(2.190)

for some sector Setd0, 8 G [0, | ) . B. We call (A, D ( A ) ) a form m-sectorial operator if it is form sectorial and for some \x G R the operator A + fi has a

68

Chapter 2 Essentials from Analysis

continuous

inverse defined on H such that

IKA + M + A ) "

1

! ^ ^

(2.191)

holds for all A e C, Re A > 0. Note t h a t (2.191) is equivalent to

H(A + ^ ) - | | < ^ - J _ _

( 2 . 192 )

for all K G C, Re K > /x. T h e o r e m 2 . 7 . 3 9 Let (q, D (q)) be a densely defined sectorially bounded closed sesquilinear form on a Hilbert space (if, (., . ) # ) . Then there exists a unique closed form m-sectorial operator A such that D(A) C D(q) and q{u,v)

= {Au,v)H

(2.193)

for all u € D(A) and v £ D(q). Moreover, D(A) is a core of q. If u € D(q), w G H and q(u,v)

= (w,v)H

(2.194)

for all v belonging to a core of q, then u € D(A) and Ku = w. Further, if q is Hermitian, i.e. q(u,v) = q(v,u) for all u, v € D(q), then A is selfadjoint. Now let us start with a densely denned linear operator ((A, D(A)) on a Hilbert space (H, (., .)#) into itself. For u , » £ D(A) we define the sesquilinear form q(u,v)

= (Au,v)H.

(2.195)

If in addition the operator A is form sectorially bounded, it follows by a result in [179], p.318, t h a t the sesquilinear form q is closable. Let us denote the smallest closed extension of q by q and by A the associated m-sectorial operator. T h e operator A is a closed extension of A, called the Friedrichs extension. T h e o r e m 2 . 7 . 4 0 The Friedrichs extension A of A is the only form extension of A with domain contained in D(q).

m-sectorial

2.8

69

Some Interpolation Theory

When working with generators of contraction semigroups and related sesquilinear forms, it is more convenient t o work with the operator —A instead of A. For the numerical range we find of course 0(—A) = —9(A). If A is a form sectorially bounded operator such t h a t G(A) c Se,d0 for some 6 G [0, f ) and d0 G R, it follows t h a t 0 ( - A ) c {z G C I |arg(2 + do) - | | < 0 } .

(2.196)

Finally, let us state a version of the Lax-Milgram-Theorem taken from [98], Theorem 1.14.1. T h e o r e m 2.7Al (H,(.,.)H)-

Let B be a sesquilinear

Suppose

form

which we have

on a complex Hilbert

space

that

\B(u,v)\
(2.197)

\B(u,u)\>7\\u\\2H

(2.198)

and

hold for all u,v G H with some 7 > 0. In addition, let I : H —>• C be a continuous linear functional, i.e. I G H*. Then there exist unique elements v,w G H such that ^ l(u) = B(u, v) = B(w, u)

(2.199)

holds for all u G H. Note t h a t if B satisfies (2.197) and with some 7 > 0 ReB(u,u)>f\\ufH holds for all u G H, then clearly Theorem 2.7Al

2.8

(2.200) is applicable.

Some Interpolation Theory

T h e main reference for the general theory of interpolation of function spaces is the monograph [299] by H. Triebel, much material is also included in his books [300]-[301]. But we will give more special references which seems to us easier to follow when the interest is limited to more concrete situations. T h e most classical interpolation result is the M. Riesz-Thorin convexity theorem.

70

Chapter 2 Essentials from Analysis

T h e o r e m 2 . 8 . 1 A. Let X be a Hausdorff topological vector space containing all spaces L P (R™) ; 1 < p < oo, in the sense of continuous embeddings. Let T be a linear operator acting from X into X and for 1 < Pi,P2,Qi,Q2 < oo suppose that T\hvj{Rn) : L P ' ( R n ) -> L ^ ( R " ) is continuous, \\T\\hPj^qj =: Mj; j = 1,2. For 8 G [0,1] define pe and qg by 1

1-6*

Pe

Pi

e

1

-\

1-6i

6

9i

92

and qe Pi

s Then T LPS(R") : V

n

- + —.

(R™) - > Lqe' ( R ) is continuous

(2.201) with operator norm Me

satisfying Me < M{- 6 Me2.

(2.202)

B. Let 1 < Pi,P2,9i,92 < oo and Zei T be a linear operator from L P l ( R " ) fl LP 2 (R") to L « ( R n ) + L * » ( R n ) satisfying H T H ^ ^ = Mj < oo, j = 1,2. Then for pe and qe as in (2.201) the operator TILPIRLM satisfies the estimate ||Tu||„ < M}-9M!\\u\\pe and therefore extends to a continuous

(2.203) operator from LPe(R™) to L 9 s ( R " ) .

P a r t A of this theorem is taken from [299] and P a r t B is taken from E.B. Davies' book [66]. We need a generalisation of Theorem 2.8.1 which is due t o E.M. Stein, for a proof see [288]. Let (ft, A, fj) be a measure space and denote for a moment by simp(fi) t h e set of simple functions on $7, i.e. all functions f : CI —>• C such t h a t there exists a finite number of measurable sets Ai £ A, I = 1,... ,k, with t h e property t h a t n(At) < oo, i\Al = a G C, and f(z) = 0 for x € O \ (Uf=i AfiD e f i n i t i o n 2 . 8 . 2 Let ( U ( z ) ) 2 e S , G = {z G C | 0 < Re z < 1 } ; be a family of linear operators U(z) : simp(fi) --> simp(fl) n L 1 ( f i , / i ) where (Q,.A,/z) is a measure space. Suppose further that for all f, g G simp(fi) the function z t-> /

(U(z)g)(s)f(*) dx

Jn

is bounded in G and analytic in G. Then ( U ( z ) ) 2 e ^ is called an analytic family of operators (with respect to G and (fi, A, jj))-

2.8

71

Some Interpolation Theory

T h e o r e m 2 . 8 . 3 Let ( U ( Z ) ) Z € Q be an analytic family of operators and for 1 < Pi,P2 < oo suppose for all f G simp(fi) ||U(z)f|| LP1 < M i U f H ^

for R e z = 0

(2.204)

||U(z)f|| LP2 < M 2 ||f|| LP2 for Rez = 1.

(2.205)

and

For pe, 9 G [0,1], defined by j-g = ±f- + £ . Then it follows for all f G simp(ft) that ||U(0)f||LPe<M^M|||f||LPe.

(2.206)

Let us discuss more detailed the complex interpolation method which we will apply later on. We follow now closely the presentation of M. Schechter [262]. For a domain G C C, i.e. a n open and connected set, and a Banach space (X, \\.\\x) we denote by W(G; X) the set of all continuous functions f : G -4 X such t h a t f is analytic in G and p u t ||f||w = s u p | | f ( Z ) | | x .

(2.207)

z€G

The space (W(G\X),

||.||^y) is a Banach space and

||f||w = m a x | | f ( z ) | | x .

(2.208)

z&dG

Further we have L e m m a 2 . 8 . 4 If G = {z £ C | 0 < Re z < 1 } and f G W(G; X) we have \\i\\w = m a x sup ||f(fc + iy)\\x.

(2.209)

fc=0,l y£R

L e t (Xo, II-IIXQ) D e a Banach space and X\ C Xo a dense subspace such t h a t there is a norm | | . | | x turning (Xi, \\.\\x ) into a Banach space satisfying Mxo

^ c\\x\\Xl

f o r a11 x e x

(2-210)

i-

Once more let G = {z G C | 0 < Re z < 1 } and denote by 7i(XQ, Xi) t h e set of all f G W(G; X0) such t h a t f(l + iy) G Xx for all y G R, and p u t ||f\\ n = m a x sup j|f(fc + iy)\\x. fc=0,ly£K

< oo.

(2.211)

72

Chapter 2 Essentials from Analysis

It follows t h a t ||f||w
foraUfG«(X0,Xi)

||.|| w ) is a Banach space. Next let 6 G [0,1] and define

Xg := [X0, X^e

:={x€X0\x

= f(0) for some f e H(X0,

X{)} , (2.212)

and \\x\\

:=

inf

||f||w.

(2.213)

fewfXo.Xj)

D e f i n i t i o n 2 . 8 . 5 PFe ca/Z ([Xo,-X"i]e, | | . | | x ) the space obtained interpolation from (Xo, \\-\\x )

by complex

an

d C^ii ll-llx )•

L e m m a 2.8.6 The space ([Xo,X\}g,

\\-\\x ) is a Banach

space.

Further we have T h e o r e m 2 . 8 . 7 Let (XQ,\\.\\Xo) and ( X i , | | . | | X i ) be two Banach spaces as an< above and further let (Yo, W-Wy ) ^ (^i> ll-lly ) be ^wo Banach spaces satisfying the same conditions as XQ and X\. Suppose that T : XQ —> Yo is a bounded linear operator such that Tx G Yfc for x G Xk, and \\Tx\\Yk

< Mk\\x\\Xk,

A; = 0 , 1 .

Then T maps Xg continuously \\Tx\\Ye

< MteMf\\x\\X9,

(2.214)

into Yg and we have the 9 G [0,1].

estimate (2.215)

So far, we interpolated norm-continuous operators. In some cases only w e a k type estimates do hold and we have to work with the interpolation theorem of J. Marcinkiewicz [220], We need first D e f i n i t i o n 2.8.-8 Let T : LP(K n ) ->• B(Rn), 1 < p < oo, be an operator necessarily linear). For 1 < q < oo we call T of weak-type (p, q), if \^{x&Wl\\Tu(x)\>a}
.

(not

(2.216)

2.8

Some Interpolation Theory

73

T h e well known Chebyshev inequality which holds for u £ L p ( R n ) , i.e. the inequality X^{\u\>a}<^J\n(x)\"dx,

(2.217)

implies t h a t an operator T : L p ( R n ) —> L 9 ( R " ) which is bounded is also of weak-type (p, q). Let us denote by L P l ( R n ) + L P 2 ( R n ) all measurable functions f : R n -> C having a decomposition f = fi + f2 with f,- £ Lp^ (R™). T h e o r e m 2 . 8 . 9 ( M a r c i n k i e w i c z ) Let 1 < r < oo andT be an operator from V-iW1) + L r ( R n ) to B(Rn) which is subadditive, i.e. |T(u + v)| < |Tu| + |Tv|. IfT is both weak-type (1,1) and weak-type (r,r), we have for all p, 1 < p < r ||Tu||LP
(2.218)

In particular we have C o r o l l a r y 2 . 8 . 1 0 Let T : L ^ R " ) + L r ( R " ) -> B ( R n ) be of weak-type (1,1) and continuous from L r ( R " ) into itself, then T : L P (R") ->• L p ( R n ) is a continuous operator for all p, 1 < p < r . A proof of Theorem 2.8.9 can be found in the monograph [284] of Chr. Sogge, Theorem 0.2.5. One should also consult E.M. Stein and G. Weiss [289], Chapter IV.

Chapter 3

Fourier Analysis and Convolution Semigroups This chapter is devoted to the study of the Fourier transform acting on functions and distributions. In particular we are interested in questions related to positivity of translation invariant operators. After handling the fundamentals we discuss more detailed convolution semigroups of sub-probability measures and continuous negative definite functions. We will work out the Levy-Khinchin formula and rather detailed Bochner's theory of subordination. Most of our considerations later take place in certain function spaces defined with the help of continuous negative definite functions. These spaces are introduced, and further we present some important results for Besov and Triebel-Lizorkin spaces. Finally we have to discuss Fourier multipliers.

3.1

T h e Fourier Transform in S(Rn)

We start with the basic Definition 3.1.1 Let u £ iS(R n ). The Fourier transform of u is defined by u ( 0 := (27r)-"/ 2 f

e-ix^u{x)

Ax.

(3.1)

Sometimes we will write Fxl_y£(u)(£) or F(u)(£) for u(£). Note that for any £ G R n the function x >-> e~ix
Chapter 3

76

Fourier Analysis and Convolution Semigroups

T h e o r e m 3 . 1 . 2 The Fourier transform F is a continuous «S(R n ) into itself.

linear mapping

from

Proof: First we will show t h a t for u G S(Rn) its Fourier transform u belongs to <S(R") too. For this we make use of the family of seminorms (patp)a,p€Ns introduced in (2.115). For a,/3 G Nft we find

= (2n)-nl2

^ 8 f ( e _ f a ' € ) u ( x ) dx

f JR™

= (27r)- n / 2 f

f(-ix)ae-ixiu(x)

= (27r)-"/ 2 f

d%{e-ix*)iW{-ix)au{x)

dx

dx

JR"

= (27r)-"/ 2 /

( - t ) l / s | + | a | e - f a ' € 5 f ( x a u ( i ) ) dx,

which already implies p Q i / 3 (u) = sup | £ ^ u ( 0 | < (27T)-"/ 2 /" C6Mn

|d£(x«u(x))| dx < oo,

->Rn

i.e. u G S ( R n ) . But now the linearity of the Fourier transform as an operator from <S(Rn) into <S(R") is obvious. Finally, we prove t h a t F : <S(Rn) ->• <S(Rn) is continuous. For this, note t h a t by Leibniz' rule (2.19) we have dP(xau(x))

= ^

(f3)(d-'xa)df}-^xx(x),

and t h a t (1 + |x| 2 ) "2 ' dx = cn < oo.

/ JR

n

77

3.1 The Fourier Transform in <S(R") Thus we get

/ la^uOc))! dx < Cn £ (p) sup ((i+|a:|2)2*1|^(a!a)| \de-Mz)\) 7
< c

max

p /3 _ 7)(7 (u),

|<7|
implying the continuity of F in <S(Rn). • Corollary 3.1.3 For a, (3 € N£ and u G <S(Mn) ^ D f u t f ) = ( - l ) H (D£(^u(.)) A ) ( 0

(3.2)

Proof: Using the calculation of the proof of Theorem 3.1.2 we find ^ D £ u ( 0 = (27r)-™/2(-l)l°l f

{-i)-Wd%(e-ix-t)xau(x)

dx

n

JR

= (27r)-" / 2 (-l)l Q l j

(-i)We-ix-tdP(xau(x))

dx

n

JM.

= (2TT)-"/ 2 (—l)'"! f

e-ix^{xau(x))

dx

JR"

= ( - 1 ) H F ^ {D2(xau(x))

(0-

•

Let P(D) = X^ia|<m a aD Q be a linear differential operator with constant coefficients aa £ C. Using Corollary 3.1.3 it follows for u € <S(R") that (P(D)(u))A(0=P(0u(0-

(3-3)

L e m m a 3.1.4 The function g(i) = e - l x l / 2 is a fixed point ofF, i.e. m

= e-'^/2.

(3.4)

Proof: Since j

e -*x.£ e -|x|V2

dx = f f / e - ^ ^ e - ^ / 2

dx

78

Chapter 3 Fourier Analysis and Convolution Semigroups

it is sufficient t o consider the case n = 1. The function g (now depending only on x £ R) belongs t o <S(R) and satisfies the differential equation iT>xg(x) + xg(x) = 0, but Corollary 3.1.3 implies for g

£g(0+*D € g(0=0, which gives g(£) = ce~^ / 2 . However, the constant c is determined by = g(0) = (27T)- 1 / 2 / e - * 2 / 2 dx = 1

D

Our next aim is to prove that F is a bijective mapping on <S(R") with a continuous inverse. D e f i n i t i o n 3 . 1 . 5 On <S(Rn) we define the inverse Fourier transform by (F-1U)(T7) := (2TT)-"/2 /

eir>yu(y)dy.

(3.5)

(F _ 1 u)(r/).

We also will use F~J^„(u)(»j) for denoting

T h e proof of the following result is almost identical with t h a t of Theorem 3.1.2. T h e o r e m 3 . 1 . 6 The inverse Fourier transform tor from <S(R") into itself.

is a linear continuous

opera-

We now justify the name inverse Fourier transform for F _ 1 . T h e o r e m 3 . 1 . 7 The Fourier transform F is a linear bijective and continuous operator from <S(Rn) into itself which has a continuous inverse given by (3.5). Thus on <S(R") we have F o F " 1 = F " 1 o F = id. P r o o f o f T h e o r e m 3.1.7: Let u, v £ 5 ( R n ) be given. It follows t h a t /

i i ( 0 v ( 0 e f a * d£ = ( 2 T T ) - " / 2 / = f

J

v(0efa*/

u(2/)(27r)-"/ 2 / u(y)(F v){y - x) dy

R"

Ju / (Fv)(z)u(z JRn

+ x)dz.

e^Mv)

*V <X

e-tv-^MOdtdy

3.1

79

The Fourier Transform in <S(R")

Taking v(a;) = e /

e

u ^ e ^

lx' / 2 it follows by Lemma 3.1.4 t h a t 1

eix^ A^ = e~n f =

e ^ u ( a : + ;z)dz

e~M2/2u(x

/

+ ey) Ay,

(3.6)

where we used the formula

F(g e )(O=e- n (Fg)0),

(3.7)

which holds for all g e «S(Mn), g e (a;) = g(ex), and follows by a substitution f

e-ixlig(ex)Ax=

ex e

[

e-^

^ g(ex)

Ax = e~n f

e-iy*/'g(y)

Ay.

Applying the dominated convergence theorem to (3.6), for e —> 0, we get /

u ( 0 e t e * d£ = /

JR" which gives

e-^l 2 / 2 u(a;) dy = (27r)"/ 2 u(a;),

7R"

u(x) = (27r)- n / 2 /" e f a * u ( 0 d£. Thus we find F o F _ 1 = id, implying t h a t F is surjective. Interchanging the role of F and F _ 1 the calculation made above also implies t h a t F _ 1 o F = id, which gives the injectivity of F. D R e m a r k 3.1.8 Note that on S(Rn)

we have F 4 = id, or F " 1 = F 3 .

T h e following lemma collects some useful properties of the Fourier transform. L e m m a 3 . 1 . 9 A. LetTa : R n ->• R™ be the translation a £ R". For u G <S(R") it follows that (uoTa)A(0=e*"«u(0

operator Tax = a + x,

(3.8)

80

Chapter 3 Fourier Analysis and Convolution Semigroups

B. Let T : R™ —>• R n be a bijective linear mapping, then for u £ iS(R") the function u o T belongs to <S(R") too, and for its Fourier transform we find (u T)AK) =

°

pTJflo(rl)t(°-

(3 9)

-

In particular, for the reflection Sx = —x we have (uoS)A(0=u(-0,

(3.10)

and for the homothetic mapping Y{\ (x) = Xx we get (uoHA)A(0=A-"u(!Y

A>0.

(3.11)

C. For u e <S(R") we have fl(0=F-1(u)(0

(3.12)

Proof: A. For a e R™ we find (u o T a ) A ( 0 =

/

e- fa -«u(i + a) dx

= (27r)-"/ 2 /

e-'fo—^ufo) dy

(2TT)-"/ 2

= e ia -«u(0. B. From the discussion in Section 2.6 we know that for a bijective linear mapping T : R" -> R" the function u o T belongs to <S(R") for any u 6 Further we have (u o T ) A ( 0 = (27r)-"/ 2 /

e- f a - £ u(Ti) dx

-ReW^""72!^^1"^^ 1

|det T|

uo^-1)*^).

C. By the very definition of u we find u(£) =

(2TT)-"/ 2

/

= F-1(u)(0-

e-

te

-«u(x) dx = (27r)- n/2 f

•

e ix «u(x) dx

3.1 The Fourier Transform in S(R")

81

In the next paragraph we will examine the Fourier transform in some L p spaces. We will use the following results: Theorem 3.1.10 For all u G <S(R") M o o < (27T)

i/2 |

(3.13)

iLi

and (3.14) hold. Remark 3.1.11 The equality (3.14) is a first version of the Theorem of Plancherel. By polarisation we get immediately (3.15)

(u,v) 0 = (u,v) 0 all\i,veS{Rn).

for

Proof of Theorem 3.1.10: For u € <S(Rn) we have e-ix^u(x)dx

f n

JR

<

(2TT)-"/

2

f n

\u(x)\ dx = (27r)- n / 2 ||u|| L1 .

JR

To prove (3.14), let u,v G S(R"). It follows that f u(x)v(x)dx

= f

JRn

u(x)(27r)-"/ 2 f

=

(2TT)-"/ 2

e-ixiv{£)

d£dx

JRn

JR"

/ JRn

f

e~ixiiu(x)v(Z)

dx d£

JR"

= I u(0v(0 d£ JRn

Now replace v by u and as F(u) = F o F - 1 u = u we get (3.14). • We have already introduced the convolution of two functions u, v G i.e. the function (u*v)(x)=

u(x - y)v(y) dy

(3.16)

which is again an element in cS(Rn). Now we can prove the convolution theorem.

82

Chapter 3

Fourier Analysis and Convolution Semigroups

T h e o r e m 3 . 1 . 1 2 Let u , v e «S(R n ). Then we have (U.V)A(0 = (2TT)-"/2(U*V)(0

(3.17)

(u*v)A(0 = (27r)"/2u(0-v(0.

(3.18)

and

Proof:

Taking the function t/i-4e

(e-*(-«u, v ) 0 = /

ty

't\i(y)

and y (->• v(y) in (3.15), we find

dy = (2TT)"/ 2 (U • v) A (£),

e-^MvXv)

and on the other hand we have by Lemma 3.1.9.^4 and (3.12) (F(e-i(-«)u),F(v))0 = ^

F y ^ ( e - ^ u ( y ) ) ( 7 7 ) F ^ ( v ^ ) ) W dy

= (27r)-"/ 2 f

[

e-iyte-iyr
dyv(-r?) drj

JR™ V / R "

= / u(£ + r])v(-T}) dr) = / u(£ - 77)^(77) dr? JR" JR™ and (3.17) is proved. Taking in (3.17) u and v instead of u and v, respectively, we get

(u.v)A(o = (27r)-"/2((ur*(vr)(o or

(u • v) A (0 = (2TT)-"/ 2 F- 1 ((U) A * (v)A)(£) Denoting by w the function a; i-> w ( - : r ) = w(x), w G <S(K"), we find F(u) ( F ~ i ( u ) ) , i.e. (u) A = u. Hence F-1((u)A*(v)A)(0=F-1(u*v), but ii * v = (u * v). Since F _ 1 ( w ) — w, we finally arrive at (3.18). •

3.2

The Fourier Transform in L P (R"), 1 < p < 2

83

T h e Fourier Transform in L p (R n ), 1 < p < 2

3.2

In this paragraph we want to study the Fourier transform in L p ( R n ) , 1 < p < 2. In particular we will consider the cases p = 1 and p = 2. We will focus our attention on more or less classical results, but we emphasise t h a t all results proved in Section 3.1 do hold for the L p -case , too (with an appropriate interpretation). Let us start with the case p = 1. Since for u € L 1 ( R n ) the function x t-t e~lx'£\i(x) is an element of L 1 ( R " ) , its integral is well defined and we set u ( 0 = (27r)-"/ 2 f

e- i x -«u(x) dx.

(3.19)

Once again we will call u the Fourier transform of u e L X (R"). Since S{Rn) C L 1 ( R " ) , by (3.19) an extension of the Fourier transform as it was defined in the last section is given. Furthermore, estimate (3.13) holds and leads to the Lemma of Riemann-Lebesgue T h e o r e m 3 . 2 . 1 The Fourier transform L ^ R " ) into Coo(R") and ||u||00<(27r)-"/2||u||L1 holds for

is a continuous

linear operator

from

(3.20)

allueL'fR").

Proof: We know already t h a t (3.20) holds for u e <S(R n ). But the density of <S(Rn) in L ^ R " ) implies (3.20) for all u € L ^ R " ) . Moreover, since <S(R") is dense in Coo(R") with respect to the norm H-H^ and the Fourier transform is bijective on 5 ( R n ) , (3.20) says t h a t the Fourier transform u of u £ L 1 ( R n ) can be approximated with respect to the norm ||.||TO by elements from 5 ( R n ) , implying t h a t u e C 0 0 ( R " ) . T h e linearity of F is obvious. • T h e set of Fourier transforms is a "small" set in Coo(R n ). For example, a recent result of K. Karlander [177] states t h a t if a closed reflexive subspace of Coo(R) consists entirely of Fourier transforms of L 1 (R)-functions, then this space must be finite dimensional. We give a concrete example of a Fourier transform which we need later on.

84

Chapter 3 Fourier Analysis and Convolution Semigroups

Example 3.2.2 Let A > —1 be a real number and denote by ^\ : R —>• R the function x >->• ^A(X) = |a;jAe—I31'. For its Fourier transform we have X+l

*A(0

2

= ^ ^ ( ^ y

cos((A + l)arctanO.

(3.21)

We refer to [111], 17.34.7. There is no problem to carry over Lemma 3.1.9 to the new situation. Thus we have with the notation of that lemma: Lemma 3.2.3 For u <E L^R") we have ( u o T o ) A ( 0 = e»'«u(0;

(3.22)

(ioT)A(0 = | ^ | f i o ( T - 1 ) t ( 0 ;

(3.23)

(uoS)A(0=u(-0;

(3-24)

(uoHA)A(0 = A - " u ^ j ,

A>0;

(3.25)

u]|) = F- 1 (u)(0-

(3-26)

Moreover, we have one part of the convolution theorem. T h e o r e m 3.2.4 For u,v e L J (R n ) it follows that ( u * v ) A ( 0 = (27r)"/ 2 u(0-v(0

(3.27)

holds. Proof: By Fubini's theorem we have (u*v)A(0 = (27r)-n/2/ JRn

= (27r)- n/2 / JR"

= u(0/ JR"

[

e-ixiu(x-y)-v(y)dydx

JRn

/

e-^x~yHu(x

- y) • e-iyiv{y)

Ay dx

JRn

e- iyf v(2/)d2/ = (27rr/ 2 u(0-v(^).

D

3.2 The Fourier Transform in LP(R"), 1 < p < 2

85

Note that for u, v € L^R") we have u * v G L^R") by Lemma 2.3.15.^1, however u • v is in general no longer an element in L 1 (R"). Moreover, since in general u G L^R") does not belong to L 2 (R"), the proof of Theorem 3.1.12 can not directly be used to prove (3.27). The following consideration shows the intimate relation between the smoothness of a function u G L 1 (R") and the decay of its Fourier transform at infinity. Definition 3.2.5 Let u G L p (R n ), 1 < p < oo. We say that u has a partial derivative with respect to Xk, 1 < k < n, in the sense of the space L p (R n ), if there exists a function g G L p (R n ) such that lim h-+0

u(x + hek) — u(x) \JR"

l/p

lW

dx

(3.28)

= 0

holds. Obviously, for u G C^R") such that u, J ^ G L P (R"), it follows that J ^ and the partial derivative of u with respect to Xk in the sense of the space L p (R n ) coincide. Theorem 3.2.6 Let u G LX(R™) have the partial derivative with respect to Xk, 1 < k < n, in the sense of the space L 1 (R n ) which we denote by g. Then we have (3.29)

g(0 = iikKi)Proof: Using (3.22) we find

g(0
h

J=g(0

{

h

-«>)*<*

-<)V(a

_/gG)_u(. + ^ ) Thus it follows with (3.20) that

m - a(0

h

<(2TT)

I/2

implying for h -> 0 the validity of (3.29). D

u ( . + /iejb) - u ( . ) g(-)

h

L1

86

Chapter 3 Fourier Analysis and Convolution Semigroups

Let u € L1(M), n = 1 is taken for simplicity, and suppose that the first derivative of u in the sense of the space LX(R) exists and denote it by g € L 1 (R). For |£| > 1 it follows from (3.29) and Theorem 3.2.1 that

m)\

< |^|g(OI < i|p

(3-30)

implying that u has to tend to zero as |£| tends to infinity (with a well determined decay) in order that the derivative exists. Of course, it is possible to iterate the result of Theorem 3.2.6, but we will not do this now. However, we will prove a type of converse to Theorem 3.2.6. Theorem 3.2.7 Let u G L 1 (R n ) and suppose that for some 1 < k < n the function x i-* xkn{x) belongs to L 1 (R") too. Then J^r- exists and we have ^

= -iFx^(xku(x))(Z).

(3.31)

Proof: Using (3.22) we find u(£ + hek) - u(£) h

1 h

u(x)

(0.

and by the dominated convergence theorem we get

h

u{x)

( 0 —>•

[-ixku(x)]*{Z),

implying (3.31). O We know already that for u G L x (R n ) its Fourier transform u is an element of C00(R71), but in general it does not belong to L J (R n ) as is easily seen by the Fourier transform of the characteristic function \Q °^ a C U D e Q = [ai>^i] x . . . x [an, bn) which is given by

xQ (o=(27T)-/2 n e " " " v ""• k=i

i£k

(3-32)

Thus, in general for u £ L 1 (R") it is not possible to define an inverse Fourier transform for u by (3.5). However, it is reasonable to consider the inverse Fourier transform on the Wiener algebra.

3.2 The Fourier Transform in LP(R"), 1 < p < 2

87

Definition 3.2.8 The Wiener algebra A(Rn) is defined by A{Rn) := {u G L^R") | u e L^K") } •

(3-33)

We want to study A(Rn) more closely L e m m a 3.2.9 For u, v G L^R 71 ) we have [

u(Ov(0 d£ = /

JR™

u(Ov(0 d£

(3.34)

JR™

Proof: Fubini's theorem implies / fi(£)v(0 d£ = ( 2 < T n / 2 / = (2TT)-"/2/

= /

(7

e - ^ u ^ ) d x ) v ( 0 d£

(/

v ( O e - " « d ^ u(z) dz

v(x)u(x) dx.

•

JR"

Now let $ G C 00 (R n ) n L^R") such that $(0) = 1 and set <j> := $, hence we have J"K„ <j>{x) Ax = (27r) _n / 2 . With e(x) = £_n(|), e > 0, we find using (3.25) ($ o H £ ) A (0 = £ " ^ ( f ) = & ( 0 -

(3-35)

Lemma 3.2.10 Let u G L 1 (R n ) and let <& and be as above. For any e > 0 we have f u ( 0 e ^ « $ ( £ 0 ^ = /" u(j/)&(j,-a:)dj,. JM.n

(3.36)

JR"

Proof: We only have to take in (3.34) instead of v the function x H-> e i y ? ( $ o

He)(0- ° T h e o r e m 3.2.11 Let $ and <j> be as in (3.35) and for u G L^R") we set M £ (u)(x) := (2TT)-"/ 2 f

u(C)e"'«$(£0 d£.

JR"

It follows that lim ||Me(u) - u|| L1 = 0 £->0

(3.37)

88

Chapter 3

Proof:

Fourier Analysis and Convolution Semigroups

By (3.36) we have

Me(u)(a:):=(27r)-n/2u*^, thus it remains to prove lim||(27r)-n/2u*
=0,

but this is just the statement following Proposition 2.3.17. D C o r o l l a r y 3.2.12 For u 6 A(Rn) u(x) = (27r)-"/ 2 (

we have for almost all i g l "

efa*u(0d£.

(3.38)

Proof: We know t h a t M e (u) converges to u in L ^ R " ) . Therefore we can find a subsequence (si)tew such t h a t M e , (u) converges almost everywhere to u. But for u e A(M.n), it follows by the dominated convergence theorem t h a t M E ,(u) converges to (2TT)-"/2/

ete-«u(0d^

for I —> oo. • From Corollary 3.2.12 we deduce C o r o l l a r y 3 . 2 . 1 3 The Fourier transform is an injective mapping from L 1 ( R " ) into C o ^ R " ) . Proof: By linearity we have to prove t h a t u G L 1 ( R n ) and u = 0, thus n u e A(R ), implies u = 0. But this follows from (3.38). D Now we can examine ^4(R") more closely. From (3.38) we conclude t h a t u £ A(W) is equivalent to an element in C 0 0 ( R n ) and in the following we will always work with t h a t representative, i.e. we consider A ( R n ) as a subset of Coo(R n ). Clearly, ^4(R n ) is a linear space and we may introduce the norm

lluIU(R") : = IMILHR") + H U IILI(K")-

(3-39)

3.2 The Fourier Transform in L p (R n ), 1 < p < 2

89

Lemma 3.2.14 A. A function u G 1/(1$™) belongs to A(Rn) if and only if u belongs to A(Rn). B. Ifu G A(Rn), then u G C ^ R " ) and ||u|| 0 O <(2 7 r)-"/ 2 ||u|U ( K „ ) .

(3.40)

C. For 1 < p < oo we /ioue ^ ( R n ) C LP(R n ). Proof: A. The inversion formula (3.38) implies u(-x)

= (27r)- n / 2 f

e" i a : «u(0 d£ = (F2u)(a;).

(3.41)

JU"

Further x M- u ( - z ) belongs to L^R") if and only if u G L ^ R " ) . Thus u G A(Rn) implies that u and F 2 u belongs to L^R"), i.e. u G A(Rn). Conversely, u G A(Rn) gives u G L^R") and F 2 u G L : (R n ) implying that u and u are elements of L 1 (R n ), i.e. u G A(R n ). B. This is an immediate consequence of (3.38). C. For a G 4(R n ) we find /

\vL(x)\"dx

< IIUH^1 /

\u(x)\"dx

< IIUII^^IUIU^).

D

Next we want to justify the name algebra for yl(R n ). Theorem 3.2.15 Let u, v G A(Rn). we have

Then u* v andu-y

belong to A(Rn) and

(u * v)A ( 0 = (27r) n/2 u(0 • v ( 0 ,

(3.42)

and ( U . V ) A ( 0 = (2TT)-"/2(U*V)(£).

(3.43)

Proof: Since u,v G L^R"), we know already (3.42), see Theorem 3.2.4. We prove next that (u * v) A G L^R"). Using (3.42) we find ll(u*v) A || L i(R-) = (2 7 r)-"/ 2 ||u.v|| L 1 ( R „ ) < (27r)-"/ 2 ||u|| 00 ||v|| L1(K „ ) < (27r)-"/2||u|| L1(Rn) ||v|| A{RB) . Further we observe that by (3.41) and (3.42) (u • v)(-a;) = (27r)- n / 2 (u * v)A(a;),

90

Chapter 3

Fourier Analysis and Convolution Semigroups

and applying the inversion formula yields (27r)-™/2 /

(u • v)(-x)eix-t

dx = (2n)-n/2(u

* v)(£),

hence (u-v)A(0 = (27r)-"/2(u*v)(0.

Since <S(Rn) C A(Rn),

•

we get immediately

T h e o r e m 3 . 2 . 1 6 The Wiener algebra is dense in L P ( R " ) , 1 < p < oo, and in Coo(]R n ). For a function u G L 2 ( R n ) the integral (27r)-n/2/

e-™*u(x)dx

(3.44)

does not in general converge. Thus the Fourier transform can not be defined for u G L 2 ( R n ) by (3.44). To get a reasonable extension of the Fourier transform from L x ( R n ) n L 2 ( R " ) onto L 2 ( R " ) , we will make use of Plancherel's theorem. For this we note t h a t the proof of (3.14) is also valid for u G L ^ R " ) n L 2 ( R " ) , i.e. we claim C o r o l l a r y 3 . 2 . 1 7 For all u G L ^ R " ) n L 2 (R") we have ||u|| 0 = ||u|| 0 .

(3.45) 2

Proof:

n

From Lemma 3.2.9 it follows for u, v G L ^ R " ) l~l L ( R ) t h a t

/ fl(0v(0 d£= [ u(0v(0 dc Now let ( U ^ ^ N be a sequence in <S(Rn) converging to u in L 2 ( R " ) and set

/„(£) := (2^)""/2 /

eix^(x)

dx.

JR" By Theorem 3.1.7 we have vv(x) = u„(a:), implying t h a t v„ converges to u in L 2 ( R " ) . Therefore, we may take the limit in the identity / JRn

\uI/(x)\2dx

= [

|u„(0|2d£,

JRn

which is derived as (3.15). •

3.2

The Fourier Transform in L"(R n ), 1 < p < 2

91

T h e o r e m 3 . 2 . 1 8 The Fourier transform as it is defined on L ^ R " ) n L 2 ( R " ) has an extension to L 2 ( R " ) . This extension is an isometry on L 2 ( R " ) which is bijective and has a continuous inverse. Proof: From Corollary 3.2.17 it follows t h a t F maps L ^ R " ) n L 2 ( R " ) into L 2 ( R n ) . Since L x ( R n ) n L 2 (R") is dense in L 2 ( R n ) we can extend F onto L 2 (R") such t h a t (3.45) remains valid. Hence, this extension is an isometry and especially it is injective. Let us denote this extension for a moment by F. Since A(Rn) C L ^ R " ) n L 2 (R") we find FXL^R") n L 2 ( R " ) ) D F ( A ( R n ) ) = F ( A ( R n ) ) =

A(Rn),

implying t h a t F ^ R " ) n L 2 ( R n ) ) is dense in L 2 ( R n ) . Since F is an isometry F ( L 2 ( R n ) ) is a closed subspace of L 2 ( R n ) . Hence it must coincide with L 2 ( R n ) . Thus F is surjective. We can determine the inverse of F on the dense set A(M.n), and we find u(x) = ( 2 < T " / 2 /

e i x ? u ( 0 d^ = (2TT)-™/ 2 /

e«-«
and on L 2 ( R n ) we have F-1(u)=f(n), which gives also the continuity of F _ 1 .

(3.46) •

In the following we will denote the Fourier transform of u G L 2 ( R " ) as it is obtained from Theorem 3.2.18 again by Fu or u. R e m a r k 3 . 2 . 1 9 Let u e L 2 ( R n ) and (v I / ) l / € N be any sequence in L 1 ( R n ) n L 2 ( R " ) converging in L 2 ( R n ) to u. Then the Fourier transform u of u is the L2-limit of the sequence ( v i , ) ^ ^ . In particular we may take the sequence v v : = X a (oi ' u > ^ = 1) 2 , . . . . It turns out that u is the \?-limit of the sequence of functions given by v „ ( 0 = (27r)-"/ 2 f e- f a '«u(x) dx. J\x\
(3.47)

So far we have proved t h a t the Fourier transform is a continuous operator from L ^ R " ) into C 0 0 ( R n ) c L°°(R n ) with norm equal to (27T)""/ 2 and it is

Chapter 3 Fourier Analysis and Convolution Semigroups

92

also a continuous operator from L 2 (R") into itself with norm equal to 1. For 9 e [0,1] we set

PO = TTe

and p e =

'

rb'

(3,48)

hence, 1 < pe < 2 and — + \ = 1. Now a straightforward application of the Riesz-Thorin convexity theorem, Theorem 2.8.1, gives the Hausdorff-Young theorem: T h e o r e m 3 . 2 . 2 0 The Fourier transform extends to a continuous linear mapping from L P e ( R n ) into LP«(R™) with norm less or equal to (2n)~n2. Many properties of the Fourier transform in L P (]R"), 1 < p < 2, can be obtained by extending F first to the space of tempered distributions <S'(R") which we will do in the following section.

3.3

T h e Fourier Transform in <S'(Rn)

We extend the Fourier transform from <S(Rn) to <S'(R n ) by duality. On <S'(R n ) we will always consider the weak-*-topology. Hence, by the consideration of Section 2.2, in particular by (2.46), we know t h a t every neighbourhood U of 0 e S ' ( R n ) contains a neighbourhood V of 0 G <S'(R n ) of the form V = {u G S ' ( R " ) | |(u, j)\k G <S(R")} .

(3.49)

Moreover, a sequence (u 1 / ) I / £ N , u„ G <S'(R n ), converges in the weak-*-topology to u G <S'(R n ) if and only if (u,) f o r a l l ^ G < S ( R n ) .

(3.50)

D e f i n i t i o n 3 . 3 . 1 Let u G <S'(R n ). The Fourier transform u o / u i s defined by (u, 4>) : = (u, <j>) for all <j> G 5 ( R " ) .

(3.51)

As usual we use also the notation Fu for u. Since F : <S(R") -> 5 ( R n ) is continuous and by definition u G S ' ( R n ) is a continuous mapping u : <S(R") -> C, it follows t h a t u : <S(Rn) —> C is continuous as a composition of continuous mappings. Hence, we have

3.3 The Fourier Transform in S'(Rn)

93

Corollary 3.3.2 The Fourier transform of u G <S'(Rn) is an element in S'(Rn). Before we continue our studies of the Fourier transform in <S'(R"), we show that it is an extension of the Fourier transform as it was defined in L 1 (R n ). Indeed, for g G L^R") C <S'(Rn) and cf> G <S(Rn) we find by Lemma 3.2.9 (g, ) = (g,
g(x)4>{x) dx =

g(x)4>(x) dx,

which shows that g in the sense of «S'(Rn) coincides with g as it is defined in L 1 (R n ). Note further that since convergence in L 2 (R n ) implies weak-*convergence in S^R"), we may deduce from Remark 3.2.19 that the Fourier transform as defined on <S'(Rn) by Definition 3.3.1 also extends the Fourier transform as defined on L 2 (R n ) by Theorem 3.2.18. On <S(Rn) we proved the following relation for the Fourier transform: F 4 = id and therefore F " 1 = F 3 .

(3.52)

Theorem 3.3.3 The Fourier transform is a continuous linear operator from S'(R") into itself which is bijective and has a continuous inverse F _ 1 . Proof: Let U be a neighbourhood of 0 in 5'(R") and let V C U be a neighbourhood of 0 G <S'(R") given by (3.49). We define W := {u G <S'(Rn) | | ( u , ^ ) | < l f o r 0 1 , . . . , ^ G < S ( R n ) } , which is a neighbourhood of 0 G <S'(Rn). From the definition of u it follows that u G V whenever u G W, implying the continuity of F : <S'(R") ->• <S'(R"). The linearity is trivial. Furthermore, we find using (3.52) that

(F 4 u,^ = (u,FV) = {ui>, i.e. F 4 u = F 3 (Fu) = u for all u G <S'(Rn) which gives the injectivity of F. In addition u = F(F 3 u), i.e. F is surjective, and F _ 1 = F 3 which also implies the continuity of F _ 1 . •

Chapter 3

94

Fourier Analysis and Convolution Semigroups

E x a m p l e 3 . 3 . 4 We know that the Dirac measure e belongs to <S'(R"). For its Fourier transform we find
[

e'^^x)

dx

1 • 4>(x) dx = (27T)-"/ 2 (1,),

hence i = F(e) = (27r)-"/ 2 . On t/ie otner hand, the function Fourier transform we get (F(l),) = (1,F(0)) = /

(3.53) x t-¥ 1 belongs to S'(M.n) too. Calculating

its

( F 0 ) ( O d£ = ( 2 7 r ) " / ^ ( 0 ) ,

i.e. F ( l ) = (27r)™/2£.

(3.54)

Theorem 2.6.9 says t h a t for u € <S'(R n ) and £ <S(Rn) the convolution u*cj> is always defined and gives an element in 5 ' ( R n ) which is a smooth function with at most polynomial growth. Thus (u * )A and u * <j> a r e defined. T h e o r e m 3.3.5 For u € S'(Rn)

and £ 5 ( R n ) we have

(u * <j))A = (27r) n/2 (^ • u

(3.55)

( • u ) A = (27r)-"/ 2 u * 4>.

(3.56)

and

Proof: By Lemma 2.6.8.2? we know t h a t 5 ( R n ) is sequentially dense in «S'(R") and therefore the theorem follows from Theorem 3.1.12. • Recall t h a t for a diffeomorphism T : R n ->• R n the pull-back of u e £>'(R") is denoted by T*u, see (2.131). T h e o r e m 3 . 3 . 6 A. Let u € 5 ' ( R n ) and T : R™ ->• R " be a bijective mapping. Then T*u G <S'(R n ) and

(T U)A =

*

ReW ( T ~ 1 ) t r , i -

linear

(3 57)

'

3.3 The Fourier Transform in <S'(R")

95

B. Let u G <S'(Rn) be rotationally invariant, i.e. R*u = u for any R G 0(n). Then u is also rotationally invariant. C. Let u G <S'(Rn) be a homogeneous distribution of order p G R. Then its Fourier transform is homogeneous of order —n — p. In particular, when u is homogeneous of order 0, then u is homogeneous of order —n. Proof: A. By (2.131) we have for G (T*u ,)

="

(3.58)

|detT| < U ' ( T • ^ v )

which gives | < T * i *,) \

<

1

c•

|Uei-L|

max

al

k a

,01-,...,l3k

paJ,pj(4>o(T *)),

implying the continuity of T*u on <S(Rn), i.e. T*u G <S'(Kn). Moreover, we have {{T*n)\4>) = (T*U,4>) = j

~ (u, (T-1)*<;

From Lemma 3.1.9.5 it follows that (T-1)>=|detT|(0oT*)A, hence <(T*u)A, $ = o T«)A> = (u, 4> o T«) = ^

( ( ( T - 1 ) ' ) ^ , 0) ,

and (3.57) is proved. B. This is an easy consequence of part A. In fact, we have for a rotation R G 0(n) that (R - 1 )* = R and |detR| = 1. By assumption u G S'(Rn) is rotationally invariant which yields

u = (^)A = p ^ ( ( ^ r u = R*u. C. Let <j> e S(Rn) and for A > 0 set as usual H ^ = (j> o HA, HA(a;) = Ax, A > 0. It follows from Lemma 3.1.9.5 that (H^) A (O = A-"H;

A«i

96

Chapter 3 Fourier Analysis and Convolution Semigroups

and since H^u = A^u, i.e. (u, H*x) = A

n

= A - n ( u , H J / A 0 ) = A - " '

l

p

\

(u, <j>), we get -n-p

or (u,H^)=A"(u, M™, x i-> Sz = -a;, we / m d / o r i/ie Fourier transform of S*u (S*u) A = S*u. E x a m p l e 3.3.8 For function of degree 2a transform exists, and which is homogeneous

(3.59) 0 < a < 1 the function x M- | x | 2 a is a homogeneous and it is also an element in <S'(R n ). Thus its Fourier it must be a rotationally invariant element of <S'(R") of degree —n — 2a. It can be shown that

F x ^ ( | x | 2 a ) ( 0 = cn,a\t\-2a~n

(3.60)

holds. Let u G 5 ' ( R n ) and g € C°°(R n ) be a polynomially bounded function. It follows t h a t g • u G iS'(M n ), hence (g • u ) A exists. C o r o l l a r y 3.3.9 Let u G S'(Rn)

and a,(3 G Nft. Then we have

( D a u ) A = f*u

(3.61)

(^u)A = (-1)1^11.

(3.62)

and

Proof: Both equalities follow from Corollary 3.1.3 and the definition of D a u and x^u, respectively. • C o r o l l a r y 3 . 3 . 1 0 LetTa : M n ->• K n be the translation operatorTax = a + x, a G R n . For u G S ' ( K n ) the distribution T*u belongs to S'(M.n) and we have (1»

A

= eia
(3.63)

3.3

The Fourier Transform in <S'(Rn)

97

Proof: For <j> G S(Rn) we find (T*u,) = (u,T*_a), which proves t h a t T*u G «S'(R"). Further we find

But T_ a (<£) = fa - a) = ( ^ e f a - ° ) A ( 0 , hence <(T» A ,) = ( u , ( 0 e f a - ) A ) -


which gives (3.63). D Note t h a t the proof of Corollary 3.3.9 yields also ( u e " ' a ) A = T!. 0 u.

(3.64)

E x a m p l e 3 . 3 . 1 1 We know already that the function sign, x >-> s i g n s , gives a tempered distribution on R and that d(sign) = 2e. Thus we find for its Fourier transform ^ ( s i g n ) A ( 0 = 2(27r)- 1 /2, or 2(27r)-V2 A m (sign)A(0 = ^ ^ v.p. 1

(I V?

w/iere u is a distribution supported at 0. But (sign) degree —1, hence u = c"£: &?/ Theorem 2.6.13, i.e. . .

,

A

2 ( 2 T T ) - 11 / 2

_

A

(sign) (0 = - i — j

5?/

2(2

/ /Ir \

m«s£ fee an odd

^ " 1 / 2 v.p. Q ) .

using i/iai (u.) (x) = u(—s), we find for the Fourier transform

v p

1^A

- - u J ; ( 0 = ^ F 1 ^ sign(0,

since sign is an odd

function.

of

v.p. ( - J + ce,

On £/ie oi/ier /land, 6y Remark 3.3.7, (sign) implying that c = 0. TTius we have (sign)A(0 =

mws£ fee homogeneous

distribution,

(3.65) of v.p.

(-)

(3 66)

-

98

Chapter 3 Fourier Analysis and Convolution Semigroups

3.4

The Paley—Wiener—Schwartz Theorem

Let us start with D e f i n i t i o n 3 . 4 . 1 Letu £ Co°(R n ). Its Fourier-Laplace transform is the function C " 3 m

(27r)-"/ 2 f

e-iz-xu(x)

dx.

(3.67)

First note t h a t (3.67) is well denned for all z € C n . In fact, we have e-izxu(x)

/

e~iz-xu(x)

dx= [

n

JU

dx

./supp u

and t h e function x M- e~lz'x\i(x) is bounded on t h e compact set supp u. T h u s the Fourier-Laplace transform of u is a function from C n into C. First we prove t h e theorem of Paley and Wiener. T h e o r e m 3 . 4 . 2 A function f : C n —> C is the Fourier-Laplace transform of u € Co°(R") with supp u C BR(0), if and only if i is an entire function and for any N £ N there exists a constant c^ such that fi|Imz|

(3 68)

'^'^(iTkF

'

n

holds for all z e C . Proof: A. We prove t h a t t h e Fourier-Laplace transform of u € Co°(lR™), s u p p u C BR(0), is a n entire function f satisfying t h e estimate (3.68). Let z S C n and x £ supp u. It follows that i(z) = ( 2 T T ) - " / 2 f

e-iz-xu(x)

dx = ( 2 T T ) - " / 2 f

JR"

e-

izx

u(x)

dx

JBR(0)

is a continuous function on C n and for any closed curve 7 C C and 1 < j < n we find / i(z) dZj = (27T)-"/ 2 / J / e - " - s u ( x ) da; I dZj J-1 J~f {JBR(0) J = (2TT)-"/2 / JBR(0)

u(a:) ( f e~iz-x {J~f

dZj X dx = 0, J

3.4

99

The Paley-Wiener-Schwartz Theorem

since Zj t-> e lz'x is an analytic function, hence J e lz'x dzj = 0. Thus z H->i(z) is analytic in C™. Furthermore, for a G NQ it follows t h a t (27r) n / 2 z Q f(2) = /

u(x)i

JBR(0)

= (-1)I Q I /

u(x)DZ(e-iz-x)

dx

JBR(O) 'Bfl(O)

-L

(D^u(x)))e-"-'

dx,

BR(0)

which yields

\znt(Z)\<(2nrn/2\\ua*hi^fRllmzl, and (3.68) is proved. B. Now let f be an entire function satisfying (3.68). We define u(x)

:= (2TT)"/ 2 / e * * f ( 0 d£, x G JM.n

(3.69)

From (3.68) it follows t h a t for any N G N the function x i-> (1 + |z| 2 ) w f(a;) belongs to L ^ R " ) , which implies u G C°°(]R n ). Next we prove t h a t for 1 < j < n oo f ( 2 l , . . . , £j + IJ7J, . . . , Zn)ei(^i

/

+ - + (^+irii)xj

+ ...+znxn)

^ .

-oo

is independent of r/j. We apply Cauchy's integral theorem taking the curve 7 : [0,1] -> C defined by

2 7(s)

^J'

0<s<\,

= <

(3.70)

-Stfa + Stf + irtf, | < « < f , 3 -4ir)°s - £° + 4177?, T < s < 1. Since f is an entire function, we find for h(Zj) = f( Z l ) ...,Zj,...,

Zny(*i*i+-+*i*i+-+*~*~)

100

Chapter 3 Fourier Analysis and Convolution Semigroups

that

I

h(zj) dzj = 0,

or f ' h & ) d& = / J h{Zi + it/}) dfc - f ' h(£° + iru) dVj J-q J-t*j Jo +

h(-#+Mfe)d7fc Jo

But (3.68) implies for z = t + iy, t,y £ l " , that + ...Wj+---+yl)1'2e-(vixl

eR{vl

|h(±^- +iVj)\

< cN

(|^|2

(i +

+

_

+ (^0)2 +

+ ...+r)jxj +

^

+

...+ynxn)

•

+ W?y/2)N

For fixed a; it follows that f ' h ( ± $ + ir,j) dVj Jo eR(y^+...+(.V°j)

-CNVj

2

+---+vl)1/2e-(yixi+...+\xj\r1°+...+ynxn)

(i + ( N 2 + ••• + (£°)2 + • • • + W\2Y'2)N

'

For £j —> oo we arrive at OO

/

roo

J — oo

-oo

and therefore we get for all ygR™ u(i) - (27T)-™/2 / f(t + iy)eix(t+iy)

dt.

Let x e R™, a; ^ 0, and y = AT|T, A > 0, i.e. £ • y = A|z| and |y| = A. Since |f(i + iy) e «('+*)| < c N e( R - | x | ) A (l +

\t\)~N,

we find for N sufficiently large |u(a;)| <

C j ve^-

|X|)A

/

{l + \t\)-N

dt.

3.4

101

The Paley-Wiener-Schwartz Theorem

But for A —>• oo it follows when R — \x\ < 0 t h a t |u(x)| = 0, i.e. supp u C BR(0). Finally, we have to show that z H-> J R „ etz'x\i(x) dx is an entire function. But for z = t G R™, (3.69) yields t h a t f(t) = (27r)- n / 2 /

u(x)e-itxdx.

(3.71)

Since the right hand side has an analytic continuation to an entire function by part A of the proof, and since the left hand side is by assumption analytic, (3.71) has to hold for all z G C n and the theorem is proved. • We know t h a t every distribution with compact support is a tempered distribution, hence it has a well defined Fourier transform. We want to determine the image of £ ' ( R n ) under the Fourier transform. In Theorem 2.6.10 it was stated t h a t for any u G V'{G) and ^ G C ° ° ( G x G ' ) , G C R " and G' C R m open sets, a C°°-function is defined by y H* (U, $(.,y)) provided t h a t $(x,y) = 0 for all x £ Kc with some compact set K C G. Now, for u G £ ' ( R " ) and any \t G C°°(G x G') we always can define the C°°-function 1/H4(u,*(.,y)>.

(3.72)

In fact, we only have to substitute in Theorem 2.6.10 the function $ by (a;, y) h-> 4>{x)^{x,y), where G Cg°(G) such t h a t <j>\ supp u — 1 • For this reason, for any u G £ ' ( R n ) the mapping ^^(27r)-"/2^u,ei(-«)

(3.73)

is defined on R n , Moreover, we have T h e o r e m 3 . 4 . 3 The Fourier transform u o / u G £ ' ( R " ) is given by u(£) = (27r)-"/ 2 ( U , e - * ( - « ) .

(3.74)

Proof: T h e function defined by £ H-> (u, e _i (-'*>) belongs to C ° ° ( R n ) . Furthermore, for any u G £ ' ( R n ) its Fourier transform u is determined by its values on Cg°(R n ). For G Cg°(R") we find using Corollary 2.6.15 t h a t ( u ® , e - j ( - . « ) = ^ u , 1^ 4>(0e-iU)

dA = J

( u , e - « ( - « ) <£(£) d£,

102

Chapter 3

Fourier Analysis and Convolution Semigroups

thus

which proves the theorem.

•

D e f i n i t i o n 3 . 4 . 4 Let u G £ ' ( R n ) . Its Fourier-Laplace transform is the function defined on C™ by z^{2Tr)-nl2U,e-^'zA.

(3.75)

T h e o r e m 3.4.5 The Fourier-Laplace function f satisfying for all z € C™ |f(z)|
of u G £'(M.n)

is an

+ \z\)NeR\lmz\,

where c > 0 is a constant, suppu C

transform

entire

(3.76)

N G No «?i 0 is determined

by the

condition

BR(0).

Proof: We know already that f is arbitrarily often differentiable, and moreover, the analyticity of z H-> e~lx'z

implies the analyticity of z M- (u, e _ t ( , z ) ) , which

follows by applying the operators -J^-, 1 < j < n, to t h a t function.

Let

ip G C°°(M) be such t h a t ip(t) = 1 for t > - \ and ip(t) = 0 for t < - 1 and consider the function

$z0r) = V>(M(fi-M)), l e i " . It follows t h a t $ z G Cg°(]R n ) and $ z = 0 for |z| > R+ Iz]'1

and $ z = 1 for \x\
By our assumptions we have supp u C BR(0),

(3.77)

hence we have

f(z) = ( 2 7 r ) - " / 2 ( u , $ z ( . ) e - i ( - 2 ) N For \z\ < 1, it follows t h a t i(z) is bounded and for \z\ > 1 it follows t h a t supp $ z C BR+\(0). Thus by continuity of u we get |f(z)|
SuP(9«($z(a:)e---))

(3.78)

3.5

Bounded Borel Measures and Positive Definite Functions

103

with some constant c > 0 and some N G No. But for x G supp 3>z we find | 5 7 e - « - z | < |z7|e*ImZ <

l^Mgffl+M-^IImz^

But now Leibniz' rule and (3.78) give (3.76). • Finally we can prove the Paley-Wiener-Schwartz

Theorem:

T h e o r e m 3.4.6 A distribution u has a compact support supp u C -BR(O) if and only if its Fourier transform u has an analytic continuation to an entire function f and for some c > 0 and N G No the estimate (3.76) holds for f. Proof: By Theorem 3.4.5 we know already t h a t for u G £ ' ( R n ) the Fourier transform u has the properties stated in the theorem. Now let f be an entire function on C™ such t h a t | f ( z ) | < c ( l + |z|) J v e f l l I m z l

(3.79)

holds. We have to find u £ £'(Rn) with supp u C BR(0) such t h a t u(£) = f(£) holds for all £ G R". Taking in (3.79) £ € R", it follows t h a t v := f|Rn is a distribution in S'(Rn). Thus there exists u G <S'(R n ) such t h a t u = v in 5 ' ( R n ) . Let j £ , e > 0, be as in Section 2.3 and set u £ := u * j e . T h e convolution theorem gives u £ (£) = (27r) - ™/ 2 v(£)j £ (£) and Theorem 3.4.2 applied to u £ implies t h a t u £ has an analytic continuation to an entire function g £ satisfying |g*(*)| < c(l + 1^1)^(1 + e | z | ) - ' n e ( i l + e ) l I m 0 I for all m G No and all z G C " . But now, for e > 0 fixed, Theorem 3.4.2 implies t h a t u £ G C°°(R n ) and supp u £ C BR+£(Q). Since s u p p u £ = supp (u * j e ) C s u p p u + Be(0), we find supp u C r i e > o s u P P u £ > i-e- s u p p u C BR(0). Thus u G £' (R n ), and u has an analytic continuation to an entire function on C™. Since this continuation restricted to R n coincides with v, i.e. with f|nn, the theorem is proved. D

3.5

Bounded Borel Measures and Positive Definite Functions

In this section we want to study the Fourier transform of bounded Borel measures fj, G M£(Rn). Since M^{Rn) is a subset of <S'(R n ) the Fourier transform

104

Chapter 3

Fourier Analysis and Convolution Semigroups

/t of /i is well denned and for <j> £ <S(Kra) we have

\

/

(3.80)

,/Rn

From the definition of we get, using Fubini's theorem, (A,
/

JUn

e-i'UWtenW)

JRn

= (27r)-"/2/ ([

e-«« M (d0W)dz

= ( W " / 2 / e-('^(d?), JRn

\

hence we have

m = (27 T ) - " /

2

/ JRn

-ix-£ jj,(dx).

(3.81)

T h e o r e m 3 . 5 . 1 Let fi £ A1jJ"(]Rn) be a bounded Borel measure. Its Fourier transform fi is given by (3.81) and it is a uniformly continuous function on

Proof:

First note that |e-ia;-|i _

e-ix-£2

rx£,2

i/

e~a dt

< 1*116-61-

(3.82)

Since /J, is bounded, we find for any e > 0 a ball BR(0) of radius R = such that ^,(BR(0)) < e. Now, we get /

e-*1-* n(dx) -

JRn

f

e-^x

n(dx)

JRn

-L < #16 -

<

[

\e-^-x

R(e)

- e~i^x\ n{dx)

JR

H{dx)

|/x(d

'Bfl(O)

JB' JB%(0)

ZMBR(0))

+ 2»(BCR(0)) < #16 - 61 IHI + 2e,

which shows the claimed uniform continuity. Obviously, we have

l£(OI<(27r)- B / 2 |M|=£(0).

•

(3.83)

3.5 Bounded Borel Measures and Positive Definite Functions

105

The next theorem summaries some properties of the Fourier transform on f n). M+(R b Theorem 3.5.2 A. For a linear mapping T : R n —>• R™ we have for the image T(/x) of the measure p, G M^(M.n)

[T(/x)] A =A°T*,

(3.84)

in particular for the reflection S : R™ —> R n , Sx = — x, it follows that [S(^)]A = JL = fi o S.

(3.85)

B. For the translation Ta : R ™ 4 R " , i H i + a, we find [T a ( M )] A = e - * a A .

(3-86)

C. For /x, v € M.£(M.n) the convolution theorem holds, i.e. (ji * i/)A =

(2TT)" /2 A

D. For y. G M£(Rn) (V®vf(£,r])

• v.

(3.87)

and v e M^(Rm)

we have

$ e i " andrjeR"1.

= M0-Hv),

(3.88)

Proof: In principle it is possible to deduce all results from corresponding results for the Fourier transform in <S'(Rn). However, there is no problem to do all the calculations straightforward for measures without using any duality, and we will follow that way. A. Since T is continuous, it is measurable and therefore T(/i) is defined and a bounded Borel measure. Further we have (T(/x)) A (0 = (27T)-/ 2 /

e-«' T * /x(dx) = (27T)-/ 2 /

= A(Tt0 = (A°T*)(0. Since /}(£) = £ ( - £ ) we get (3.85). B. For a G R n we find (T a (/.)) A = (27r)-"/ 2 [

e - ^ * + a ) p(dx) =

e-*afL(t)

e " ^ * »(dx)

Chapter 3 Fourier Analysis and Convolution Semigroups

106

C. From Definition 2.3.5 and (2.54) we get for any f G C 6 ( R n ) /

fd(/z*^)=/

jRn

JR"

/

i(y +

x)n(dy)u(dx),

7R"

and taking f(a;) = e~l^'x it follows t h a t (/x*^)A(0 = ( 2 7 r ) - n / 2 /

e-^+z)

/

JRV-

M (dy)

i/(dz)

JR"

= ( 2 T T ) " / 2 A ( € ) • i>(0-

D. A straightforward calculation yields (/x ® i/) A (£, r?) - (2TT) ^ ^ = (2TT)-^2

f [

e-^-t+v^

fi{dx)

u{dy)

e~ix-t n{dx) • ( 2 7 r ) - m / 2 f

JRn

e~iyr>

v(dy)

JRm

= Ki) • Kn)-

n

To characterise the Fourier transforms of bounded measures we need the notion of positive definite functions. D e f i n i t i o n 3.5.3 A function u : R n —¥ C is called positive definite if for any choice of k G N and vectors £ * , . . . , £fc G R " the matrix ( u ( ^ — £l))j,i=i,...,k is positive Hermitian, i.e. for all A i , . . . , Afc G C we have k

53 u (£ j - £')*A > 0.

(3.89)

3,1=1

L e m m a 3 . 5 . 4 Let fi G .MjJ"(Mn). TTien fi is a positive definite

function.

For k G N and £ \ . . . , £fc G E n we find with A X l . . . , Afc G C

Proof:

k

£

k

A,-A7£(? - £') = ( 2 v r ) -

n/2

/

5] A^e"*^-^)-

M (dz)

j,'=i

(2 )_n/2

" L (x>~*"") • (x>- i e , x ) M*^

(2n)-n/2 [ JR"

/x(dx) > 0.

D

107

3.5 Bounded Borel Measures and Positive Definite Functions

Let u : R n —»• C -be a continuous positive definite function. Taking k = 2, £ = 0 € R n and ( 2 = ( 6 M", the positive Hermitianness of the matrix x

u(0) u ( 0 u(-£) u(0) implies that u(£) = u ( - 0 and u(0) > 0,

(3.90)

and therefore |u(a;)|
(3.91)

Thus any continuous positive definite function is bounded, and therefore it belongs to <S'(Rn). We give a useful characterisation of positive definite functions. Lemma 3.5.5 Let u : R™ —> C be a bounded and continuous function. function is positive definite if and only if for all G <S(Rn) we have

I

I u(£- v)W)0.

This

(3.92)

Proof: Suppose that u is positive definite. The integral (3.92) is the limit for R —> co of the integrals /

/

JQR(0)

u ( £ - , 7 ) ^ . ( 0 d£d7?, R>0,

(3.93)

JQR(0)

where Qii(O) := { i 6 l " \-R<

Xj
for 1 < j < n} .

Furthermore, each of the integrals in (3.93) is the limit of the Riemann sums k

£ u(^" - tl)Wi
By our assumptions these sums are non-negative, which implies that (3.92) holds for a continuous positive definite function. Conversely, suppose that for

108

Chapter 3 Fourier Analysis and Convolution Semigroups

a bounded and continuous function u : Rn —>• C the integral (3.92) is for all <j> G S(Rn) non-negative. Given k G N and £ \ . . . , £k e R " and A 1 ; . . . , Afc G C. Further, let ( ^ m ) m 6 N , <j>m G Cg°(R Tl ), be a sequence converging to the complex-valued measure X)j=i ^j£& m t n e vague topology. Such a sequence is easily constructed with the help of (2.81). It follows t h a t (3.92) converges for 4> = <j>m to k

u ( e - il)XjXi > 0,

£

which proves the lemma.

•

R e m a r k 3 . 5 . 6 The proof of Lemma 3.5.5 implies the assertion already for G C 0 (M n ).

of this

lemma

T h e following theorem of Bochner is fundamental in the theory: T h e o r e m 3 . 5 . 7 A function u : R n —• C is the Fourier transform of a measure (j. G Ml(Rn) with total mass ||/x||, if and only if the following conditions are fulfilled 1. u is

continuous;

2. u ( 0 ) = / i ( 0 ) = ( 2 7 r ) - " / 2 | | M | | ; 3. u is positive Proof:

definite.

For /i G Ai^(M.n)

|M| = /

1 /i(dx) -

we know 1. and 3. for its Fourier transform. But /

e - * * /i(cb) = ( 2 T T ) " / 2 A ( 0 ) ,

thus the Fourier transform of a bounded Borel measure satisfies 1-3. We prove now the converse. We know already t h a t u is an element of <S'(R™) and therefore there exists a tempered distribution v such that u = v. We will prove t h a t (v, <j>) > 0 for all <j> G <S(En) such t h a t <j> > 0, which implies by Theorem 2.6.13.5 t h a t v is a measure, i.e. (v, ) — JRn <j>{x) v{Ax). Then, by our assumptions on u we find for this measure t h a t v(0) = J>(0) = u(0), implying t h a t i/(E") = (27r)"/2z>(0) is finite, i.e. v G M j " ( R " ) . Now, we prove

109

3.5 Bounded Borel Measures and Positive Definite Functions

{v,4>) > 0 for 4> > 0, 4> G S(Rn). Denote by <j> the function 4>(z) = cj>{-z). Since u is continuous and positive definite we know that /

/

u(x — y){y){x) dx dy = /

JR" JR"

u(z)(<j> * 4>){z) dz > 0.

JR"

For G <S(Rn) the function 4>* <j> belongs also to <S(R"), and since u € «S'(Rn), we have that

In, 4>*A>0.

(3.94)

However, this yields

o< (u,4>*4>) = (y,4>*4>) = / v , ( ^ * ^ ) A \ = (27rr/ 2 (v,|^| 2 ), thus (v, |^>|2) > 0 for all <j> € <S(]R"), which is equivalent to 'v, \j>\2) > 0 for all <j> G S(Rn).

(3.95)

By the remark following Theorem 2.6.13, it follows from (3.95) that (v, ij)) > 0 for all ip G <S(Mn), i[> > 0, which proves the theorem. • Definition 3.5.8 A distribution u G V'(Rn) is called positive definite if for all(f>eC^'(Rn) (u,4>*\>0

(3.96)

holds. Using this definition there is a generalisation of Bochner's theorem, usually called Bochner-Schwartz-Theorem. Since we do not need this generalisation later on, we refer to the literature [104], p. 157, or [247], p. 14. The rest of this paragraph is devoted to a more detailed study of positive definite functions. Many of the following considerations are close to that in the monograph of Chr. Berg and G. Forst [25]. First let us note an auxiliary result. Lemma 3.5.9 Let (a.ki)k,i=i,...,m and (bki)k,i=i,...,m be two positive Hermitian m x m-matrices. Further let (cki)k,i=i m be the matrix with elements Cki = a-kibki- Then (cki)k,i=i,...,m is positive Hermitian too.

110

Chapter 3 Fourier Analysis and Convolution Semigroups

Proof: For B = (bki) there exists a positive Hermitian matrix D such that B = DD*, hence b^i = Y^T=i dkjdij. Let A i , . . . , Am G C, it follows that m

^

m I m

\

akibki\k\i = 5Z I 5Z aki(dkj\k)(dij\j)

k,l=l

j=l \k,l=l

> 0.

D

)

Now denote by P(R n ) the set of all positive definite functions on R™ and CP(R n ) is the set of all continuous positive definite functions on R n . For u G P(R n ) we have already proved u(|) = u(-£) and u(0) > 0,

(3.97)

| u ( 0 | < u ( 0 ) , hence sup | u ( 0 | = u ( 0 ) . £eRn

(3.98)

and

Let us use once more the notation u(0 = u ( - 0 -

(3-99)

We will prove some useful inequalities for positive definite functions. Lemma 3.5.10 For any u € P(R n ) we have |u(0 - u(r?)|2 < 2u(0)(u(0) - Re u(£ - rj)),

(3.100)

and when u(0) = 1 we get in addition K £ + rj) - u(0u(r/)| 2 < (1 - |u(0| 2 )(l - |u(r?)|2). Proof: For

(,IJ£R"

A^,rj)=

such that u(£) ^

U(T?)

we consider the matrix

/u(0) u ( 0 ufo) u(Q u(0) u t f - r ? ) \VL(V)*(£-V)

u(o)

and the related quadratic form (Au(£,r])c, c) with

AKQ-u^l C l = A

'

C 2 =

u«)-u(,) '

_ C3

-"C2'

(3.101)

3.5 Bounded Borel Measures and Positive Definite Functions

111

and A € R is arbitrarily given. Since (Au(£,r))c,c) > 0, we find by a straightforward calculation u(0)(l + 2A2) + 2A|u(0 - U(JJ)| - 2A2Re u(£ - 77) > 0 for all A € R. This however implies that 4|u(0 - u(7y)|2 < 4(2u(0) - 2Re u(£ - ^)u(O), or (3.100) for u(£) ^ u(r?). But for u(£) = u(rj), it follows that u(0) - Re u(£ 77) > 0, thus (3.100) holds for all £,77 £ Rn. /lab\ B. Suppose that for a, b, c G C the matrix l a i c I is positive Hermitian.

\b-clj Then its determinant is non-negative, which gives l + abc + abc> \a\2 + \b\2 + \c\2, which is equivalent to \c-ab\2

< (l-|a|2)(l-|6|2).

Thus, when u is a positive definite function with u(0) = 1, we find | u « - „) - u(0u(r?)| 2 < (1 - |u(0| 2 )(l - |u(i,)| 2 ), which gives by (3.97) estimate (3.101). D Corollary 3.5.11 Let x G R™ be fixed, then £ i-> e~lx'* is a continuous positive definite function. Conversely, let u : R™ —» C be a continuous positive definite function such that |u(£)| = 1 for all £ € K n . Then there exists a vector i £ E " such that u(£) = e~ix<. Proof: For the function £ >->• e~ix^ we find with A i , . . . , ATO e C

k,l=l

k,l=l 2

E«^ fc=i

>o,

Chapter 3 Fourier Analysis and Convolution Semigroups

112

hence, £ H* e~ix'^, x G Rx fixed, is positive definite. Now, for u G C P ( R n ) such t h a t |u(x)| = 1, it follows from (3.101) t h a t u has to satisfy the functional equation of the exponential function, i.e. u(£ + rj) — u(£)u(?y). T h e continuity of u implies t h a t u must be exponential and the fact t h a t |u(£)| = 1 for all £ G R n implies t h a t u is of the form £ i-> e~lx'^ for some x G R n . • The next lemma collects some obvious results for the set of positive definite functions. L e m m a 3 . 5 . 1 2 A. The set P ( R n ) is a convex cone which is closed with respect to the pointwise convergence. B. If u G P ( R " ) , so are u and Re u. C. If u, v G P ( R n ) , it follows that u • v G P ( R n ) . D. The set CP(M.n) is a convex cone which is closed with respect to uniform convergence on compact sets. Proof: T h e assertion A, B and D are obvious, C follows directly from Lemma 3.5.9. • L e m m a 3 . 5 . 1 3 Let u G P ( R " ) and suppose that R e u is lower at 0 G R™. Then u is uniformly continuous on R n . Proof:

semicontinuous

Since Re u(£) < |u(£)| < u(0) it follows t h a t for any e > 0 {£ G R " | Re u ( 0 G (u(0) - e, u(0) + e ) } = { ( € « " | Re u ( 0 > u ( 0 ) - e } ,

hence, Re u is continuous in 0 G R™ if it is lower semicontinuous at 0 £ 1 " . But now (3.100) implies the uniform continuity of u on R™. • T h e set .MjJ~(Rn) is also a convex cone and .M£(R n ) C .MjJ"(R™) is a convex subset. Bochner's theorem states t h a t the Fourier transform is a bijective mapping from M£(Rn) onto C P ( R " ) and .A/f£(Rn) is mapped onto the convex set {u G C P ( R n )

u(0) = (27r)-"/ 2 } .

We have even a stronger result: T h e o r e m 3 . 5 . 1 4 The Fourier transform F is a bicontinuous mapping from M^(Rn) equipped with the Bernoulli topology onto CP(Rn) equipped with the topology of uniform convergence on compact sets.

3.5

Bounded Borel Measures and Positive Definite Functions

113

T h e proof of Theorem 3.5.14 requires L e m m a 3 . 5 . 1 5 Let <j) £ Co(R n ) and e > 0 be given. function

n

f G Co(R ) such that

>-f

Then there exists a

<£.

Proof: First we choose ip £ Cg°(R n ) such t h a t \\(f> - ipW^ < | . Since F " 1 ^ £ S(Rn) we may select f £ C 0 ( E " ) , in fact, we may take f £ Cg°(R"), such t h a t | | F - 1 ^ — f||L1 < (27r)™/ 2 |, see Section 2.3. Now, the Riemann-Lebesgue lemma implies

-l

\\aD + V>-f < | + (27r)-"/ 2 ||F-V - f ||L1 < e. •

P r o o f of T h e o r e m 3.5.14: Let (/Xj)j 6 / be a net of measures /ij £ M^(Rn) which converges in the Bernoulli topology to /x G M^(U.n). Since x t-¥ e~%x'^, £ £ R™, is continuous and bounded, it follows t h a t (3.102)

lim£j(£) = A(0

for all £ G R n . We prove t h a t this convergence is uniform on compact subsets. First we claim: For any e > 0 there exists a neighbourhood U of 0 G R n and j 0 6 / such t h a t for all j £ l , j h jo, and all &, £> G Rn with & - 6 € ?7 (3.103)

(27r)-"/2|/ij(ei)-A,(6)l<£

holds. Let £ > 0 and take S > 0 such t h a t (5(3 + /x(R n )) < e. Further let <j> G Cg°(R n ), 0 < 4> < 1 and L„ (1 - <j>) n(dx) < S. Since l i m / ^ = y, in the J'€/

Bernoulli topology, there exists jo £ I such t h a t /Zj(K n ) < i i ( R " ) + l and

/

(1 - 0) ^ ( d z ) < J

for all j £ I, j h jo- We define {/ to be the set [ / : = { ? e l n I |1 - e~ix-t\ < 6 for all x G supp 4>) .

114

Chapter 3 Fourier Analysis and Convolution Semigroups

For j £ I a n d £ 1 , 6 € R n such t h a t j y j (2TT)"/ 2 IAJCCI) - A , ( 6 ) | <

/

<

J

0

and 6 - £ 2 G J7 it follows t h a t

|e-»-«' -

I _

J <sf

e - ^ \ ^ ( d x )

e-i(t\-(2)-x

I _

<{>(x) fJ-j(dx)

e-i(ti-t2)-x

(f>(x)fij(dx)+2

[

(l-
/j,j(dx)

(I - 4>{x)) iij{dx)

< <5(/i(M") + 1) + 26 < e, hence we have proved (3.103). For K c R™, K compact, a n d e > 0 we can find jo G 7 and U, a neighbourhood of 0 G R n , such t h a t for t h e limit measure fi we get

(27r)-"/ 2 |A(6)-A(&)l<£ for all £1,62 € R™, £1 — £2 £ C- T h e compactness of K implies t h e existence of finitely many points £ * , . . . , £fc G iC such t h a t

Therefore there exist j ; , 1 < Z < k, such t h a t (27r)"/ 2 !£,(£') - £ ( £ ' ) | < £ for j >: j , , Z = 1 , . . . , k. Now let j * G 7, j * >r j ; , Z = 1 , . . . , k. For £ = £/ + U a n d j ^ j * we get

|£i(0 - A(0I < l&(0 - Ai«')l + l£i(0 - £(£')! + IA(0 - £(01 < 3(27r)- n / 2 e thus sup|AJ(0-A(OI<3(27r)-"/2£, which proves t h a t (Aj)j'ei converges uniformly on compact sets t o £ .

3.5

Bounded Borel Measures and Positive Definite Functions

115

Next we prove the converse. Let (fJ-j)jei be a net in .Mj|"(R") and /x G •MjJ"(Rn) such t h a t (Aj)je/ converges uniformly on compact sets to /x. In particular, it follows t h a t l i m / x ^ R " ) = lim(27r) ri / 2 A J (0) = (2TT)"/ 2 /X(0) = jei

' ) •

j€i

Thus, in order to prove t h a t (fij)j^i converges to /x in the Bernoulli topology, it is sufficient, see Theorem 2.3.7, to prove t h a t (/Xj)^/ converges vaguely to /x. For (j> G C 0 ( R n ) , > 0, and e > 0 take f G C 0 ( R n ) such t h a t U - f I< £ , see Lemma 3.5.15. Now we find /

<j> d/Xj — /

4> dfi

I (0-f)d/x,- + [ idfXj- f {dfi + f (f-^)d/i 7K" JM JM. n

JM.n

n

< efa(Rn) + /x(Rn)) + f |Ai(0 - £(01 |f(0l d£, where we used the equality

/ f(o z,(d0 = / *(0f(0 d£, JR"

(3.104)

JR™

which holds for i/ G X ^ " ( R " ) and f G L x ( R n ) by Fubini's theorem. T h e compactness of supp f and flj —> fi uniformly on compact sets imply by the dominated convergence theorem t h a t lim

/

JRn

4> d/ij -

/

> dxx < 2£ii(R"),

JR'

hence the convergence of JRn d/x for all <j) G Co(R"), which proves the theorem. • A stronger result is given by Levy's continuity

theorem:

T h e o r e m 3 . 5 . 1 6 Let (/x.,). N be a sequence in M^(M.n), function which is continuous in 0 G R n . / /

and u : R " - > C f l

lim /xj(0 = u(£) for all^G R™, j—»oo

£/&en t/iere exists a measure \x G .Mj]~(Rn) SMC/I i/mt (/Xj). t/ie Bernoulli topology.

N

converges to /x in

116

Chapter 3 Fourier Analysis and Convolution Semigroups

Proof: Since the pointwise limit of positive definite functions is positive definite, the function u is an element in P(Rn). But Lemma 3.5.13 implies t h a t u is continuous since by our assumptions u is continuous in 0 G Rn. Now, by Bochner's theorem, Theorem 3.5.7, it follows t h a t there exists a measure fj. G A4~l(M.n) such t h a t fi = u. To prove the convergence of (/X,). N to /x in the Bernoulli topology, it is once again sufficient to prove

lim / |&(0-£(OWOd* = 0 for all 4> € Co(M n ), > 0. But this follows by the dominated convergence theorem. D We want to discuss several examples of concrete positive definite functions. Our first two examples show how to obtain new positive definite functions from a given one. P r o p o s i t i o n 3 . 5 . 1 7 Letu be a continuous positive definite function such that u(0) < R. Further let i(z) = Y^'jLo a3z^ ^e a Power series with non-negative coefficients aj > 0 converging in the disc \z\ < R. Then the function f o u is positive definite. Proof: From (3.91) it follows t h a t f o u is well defined on Rn and is given by H i l o a - ? ( u ( £ ) ) J ' - Further, this series converges uniformly on compact subsets. Hence, it is sufficient to prove t h a t for any m the partial sum YllLo a j'( u (0)' 7 ' is a positive definite function. But Lemma 3.5.9. implies t h a t £ i-»- ( U ( £ ) ) J is positive definite for any j € No, which proves the proposition by Lemma 3.5.12.J9. • P r o p o s i t i o n 3 . 5 . 1 8 Let u : R™ -> C and v : Rm ->• C be two continuous positive definite functions. Then the function (£, rj) — i > u(£)v(?7) is a positive definite function on Rn x Rm. Proof: Bochner's theorem implies the existence of two measures /J, G ./Vf ^"(Mn) and v £ M^(Rm) such t h a t /i(£) = u(£) and £(£) = v(£), respectively. B u t the measure [i®v belongs to M.^{Rn x Rm) and its Fourier transform is given by (£,?7) ^ u ( 0 v ( ' ? ) i s e e Theorem 3.5.2.P. Hence, a further application of Bochner's theorem gives the result. •

3.5

117

Bounded Borel Measures and Positive Definite Functions

Thus, by Proposition 3.5.18 we can construct many examples of continuous positive definite functions on R™, once we know continuous positive functions onR. T h e first of the two following tables is taken from [25], p.27, but we changed the normalisation of the Fourier transform according to our convention. T h e last item is taken from [91], p.503. Table 3 . 5 . 1 9 Probability measure on R Degenerate distribution, a<=R Symmetric degenerate distribution, a G R Binomial distribution with parameter p e (0,1), q=l-p Poisson distribution with parameter a > 0 Uniform distribution on the interval [—a,a], o > 0 Laplace distribution with parameter a > 0 Normal distribution with parameter t > 0 Cauchy distribution with parameter t > 0 Gamma distribution with parameter t > 0 Triangular distribution on the interval [—a,a], a > 0 —

M ea

(27r)^2 • m

i(e a + £_ a )

cos(a£)

e~lai

zr=0 (Tyv"-^

(q +

pe-^r

exp(
sin(a£)

\')*

at

£exP(-y)A«

(l+^er1

jzM-^)*"

exp(-t£ 2 )

±(t* +

x>)-i\m

m**-1*-*^-)^ U 1 " ^ 1 ) *[-..] <->A(1) 1 f l-cos(ax)\

x(l)

e-"«l ( 1 + <€)-* 2(l-cos(a£))

(l-^XH^tf)

In the following table the first four items are constructed by using Tabel 3.5.19. A calculation giving the 5 t h item can be found in [94], p. 105-106, for the 6 t h item we refer to [289], p.6-7, for r = 1. Note t h a t in case of the 5 t h and 6 t h items the measures are not normalised, i.e. these measures are not probability measures.

Chapter 3 Fourier Analysis and Convolution Semigroups

118

Table 3 . 5 . 2 0 Measure on R n Degenerate distribution,

(27r)n'2m

M Sa

oel" Symmetric degenerate distribution, a £ R n Uniform distribution on the rectangle X ?=l[-a3taj] Normal distribution with parameter t > 0

cos(a • £) •p-rn

i ( e a + e_ 0 )

(4^t)"/2C

i

11.7 = 1

a ^

A

r-^-\M

(ifr) e

sinaj£,-

-t|.|

A

(n)

2

J^C-l^i)

n/2r(n±1)

) i > 0

(Jn is the Bessel function of first kind and order ^ ) . We will need P r o p o s i t i o n 3 . 5 . 2 1 A continuous isfying

positive definite function

u £ CP(M)

(3.105)

l i m ^ - ^ = 0 «-+o

e

is a

sat-

constant.

Proof:

Using t h e inequality (3.100) we get for any unit vector rj GRn

u(g + fa7)-u(fl h

< ^2u(0)(u(0)-Reu(/i7?)) , >n /u(0)-u(H\ = 2u(0)Re ^ W h2 j < 2u(0)

u(0) - u(hrj)

a n d all

3.5

Bounded Borel Measures and Positive Definite Functions

119

thus we have l i m

^

+

M-u(0=a

Thus for all 77 G R™ the directional derivative 4 u ( ^ ) exists and is 0, implying t h a t u is constant. • T h e next result is often called the theorem of Polya. T h e o r e m 3 . 5 . 2 2 Let u : K —• R+ U {0} be a continuous function which is even, i.e. u(—£) = u(£), and which is decreasing and convex on R + . Then u is positive definite. Proof: From Table 3.5.19 we conclude t h a t the function V(C) : = (1 _ l£l)Xr_i 11 (£) is a continuous positive definite function which is even a n d its restriction to [0,00) is convex. Take a\,..., a,k G [0,00), a\ < a-i < ... < ak, and numbers 0 < pi,.. • ,Pk such t h a t ^ 7 = 1 Pi = 1 a n c ^ c o n s i d e r the function

*(0 == I>V (P\ •

(3.106)

It is easy to see t h a t TT is a convex polygon which is even and 7r|[ 0oo ) is convex. Moreover, since Pj"4>\^-) is a continuous positive definite function, the same holds for TV. But any continuous function u : R —> R + U {0} which is even and whose restriction to [0,00) is decreasing and convex is the pointwise limit of inscribed convex polygons of type (3.106), hence by Lemma 3.5.12 the theorem is proved. G We borrowed the proof of Theorem 3.5.22 from W. Feller [91]. For a recent generalisation to radially symmetric functions in R 2 we refer to T. Gneiting [107]. E x a m p l e 3 . 5 . 2 3 From Theorem 3.5.22 it follows that for a G [0,1] the function £ i-> e~l£l is positive definite. Later we will generalise this example to a G [0,2]. But from Proposition 3.5.21 we may conclude that £ (->• e~^ , a > 2, will never be positive definite since for a > 2 we have ,

llm

l-e"l«la n 1.12 = °-

120

Chapter 3 Fourier Analysis and Convolution Semigroups Finally, we prove the result due to A. Khinchin.

Theorem 3.5.24 Let <j> e L 2 (R"). Then the function u ( 0 := TTTTTa / {x + Z)W)te ||0|| o iK"

(3-107)

is continuous and positive definite. Proof: Obviously, we have u(0) = 1. First we prove that u is continuous:

|u(£ + h)- u(0| < TTL / W* + £ + h) -
£

4GC l * & ' + * ) -* < " |, ' l *) 1/, •

lim ||0(. + h) — (.)|lo = 0 f° r

but

an

Y ^ £ L 2 (R n ). In order to prove that u is

h-+0

positive definite we can now use Lemma 3.5.5. For ip G 5(R n ) we find \\
JR"

= i

J&n

JRn JRn

JRn

I I (v)
JRn JK"-

= [ ([ JE"

\JR"


\JRn

(y-t)W)<%)*y>o, /

hence, u is a positive definite function. • Remark 3.5.25 In T. Kawata [181], p.379-380, it is shown that (for n = 1) any continuous positive definite function has a representation as in (3.107).

3.6

Convolution Semigroups and Negative Definite Functions

We begin with

3.6

121

Convolution Semigroups and Negative Definite Functions

D e f i n i t i o n 3 . 6 . 1 A family (fit)t>o of bounded Borel measures on R n is called convolution semigroup on R n if the following conditions are fulfilled /x t (R n ) < 1 for allt>0;

(3.108)

fis * Ht = Vt+s s, t > 0 and fi0 — £ 0 ;

(3.109)

fit —> £o vaguely as t —> 0;

(3.110)

(Note that some authors do not require the normalisation

fio = EQ.)

L e m m a 3 . 6 . 2 Let (fit)t>o be a convolution semigroup on R™. Then the mapping 11-» fit is continuous at t — 0 with respect to the Bernoulli topology. Proof: (3.110)

For cj> € C 0 ( R " ) , 0 <

(0) = 1, we find by (3.108) a n d

1 = <£(()) = lim /


hence lim^t(Rn) = l = e0(Rn) t->o and by Theorem 2.3.7 the lemma is proved.

•

Let (fit)t>o be a convolution semigroup on R n . It follows t h a t the family (At)t>o of the Fourier transforms of fit, t > 0, consists of continuous positive definite functions on R™ satisfying |At(£)l — (27r) - T l / 2 . Our aim is to study this family more carefully. In particular, we want to show the existence of a unique function V : R n -> C such t h a t £ t (£) = ( 2 7 r ) - n / 2 e - t ^ holds. Combining Lemma 3.6.2 with Theorem 3.5.14 we find as a first result L e m m a 3 . 6 . 3 For any convolution semigroup (fit)t>o on R™ the mapping t Mfit is continuous from [0, oo) into ,Mjj~(Rn) equipped with the Bernoulli topology. Proof:

For t, t0 > 0 and f € R™ we get

MO - /WOI < (27r)-"/2|Ato(OI IA|t-t0|(0 - {2*rn/2\


122

Chapter 3

Fourier Analysis and Convolution Semigroups

but by Lemma 3.6.2 we conclude t h a t /i| t _ t o |(£) —>• (2TT) n / 2 uniformly on compact sets as t —> to- Now, Theorem 3.5.14 implies t h a t l i m Ht = fJ-to

(3.111)

t-+t0

in the Bernoulli topology.

•

T h e o r e m 3 . 6 . 4 Let (/J,t)t>o be a convolution exists a function ip : K™ —> C such that

semigroup

on W1.

Then

A t ( 0 = {2Tr)-n'2e'^^ holds for all ( e R Proof:

B

there

(3.112)

and t > 0.

For ( e l " fixed we consider the mapping <j>£ : [0, oo) -> C defined by

fc(t)

:= (27r)"/ 2 /} t (0, t > 0 .

(3.113)

By Lemma 3.6.3 this mapping is continuous and the convolution theorem gives & ( * + *) = & ( * ) & ( * )

(3.114)

l i m ^ ( i ) = l. t->o

(3.115)

and

It follows the existence of a unique complex number ip(£) such t h a t ^(t)=e-t*«),

i>0.

(3.116)

Note t h a t the mapping £ H-» e _ t ^ ) must be positive definite and T/>(0) > 0. • D e f i n i t i o n 3.6.5 A function

ip : R n -> C is ca/fed negative definite i/

V»(0) > 0

(3.117)

and f H+ ( 2 7 r ) - " / 2 e - t v , ( 4 ) is positive definite fort

> 0.

(3.118)

3.6 Convolution Semigroups and Negative Definite Functions

123

Theorem 3.6.4 says that for any convolution semigroup (fit)t>o there exists a negative definite function ip : R n —> C such that (3.112) holds. We want to study negative definite functions more closely and for this reason we introduce the classes N(Rn) and CN(R n ). Definition 3.6.6 A function ip : R n ->• C belongs to the class N(Rn) any choice of k G N and vectors £*,..., £k G R™ the matrix (W)

+ V>(£') - W

- Z%,i=i,...,k

if for (3.119)

is positive Hermitian. Further we set CN(Rn) := JV(Rn) n C ( R n ) . For ip G N(Rn) we have obviously -0(0) > 0,

(3.120)

and since for ^ £ l " the matrix

( <MO+W)-v>(o) m+m_-wo\

,

n

is positive Hermitian we find

iP{Z) + v(0) - V(0 = W + V(0 - ^ R ) , i.e.

HO = H-t) or V(0 = m

(3.122)

where we used the notation from (3.99). Furthermore the determinant of the matrix (3.121) must be non-negative, implying that Re V(0 > V(0) for all £ G R™.

(3.123)

From Definition 3.6.6 we derive immediately Lemma 3.6.7 A. The setN(Rn) is a convex cone which is closed under pointwise convergence. B. For ip G iV(Rn) it follows that ip and Re ip belong to N(Rn) too. C. Any non-negative constant is an element of N(Rn). D. For ip G N(Rn) and A > 0 the function £ M- i/j(\£) belongs to N(Rn). E. The set CN{Rn) is a convex cone which is closed with respect to uniform convergence on compact sets. F. For ipj G N(Rni), j = 1,2, it follows that ip{£,r]) •= V>i(£) +i>2{v) defines an element in iV(Rni+™2).

Chapter 3 Fourier Analysis and Convolution Semigroups

124 Further, we have

L e m m a 3.6.8 A function ip : R n —• C is an element in N(M.n) if and only if the following three conditions are fulfilled V>(0) > 0;

(3.124)

tp = i>\

(3.125) 1

and for any k G N and any choice of vectors ^ , . . . , ^ numbers c\,... ,ck k

^

fc

6 R

n

and

complex

k

ip(£j - £l)cjCi < 0.

Cj = 0 implies that ^

3=1

(3.126)

3,1 = 1

Proof: For any ip G N(Rn) we know t h a t (3.124), (3.125) hold. Further, for fceN,^,...,^ G R n and c i , . . . ,c fc G C such t h a t Ylj=i C3 = 0 it follows that k

o < E fa?)+WJ- Mj - O) w 3,1=1

k

( k

\

k

/ k

\

k

i=i

\j=i

J

j=i

V=i

/

j,i=i

= X> E^(^>,- + X> E ^ > n - E W -^)) c>« 3,1=1

Conversely, suppose t h a t the function ip : R " —>• C fullfills the assumptions (3.124)-(3.126). In addition let k G N, £ \ . . . , £ f c G R n and cx,...,ck G k 1 C. We consider the vectors Q,(},...,£ G W and the complex numbers (c, cu ..., cfc) G C f c + 1 , where c = - £ * = 1 Cj. From (3.126) it follows t h a t k

k

k

j=l

1=1

3,1=1

^(o)ici2 + E^)c^+E^(-^)cc[+ E ^ ' - *')c^ ^ °But using (3.124) and (3.125) we conclude t h a t

Efc (v^)+v^) - W - i1)) c& > HO) 3,1=1

3.6

Convolution Semigroups and Negative Definite Functions

which proves the lemma.

125

•

Note t h a t some authors call functions satisfying (3.124)-(3.126) ally positive definite. C o r o l l a r y 3 . 6 . 9 For tp G N(Rn) element of N(Rn).

the function

condition-

£ >-> V ( 0 - V>(°) is also an

Proof: Let k G N, £ \ . . . , £fc G Rn and a, ..., ck G C such t h a t ] £ * = 1 Cj = 0. T h e n we find k

k

l

£ W - i ) - v-(o)) cjci = Y^ *Ktj - tl)w ^ °. j,k=l

i,^=l

whereas the conditions (3.124) and (3.125) are obviously satisfied for the function £ i-> V>(£) - V>(0)-

D

C o r o l l a r y 3 . 6 . 1 0 Let u : i n -4 C k a positive definite function. function £ >-> u(0) - u(£) is m JV(R n ).

Then the

Proof: Suppose t h a t k € N, £ \ . . . , £k G R n and cu ..., ck G C such t h a t X3j=i c j = 0- Since u is positive definite it follows t h a t k

£

k

(u(0) - u ( ? - ?)) cjci = - ^

u ( ? - ^ ) c j C 7 < 0.

Moreover, (3.124) and (3.125) from Lemma (3.6.8) are satisfied. D Now we can prove Schoenberg's theorem stating t h a t N(Rn) the set of all negative definite functions. T h e o r e m 3 . 6 . 1 1 A function if it is negative definite.

coincides with

tp : R™ ->• C is an element of N(Rn)

if and only

R e m a r k 3 . 6 . 1 2 From now on we will denote the set of all negative definite functions by N(Rn), and CN(Rn) is the set of all continuous negative functions. P r o o f of T h e o r e m 3.6.11: First let tp G 7V(R"). It follows t h a t -0(0) > 0. For k G N and £ \ . . . , £fc G Rn the matrix \

/

J,l=l7...yK

126

Chapter 3 Fourier Analysis and Convolution Semigroups

is positive Hermitian and Lemma 3.5.9 implies that the matrix V

\

/ /

j,l—l,...,k

is positive Hermitian too. For c i , . . . , Cfc € C we find with c'j := exp(—i/>(£J')) Cj G C that k

Y, exp (-M?

- e))

Cjci

fc

= J2 exp (v'(e')+W) - we - *')) exp (-v(e')) x exp (-V'(^))

c

^

= J ] exp (V(^) + V^1) - V(^ - £')) c ^ > 0. Thus the function £ H-> e - 1 ^ ) is positive definite. But for £ > 0 it follows that tip £ iV(R n ), hence we conclude that for t > 0 the function £ H-> (27r) _n / 2 e _t1 ''^^ is positive definite, hence i/;: R™ —» C is negative definite. Conversely, let tp : R™ —>• C be a negative definite function. Since ip(0) > 0 we find e - *^ 0 ) < 1 for all t > 0 and by Corollary 3.6.10 it follows that the function t » \(1 - exp(-tiKO)) = j ( l - e-**W) + J (c-**W belongs to N(Rn).

C -**«>)

Applying Lemma 3.6.7.^4 we get that

£ ^ V ( 0 = t-*o l i n ti 7 ( l - e x p H V ( 0 ) ) belongs to N(Rn).

•

Corollary 3.6.13 Let ip : R™ -> C be a negative definite function. Then is for all e > 0 a positive definite function.

^j

Proof: From (3.123) it follows Re V(£) > V'(O) > 0 for all £ e R", thus it is sufficient to prove the corollary for all ip such that ip(0) > 0. For t > 0 the

3.6

Convolution Semigroups and Negative Definite Functions

function £ H-» e ^ ^ e

127

is positive definite and we have

-t^«)|<e-^(o)i

t>0and£eRn.

Hence, it follows t h a t /•OO

V>(0

/ Jo

(6RB,

e-WQdt,

which implies the corollary as one sees immediately when approximating the integral. • C o r o l l a r y 3 . 6 . 1 4 Let ijj : R n —>• C fee a negative definite function (3 > 0. TTie function £ H->- a _ ^ 9 ( f , is afeo negative definite.

and a > 0,

Proof: We know already t h a t the function a + (3ip is negative definite, and a+{3tp(p) > 0. Hence, by Corollary 3.6.13 the function £ i->- a+g^ie\ is positive definite, and therefore, by Corollary 3.6.10, the function £ i—• a+g1i,(0) — a i i i , M is negative definite, hence

a + f3ip(.)

v

a

y

However, it follows t h a t

V'G)-V'(o) + m = (i + atW\ a + /?V(.) implying t h a t ^

a

V

G JV(R").

a

^(-) J a + /?
•

R e m a r k 3 . 6 . 1 5 Let tp G iV(R"). Corollary 3.6.14 says that we can approximate %[> pointwise by bounded negative definite functions since lim x~j = ip. Next let us establish a one-to-one correspondence between convolution semigroups on R" and continuous negative definite functions. T h e o r e m 3 . 6 . 1 6 For any convolution semigroup (fit)t>o on R " there exists a uniquely determined continuous negative definite function ip : R™ —> C such that n/2 t Vt {£_) = {2-K)- e- ^\

t>0and^eRn,

(3.127)

Chapter 3 Fourier Analysis and Convolution Semigroups

128

holds. Conversely, given a continuous negative definite function ip : R™ —> C, then there exists a unique convolution semigroup (p>t)t>o on R n such that (3.127) holds. Proof: By Theorem 3.6.4 we know the existence of a negative definite function ip : R™ —¥ C such t h a t for a given convolution semigroup (pt)t>o the equality (3.127) holds. We want to prove t h a t ip is a continuous function. For this let us consider the measure p £ M^(M.n) defined as a linear positive functional on C 0 ( R " ) by

JR™

\JRn

JO

/

Obviously, we have \\p\\ < 1. Since p € <S'(R™) we may calculate its Fourier transform, and for <j> £ Cg°(R n ) we find

{p, ) = (p, fy = I

e - t (fit, j>) di

/•OO

= /

/>00

e " ' (At, 4>)dt=

Jo

= f

f

e~* / Jo

JRn

£ t (0<M0 <% dt

/"e^AtCOdt^Odf,

JK" Jo and we find

P(0 = ^°° c-*A.(0 d* = (2-)- / 2 /°° e-t(1+^(£)) d* =

f^.

But p is continuous by Theorem 3.5.1, hence ip is a continuous function. Since the Fourier transform is uniquely determined, it follows t h a t ip is unique. Now let ip G C N ( R " ) . For any t > 0 the function £ i-> ^ T T ) - " / ^ - * ^ is continuous and positive definite. Therefore, there exists a non-negative bounded Borel measure p,t such t h a t

MO = (27T)' -n/2.-tV(£) We claim t h a t (/it)t>o is a convolution semigroup on R n . Since ip(Q) > 0 it follows t h a t

/M

= (27r)'l/2/it(0)=e-t,/'W < 1 .

3.6

Convolution Semigroups and Negative Definite Functions

129

Further, using the convolution theorem we find At(OA.(0 = (27r)-n/2e-^(«)(27r)-"/2e-^(« = (27T)-"e-(t+SW«

=

(27T)-"/2£t+S(0,

implying t h a t fH*fis = Ht+s- Since ip is continuous, hence bounded on compact sets, we have l i m / 2 t ( 0 = (2n)-n/2

lime-**«> = ( 2 T T ) - " / 2

t->0

t-*0

uniformly on compact sets in R n . Now, Theorem 3.5.14 implies t h a t lim/x t = t-»o £o in the Bernoulli topology, which finally proves the theorem. • T h e last theorem and the Theorem of Schoenberg immediately give C o r o l l a r y 3 . 6 . 1 7 A function if and only if

ip : R n —> C is continuous

and negative

ip(0) > 0

definite

(3.128)

and £ H-> e - * * 1 ^ , t > 0, is continuous

and positive definite.

(3.129)

Let fit = gtA^™), gt(x) = (4nt)~n/2e « , be the Brownian semigroup in R . By Lemma 3.1.4 its Fourier transform is given by n

A t ( 0 = (27r)-"/ 2 e-'l«l 2 , hence, the function £ M- |£| 2 is a continuous negative definite function. E x a m p l e 3 . 6 . 1 8 Any non—negative symmetric quadratic form q : R™ x R n —• M. is a continuous negative definite function. Note, that we do not assume q to have full rank. A convolution semigroup with q as corresponding continuous negative definite function is called a Gaussian semigroup. Proof:

We write q(f) instead of q(£, £) and find for £, r) £ R™ t h a t

2q(0 + 2q(r?) = q(£ + v) + q(£ - v),

130

Chapter 3 Fourier Analysis and Convolution Semigroups

hence q(0) = 0 and q(£) = q(-£). Since q(£) G R we have q = q. Let us consider the bilinear form r : R" X 1 " ^ 1 r(£,»?) = q ( 0 + q ( » 7 ) - q ( £ - » 7 ) It follows that r(£,0=2q(0-q(0)>0 and r(£, V) = q ( 0 + q(»y) - q(£ - r?) = qfa) + q ( 0 - q(r? - 0 = r(»7, 0 For c G C define r(c£,?7) = cv(£,,r)) and r(£, C77) = ci(£,r)). In particular, we have r(c£,c£) = |c| 2 r(£,£) > 0. Now, let k G N, £ \ . . . ,£fc G R n and c i , . . . , Cfc G C. For £ = £ * = 1 Cj£J we get

on = EE c ^ r (^) fc

- E ^

+ q^y-q^'-O)^,

thus q is a negative definite function and obviously continuous. • Let (/it)t>o be a Gaussian semigroup with corresponding continuous negative definite function q G CN(R n ). We want to determine the measures /xt, t > 0, more precisely. For this suppose that q has rank k < n. Let us denote by Q the matrix corresponding to q. By a coordinate transform we may assume Q to be of the form (

), where Qi is a k x /c-matrix of rank k. For fit

this implies that Attt) = AtfoO = (27r)- l / 2 e-*W«-« = (27r)- f e / 2 e- t «^-") • ( 2 T T ) ^ , where £ = (77, £) G R™ = Rfc x R n - f c . Taking the inverse Fourier transform we find with x = (y, z) € R n = Rfe x Rn~fc Mt (dx)

= Mt(dy,d«) = /4"°(dy) ® £<"-*>(dz),

131

3.6 Convolution Semigroups and Negative Definite Functions

where e^n~k\dz) is the Dirac measure on R n - f c and/4 (dy) = gQi(2/)^^(^2/)> where gQ1(y) is the inverse Fourier transform of the function rj H-> (2n)~k/2 e-t(Qir),ri) _ The latter Fourier transform can be calculated explicitly and we may proceed as in Lemma 3.1.4 to get gQl(y) = ( 4 7 r f ) - f c / 2 - = i = e ^ V ^ .

(3.130)

Thus we have fit(dx) = (47rtyk/2—^e-l-S±^\(k\dy)®e(n-k\dz). VdetCth

(3.131)

Example 3.6.19 Let I : R™ —> R be a linear functional and define il>(Q '•— il(£). Then the function tp is a continuous negative definite function. Moreover, whenever ip(£) = il(£), I '• R™ —> R, is a continuous negative definite function, then I must be linear. Proof: For k € N and £ \ . . . , £k G R™ it follows that

=

i(l(e)-K?')-l(?-?'j)=0.

Since any linear functional on R™ is of the form l(£) = h • £ with some h G R™, it follows that for any h G R™ the function f M- ih • £ belongs to CN(R"). To prove the second statement assume that ^>(£) = il(£) , I : R™ —»• R, is a continuous negative definite function. Then the function t(£) = e~t^^\ t > 0, is positive definite and |t(OI = 1- Therefore, by (3.101) we have e-ni(£+v) = e-W(Z+v)

- e-t4>(£)e-tip(v)

= e-HKOe-HKv)

=

e-imt)+lW)

and for k G N it follows that

( e -*i'«)) = e-*"(l+-+l) = (e-**'<*>) , implying that I is additive and l(f£) = fl(£) for all f G Q. The continuity of I implies now that I must be a linear functional on R". •

132

Chapter 3 Fourier Analysis and Convolution Semigroups

Combining the last two examples and Lemma 3.6.7 we find t h a t for any c > 0, h £ R n and symmetric non—negative definite quadratic form q the function £ >->• q ( £ ) + i / i • £ + c

(3.132)

is an element of C N ( R " ) . For the corresponding convolution semigroup (/it)t>o on R™ we find

MO = (2*

:-n/2

-tqm-ith-^-tc

Denoting by (//J?)t>o, {fJ-t)t>o and (ix£)t>o the convolution semigroups associated with q, h and c, respectively, we find

and the convolution theorem yields

fit = $ * /41 * fi\. E x a m p l e 3 . 6 . 2 0 Since for / i e R " the function £ \-> e~%h'^ is positive definite and e° = 1, it follows from Corollary 3.6.10 that £ H* ( l — e~lh'^) is a continuous negative definite function implying that £ \-t (1 — cos(/i-£)) is an element in CN(R.n) too. For h € R, h > 0, and t > 0 let us consider on R the measures

^ = S e_t jfe! £fcfc -

(3 133

-

)

fc=0

Taking the Fourier transform

fc=0

•

of fit we get

fc=0

* c* -l/21/2 -t V- ^ = (27r)-1/2e-* J2 \f~i fikhi = (^)e^ Yl — e

=

fc=0

'

(27r)

e

'

k=0

= (27r)- 1 /2 e -*(l-e-^) )

implying that £ i-> 1 — e lh^ is a continuous negative definite function and that (Mt)t>o is a convolution semigroup on R, called the Poisson semigroup.

3.6 Convolution Semigroups and Negative Definite Functions

133

We will give a table of continuous negative definite functions and the corresponding convolution semigroups in Section 3.9 after we have discussed Bochner's theory of subordination which gives a method to construct new (continuous) negative definite functions from old ones by composition with certain functions. In this section we continue by giving some properties of negative definite functions. L e m m a 3.6.21 For any ip € N(Rn)

we have

vm+v)\ (3.134)

\V\m\-y/\m\ {£ + V)\ < 2(Re V ( 0 ) V 2 • (Re V-(r/))1/2Proof: For £, rj G W1 the determinant of the matrix

is non-negative. Since ip — ty and V'(O) > 0 it follows that

hKO + VM -W-v)\2

< 4Re V(0Re 1>(V) < 4|V(0I \4>{v)\

We may also take —77, and observing that \ip(r))\ = \ip{—TJ)\ we get

|V(0 + 1>{±n) - V>(£ ± v)\2 < 4|V(0I \1>iv)\, and further we have

m±v)\-MQ\-m±ri)\ < Mt±v)\-mt)+i>(±r>)\ < Mt) + 'K±v)-'Kt±v)\

< 2|V(0l1/2IV'(r?)r/2

which yields ^ ( £ + 77)1 < ( | ^ ( 0 | 1 / 2 + \1>(v)\1/2)2 and

V\m\-VWW\ =\m\+\m\-2V\m\VW\<m-v)\,

134

Chapter 3 Fourier Analysis and Convolution Semigroups

implying (3.134). The third estimate follows from the same calculation when using the stronger estimate W O + 1>(V) - 4>(Z + V)\2 < (2Re V(0 • 2Re ^ ) ) 1 / 2 .

D

Lemma 3.6.22 For any locally bounded negative definite function tp G there exists a constant c^ > 0 such that for all £ G M.n h K O I < < v ( l + |£| 2 ).

(3.135)

Proof: Since ip is locally bounded, it is sufficient to prove

WOI < c'kl2 for all ( e R " \ 5i(0) with some c' > 0. By Lemma 3.6.21 it follows that V\^(mv)\

< my/\ip(r))\

holds for all J ) £ l " and m G N. Thus for r\ = ^ we have

MOI <

1> m

for all £ G R™ and m G N. Let c' = sup \ip(r))\. For £ G R", |£| > 1, there hl<2 exists mo G N such that £ G [mo,mo + 1), and we find

m)\<m20

V>( —

< m20c' < c'\£\*.

a

m0 Note that Lemma 3.6.22 holds in particular for any continuous negative definite function. The next result is an inequality which in case of the function £ i->- |£| 2 is often called Peetre's inequality. The validity of this inequality for general negative definite functions was pointed out to us by W. Hoh. Lemma 3.6.23 Let ip : R n -»• C be a negative definite function. have

i + MOl < 2 ( l + hK£-r7)|). i + IV'WI

Then we

(3.136)

3.6 Convolution Semigroups and Negative Definite Functions

135

Proof: For 77, < € R n we find 2(l + hM77)|)(l + |V(C)l) = 2 + 2|V(r?)| + 2|V(C)|+2|V(r?)||V(C)l

= (1 + |V(r?)| + |V(C)I + (IVMI + MOD) + (1 + 2|V(T?)V(0I) > 1 + |^(77)| + MOI + 2^(77)^(01 = 1+ (V / I^0T)T + \ / | V ; ( O I ) 2 where we used the estimate 2y/\bib2\ < \bi\ + j&21. Using the subadditivity of V l~* \/WW)\ ^ follows that

2(1 + |v>(7?)|)(i + MOD > (1 + VM^ + OI ) = 1 + IV-fa + 01Taking £ = £ — 77, we finally get

Corollary 3.6.24 For ip £ 7V(Rn) o i w i s e R we have

(rrjlii)^2''^^1^-^^1-

(3 137)

-

Since in (3.136) we may choose £ and 77 arbitrarily, we can derive further inequalities, namely

1 + M£ - V)\ < 2(1 + |tffa)|)(l + MOD

(3-138)

I±j||i< 2(1+

(3-139)

or | ^ + ,),).

The following inequality is a certain refinement of Peetre's inequality, and it it is due to R.L. Schilling. Lemma 3.6.25 Let tp be a negative definite function. Then we have

1 + M£ ± 77)| < (1 + MOD (l + V l W l ) 2 •

(3-140)

136

Chapter 3 Fourier Analysis and Convolution Semigroups

Proof:

For £, 77 G R n we find using Lemma 3.6.21 -2

1 + |V(£ ± »7)| = 1 + v / l ^ + ^l 2 < 1 + (y/\W\ + v W l ) '

= 1 + MOI + w(v)\+zVmfiVWWi < 1 + MO I + IV>fa)l + 2 v W I ( i + HKOI), where we used the estimate -^/|6| < 1 + |6| for all b G C. Thus we get

1 + wt ± 77)1 < 1 + MOI + IV-fa)l +WO I \Hv)\ + 2 v ^ ) [ ( l + MOD = (1 + MOD -(i + |VWI + 2 ^ M ) = (I + M 0 ! ) - ( I + N ^ M ) 2 , which proves (3.140). • Taking in (3.140) instead of £ the vector —£ and using ip = tp, we find

1 + IV^ - 01 < (i + WOI) (1 + VT^faJI)2 -

(3-i4i)

and substituting £ H-» £ + 77 we get

1 + 1-0(01 < (1 + Mt + ri)\) (l + VWMf

,

(3-142)

where we used once again ip = tp. C o r o l l a r y 3 . 6 . 2 6 Any negative definite function tinuous in 0 € R™ is continuous on R™.

ip : R n —>• C which is con-

Proof: By Corollary 3.6.9 the function £ i-> VK0 - V'(O) i s negative definite too, and T/> is continuous if and only if ip{.) — V'(O) i s continuous. Thus we may assume V(0) = 0. From (3.140) and (3.142) we deduce

^H^iM-^r For 77 —> 0 we find ^ J u ^ n f 1 — • 1> which gives the corollary. •

3.6 Convolution Semigroups and Negative Definite Functions

137

Let ip : R n -> C be a negative definite function and T : R™ —>• R™ a linear bijective mapping. We claim that ip o T is negative definite. For this let k G N and £ \ . . . , £fc G N be given. Further let C\ • • •, Cfc £ R" such that £»' = T " 1 ^ ' . For 1 < j , I < k we find j l

V W ) + V
implying that ip ° T G iV(R n ). In particular, for any rotation R^ G O(^) it follows that ^ o R ^ is negative definite if ip is negative definite. Obviously, we have ip o T G CN(R n ) for V G CN(R"). Lemma 3.6.27 Let ip '• K n —> C 6e a negative definite function. A. If ip(0) > 0 ; i/ien i/ie function ip has no zeroes. B. The set of all zeroes of ip form a subgroup o / ( R n , + ) . Ifip is continuous, then this subgroup is closed. Proof: A. This follows immediately from the estimate ip(0) < Re ip(£) for all £ G R™, see (3.123). B. We know already that ip will have zeroes only if ip(0) = 0. Now let £ \ £ 2 G R n be two further zeroes of ip. Since ip = ip it follows that

det (^ e

+

* 2 )+^ 1+ * 2 ) - ^(°) M? + & ±W) ~ ^ ) \

\ip(e)+Tp(e+e)-ip(-e) =

^ l )+^ i )-^(o) )

-k(e+e)\2>o,

implying that V'^ 1 + £2) = 0. Further, for any zero t;1 it follows that ip{—£x) = ip(£}) = 0, hence, — £} is a zero of ip. Clearly, if ip is continuous, {£ G R" | ip(£) = 0} is closed. D Lemma 3.6.28 Let ip : R —• C be a negative definite function such that ^[a^p] = 0, a < (3. Then ip is identical equal to 0. Ifip is continuous, it is sufficient to require that ip\i = 0 for a dense subset I C [a, 0\. Proof: From Lemma 3.6.27 it follows that ip is identically equal to zero on [a- P,p-ct\. Let 7 G [f3 - a, 2{3 - a]. Then there exists S G [0,/3-a] such that 7 = P + S. Hence, by Lemma 3.6.27 we have ^(7) = 0, implying (by induction) that ip\[a-pt0o) = 0. Since ip(-£) = V>(£)> i* follows that ip = 0. The supplement in case of continuous functions is trivial. • We continue our study of negative definite functions in the next section by discussing the Levy-Khinchin formula.

138

Chapter 3

3.7

Fourier Analysis and Convolution Semigroups

The Levy-Khinchin Formula for Continuous Negative Definite Functions

This paragraph is devoted to the Levy-Khinchin formula which states t h a t every continuous negative definite function ip ; R n —¥ C has the representation

+

i

(3i43

u( -^-^y-^^

->

with a non-negative constant c > 0, a vector d G R n , a symmetric positive semidefinite quadratic form q, and a finite Borel measure fi on R71 \ {0}. T h e function ip is uniquely determined by (c, d, q, /x) and any such quadruple defines via (3.143) a continuous negative definite function. T h e proof we will give follows our joint paper [172] with R.L. Schilling. For a discussion of other proofs we refer to the notes at the end of this chapter. Note t h a t by Example 3.6.19 and Example 3.6.20 for every x G R n the function

£-Ml-e

i x

-ix-i _

-i

l + bl 2

is negative definite, and taking into account (3.132), the Levy-Khinchin formula has the interpretation t h a t every continuous negative definite function is a superposition of elementary continuous negative definite functions. We need some preparations to prove (3.143). Denote by <So(R") the set <S0(R") := ( v G <S(Rn) = jveS(Rn)

v(0) = J^-(O) = 0 for 1 < j < n \ v(0) = J ^ - ( 0 ) = 0

fori

< j < n j .

L e m m a 3 . 7 . 1 Let f, g : K™ -> C be two measurable functions nomially bounded. If for all G <S0(Rn) /

f(x)0(x)di=

f

(3.144)

which are poly-

g(x)(f>(x)dx

(3.145)

holds, then there exists a constant c G C and a vector d G C™ such that i(x) = c + d- x + g(x).

(3.146)

3.7

The Levy-Khinchin Formula for Continuous Negative Definite Functions

Proof:

139

Clearly, both f and g induce elements of <S'(R n ). Thus, (3.145) implies


(3.147)

( U ) = (g,£)

(3-148)

and

for all 4> e <S0(Rn)- B u t (3.148) leads to supp ( f - g )

C {0}, implying by

Theorem 2.6.13.4 t h a t

cadae0,

f - g = ^2

(3.149)

|a|<m

where en is the Dirac measure at 0. Since for \a\ > 2 there are functions <j> E <S0(Rn) with da(j>(0) ^ 0, we have necessarily t h a t m < 2 in (3.149). T h u s u := f — g is given by

E

d

(3.150)

Cj—-£0.

Using Corollary 3.1.3 and the fact t h a t in = (27r) t h a t (3.146) holds. D

n 2

/ we conclude from (3.150)

Let (nt)t>o be the convolution semigroup associated with the continuous negative definite function tp by At(0 = (27r)-"/2e-^(«.

(3.151)

On <S(R™) we can define the operators T t u ( i ) = (27r)-™/2 f

e f a * e - * * « > u ( 0 d£,

(3.152)

and the convolution theorem yields Ttu(z) = /

u{x~

y)

(3.153)

m{dy).

For u G <S(Rn) we find uniformly in x G R n t h a t lim t->o

Ttu(a:) U(a;)

-

+- (2U)-/

= lim(27r)- n / 2 t->o

2

/ e<~mmo <* JRn

e -*V-«)

_ i

+ V(0 u(0
• 0.

140

Chapter 3 Fourier Analysis and Convolution Semigroups

Denoting by Au t h e strong limit (in t h e norm H-H^) ,. T t u - u lim-^ , t->o t

, (3.154)

we have for u G <S(R") =-(2n)~n/2

Au(x)

eix^(0u(0de

[

(3.155)

JUL™

But from (3.153) it follows t h a t -(27T)-"/2 /

e « ' « V ( 0 u ( 0 d£

JM."

= lim - ( / u ( x - y) m(&y) - / u(x - y) e0(dy) ) . t->0 t \ 7 u n ,/Rn J

(3.156)

for i £ l " and u G <S(R n ). Our way of proving t h e Levy-Khinchin formula is to examine t h e right-hand side of (3.156) more carefully. L e m m a 3 . 7 . 2 There exists a finite measure fi on R n such that we have in the sense of weak convergence of measures 1 Id2 ' „/xt(ds) — > / i ( d a ) ast—-»0. t 1 + |a;|z / n particular, mass. Proof:

(3.157)

the measures in (3.157) have uniformly

(in t > 0) bounded total

We want to apply Theorem 3.5.14. Therefore we have to calculate t h e

Fourier transform of t h e measures | ^ J , 2 /it(d:r). For this note t h a t

1 + M2

2 J0 \ = -(°°

I

J ( 2 T T ) - " / 2 (1 - e - » « ) e-l«l 2 /2^ e -A/2 d £

d A

2 Vo JR" = /" (1 - e - " « ) g ( 0 ^ = /

(l-cos(x-O)g(O^,

where g(£) = i /

27o

(27rA)-"/ 2 e-lfl 2 / 2 A e- A /2 d A.

(3.158)

3.7 The Levy-Khinchin Formula for Continuous Negative Definite Functions

141

It is easy to see that / JR"

g(£) d£ < oo and [ |£| 4 g(0 d£ < oo. JRn

(3.159)

Hence, we find for the Fourier transform of \ 1^fjLa A*t(d^c)

t JRn

-I- + \x\

l

=

JRn

JRn

if L-WW t yKn \

_ e-*M+v)\

g (£) d £.

/

Using a Taylor expansion we obtain

*7R»

i + i^r

JUL"

2

where

with TJ-R(£, rj) being the remainder of the Taylor expansion of tip(rj) H-> e '^M up to order two. We shall prove that for any h > 0 sup sup |I(i,7?)| < oo.

(3.160)

t>0 \r)\
Indeed \I(t,r,)\<

f

(|R(i,77)| + |R(i,e + ^ | ) g ( 0 ^

JHLn

< In (l^)| 2 + l ^ + »7)|2)g(0^ jR

jRn

and with (3.159) the estimate (3.160) follows. Thus we find that the Fourier transforms of j i+\x\i Vtidx) converge uniformly on compact sets to a continuous positive definite function, and by Theorem 3.5.14 there exists a finite measure p. such that (3.157) holds. The uniform boundedness of the total masses follows from the above calculations if we set rj = 0. •

142

Chapter 3 Fourier Analysis and Convolution Semigroups

Lemma 3.7.3 Fix a function x G Co°(R n ) such that xlsi(o) — 1, 0 < x(x) < 1 and xlsc(o) = 0- For k,l = 1 , . . . ,n there exist signed measures vki on W1 with finite total mass such that —

• ,2x(x)nt{dx)—>vkl(dx)

as t—> 0

(3.161)

holds in the sense of weak convergence of measures. Proof: Using the identity 2xkx\ = (xk + x{)2 — x\ — xf it is sufficient to show that j i^\x\2 x(a;)Mt(da;) —>• pki as t —> 0 for all k, I = 1 , . . . , n in the sense of weak convergence of measures. Clearly, the measure Vki is then given by \pki — \pkk — \pu a n d has finite total mass. Since ^TTTWXW 1 + |a;|2

^ 71^+T l\x\ 2 z'

l<M
Lemma 3.7.2 shows that the measures \ \+Zl2 x(x)^t(dx), t > 0, have uniformly bounded total masses with respect to t, k, I and x- Again, we want to apply Theorem 3.5.14, and for this reason we compute the Fourier transform of these measures. Note that e

1 + \xr l(Xk Xl)2

-

-x(.x)^t-eo){Ax).

1+ xIf $n denotes the function

§ (l):=e

'

" k '^W x ( l ) '

(3 162)

'

we have $,, £ <S(R"). For any ip € S(Rn) have [

rl>{x) n(dx) =

(2TT)-"/ 2

/

/

and bounded Borel measure fj, we

eix^(0

d^n(dx)

(3.163)

3.7

The Levy-Khinchin Formula for Continuous Negative Definite Functions

143

Thus we find

-I

l + \x\2

X(x)nt(dx)

WE"

= 7/

$,(o(A(0-^o(0)dS

As in the proof of Lemma 3.7.2 a Taylor expansion yields 1

f

-ix.r,(xk+Xl)2

where

with the remainder term R(i, £) of the Taylor expansion up to order two. Now, (27T)"/2 I(t, 77)

JRn

JR"


|$0(0

(1 + 1 ^ + |£|4)

^

which implies t h a t sup sup I(t, 77) < 00 t>0\r,\

(3.164)

I

for all h > 0. Thus, the Fourier transform of j^j!n%$-x(x)Pt(dx) converges as t —> 0 uniformly on compact sets, and by Theorem 3.5.14 the assertion follows. •

Chapter 3 Fourier Analysis and Convolution Semigroups

144

Corollary 3.7.4 Let U be an open neighbourhood of 0 € R n such that 2U C Bi(0) and let §u e Cg° (R n ) be a function satisfying $t/|t/ = l , 0 < $ i / < l and $>u\(2U)c = 0- Then lim-/

xkxi^u(x)

(j,t(dx) = /

$u(x) vkl(da;) + 0((dianu7) 2 ) (3.165)

t

is valid. Proof: Let x be the function introduced in Lemma 3.7.3. For supp $[/ we find supp $[/ C { i £ 1 " | x(x) = 1 }i which gives - /

Xfca^t/ta;) /xt(dx) = - /

(l + \x\2)$u{x)

X

**1 x{x) m{&c)

and since (l + \.\2)®u S C 0 (R n ), it follows from Lemma 3.7.3 that lim - /

xkxi<&u{x) fit(dx) = /

(l + \x\2)$u(x)

vki(dx).

Let |i/fe/|(da;) denote the modulus of the signed measure uki(dx). Then /

\x\2<$>u{x) vH{dx) < /

\x\2 \vH\{dx) < (diam2C/)2

J2U

max

\vki\(M.n),

l
and the assertion follows since the vki are of bounded total mass. • Let <S0(R") be defined as in (3.144) and define

T := [f: R n \ {()}-»• 1 -f- Ice'2

f(x)

v(a:) for some v € <S0(Rn) \

Lemma 3.7.5 Let f G T, i(x) = ^$-v{x)

(3.166)

with v £ <S0(Rn). Then we have

k,l=l

In particular, the left-hand side of (3.167) has an extension onto W1.

3.7

The Levy-Khinchin Formula for Continuous Negative Definite Functions

Proof:

145

By definition we have for x £ R™ \ {0} r f(.)

l + l-l2

= v(.)€50(R")

which is a function defined on R™. T h e Taylor expansion of v at 0 G R n yields 1

d2

"

dxkdxi

k,l=l

where we can estimate the remainder by

d3

|R(a;)| < — max 6 l
= ^^-v(x)

and v £ S0(Rn),

we have

- ( 2 T T ) - " / 2 / V ( 0 v ( 0 d£ = J2 9 « « ^ 5 - v ( 0 ) + / i(x) •/»« 9xkdxi 7»"\{o} I ^ 1 where (qki)k,l=i,...,n is a symmetric, positive semidefinite measure constructed in Lemma 3.7.2. Proof:

matrix,

fi(dx),

and /i is the

Let f and v be as above. Then

1 f 7 / 1

\x\2 1 f *(*) i • ui2 ^ ( d a ; ) = 7 /

JR" =

•!• T \X\ - [

v(x)

v(x)

fh(dx)

I Jftn (nt

- e0)(dx)

= - j

v(x)

(&(£)

-

(2TT)-"/2)

d£,

where we used (3.163), the fact t h a t F 2 v(£) = v ( - £ ) , and t h a t V ( 0 = V'(-C)Therefore

Since | e - ' ^ ) - l | < *|V»(0I < c ^ l 1 + l£l 2 ) limit under the integral sign and get lim if t-+o

and v

f(x)T^i2Mt(dx) = -(27r)-"/2/

e 5 ( R n ) we can pass to the

v(0V(0^.

(3.168)

146

Chapter 3

Fourier Analysis and Convolution Semigroups

W i t h $ [ / as in Corollary 3.7.4 it follows t h a t

if

{(X)

I TW ^t{dx)

= 1\ l

7R"

^u(x)i(x)-^-fit(dx)+x •>•

T

[

\\

Jut"

( l - S ^ f ^ J ^ ^ d z ) X -|- |a:|

= Ii(t)+I2(t). Since (1 - $[/)f e C&(R n ), Lemma 3.7.2 gives I2(*)—• /

(1 - $u(x))f{x)

n(dx)

asi—>0.

(3.169)

As in the proof of Lemma 3.7.5 we see

IlW =

\L *U{X) £ l2XkXldx~dx-im "t{dX) + - /

$u(x)R(x)

fj.t{dx),

where R is a continuous function satisfying |R(a:)| < c|:r| 3 . Therefore

f R(a;)(l + N 2 ) R(ar) := 0,

x^O x = 0

is also continuous on R " and we get - i

<S>u(x)R(x)fit(dx)

= U r ./R»

UR»

$u{x)R(x)-^-&(**)• l + \x\

Since $c/(-) • R{-) is bounded and continuous we find t h a t the limit lim - /

$u(x)R(x)

fn(dx)

= hv € R

exists. But supp [/ C 2U C 5 i ( 0 ) , thus i / $c/(a;)R(a:) lh(&c) *7K"

< £ / |z|3 /^(cb) t J2U

<~cd^{2U)\j^^_»t{dx).

(3.170)

3.7

The Levy-Khinchin Formula for Continuous Negative Definite Functions

147

T h e last t e r m is, however, uniformly bounded in t > 0. This gives | M < c'diam(2£/),

(3.171)

and by Corollary 3.7.4 1

n

t

limli(t) = - V

t-+o

'i

/

^ ,

rfi"

Jr

$u{x) m{dx)

a

I

dxkdxi

{0)+hu.

So far we have proved 92v(0)

i1 Jv-^ L /• 1 /" I d.122 f lim - / f(g) ' /z t (dz) = ^22 *v(x) t-*0 t JRn 1 + \X\* I £j^1 yRn

+ f

v dx

^ ) dxkdxi

{l-^u(x))i(x)fi{dx)+hu.

Since the measures Vki, 1 < k, I < n, are finite we can pass to the limit U 4- {0} to obtain

lim

7/

^TXM*

t-yo tj^n 1 + \x\ or in view of (3.168) -(27T)""/2/

*(da:)4 £ ^ ( { ° } ) 3 + /

z

2^ ^

dxkoxi

f

(*) »(**)

y K \{ 0 }

V(0v(0^

= 5 E - ( ( » » | S + /„U0)'W^)

(3-172)

It remains to show t h a t the matrix {qki)k,i=i,...,n = (^ki({0}))k,i=i n is symmetric and positive semidefinite. But for any choice of ^ s R, 1 < k < n, n

-

n

Y] qutkb = Y]

lim

o/

„

$

u(a;) m{dx)Zk&

1 ™ 1 /" = - Y ] lim lim - / $u(x)xkxi£kti ^ fc^! c/4-{o} *->o t 7R« = - lim lim - / $c/(z) I Yxk£k * C/|{0} t->0 * JRn \j^ and the symmetry is obvious. •

Mt( da; )

I /xt(da;) > 0, J

Chapter 3 Fourier Analysis and Convolution Semigroups

148

Now we can prove the Levy-Khinchin formula. T h e o r e m 3 . 7 . 7 Let ip : R™ —> C be a continuous negative definite function. Then there exists a constant c > 0, a vector d G R™, a symmetric positive semidefinite quadratic form q on R™ and a finite measure /J, on M.n \ {0} such that

+ /

(l-

e-™*

7^4) lJT$- ^*) 1 + \x\2 I \x\2 v

(3-173) '

;

v

holds. Proof: Let f € F, i(x) Lemma 3.7.2 we find

and v e S0(Rn).

= ^ff-v(x)

W i t h p as in

/ f(*) M(di) JR"\{0}

1 + M2

f

= /

M

2

v(a;) ^(da;)

= (27r)-"/2/

/

./R"\{o} 7 R "

= (2*)W AW x

e ^ v (0d^i±i^Md^) FI

/ (C^-1

+ I ^ij)v(0^

1 + \x\2 - £ 1 2 - ^ )

(3.174)

because v € <S0(Rn) implies /„„ v(£) d£ = / R n ^-v(£) d£ = 0 for 1 < j < n. Since -ix-£

-1 +

ix • £

< \e~ix^

- 1 + ix • ^\ + ix • £

l + \x


x'

1*1 Kl

we find for Ixl < 1 t h a t -ix-i _ J _|_

ix • £ \ 1 + M'

l + \x\

< 2 ( 1 + |£| 2 ),

ix • £ 1+

x[

3.7

The Levy-Khinchin Formula for Continuous Negative Definite Functions

149

and for |a;| > 1 we have ,-ix-e.

ix • £

i + !£!!<4 + K|»<4(i + K| a ).

1 + l + |z|2

Both cases together yield ,-ix-i

ix • £

1+

S4 1+

( i«i 2 )rrU-

i + M:

(3.175)

see also W. Hoh [147]. We may, therefore, change the order of integration in (3.174) and obtain

f

f(*) M(dx)

./K"\{0}

'-'•'S-' + I ^ ) 1 ^ ! ^ ) v « ) d { .

JUL" 7R"\{0}

By Lemma 3.7.6 we have I

V ( O v ( 0 d£ = - ( 2 T T ) " /

y»"

2

d2v(o)

9 f c i | ^ - - (SO"/2 /

£

dxkdxi oxkdxi

J^J

x

= M E ««&&+ / •/IK" [ ^ J ^

f -'

-ix-£

K "" \\ {{ O O }} 77 K

f(x) /z(dz)

ix • £ \ 1 + \x\2

/i(dx)

Mt)d£

•/R«\{0}\

Applying Lemma 3.7.1 we find n

V>(£) = c + t(d • o + X ) *"&& fc,J=l

-«•€ _ ix-Z \ 1 + N a VR"\{O} V

i + Nv

M2

with some c e C and d £ Cn. For £ = 0 we know i/>(0) > 0, hence, c € R and c > 0, and, since ^ ( £ ) = ip(—£), we see t h a t d s M n . • T h e o r e m 3 . 7 . 8 Let-ip € CN(Rn) with Levy-Khinchin representation (3.173). Then c, d, q and fi are uniquely determined by I/J. Moreover, given c, d, q and fi as in Theorem 3.7.7, then the function defined by the right-hand side of (3.173) is continuous and negative definite.

150

Chapter 3 Fourier Analysis and Convolution Semigroups

Proof: The constant c is given by ip(0). Lemma 3.7.2 it follows that m

= [

Further, from the proof of

Mv + 0 - V-(O)gfa) drj,

with g as in (3.158). Hence, p, is determined by ip, implying that /x is determined uniquely by i/>. For r > 0 we find

/

2

/R"\{0} rr ^

l + |a:| 2 ;

V

|z| 2

K

h

Since £ ( l - e~ i x '^ - j g f j ) - 4 0 a s r ^ o o , and further lX

•r£ i + H:

1 _ e-ix-rt

l + \x\2 „ 12
we may apply the dominated convergence theorem to see that •|2

2

r->oo./]R»\{0} r

V

1+ M

2

/

|x|

2

from which we conclude that q ( 0 = lim - j ^ K ) r—•oo ^

Now, c, g and ^z are uniquely determined by ip, which implies that d is also uniquely determined by tp. In order to see that the right-hand side of (3.173) defines a continuous negative definite function, it remains to prove that

7K"\{o} V

1+ M V

\x?

is negative definite. The function £ t-> e~lx'^ is positive definite, see Corollary 3.5.11, hence, by Corollary 3.6.10 the function ( l - e~ix< - iqq^r) is negative definite. It follows immediately that for k £ N each of the functions

jR"\Bl/k(o)\

i + Fr/

\xr

3.7

The Levy-Khinchin Formula for Continuous Negative Definite Functions 151

is an element in C N ( R n ) , and these functions converge to ip, thus ij) is negative

definite. • Let us state the Levy-Khinchin formula for real-valued continuous negative definite functions explicitly. C o r o l l a r y 3.7.9 Let ip : K™ —>• M be a real-valued continuous function. Then we have the representation

negative

V>(0 = c + q ( 0 + / (1 - cos(x • 0 ) n 4 r - f*(*c) JK"\{0} \x\

definite

(3-176)

with c, q and /z as in Theorem 3.7.7. Using the Levy-Khinchin formula we get another expression for the operator A introduced in (3.155), i.e. Au(z) = -(2n)-n/2 In fact we find for u G

An(x)

f

e " ^ ( £ ) u ( 0 d£.

S{Rn)

= -{2ir)-n'2

f •

/RB

eix< ( c + j(d • £) + V % , , & £ ] u ( 0 d£ V *,j=i /

- (27r)-"/2 f eixt f

i _ e-iy« _

*yz l + |j/|:

•>\{oy V

1 + M"

/x(dy)u(£)d£ \y\ = L(x, D)u(a;) + S(a;, D ) u ( i ) .

(3.177)

T h e first term yields by Corollary 3.1.3 L(x,D)u(x) = -cu(x) + f

f=i

>

^

dx

*

+ J2 KiPir--

k%

dxkdxi

From the proof of Theorem 3.7.7 we know the estimate

\-e-i*i.

ivt \ 1 + M: „|2 1 + 12/12 // l\y

< 4 + | £ | + |£|2,

(3-178)

152

Chapter 3 Fourier Analysis and Convolution Semigroups

which implies t h a t for u G <S(Rn) the function

(v-O-V f i - e - * * - 1T+TI2/TI l/ * M )n#*(fl^ is integrable with respect to the measure /i(dy)A(n)(d£). Thus, an application of Fubini's theorem is justified and we get for the second term in (3.178) S(a; ) D)u(a;)

= -(27T)-"/2 / e*« / JR"

(l - e~^ - J ^ ) i + M ! Mdy)u(0 d£

7K"\{O> V

J- + \y\ J

\y\

- Um {2'r""L e"( ('"""'+ni?) *(0 d | T <*> - U ^ - ' - ^ - g l ^ ^ ) 1 ^ ^ ) . (3.17.) R e m a r k 3 . 7 . 1 0 There are several other ways to choose the measure \i in (3.173). Often one takes a measure v on R™ such that £({0}) = 0 and J R „ (|a:|2 A l ) i/(dx) < 00. In this case one has as Levy-Khinchin representation of a continuous negative definite function ip '• R™ —• C M)

= c + i(d, • 0 + q ( 0 + j[^ B ( l " *"**'* - i q q ^ ) *(<**),

or, using as cut-off function

xB

^ ( 0 = c + »(da-0 + q ( 0 + / or, with the cut-off

(

1,

(3- 1 8 °)

l(y)

( l - e - ^ - i z - O x - f o J a O ^ d a O , (3.181)

function 0 < |cc| < 1

2 - |aj|, 1 < |a;| < 2 , 0, 2 < |x| we have t/>(Z) = c + t(d 3 • 0 + q ( 0 + /

(1 - e " " « - ire • £9(a:)) j/(dx)

(3.182)

./Vote that changing the cut-off function causes in general also a change in the vector of the linear part of the representation formula.

3.7

The Levy-Khinchin Formula for Continuous Negative Definite Functions

153

For probabilistic considerations it is helpful to introduce the Levy measure D e f i n i t i o n 3 . 7 . 1 1 Let fi be the measure in the Levy-Khinchin of the continuous negative definite function ip : R™ —>• C. The

u{dx)

representation measure

1 4- I d 2

(3.183)

= -^T^dx)

defined on B(Rn \ {0}) is called the Levy measure associated with t/j. Note t h a t the Levy measure satisfies / K ^ w 0 \ (\x\2 A l ) v(dx) < oo. In order to have no break in our presentation when switching from analysis to probability theory, we will from now on often use the Levy measure when working with the Levy-Khinchin formula and write

JR«\{0} V

i

- + \x\

/

where v integrates x H-> (|a;|2 A l ) . In this case (3.179) reads as

S(l,D)uW - J^m (u(, - y) - »(*) - ± j f ^ ^ J •*«. (3.185) Let ip : R™ - > C b e a continuous negative definite function. For its L e v y Khinchin representation it is clear t h a t further smoothness properties in the sense of classical differentiability depends only on the measure //. First let us give a class of examples of continuous negative definite functions on M. which in general have no further smoothness properties. P r o p o s i t i o n 3.7.12 A continuous function ip : R —> [0, oo) which is even and for which IJJ\R+ is increasing and concave is negative definite. Proof: For k € N the function Vfc := ip /\ k is continuous and even on R, and increasing and concave on R + . Therefore the function fa := k — fa is continuous, even and non-negative on R, and it is decreasing and convex on R+. By Polya's theorem, Theorem 3.5.22, it follows t h a t fa is positive definite, hence by Corollary 3.6.10, the function

V"* = Vfc - V-fc(o) + Vfc(o) = fa(o) -fa + fa{o)

154

Chapter 3

is negative definite.

Fourier Analysis and Convolution Semigroups

Since ip(£) =

lim ipk(€), it follows t h a t ip is negative k—»oo

definite.

•

Let ip : R™ —• R be a real-valued continuous negative definite function and denote by v its Levy measure. It follows t h a t 2

/

B

1 + | x p "(**) < °°>

./R"\{0} and V> has the representation ^(0=c + q ( 0 + /

(l-cos(x-£))Kdz).

(3.186)

7R"\{0}

T h e following theorem is due to W. Hoh, [145]. T h e o r e m 3 . 7 . 1 3 Let ip : R™ —*• R 6e a continuous negative definite function with Levy-Khinchin representation (3.186). Suppose that for 2 < I < m all absolute moments of the Levy measure v exist, i. e. Mi := / \x\l v(dx) < oo, 2 < I < m. JR"\{0} Then, ip is of class C m ( R " ) , and for a G N Q , \a\ <m,

( V>(0, a=0 1/2 |«9^(OI
U,

(3.187) we have the

estimate

(3.188)

M>2

wii/i co = 1, ci = ( 2 M 2 ) 1 / 2 + 2A 1 / 2 , c2 = M2 + 2A and Q = Mh 3 < / < m, where A is £/ie maximal eigenvalue of the quadratic form q in (3.186). Proof: For a = Owe have nothing to show. Let \a\ > 1, but \a\ <m. We may consider the terms in (3.186) separately. T h e constant term is trivial and the estimate (3.188) is well known for the quadratic form with c\ = 2A 1 / 2 , c2 = 2A and ci = 0 for all I > 2. Now we handle the function JRn*\{0} \{0} Since the moments Mi, 2 < I < m, are bounded, we may interchange differentiation and integration to get for a € N Q , \a\ < m,

9?
x<x da cos x

(

)(

• 0 "(<**)>

3.8 Laplace and Stieltjes Transform, and Completely Monotone Functions

155

which gives for \a\ = 1 by the Cauchy-Schwarz inequality

| ^ ( 0 | <(

[ N * "(<**)) \Jw>\{0} J

<([ \x\2u(dx)) yr\{o} /

( / sin2^ • 0 \JR"\{0}

u(6x)) J

(2/ (1 - cos(a: • 0 ) K ^ ) ) \ iE"\{o} J

= (2M2)1/2^/2(0. For 2 < \a\ < m we find IW0l<

/

1^1

\(dacos)(x-Z)\v(dx)

JK"\{0} JR"\{0}

Combining the estimate for 0 and q we get (3.188). •

3.8

Laplace and Stieltjes Transform, and Completely Monotone Functions

We encountered already in Section 3.4 the Fourier-Laplace transform. This section is devoted to a more detailed study of the Laplace transform. Definition 3.8.1 Let u G Ljoc(R), supp u c [0,00). Its Laplace transform is defined by /•OO

£(u)(z) := / e- 2 t u(t)d(, 2 6 C . (3.189) Jo Note that some authors call C the one-sided Laplace transform. Clearly, we need conditions in order that the integral (3.189) converges. For this let z = x + iy, hence |c-rfu(t)|=e-at|u(t)|. It follows that the convergence of the integral in (3.189) depends only on x = Re z. Further, when the integral converges for Re z = x, it converges for

156

Chapter 3 Fourier Analysis and Convolution Semigroups

all z = x + iy such t h a t Re z = x > x since |e-2tu(*)| = e-xt|u(t)| = e - ^ - ^ ' e - ^ K * ) ! < e-*'|u(t)|. We conclude t h a t there exists a number xo G R = [—00, 00] such t h a t t H-» e~ztu(t) belongs to L 1 ( R + ) for Re z > XQ, and for Re z < xo this function does not belong to L 1 ( R + ) . Here we use the convention t h a t XQ = — 00 when 11-» e~ztu(t) belongs to L 1 ( R + ) for all z e C , and xo = +00 when there is no XQ G R such t h a t t \-> e~Xotu(t) is an element of L 1 ( R + ) . This number XQ G R is called the abscissa of absolute convergence of the Laplace transform of u. E x a m p l e 3.8.2 Let u G Ljoc(R), have

supp u C [0,00), such that for some M we

|u(t)| < ceMt.

(3.190)

For any z G C, z = x + iy, such that x > M it follows o-Zt,

*u(t)| = e "

:ct

|u(i)|

that

x M


thus the abscissa of absolute convergence of the Laplace transform or equal to M.

of u is less

T h e o r e m 3 . 8 . 3 Let u G L; 1 oc (R), supp u c [0,00), have xo G R as abscissa of absolute convergence of its Laplace transform. Then z H-» £(U.)(Z) is analytic in the half plane Re z > XQ • Proof: Let z = x + iy, x > xo- T h e function t t-¥ e~ztu(t) is integrable for all these values of z and the function z H-» e~zt\i(t) has the partial derivatives & ( e - * * u ( t ) ) = (-t)e-tzu(t) and ^ ( e " r f u ( i ) ) = (-it)e-ztu(t). Further we have

I («-"<'»

< te-Xot\u{t)\

and

£(.-«««))


thus both functions are dominated by an integrable function independent of z. From Lemma 2.3.23 it follows t h a t /•OO

dz(C(u)(z))

= / Jo

implying t h a t z H> C(U)(Z)

(dze-zt)u(t)

dt = 0,

is analytic in the half plane Re z > XQ.

•

By repeating the arguments in the proof of Theorem 3.8.3 we find immediately

3.8 Laplace and Stieltjes Transform, and Completely Monotone Functions

157

Lemma 3.8.4 Let u G L/1oc(R), supp u C [0, oo), have x0 G R as abscissa of absolute convergence of its Laplace transform and let m G N. Then we have = (-l)mC(tmu(t))(z),

^£{u)(z)

Re z > x0.

(3.191)

On the other hand, we find for u G L;1oc(R), supp u C [0, oo), with i o € l a s abscissa of absolute convergence of its Laplace transform and with the property that u|[oi00) € C m ([0, oo)), by partial integration that /jra

m 1

~

\

C( ^ u ( * ) ) ( * ) = zmC(n)(z) V /

- 2 ^ " ^ ' ( O ) , j=o

Rez>x.

(3.192)

Next we want to study the relation of the convolution and the Laplace transform. For this let u, v G L/oc(R) such that supp u C [0, oo) and supp v C [0, oo). Further suppose that the abscissa of absolute convergence for the Laplace transform of u is io G R and that of v is i i G R. The convolution of u and v is given as usual by (u*v)(£) = / u(i — s)v(s) ds. JR

However, for s < 0 it follows that v(s) = 0, and for s > t we have u(i — s) = 0, hence we find (u * v)(i) = / u(i - s)v(s) ds=

u(t - s)v(s) ds.

(3.193)

JO

JUL

Obviously we have supp (u * v) c [0, oo). Moreover, since for any T > 0 / |(u*v)(i)|d*= / Jo Jo

/ u ( t - s ) v ( s ) d s dt Jo rp

< J Jo

rp

f

Jo

|u(t-*)||v(*)|d*di,

the function u * v belongs to L;1oc(R). Theorem 3.8.5 Let u and v be as above. The function u * v has xo V 2:1 as its abscissa of absolute convergence for the Laplace transform and £(u * v)(z) = C(v)(z) • C(v)(z) holds for all z e C , Re z > XQ V x\.

(3.194)

158

Chapter 3 Fourier Analysis and Convolution Semigroups

Proof:

For Re z > x0 V x\ we may interchange the order of integration to find pOO

I

/»oo

e-zt{u*v)(t)dt=

/

Jo

ft

e-ztu{t

/

- s)v(s) ds dt

Jo Jo /•OO

=

ft

e-z^-s\(t-s)e-zsy(s)dsdt

/ Jo

Jo />00 poo

/>C />oo

e-*(.t—>u(t JO 7s = C(u)(z)-C(v)(z).

- s) dt e ~ " v ( s ) ds a

Next we want to give an inversion formula for the Laplace transform. In fact we will give two such formulas. T h e first uses only elementary properties of the Laplace transform and it holds for continuous functions. We have taken the proof from D. Widder [311], p.290. T h e o r e m 3 . 8 . 6 Let u G C(R), supp u C [0,oo), have XQ G K as its abscissa of absolute convergence for the Laplace transform, and let f(z) = C(u)(z) be its Laplace transform. It follows that

u{x) =

iim izllV) (*) .(£.) +\ fc_,oo k\

\XJ

x>

o.

(3.195)

\XJ

Proof: For shorthand let us denote the right hand side of (3.195) by Using Lemma 3.8.4 we find for x > 0 such t h a t - > XQ

C(f)(x).

fc+i C(i)(x)

= lim f: ( - ) Lfc+l

e-^sku(s)

ds

fOO

^lim^-/ fc->00 k\

/

(e-H)u{xt)dt.

(3.196)

JQ

From (3.196) it follows t h a t we only have to prove t h a t i.fc+1

r°o

lim ^ — - / fe-»-oo k\ JQ

(e-H)ku(t)

dt = u ( l ) ,

(3.197)

i.e. only the case x — 1 is to show. In fact, for any other x we replace u(xt) by v(i) := u(xt). Since v fulfils the same assumptions as u, we may apply (3.197)

3.8

Laplace and Stieltjes Transform, and Completely Monotone Functions

159

for v. Now, let us prove (3.197). Using the definition of the T-function we find kk+ k\

1

/*oo

dt = 1, Jo

hence it follows t h a t we have to prove t h a t k+i /-oo kl.fc+1 m —— / lim (e-H) kfc-»oo

"••

(u(i) - u ( l ) ) dt = 0.

(3.198)

Jo

Let £ > 0 and choose a and b, 0 < a < 1 < b < oo, such t h a t \u(t) - u ( l ) | < £ for a < i < b.

(3.199)

We split the integral in (3.198) into three integrals 7i, I2, I3, where for I\ we integrate from 0 to a, for IQ, we integrate from a to b, and ^3 is just the integral from b to 00. Now it follows t h a t

\h\<(e-aa)k

[a\u(t)-u(l)\dt,

Jo

thus kk+l k\

h

kk+i

< cu-

(e-a)'

k\

But kk+1

t.

fc—>oo "<•

Using (3.199), we get fck+l

kk+i

k\

<

rb

-I. ^

~k\

edt

<e.

To handle ^3, note t h a t b > 1 and therefore the function t i ^ e on (6,00). It follows t h a t kk+1 k\

<

kk+1 k\

(e-bb)k-k0

e-fc0V°|u(i)-u(l)|di

*t is decreasing

(3.200)

160

Chapter 3

Fourier Analysis and Convolution Semigroups

for all k > ko > XQ. Thus the integral on the right hand side of (3.200) converges, and since kk+i

lim V ( ^ ) f c " ° = 0 , fc—»oo

""

the theorem is proved. D In order to get the second formula for the inversion of the Laplace transform we establish first its relation to the Fourier transform. For this let u e L 1 (R), supp u C [0, oo), such t h a t |u(f)| < ceMt for c > 0 and some M E R. For z e C, Re z > M, we have /•OO

£(u)(z) = / Jo

/>00

e-ztu(t)dt=

e-iyte-xtu(t)dt Jo

= f

e~

iyt

(e-^-^vLQJ)

J — oo

^

dt, '

thus we may extend u from R to the half plane {—iz | Re z > M}

by

:= ( 2 T T ) - 1 / 2 £ ( U ) ( Z ) .

u(-iz)

(3.201)

For z = iy we find t h a t u(y) = (2Tr)-1/2C{u)(iy) = (2TT)- 1 /2 J ° ^ e - ^ * u ( t ) dt, note t h a t supp u C [0,oo). From Theorem 3.8.3 it follows t h a t the function z H-> u(—iz) is analytic in {—iz | Re z > M}. However, we may also use this calculation to define £ with the help of the Fourier transform: £(u)(z)

= (2n)1/2u(-iz)

= (27r)1/2Ft^y(e-xtu(t))(y),

(3.202)

and it follows t h a t (3.202) is defined for z <E C, Re z > M, and u £ L J ( R ) , supp u C [0, oo), such t h a t |u(t)| < ceMt. T h e o r e m 3 . 8 . 7 Let u e Ljoc(M), supp u C [0, oo), and assume that for some 7o G R the function x i-» e~loXu{x) is an element of A(W). It follows that 1

/•70+ioo

u{x) = — : /

ezxC(u){z)

dz.

(3.203)

2,-Kl ' 7 oJy —n-ioO too

Proof:

Let R > 0 and consider the integral 1

— / 2-Ki

p-yo+iR

ezxC(u){z)

-|1

dz = —l

ryo+iR fio+iR

roo

ezx

e~ztu(t)

dt dz.

3.8 Laplace and Stieltjes Transform, and Completely Monotone Functions

161

Substituting z = 70 + iy we find 1

flo+iR /•-yo+iR

— / 2m ™Jl0-iR

j0x eplax

ezxC(u)(z) dz = ^—

27T

fR fR

roo /-oo

/

eixye-iyt{e-iotVi{t))

dt dy.

J-RJO

We are allowed to take the limit R —>• 00 and we find by Corollary 3.2.12 and because of supp u C [0, 00) that 1

/>oo

--/

/-oo

eixve'ivt(e-yotu{t))dtdy

/ -1

/>00

= — /

/.00

eixye-iyt(e-"
/

2TI"7-OO

dt dy =

e'^^x),

J-00

hence, (3.203) is proved. • Formula (3.202) gives also an idea how to define the Laplace transform for a distribution u £ V(R), supp u C [0, 00). If u is such that e -i ( x '-)u(.) e 5'(R), we can define the Laplace transform of u for suitable values of z G C by £(u)(z) := (27r) 1 / 2 F^ v (e-(*->u)(y).

(3.204)

In particular, when /i is a bounded measure supported in [0,00) its Laplace transform would be given by /•OO

/-OO

e-xte~iyt

C{fi){z)= Jo

n(dt) = / Jo

e~zt fi(dt).

(3.205)

Since we will later on need only the Laplace transform of measures supported on [0, 00) we will only treat this case. The theory of the Laplace transform for distributions is handled in detail in the book [237] of B. Petersen and originates in the work of L. Schwartz [278]. Definition 3.8.8 Let n be a measure on R and assume that supp /J, C [0, 00) as well as JQ e~xs /x(ds) < 00 for all x > 0. Its Laplace transform is defined by /•OO

C(fi)(z):=

/ Jo

e~ztfx{dt).

(3.206)

From the assumption it follows that for Re z > 0 the integral in (3.206) converges absolutely. As in the case of functions we have

162

Chapter 3 Fourier Analysis and Convolution Semigroups

T h e o r e m 3.8.9 The Laplace transform of a measure /x satisfying the assumptions of Definition 3.8.8 is an analytic function in the half-plane Re z > 0. / / in addition fi is bounded, its Laplace transform has a continuous extension to the half-plane Re z > 0. Proof:

We can proceed as in the proof of Theorem 3.8.3 to find t h a t /•OO

dzC(jj.)(z)

= / Jo

(dze~zt)

/i(dt) = 0.

T h e continuity of £(/x) in Re z > 0 follows by a straightforward application of Lemma 2.3.22, provided xx is bounded. • Let fj,, v be two measures as in Definition 3.8.8. It follows t h a t /x * u is defined and we find /•oo

/ Jo

e-t2(iz*z/)(di)=

/>oo

/ Jo

/-oo

/ Jo

e-
/•oo

= / Jo

fi(dt)

v(ds)

/*oo

e-ztfi(dt)

e~zsv{ds) Jo

Thus we have £ ( / i * i/) = C(ji) • C(v).

(3.207)

For a measure /x as in Definition 3.8.8 it follows t h a t (3.204) or (3.205) makes sense for all z, Re z > 0. In particular, when we fix some xo > 0 we find t h a t £(/x)|R e z=Xo is uniquely determined by ix. Since £(/^) is analytic in Re z > 0, we finally see t h a t the Laplace transform is injective when defined on the set of all measures satisfying the assumptions of Definition 3.8.8. Our aim is to give a complete characterisation of the Laplace transform of positive measures satisfying the assumptions of Definition 3.8.8. For this we need D e f i n i t i o n 3 . 8 . 1 0 A real-valued function pletely monotone if ( - 1 ) ^ > 0

f G C°°((0, oo)) is said to be com-

(3.208)

holds for all k € No. Using Leibniz' product formula, it follows immediately t h a t the product of two completely monotone functions is also a completely monotone function.

3.8

Laplace and Stieltjes Transform, and Completely Monotone Functions

163

L e m m a 3 . 8 . 1 1 Let fi be a positive measure such that the assumptions of Definition 3.8.8 are fulfilled. In this case the Laplace transform of JJL restricted to (0, oo) is a completely monotone function. Proof: We know already t h a t we may differentiate in (3.206) under the integral. Thus for x > 0 we find

(e rt)M(df)

^)^=r^ " /•OO

= / Jo

(-lftke-xt

fi(dt),

implying t h a t ( - l ) f c ^ £ ( / x ) ( a ; ) > 0 for all

fceN0.

•

In order to prove the converse of Lemma 3.8.11 we need some preparation. Let (afc) fceN be a sequence of real numbers, and set as in Section 2.4 5ak := afc+i — a t .

(3.209)

Further, for I € N we define 6lak = 6(6l-1ak).

(3.210)

Recall t h a t a sequence (afc) fceN is called completely monotone if for all k, I 6 No {-l)l5lak

> 0

(3.211)

holds, see (2.103). L e m m a 3 . 8 . 1 2 Let f £ C°°((0,oo)) be a completely monotone function 7] > 0. Then the sequence (t(kr]))fceN is completely monotone. Proof: Let us set g(:r) = i(r]x). Moreover, the function

It follows t h a t g is completely monotone.

-<5g(a;) = g(x) ~ g(x + 1) is completely monotone too. In fact, by the mean-value theorem we have ( - l ) m ( g ( m ) ( z ) - g(m\x

and

+ 1)) - (_l)™+i g (™+i)(0) > 0

164

Chapter 3 Fourier Analysis and Convolution Semigroups

for some 6 € (x, x+1). Iterating this result fc-times we find t h a t x H-> (—S)lg(x) is a completely monotone function, hence non-negative on [0, oo) for all I € N. In particular, we have (-l)'(j'g(A:) = (-l)lSli(krj)

> 0

for all k, I G No, which proves the lemma.

•

Now, we can use Hausdorff's moment theorem, Theorem 2.4.5, to prove Bernstein's theorem, which gives a converse to Lemma 3.8.11. T h e o r e m 3 . 8 . 1 3 A real-valued function f G C°°((0, oo)) is completely monotone if and only if there exists a unique positive measure /x on R supported on [0, oo) such that /•OO

e~xs /i(ds), x > 0.

i(x) = / Jo

(3.212)

Proof: We only have to prove the necessity. From our considerations made above it is clear t h a t if f has the representation (3.212), it extends to the halfplane Re z > 0 as an analytic function being the Laplace transform of the measure \x. Thus \x is uniquely determined by f. Now, consider the sequence (f(m))fcpN ) m e N. By Lemma 3.8.12 this sequence is completely monotone. From Hausdorff's moment theorem, Theorem 2.4.5, we conclude t h a t there exists a seqiience of measures / i m on [0,1] such t h a t f

( m )

=

/

*fc/im^

(3 213)

-

holds. Denoting by T m : (0,1) -» (0,1) the mapping t h + ( m , m e N , it follows that f(fc)= / i f c m / i m ( d t ) = f tkTm(fim)(dt)= Jo Jo

[ Jo

tk^{At),

implying t h a t T m ( / x m ) = (ii for all TO G N. Therefore we get

f ( ^ ) = ^ V T" V ) ( d i ) = ^ ifc/m Mi(di). Defining now T : (0,1) -> (0,1), t M- - l n t , we find /•l

/ thlm^{dt)= Jo

fl

Jo

/"CO

e-"TWMi(d*)=

/ Jo

e " # Tt/iiXdt).

3.8 Laplace and Stieltjes Transform, and Completely Monotone Functions

165

Thus, setting (i := T(fXi), we have for all rationals r £ (0, oo) that /•OO

f(r)=/ Jo

e-rt»(dt)

holds. By continuity we deduce that (3.212) holds. • Remark 3.8.14 In the paper [221], L. Mattner gave an "elementary" proof of Bernstein's theorem using Helly 's selection theorem. Further, in [106] T. Gneiting proved that one may substitute the condition that (3.208) holds for all k £ No by the condition that it holds for infinitely many k £ No. His proof shows that the validity of (3.208) for infinitely many k £ No implies (3.208) for all k £ No and it is based on a careful analysis of the Taylor formula of f. Moreover, he gave a counterexample that it is not sufficient to require (3.208) only for finitely many values of k £ No. Let f be a completely monotone function having the representation (3.212). By the monotonicity of f it follows that lim i(x) exists, and from (3.212) we x->0

get limf(ar) =/x([0,oo)),

(3.214)

x-s-0

implying that lim f(a;) is finite if and only if /J, is a bounded measure. Thus we have

x-»0

Corollary 3.8.15 A. Every completely monotone function f has an analytic extension to the half-plane Re z > 0 with continuous extension to the halfplane Re z > 0 if and only if lim i(x) < oo. B. The Laplace transform is x-»0

a bijection of the set of all measures /x on [0, oo) for which J0°° e~xs n(ds) is finite for all x > 0 onto the set of completely monotone functions. Further, the Laplace transform is a bijection of the set of all bounded measures on [0, oo) onto the set of all completely monotone functions for which lim i(x) < oo. x->0 e a

Proposition 3.8.16 Let (ik)keN ^ pointwise converging sequence of completely monotone functions with associated measures (/Jfc)fcGN on [0, oo). Then the limit i(x) := lim ik(x), k—too

x > 0,

(3.215)

Chapter 3

166

is completely monotone.

i k { x )

^ ^

=

Fourier Analysis and Convolution Semigroups

Furthermore,

^

for all I G N we have (3.216)

i { x )

and lim fik = M vaguely on [0, oo), where fi is the representing

(3.217)

measure of the completely monotone

function

f.

Proof: First let us prove t h a t the sequence (/i/b)fceN is vaguely bounded. For G Co([0,oo)), s u p p ^ C [a^,6^,], a^ > 0, we find /•OO

/

/.OO

{s) fik(ds)

=

/

es(j){s)e-s /xfc(ds) /•OO

<e6*Moo/ Jo

e-Mfc(d*) = ^ 1 1 0 1 1 ^ ( 1 ) .

Since ffc(l) converges to f(l), it follows t h a t /•OO

sup / fceN 7o

(s) //fe(ds)<

OO

for all 4> G Co([0, oo)), hence (^fc)fcgN *s v a g u e l y bounded and therefore by Theorem 2.3.10 vaguely relatively compact. Let (MfcJjgN ' De a n y v a g u e t y convergent subsequence with vague limit /x. Fix x > 0 and for a > 0 take ^>0,m G C 0 ( [ 0 , CO)), 771 G N , s u c h t h a t 0 < 0 a , m < 1, a,m < a,m+l, 4>a,m\[0,a] = 1

lim 4>a,m{x) = 1- Since for s > a we have e

_;srE

_

< e ~2~e~~2", it follows t h a t

771—^OO

ikl[x)

- e-*'f fcl ( | ) < &,(*) - j™ e~sx »k,(ds)

f

Fori

oo we get i(x) - c - * f ( | ) = f

h,mm(ds)

<

Jo

e-sxcj>a,m(s)

a n

fx(ds) < i(x),

= j \ ~ ' * hi(x).

»kl(ds)

d

3.8

Laplace and Stieltjes Transform, and Completely Monotone Functions

167

which yields for m — • oo f(x) - e-Sf ( | )

e'sx

- f

fi(ds) < i(x),

(3.218)

and finally as a —> oo we get /•OO

f(z) = /

e~sx n{ds).

(3.219)

JO

This shows t h a t f is completely monotone. Since the measure fj, is uniquely determined by (3.219), we find further t h a t lim /ik = A4 vaguely. Note t h a t lim sle~as

= 0 for all a > 0 and Z € N. Hence, using t h a t lim /ifc = A* v a g u e ly>

«—KX)

fc—VOO

we may argue as above to get (3.216). • Let /x be a positive measure on R with supp /x c [0, oo). For i > 0 w e find /•OO

-I

/-OO

I

/

/>00

\

)

e~xsC(iJ,)(s)

ds =

£2{fi)(x).

/o

Jo

This transform had been considered by T.J. Stieltjes in [291]-[292]. D e f i n i t i o n 3 . 8 . 1 7 A function f : (0, oo) —»• R is called a Stieltjes transform, if there exist an a > 0 and a measure /x on R, supp /i C [0, oo), such that ,oo

1

i(x) =a+

fi(dt)

Jo

forx

> 0.

(3.220)

x+t

From our calculation made above and from the fact t h a t lim i(x) = a x—>oo

it follows t h a t the pair (a, /i) is uniquely determined by f. Let us denote by <S the set of all Stieltjes transforms. Elements of <S will sometimes be called Stieltjes functions. L e m m a 3 . 8 . 1 8 Every Stieltjes function

is completely

monotone.

168

Chapter 3

Fourier Analysis and Convolution Semigroups

Proof: This follows immediately from the calculation made above and properties of the Laplace transform. • P r o p o s i t i o n 3 . 8 . 1 9 The set S is a convex cone which is closed under pointwise convergence. Let (ffc)fcgN be a sequence in S converging pointwise to f. Then the sequences (flfc)fceN and (/ifc)fc€N, (ak,(ik) associated with ft satisfy lim [ik = A4 vaguely, where (a, [xj is the pair associated with f. / / in addition k—t-oo

lim ak = a then for every x > 0 the measures j r r j / ^ d s ) converge with respect k—>oo

to Coo to

^/j,(ds).

Proof: Obviously <S is a convex cone. Next we claim t h a t the sequence (Hk)kaN is vaguely bounded. Indeed for <j> G Co([0, oo)), supp <j> C [a > 0, we find • /•OO

/ Jo

-|

/"OO

(s) fik(ds)

= / Jo

(1 + *MB)——/ifc(ds) I T S

= (l + ^)Woo(ffc(l)-«fc)

Since ffc(l) converges to f(l), we get /•OO

sup / ken Jo

(s) Vk{ds) < oo

for all £ Co([0, oo)). Hence, a subsequence (MfcJ/eN of (Mfc)fceN converges vaguely to a measure fi. Now fix x > 0. Since lim

(/

^-gMds)

we may conclude t h a t

lim f l—K>o JO

+ ak ) =f(x),

^/Xfc(ds)

W, (°^

X+ S '

' '

'

< c(x) < oo. Since

#*) JJo 0

fj,(ds)

X+ S

for all <j> € Co([0, oo)), because s i-» ^ £ belongs also to Co([0, oo)). T h e space CQ([0, oo)) is dense in Coo([0, oo)), therefore by Theorem 2.3.6 we see t h a t

3.8

Laplace and Stieltjes Transform, and Completely Monotone Functions

169

j ^ / i f c , ( d s ) converges weakly to some measure v, i.e. for all € Coo([0, oo)) we have

Jo

x

+s

Jo

T h e inclusion Co([0,oo)) c Coo([0,oo)) implies now

"(da) = ^+~s^ds): i.e. -L-pkl(ds) —> - j - / i ( d s ) a; + s a; + s with respect to Coo- Since /•°o lim l'kl{x) = lim / i-yoo

i->ooJ0

(3.221)

_i V

x

2

/i fc ,(ds)

i *)

a n d s i-» ^ j belongs to Coo([0, oo)) we find f°°

-1

k {x)

¥TW{ds)-

}rJ < =Jo

From Lemma 3.8.18 combined with (3.216) we deduce t h a t ,oo 1 i(x) = a + — — fi(ds),

Jo

x

+s

implying t h a t f is a Stieltjes function. T h e uniqueness of the representation of f implies immediately t h a t fj,k —> /J, vaguely. Moreover, if lim ak = a, it follows t h a t -/Xfe(ds) —> —•—/x(ds) x + s x+s with respect to Coo, and the proposition is proved.

•

From Theorem 2.3.11 we know t h a t the discrete measures on [0, oo) are vaguely dense in the set of all measures on [0, oo). For any a > 0 and any discrete measure S = ^ i = 1 aieXl on (0, oo) the function /•oo

ia,s{x):=a+

-I

M

^^y^aieXl{ds)=a

M

+ Yj~^—

(3.222)

170

Chapter 3 Fourier Analysis and Convolution Semigroups

is a Stieltjes transform and by Proposition 3.8.19 it follows t h a t any Stieltjes function f is the pointwise limit of functions of type (3.222). L e m m a 3 . 8 . 2 0 For f e S and A > 0 we have always j f ^ 0, then x K-> - T W belongs also to S. Proof:

^ G S, and i / f 6 5 ,

It is sufficient to prove the lemma for functions of the form

M i(x) = a + >

, a, a > 0 and xi > 0.

It follows t h a t x i-» f(z)(l + Af(a;)) _1 is a rational function which admits a representation M

W + *<*»-'= I ^ + E ^ with suitable a[ > 0 and x\ > 0. But the same argument applies to the function x H-» 77JY, provided f ^ 0. • P r o p o s i t i o n 3 . 8 . 2 1 Let i be a completely function

monotone

is an element o / L / o c ( R ) . Then the Laplace transform li:=ae0

{1

+ f0(.)\ \

function

of the

a>0,

belongs to S, and every element of S is the Laplace transform of type (3.224). Proof:

such that the

measure (3.224) of a measure /J,

For fi as in (3.224) we find /•OO

C(fi)(x)=a+

/ Jo

e-sxi(s)ds.

But f is the Laplace transform of a positive measure with support in [0, oo), see Theorem 3.8.13, and it follows t h a t £(/x) € S. Now let f e <S with representation

f

a+ Kdi)

^= r^

3.8

Laplace and Stieltjes Transform, and Completely Monotone Functions

for a > 0 and a measure u supported in [0, oo). P u t t i n g g = £(v), t h a t the function

g

°

( )

-"l0;

171

it follows

x<0

is an element of L/ oc (K) and that f=£(a£o+go(.)A(1)).

a

We will continue to discuss Stieltjes functions in the next section. We end this section with three examples of Stieltjes transforms. E x a m p l e 3 . 8 . 2 2 A. For a € (0,1) the function transform and we have n x-a

=

x i->- x~a,

x > 0, is a Stieltjes

sin(o!7r) f°° t~a , —v.—/ / dt 7T J0 X+ t

B. The function

, (3.225) V '

x i-» log ( l + A), x > 0, is a Stieltjes transform

since

holds, C. In [274] W. Schneider proved that the generalised Mittag-Leffler function Fap(x) := r(8)Ea^(-x), where ~Eap(x) = J2T=o r(ak+0)> is completely monotone if and only if0 1 we have ^(d8)

=

(l-{l-s)f-lXm(8)XW(d8),

which gives Fip(x)=

f e-x'(l-(l-8)f-1da, Jo

/3>1.

172

Chapter 3 Fourier Analysis and Convolution Semigroups

3.9

Bernstein Functions and Subordination of Convolution Semigroups

In this section it is sometimes rather important to take care of the difference of JQ00 i(t) fi(dt) and J0°^ f ^ ( d i ) . In the first case fi is a measure on [0, oo), in particular, it might happen t h a t n({0}) = c ^ 0. In the second case either fj, is a measure on (0, oo) or /x is a measure on [0, oo) but the integration is over the set (0,oo), i.e. with respect to the measure x(Q \A*- I n c a s e M({0}) = c ^ 0 we find / Jo

i(t) n(dt) = cf(0) + / f(t) /i(dt). Jo+ '0+

Clearly, for /i({0}) = 0 we have /»oo

/ Jo

/-oo

f(i) /i(di) = / i(t) K d t ) . Jo+

This holds especially when fi has a density with respect to A ^ l ^ o o ) D e f i n i t i o n 3 . 9 . 1 A real-valued function function if f > 0 and ( - l )

f c

f € C°°((0, oo)) is called a Bernstein

^ir < 0

(3.227)

holds for all k g N. A Bernstein function is positive, increasing and concave. Moreover, the set of all Bernstein functions forms a convex cone containing the positive constants. There are obviously close relations of Bernstein functions and completely monotone functions. If f is a Bernstein function, then f is completely monotone. Conversely, if g' is a completely monotone function, it follows t h a t g is a Bernstein function. T h e next result gives a closer link between these two classes of functions. P r o p o s i t i o n 3 . 9 . 2 For a function are equivalent: 1. i is a Bernstein

function;

f : (0, oo) —> K the following two

assertions

3.9 Bernstein Functions and Subordination of Convolution Semigroups 2. f > 0 and for allt>0

173

the function exp(—if) is completely monotone.

Proof: Let f be a Bernstein function and i > 0. The function 4> '• (—oo, 0) -» R, (x) = — if(—x) is defined for x € (—oo,0), arbitrarily often differentiable, (x) < 0 and (*) > 0 for all k £ N. For g := exp() it follows that

d

m! d "V(*> = s . " dx ^ i!j! • . . . •fc!dy m

n

y=«(-)V

1!

/

""

V

«

where the summation goes over all solutions in No of the system i + 2j + 3/i + . . . + Ik = m i + j + h+...

+ k = n,

see (2.22). Thus we have for all A; e N that ^p> 0. However, this implies that x H-> g(—x) = exp(—ti(x)) defined on (0,oo) is completely monotone. Now suppose that f > 0 and that for all i > 0 the function exp(—if) is completely monotone. Since exp(—if) is arbitrarily often differentiable, it follows that f itself is a C°°-function too. For k £ N, x > 0 and i > 0 we find

° * (-l)V ( e X p ( ' i f ( a ; ) ) ) = E(-1)fc+J'7Td^(f(a;))J'Dividing by i > 0 and taking the limit i —> 0, we get

(-i)1+*^!tf*)>° or (-1)^(*)<0, proving the proposition. D Corollary 3.9.3 The convex cone of Bernstein functions is closed under pointwise convergence. Proof: This follows immediately from Proposition 3.9.2 and Proposition 3.8.16. • Now we prove an integral representation for Bernstein functions.

174

Chapter 3 Fourier Analysis and Convolution Semigroups

T h e o r e m 3 . 9 . 4 Let f he a Bernstein function. a,b>0 and a measure fx on (0, oo) verifying f°° s / Jo+ 1 + such that

s

Then there exist

n(ds) < oo

constants

(3.228)

POO

(1 - e~xs)

i(x) = a + bx +

/x(ds), x > 0.

(3.229)

Jo+ The triple (a, b, fi) is uniquely determined by f. Conversely, given a, b > 0 and a measure fi on (0, oo) satisfying (3.228), then (3.229) defines a Bernstein function. R e m a r k 3.9.5 A. Note that (3.228) is fulfilled if and only if /•l

/»oo

/ s (J,(ds) < oo and / Jo+ Ji

fJ-(ds) < oo

(3.230)

holds. B. From (3.229) it follows that a Bernstein function f with representation (3.229) is bounded if and only if b = 0 and /i((0,oo)) < oo. In fact, for 6 = 0 and /x((0, oo)) < oo we find /•OO

(1 - e~xs)

f(z) =a+

n{ds)
10+

Conversely,

if i is bounded, we deduce that b = 0, and Fatou's lemma poo /•OO

a+

yields

/-OO

l/x(ds) < a + liminf / J0+

a;->oo

( l - e~sx)

/x(dx)

7o+

= supf(a;) < oo. x>0

P r o o f of T h e o r e m 3.9.4: Let // be a measure on (0, oo) satisfying (3.228). For all i , s > 0 w e have 1 - e~xs < xs and 1 - e~xs < 1, implying by (3.230) that /•OO

g(aO = / (1 - e~*') Jo+ '0+

fi(ds

is finite for x > 0. Further, we find for x > 0 and h > 0 t h a t i ( g ( x + /i) - g(z)) = ^

^ e " " ( l - e-hs)

Kds),

3.9

Bernstein Functions and Subordination of Convolution Semigroups

175

which yields by the dominated convergence theorem lim i ( g ( a ; + h) - g(x)) = / se~xs h->-0 n Jo+

M (ds).

(3.231)

For x > 0, h < 0 such t h a t x + h > 0 it follows t h a t

i (g(a: + h)- g{x)) = J°° Ie-<*+fc>' ^x~h>

- l) »(ds)

and a further application of the dominated convergence theorem gives also for h < 0 such t h a t x + h > 0 lim -(g(a; + h)-

g(x)) = /

s e - x s fi{ds).

(3.232)

Thus g is differentiable, and for x > 0 we have /•OO

d

g(z) = / se~xs da; jJo+ + 0

fi(ds).

From Bernstein's theorem, Theorem 3.8.13, we deduce t h a t g | is completely monotone and therefore g is a Bernstein function. Thus x>-+ i(x) = a + bx + g(x),

x > 0,

(3.233)

is a Bernstein function. Suppose now t h a t f is a Bernstein function. Then g£ is completely monotone, and by Bernstein's theorem, Theorem 3.8.13, there exists a measure v on R with supp v c [0, oo) such t h a t /•OO df -(x) = / e~xs i/(ds), da;'• Jo

x>0.

T h e measure v can be written a s i / = feo + &, where b = ^({0}) and a is the restriction of v to (0, oo). It follows t h a t

r°°

df -(x)=b

+

l+e-a(ds).

Setting a := lim f(a:), we find for x > 0 t h a t

i(x)=a+

/•oo j f —(t)dt Jo+ dt

e~xs

f°° 1 = a + bx+

a(ds). J0+

s

176

Chapter 3 Fourier Analysis and Convolution Semigroups

Since C°° 1 — p~s

f°° 1

we find t h a t

i:

- a(ds) < oo.

We define the measure /J, on (0, co) as /Lt(ds) = ^cr(ds), implying t h a t /x satisfies (3.230), hence (3.228). Thus we have proved that f has a representation of form (3.229). Using the formula (3.229) for a Bernstein function f we find a=

lim i(x)

and b = lim -L—^-.

x—t-0

x—>oo

%

Since ir

— (x)=b

roo

se-sxfi(ds),

+ J

the measure beo(ds) + s/i(ds) is uniquely determined by Bernstein's theorem. Thus fi is uniquely determined by f. • From the representation formula (3.229) it follows t h a t a Bernstein function f ^ 0 is strictly positive for all x > 0. Hence, we may consider the function x M- jT^y. Applying formula (2.22), or the identity j = J0°° e~ t f di, we see t h a t this function is completely monotone. A further application of (2.22) shows t h a t for a non-zero Bernstein function f and any completely monotone function g the function g o f is also completely monotone. Moreover, it follows from the very definition of Bernstein functions and completely monotone functions, respectively, t h a t for any Bernstein function g the function h(x) = Si5i j s completely monotone. Now, let f be a further Bernstein function being strictly positive on (0, co). T h e n we find t h a t h o f = %j- is completely monotone for any Bernstein function f > 0. Let f : (0, oo) —» R be a Bernstein function with representation (3.229). Since for s > 0 and z G C, Re z > 0, the inequalities |1 — e~zs\ < s\z\ and |1 — e~zs\ < 2 hold, see Lemma 2.1.1, it follows t h a t we may extend f using (3.229) into the half-plane Re z > 0, i.e. we can define POO

i{z)=a

(l - e-sz)

+ bz+ ./o+

/x(ds), Re z > 0.

(3.234)

3.9 Bernstein Functions and Subordination of Convolution Semigroups

177

As in the case of completely monotone functions it is easy to see that dzi = 0 for Re z > 0, i.e. f is analytic in Re z > 0, and, moreover, f is continuous in Re z > 0. Next we want to relate Bernstein functions to certain convolution semigroups of measures. Definition 3.9.6 Let (vt}t>o be a convolution semigroup of measures on R. It is said to be supported by [0, oo) if supp nt C [0, oo) for all t > 0. T h e o r e m 3.9.7 Let f : (0, oo) —> R be a Bernstein function. Then there exists a unique convolution semigroup (r]t)t>0 supported by [0, oo) such that C(r]t)(x) = e~ti(x\

x>0andt>0,

(3.235)

holds. Conversely, for any convolution semigroup (?7t)t>0 supported by [0, oo) there exists a unique Bernstein function f such that (3.235) holds. Proof: Let f be a Bernstein function. From Proposition 3.9.2 it follows that the function x M- exp(—ti(x)) is completely monotone for all t > 0, and since lim i(x) > 0 we find that x->0

lim e - ^ < 1. From Bernstein's theorem, Theorem 3.8.13, we deduce the existence of a measure rjt on [0, oo) such that r)t([Q, oo)) < 1 and C{r]t){x) = e~ti{x\

x > 0 and t > 0.

(3.236)

However, (3.236) holds for all z in the half-plane Re z > 0. Therefore, having in mind the relation of the Fourier transform to the Laplace transform, see (3.204), we find for z = iy, i.e. Re z = 0, {2^)1'2flt{y)

= C{m){iy) = e-u^\

y € R.

(3.237)

Now, Schoenberg's theorem, Theorem 3.6.11, says that the function y H-> f(iy) is negative definite and obviously this function is continuous. Hence, by Theorem 3.8.17, (r?t) t>0 is a convolution semigroup. Now, let {r)t)t>0 be a convolution semigroup on R supported by [0, oo). For x > 0 fixed we consider the function <j>x : (0, oo) —• R defined by <j>x(t) :=

178

Chapter 3

Fourier Analysis and Convolution Semigroups

C(r)t)(x). We have cf)x(t) > 0 and by Theorem 3.8.5 we find x(s + t) = 4>x{s)(j)x{t). Moreover, from Lemma 3.6.3 it follows t h a t x is continuous and t h a t lim <j>x(t) = 1. Therefore, there exists a uniquely determined real number t->o i(x) such t h a t

Mt)=C{rk){x)

= e-tt^,

t>0.

Since <j>x{t) < 1 for x,t > 0, it follows t h a t f(a;) > 0 for x > 0, and from Proposition 3.9.2 we conclude t h a t f is a Bernstein function. • R e m a r k 3.9.8 Since (3.235) holds in the half-plane 77t([0,oo))=£(7 ? t )(0) = e - t f W ,

Re z > 0 we have

t>0.

(3.238)

Thus r/t is a probability measure if and only if i(Q) = 0. Further note that the continuous negative definite function associated with the convolution semigroup r s ( ?*)t>o * given by y i-» i(iy). In particular, this means that for any Bernstein function f the function £ — i > V"(0 :== f(*0 *s negative definite and continuous. From (3.229) we find the Levy-Khinchin representation of ijj to be /•OO

^)=a

(l-e- t e € )M(dO,

+ ibi+

( 3 - 239 )

Jo+ where fj, is the measure from (3.228). Now we come to the central observation of this section. Let (fj,t)t>o be a convolution semigroup on K™ with associated continuous negative definite function tp. Further, let f be a Bernstein function with associated semigroup (77t)t>o supported on [0,oo). For f we have the representation (3.229), and since Re %j> > 0 we may consider the function f o ip: (f o V>)(0 = a + W ( 0 + ^

( l - e - * « > ) fx(ds).

(3.240)

T h e function ^ H-> e~s^) \s positive definite, hence ^ M- 1 - e~s^) j s negative definite by Corollary 3.6.10, implying t h a t f o ijj is negative definite. Thus we have proved L e m m a 3.9.9 For any Bernstein function f and any continuous inite function ip : W1 -> C, the function f o tp is also continuous definite.

negative defand negative

3.9

Bernstein Functions and Subordination of Convolution Semigroups

179

Later on we will characterise the cone of Bernstein functions as those functions which operate on C N ( R n ) for all n. Now let ip and f be as in Lemma 3.9.9. Since f o^> e C N ( R n ) , there exists a convolution semigroup ( / 4 ) t > 0 associated with f o tp. P r o p o s i t i o n 3 . 9 . 1 0 Let tp € CN(M.n) be a continuous negative definite function with associated convolution semigroup (nt)t>o on R™. Further let f be a Bernstein function with associated semigroup (?7t)t>o supported on [0, oo). The convolution semigroup (/4)t>o on ^™ associated with the continuous negative definite function i o tp is given by [

4>(x) / 4 ( d i ) = / J0

JW"

Proof:

/ JRn

^eC0(Rn).

4>(x)^(dx)rh(ds),

(3.241)

For t > 0 and <j> <E C 0 ( K n ) the mapping


/

4>(x) fJ-s(dx) T)t(ds)

JR"

is a positive linear form on C o ( R n ) . Hence, there exists a measure ut on R n such t h a t (p{x) vt(dx)

= j

/

Jo Jw

(p(x)

fj,s(dx)r)t(ds).

Obviously we have vt(M.n) < 1, and for the Fourier transform of vt we find vt{£) = (2n)~n/2

(

= (2TT)-"/2 ^ Jo

e-«-«

[ e~ix< »s{dx) Jun

rjt(ds) /.oo

roo

= / Jo

ut(dx)

A,(Om(dS) = (27r)-"/2/ Jo

e-°^r,t(ds)

= ( 2 7 r ) - " / 2 £ ( 7 7 t ) ( ^ ( 0 ) = (27r)-"/ 2 e- t f ^(«)), implying t h a t ut = fi\.

n

R e m a r k 3 . 9 . 1 1 Instead of (3.241) we will sometimes

write

f°° Ht=

fxsr}t(ds) Jo

vaguely.

(3.242)

180

Chapter 3

Fourier Analysis and Convolution Semigroups

D e f i n i t i o n 3 . 9 . 1 2 In the situation of Proposition 3.9.10 we call the convolution semigroup ( / 4 ) t > 0 the semigroup subordinate (in the sense of Bochner) to {nt)t>o with respect to (r)t)t>0 Before proceeding further with the general theory we want to give examples of Bernstein functions and associated semigroups. E x a m p l e 3 . 9 . 1 3 The function x *-+ a, a > 0, is a Bernstein function with corresponding semigroup (e~ a t £o) t >o- Further, the function x i->- bx, b > 0, is a Bernstein function with associated semigroup (£bt)t>oE x a m p l e 3 . 9 . 1 4 For s > 0 the function i(x) = 1 — e~xs is a Bernstein tion. It corresponds to the Poisson semigroup with jumps of size s, i.e.

Vt = J2e~t~k\£sk>

t

o-t.

func-

( 3 - 243 )

-°-

fc=o

E x a m p l e 3 . 9 . 1 5 The function

i(x) = log(l + x) is a Bernstein

function

since

/•OO

(l-e-xs)s_1e_sds,

log( l + x)=

x > 0.

(3.244)

Jo The semigroup associated with this Bernstein function which is given by

is called the r-semigroup

where Zt{x)=X{0,oo){x)^)xt-1e-*.

(3.245)

E x a m p l e 3 . 9 . 1 6 For any a € [0,1] the function ia(x) = xa is a Bernstein function. For a = 0 and a = 1 this is trivial but for a € (0,1) we have /"OO

xa = — r r(i-a)y0

(l-e-x')3-a-1d3,

x>0.

(3.246)

This Bernstein function corresponds to the one-sided stable semigroup of order a which we denote by (a")t>0. For a — 0 we find of = e _ t £ o , and for a = 1

3.9 Bernstein Functions and Subordination of Convolution Semigroups

181

it follows that a\ = £f In case where a = ^ one knows explicitely a formula for the density of at with respect to X^1': - t 1 / 2 = g t (.)A« with

We will use some of these examples now to construct more examples of continuous negative definite functions. We know from Example 3.6.18 that any non-negative symmetric quadratic form q : R™ x R™ —¥ R, £ \-t q(£, £) =: q(£)i gives a continuous negative definite function. In particular £ M- |£| 2 is such a function. Combining this observation with Example 3.9.16 and Lemma 3.9.9 we get Example 3.9.17 For any a G (0,1] the function £ M- \£\2a, £ G R", is a continuous negative definite function. The corresponding semigroup (Mt*)t>o *s called the symmetric stable semigroup of order 2a. Each of the measures /xf has a density gf with respect to the measure \(n', St?(x)

= (27r)-n I JM.

e ^•« e -'l«l

2c,

d£.

n

In general gf is not known in a more concrete form. However, for a = | one finds 1/2/ ^ ^fn l\ t gt'W = r - + ^r2 2 2 V /(7r|:r| +t2) — In particular, for n = 1 we find

"•" , -;?V 1 , <' t a >The semigroup (/V )

(3.248)

P-249>

on R is called Cauchy semigroup.

For general a G (0,1) some asymptotic formulas are known for gf. Let us quote just one result due to R.M. Blumenthal and R.K. Getoor [35], see also the paper [20] of A. Bendikov for a short proof, Sta(z) ~

C

"'"

n+2a

(x*+tiy

as xH~i

—> oo,

(3.250)

182

Chapter 3 Fourier Analysis and Convolution Semigroups

where c„, a := a 2 2 a 7 r - t - i s i n ( a 7 r ) r Q + a ) r ( a ) . In the one—dimensional case many asymptotic results are given in the monograph [316] by V.M. Zolotarev, we refer also to the recent paper of V. Kolokoltsov [187]. A direct and remarkable proof t h a t £ \-+ \£\2a, 0 < a < 1, is negative definite was given by A. Beurling in [29]. E x a m p l e 3 . 9 . 1 8 For m > 0 the function £ i-> (|£| 2 + m 2 ) 1 / 2 - m, £ <E R " , is obviously a continuous negative definite function. This function is the symbol of a relativistic Hamiltonian and it was D. Bakry who pointed out that this operator is related to subordination. The corresponding convolution semigroup (Pt)t>o consists of measures each having a density ht with respect to \(n) which is known to be ht(x)

= 2 • {2^)-^m^emtt(\x\2

+12)'^

xKn±i(m(|z|2+i2)1/2V

(3.251)

where Ku is the modified Bessel function

of third kind of order v. A related

2

2

example is given by £ M- (|£| + m ) (Pt)t>o wtih ht(x)

which leads to a convolution

semigroup

densities = 2 • ( 2 7 r ) - ^ m 2 * i (|*f + ^ " ^ K ^

(m(|z|2 + t 2 ) 1 / 2 ) .

Formula (3.251) is taken from T. Ichinose [156], the modification for ht is obvious. For n = 3 the formula for ht was given in [201] by E.H. Lieb and H.-T. Yau, we refer also to I.W. Herbst and A.D. Sloan [121]. Note that for m —>• 0 the function h* (x) is just the density of the Cauchy semigroup. T h e following table, except the last item, is taken from the monograph [25] of Chr. Berg and G. Forst a n d it summaries some of our one-dimensional examples of continuous negative definite functions and the associated convolution semigroups. T h e hyperbolic distribution was investigated by O. Barndorff-Nielsen and Chr. Halgren [16]. Recently, this semigroup was used by E. Eberlein and U. Keller [83] for modelling some problems in finance mathematics. See also [82] and [84].

3.9

Bernstein Functions and Subordination of Convolution Semigroups

183

Table 3 . 9 . 1 9 convolution semigroup Degenerate semigroup

fit = e~ateo,

Translation semigroup with speed 6 6 R Poisson semigroup with jumps of size s > 0

negative definite function a

(/tt)t>0 a > 0

ib£

Mt = £bt

»t = T,ZL0e-t£esk

e-ist

1-

One-sided stable semigroup of order a e [0,1]

fit = <xf

(itr

F-semigroup

/it(dx) =

l o g ( l + f2) + + i arctan £

/tt(dx) =

€2

Brownian semigroup 1

2

r~~Ki AX ^ V r l r^l

(4^t)l/2 C

a/2

Symmetric stable semigroup of order a g (0, 2)

/it = / V

l€| a

Cauchy semigroup

Mt(di) = / i J ' 2 ( d x ) = |(t2+x2)-1AW(dx)

\t\

Hyperbolic semigroup

/x t (di) = hyp t (x)A( 1 )(dx)

, /

x

^'W

KllVl+ l^)X V/I+I«I 2

;

(Ki is the modified Bessel function of third kind with index 1)

Our aim is to prove t h a t the Bernstein functions are the only functions operating on the cone of all continuous negative definite functions in t h e following sense: T h e o r e m 3 . 9 . 2 0 Let f be a function defined on the half-plane R e z > 0 such that for allneN and any ip e CW(R n ) the function f o tp belongs to CN(Rn) too. Then £ is a Bernstein function. This result is due to K. Harzallah [116], we follow the presentation of R.L. Schilling [263] and we need some preparation. Let K S {K, C } and

184

Chapter 3

Fourier Analysis and Convolution Semigroups

introduce the following sets: C N + ( R " ; K ) := {i/> G CN(R"; K ) | V(0) > 0 } ,

(3.252)

CN0(Rn;K) :={V>GCN(R";K) | ^ ( 0 ) = 0 } ,

(3.253)

and C + ( K ) : = { f : K - > K | f ( C N + ( R n ; K ) ) C C N ( R n ; K ) for all u G N } .(3.254) L e m m a 3 . 9 . 2 1 The set 0 + ( R ) is convex and closed under locally uniform convergence. Moreover, for f G 0+(R) the function f|(o,oo) is continuous. Proof: Since C N ( R n ; R ) is convex and closed under locally uniform convergence, it follows t h a t C + ( R ) must have these properties too. Now, for e > 0 the function £ M- |£| + £ is an element of C N + ( R " ; R ) implying t h a t for f G C + ( R ) the function £ i-> f(|£| + e) is a n element in C N + ( R n ; R ) , thus f|(e,oo) G C((e, oo)) for any e > 0, hence f|(o,oo) is continuous. • L e m m a 3 . 9 . 2 2 Leti G 0+(M) andc > 0. It follows that f|(o,oo) *5 a™ isotone, subadditive and concave function, and the function x H-> Tci(x) := f(:r + c) — f(c) as well as the function f — r c f belongs to 0+ (R). Proof: Let f G C + ( R ) . For any c > 0 the function £ M- |£| + c is an element of C N + ( R n ; R) and therefore £ .-> f(|£| + c) belongs to CN(R"; R). But (3.123) implies t h a t f(c) 0 and £ G R™, hence, f|(0iOO) is isotone. Now, for -0 G C N + ( R n ; R ) and c > 0 we find for all £ G R n

r c W0)=W0+c)-f(c) = ( f ( ^ ( 0 + c) - f(^(0) + c)) + (f(V>(0) + c) - f(c)). T h e function £ !-»• f(V>(0 + c) - f(-0(O) + c) is negative definite whereas t h e isotony of f implies t h a t f(ip(0) + c) - f(c) > 0, hence, (r c f) o -0 G C N ( R n ; R ) or TC{ G C + ( R ) . Next let us prove t h a t f - r c f G C + ( R ) , c > 0. For this we apply Lemma 3.6.8 to prove t h a t for any tp G C N + ( R n ; R ) t h e function

3.9 Bernstein Functions and Subordination of Convolution Semigroups

185

£ ,_> f(V>(0) + f(c) - f(V>(0 + c) belongs to CN(R n ; R). Let £ \ . . . , £k G R n and A i , . . . , Afc G C such that E J = i ^j = 0- We claim that the real-valued function k

c M. 0(c) := £

(fty(£' - ?) + \c\) - f{il>(? - tj)))\i\]

(3.255)

1,3 = 1

is negative definite. Obviously we have 0(0) = 0 and 0(c) = 0(—c). Moreover, for any choice of c 1 , . . . , c m G R and fii,..., /xm G C such that Yl^Li M = 0 we have m

m

k

p

£ 4>{c - ci)nPwq = E E W * ' - O + \°v p,q=l

cq x Y

\) i 3»PN

P,q=l 1,3 = 1 m

k

- E EfW£'-^'))A
m

= E EW*' l,j=i

- O + icP - C9I)A^PV;

p,q=i

<0, since (£, c) H-> f(-0(O + l c l ) i s a n element in CN(M™+1) and YALI E ^ = I A 'MP = °The function c i-» 0(c) is real-valued and 0(c) > 0(0) = 0, hence, k

E (f M*' - f) + lcD - f M * ' - ?)))>*~>* > 0Applying Lemma 3.6.8 to the function

t -> g(0 = f(V>(0) + f(^(o) + c) - f(V(0 + c) - f(V(o)), we find, since g(0) = 0, g(£) = g(—£) and with Xj and £J above

n

= E (W** - O) - f (v>(? - ^')+c)) A4Aj < o, i,3 = l

(3-256)

Chapter 3 Fourier Analysis and Convolution Semigroups

186

hence, g e CN 0 (R n ;R) for any choice of tp € CN+(R n ;R). In particular, we have for c > 0

W O + c) < f(V(0) + W o ) + c) - f(v»(o)) < fty(0) + f(V>(o) + c). Taking V>(£) = |£|+£, £ > 0, it follows for e —> 0 the subadditivity off. Hence, f(V>(0)+c)-f(V>(0)) < f(c) and therefore we have g-f(V>(0)+c)+f(V>(0))+f(c) € CN(R";R) implying f - r c f € CN(R n ;R). It remains to prove that f|(0|OO) is concave. For this let c, d > 0. It follows that r c (f - T^f) € C+(R) and we have 0 < r c (f - rdi){d) = 2f(c + d) - f(c + 2d) - f(c),

^(i(c + 2d) + i{c))<{(c + d). The continuity of f implies that f is concave. • Proof of Theorem 3.9.20: First note that it is sufficient to prove that any f S C + (R) is a Bernstein function. In fact, we are done if it is known that C+(C) is a subset of all Bernstein functions. But for f S C+(C) and the constant continuous negative definite function ip(£) = c > 0 it follows that £ i-)- f(•;/>(£)) = f(c) is a constant negative definite function, hence f|(o,oo) must be real-valued and can be considered as an element of C + (R). Now, since C + (R) is a convex cone by Lemma 3.9.21 and all elements of O-)-(R) are concave functions by Lemma 3.9.22, it follows that 0+(R) C L1((0,oO),e-*dx), Ll((0,<x>),e-xdx)=L1(R,x(0oo)(x)e-xdx). The set

f

B:=UGO+(R)

{{x)e~x dx = 1 1

is a convex base of that cone. For f € B it follows that /•OO

{(x)e~x = f(x) /

/•OO

e" 1 dt<

JX

f(t)e _ t dt < 1. JX

1

We claim that B is compact in L ((0, oo), e~xdx). For this note that su

Pl|f|lL1((01oo),e-a:dx)

= 1,

(3.257)

3.9 Bernstein Functions and Subordination of Convolution Semigroups

187

and that for all x, y > 0 in a bounded set such that |x - y\ —>• 0 we get with lllflll

: =

ll I 'llL 1 ((0,oo),e-*da;)' />oo

|||f(.+a;)-f(.+3/)|||= / Jo

+ y)\e-tdt

\i(t + x)-i(t

(3.258)

/•OO

= exAy

|f(t + a ; V j / - a ; A y ) - f ( t ) | e - * d t JxAy /•OO

<exAy

{iit +

Jo

lx-yD-mie^dt

= e**v (e\*-v\ [

V x

f(i)e-*dt-l)

J\x-V\

< e^vL\ -y\

-{\

J

—y 0 as x —>y.

(3.259)

Hence, by Theorem 2.3.18 and the following remark we deduce that B is conditionally compact, hence, it is compact. Now, the Theorem of Krein and Milman, Theorem 2.4.1, says that B is the closed convex hull of its extreme points. Let us determine all extreme points of B. For this let f e B be extreme and set />oo

A,

/»oo

/ Tci{t)e~tdt= Jo

(f(t + c) - f(c))e-* dt < 1, Jo

where we used the notation of Lemma 3.9.22. It follows that for Ac ^ 0 and Ac ^ 1 we have A~1rcf e B and (1 — A c ) _1 (f — r c f) € B. Therefore, we have f = A^A-Vcf) + (1 - A c )((l -

Xc)-\i-rci)).

But f was supposed to be extreme, and it follows that A~1rcf = (1 — A c ) _1 (f — Tcf) implying that TC{ = Acf with Ac £ (0,1). For Ac = 1 we find using Lemma 3.9.22 that r c f < f and therefore /-OO

POO

/ Jo

(f(t) - Tcf(t))e-* di = / Jo

f(t) e -* dt - Ac = 0,

hence, Acf = f = rcf. Analogously, for Ac = 0 we find using the monotonicity of f that /•oo

poo

/ r c f(t)e-* di = / Jo Jo

(f(t + c) - f(c))e _t dt = 0,

Chapter 3 Fourier Analysis and Convolution Semigroups

188

thus Acf = 0 = Tci. Therefore, when f is extreme, there exists Ac € [0,1] such that TJ = Acf

(3.260)

holds. Now, we may distinguish three cases: 1. f is constant, then it follows t h a t i(x) = 1 and Ac = 0; 2. f is unbounded, then by the monotonicity it follows for all c > 0 t h a t Tci(x) = i(x + c) - f(c) = A c f(z) < \ct(x

+ c)

implying t h a t (1 - Ac)f(a: + c) < f(c) < oo for all x > 0, hence Ac = 1. We find t h a t f is continuous and additive, hence linear, and by our normalisation we conclude t h a t i(x) — x; 3. f is bounded but non-constant. Let m = P H ^ . Using (3.260) we get for c> 0 m — f(c) =

sup (f(z + c) — f(c)) = Ac 0
sup

f(a;) =

Xcm.

0<x

Let us consider the function C H > A C defined on (0, oo). It follows t h a t Xc+d = 1 - l f ( c + d) = 1 - ^ ( A c f ( d ) + f(c)) = ( l - -i(c))

\

m

J

- -XJ(d)

= \c(l-

m

\

-f(d))

m

J

= XcXd.

Since c i->- Ac = 1 — ^ f ( c ) is continuous and strictly positive, we find t h a t c h-> Ac = e"JC, 7 > 0, i.e.

l{x)=ly{x)=

m(l-e-^x).

But f e B implies

1 - = m

r°° (1 - e-*)e-* Jo

dt

7

=1+

7'

3.9

Bernstein Functions and Subordination of Convolution Semigroups

189

Thus we find the set of extreme points of B to be included in the set ^^{fi.f^ujt,

f 7 (z) = ^ ( l - e - ^ ) ,

7

e ( 0 , o o ) j ,

(3.261)

where fi (x) = 1 and f2 (x) = x. T h e Theorem of Krein and Milman, Theorem 2.4.1, implies t h a t 0 + ( R ) c Vri I r G (0,oo) and f G c o n v ( £ ) j , where conv(E) of the form i{x) =r(a

denotes the closed convex hull of E. But any f G r • conv(i?) is

+ (3x+

f°° i ± 2 ( l - e" 7 *) / x ( d 7 ) )

(3-262)

with a bounded measure fi supported on [0, oo), /x({0}) = 0 and a, f3 > 0 such t h a t a + P + /i((0, oo)) = 1. Hence, f is a Bernstein function and the theorem is proved. • E x a m p l e 3 . 9 . 2 3 By Corollary 3.6.14 the functions £ n - a ,^j5,ygs are continuous negative definite function provided a > 0, j3 > 0 and ip G CN(M.n), n G N. Hence, the function x i-> aZnx is a Bernstein function. R e m a r k 3 . 9 . 2 4 In order that f o V> G C7V 0 (K n ;C) / o r all ip G CA^ 0 (K";C), it is necessary and sufficient that f is a Bernstein function with the property that f(0) = 0. A proof of this non-trivial result is given by R.L. Schilling in [263]. T h e function £ M- |£| 2 is an element of C N ( R n ) for any n G N. Therefore, for any Bernstein function f the function £ H» f(|£| 2 ) is a radial symmetric continuous negative definite function. Using a modification of the proof of Theorem 3.9.20 we can prove T h e o r e m 3 . 9 . 2 5 If for all n G N the function CN(M.n), then i is a Bernstein function.

£ i->- f(|£| 2 ) is an element

in

This result was in principle proved by I.J. Schoenberg [275], we follow an unpublished proof of R.L. Schilling [265], and refer also to the lecture notes [176] of J.-P. Kahane. Denote by O :={{-. R - > R | f o | . | 2 G C N ( R " ) f o r a l l n G N } .

(3.263)

190

Chapter 3 Fourier Analysis and Convolution Semigroups

It is straightforward to see t h a t Lemma 3.9.21 and Lemma 3.9.22 do also hold for O instead of 0+(W). Hence, O is a convex cone which is closed under locally uniform convergence, such t h a t for all f € O we have t h a t f|(o,oo) i s continuous, TCA, f — TCI{ G O, and any f G O takes only values in [0,oo), is increasing, subadditive and concave. P r o o f of T h e o r e m 3.9.25: As in the proof of Theorem 3.9.20 we conclude t h a t f G O belongs to L ^ O , oo), e~xdx). Set B

:= li G O J l{x)e~x dx = 1 j

Once again we see t h a t B is a convex base of O which is compact in L 1 ((0, oo), e~xdx). From this we conclude literally as in the proof of Theorem 3.9.20 t h a t f is a Bernstein function. • T h e next result is a converse to Theorem 3.9.25. We take this result once again from the unpublished note [265] of R.L. Schilling. T h e o r e m 3 . 9 . 2 6 A continuous negative definite function ip : M.n —>• C is subordinate to £ H* |£| 2 if and only if its Levy-Khinchin representation takes the form V>(0 = V-(O) + b\£\2 + f (1 - c o s * • £)m(|£| 2 ) d£, JRn\{0} where m is a completely monotone

(3.264)

function,

/•OO

m(r) = / e - " i/(ds), r > 0, Jo+

(3.265)

and v is a measure on (0,oo) such that JQ s~n^2v(ds) < oo and j (ds) < oo. The subordinating Bernstein function f is then given by

l

s-?-1^

POO

(1 - e~rs) (4TTS)" / 2 $ ( I / ) (ds),

f(r) =ij(0)+br+

(3.266)

where 3>(f) is the image measure of v with respect to the mapping s !->• $ ( s ) = 4s'

Proof:

Assume first t h a t VKO = g(|£| 2 ) with some Bernstein function g, /»00

g(r) = a + br+

(l - e " " )

Jo

A*(dr), M ( { 0 } ) = 0.

3.9

Bernstein Functions and Subordination of Convolution Semigroups

191

Then we find ( l - e~rm)

M(dr)

= a + b\£\2 + ^ [ (1 - e - i x « ) ( 4 7 r r ) - n / 2 e - ^ d£ ji(dr) Jo JR™ = a + b\£\2 + f (1 - e~ixt) f Jut" JO

( 4 7 r r ) - " / 2 e - ^ fi(dr) d£.

Since

/ T^r^O-^e-^Mdr^ 7R"

i + Kr Jo

2-W 2 p [°° s2 _£ 2 • 7 r-- n ™/ /22

, 0 00 0 ,o r u 2 Z" Zo100

r(f)Jo

X i+ w

2 • 7T-"/ 2 f fl

+

/

/

fx

r

n-1

/[/(dr)e ,, ,

* u™ d u „»

n

, ,

u ( d r ) e ~ T U ™ + 1 du I < oo,

the function f(|£| 2 )

:

= / Jo

( 4 7 r s ) - " / 2 e - i ^ /x(ds) - T T " " / 2 / Jo

sn/2e~s^2

$(/x)(ds)

is well-defined and the Laplace transform of the measure s n / 2 $ ( / i ) ( d s ) . Moreover, f(|.| 2 )A( n ) is a Levy measure, i.e. is a measure being allowed in the L e v y Khinchin formula. Moreover, /•oo

/»oo

/ s-n/V/2$(//)(ds)+ / Jo+ Ji /•oo

=

s-t-1s"/2$(/i)(ds)

/"1/4

/ /i(ds) + / 4s fx(ds) < oo, Ji/4 Jo+

since /< is a measure representing a Bernstein function. Conversely, assume t h a t ip is of the form (3.264), (3.265). By the calculation made above, it is now sufficient t o verify t h a t (3.266) is indeed a Bernstein

192

Chapter 3

Fourier Analysis and Convolution Semigroups

function. But /•OO

(47T)"/2 /

/.OO

-^

S

2

"/

< (47r)"/ 2 f I implying the assertion.

$(!/)ds = (4TT)"/ 2 /

-i

— ^ — ( 4 s ) - n / 2 i/(ds)

( 4 s ) - " / 2 i/(ds) + /

( 4 s ) " ? - 1 u{As)\

< oo,

•

For later purposes we need a subclass of the Bernstein functions. D e f i n i t i o n 3 . 9 . 2 7 A function f : (0, oo) —> R is called a complete Bernstein function if there exists a Bernstein function g such that i{x) = x2£(g)(x)

(3.267)

holds for all x > 0. R e m a r k 3 . 9 . 2 8 Assume that the Bernstein function

g has the

representation

/•OO

(1 - e~sx)

g(x) =a + bx+

/z(ds).

Jo+ It follows that f is given by f(x) = x2C(g)(x)

= x2 r e ~ /•OO

= ax + b + x2

x t

(a + bt+

f

( l - e~st) fi(ds)\

/-OO

/ e~xt (1 - e~st) /x(ds) di. Jo+

Jo However, we have /•OO

x2/ Jo

/-OO

/ e-xt(l-e-st)M(ds)di Jo+

/•OO

/*00

/

3 - 3 ( 1 - « " " ) ) ( ! - « - " )dfM(d») /•OO

/*00

= / / 7o+ Jo /•oo

= / Jo

(l-e-xt)s2e-s'di/x(ds)

(1 - e~xt)

/-oo

/ s 2 e - s t /z(ds) di. Jo+

di

3.9 Bernstein Functions and Subordination of Convolution Semigroups!

193

Hence, the function f has the representation /•OO

i(x) = b + ax +

(1 - e~xt) p(dt)

Jo

where p is the measure given by POO

/ s2e~st Jo+

p(dt)=

p(ds)\^\dt).

It follows that f itself is a Bernstein function. Complete Bernstein functions are sometimes called operator monotone functions and were studied by several authors in different contexts. The following theorem is taken from R.L. Schilling [263], but we refer to the notes of this chapter for a more appropriate account and historical credits. Theorem 3.9.29 For a function f : (0, oo) -» M. the following assertions are equivalent 1. f is a complete Bernstein function. 2. The function x M- - ^ is a Stieltjes function with representation measure p satisfying /-oo -i

/*oo

jo+

P (dt)+y i

-tp(dt)

< CO.

3. f has an analytic continuation onC\{a; £ R | a? < 0}- with real values on (0, oo) and existing limit f(0) := lim f(a;) > 0.

Moreover, we have i(z) = f(z) and Im z • Im f(.z) > 0 for all z G

C\{x e R | z < 0 } . 4- f has a unique representation as f{z) = a + /3z+

f°° tz — 1 cr(d£), z e C \ { a ; e K | x < 0 } ,

Jo

t

+z

Chapter 3

194

Fourier Analysis and Convolution Semigroups

where the measure a is finite on [0, oo) such that oo, Jo

t

J\

and positive constants a, j3 with a > f£° J p{dt). 5. f has a unique representation {(x) = a + Px+

as

r°° tx - 1

t+x

Jo

cr(dt),

x>0,

where the finite measure a on [0, oo) fulfils / - a(dt) + / Jo * J\ and a,P

a(dt) < oo,

are positive constants

6. i is a Bernstein

function

such that a > JQ

having the

jcr(dt).

representation

/•OO

(l-e~sx)

i(x) = a + bx+

n{ds),

x>0,

Jo+ where a and b are positive constants and the measure /x is given fi(ds) = m(s)AW(ds). The density m is given by /•OO

m(a)= /

e~tsT(dt),

s > 0,

Jo+ where T is a measure on (0, oo) r1 1 Jo -tr(dt)

f°° 1 + J^

satisfying

-^r(dt)
7. x H-> j ^ r is a complete Bernstein Proof:

function.

We prove the following implications

3. =>• 4. =* 5. =» 6. =>• 1. =» 2. =» 3. and 1 . & 3 . =5>7. =>• 1.

by

3.9 Bernstein Functions and Subordination of Convolution Semigroups

195

3. => 4. Let H := {z G C | Im z > 0} and E := #i(0) C C. The Cayley mapping h : H —• E, z \-t | = | is a biholomorphic map with inverse given by w i->- j ^ - Further, it has a continuous extension to dH such that h(dH) = dE. Suppose that f satisfies 3. Then on E we have the analytic function -i(h~1(w))

w t-¥ (w) := i

with the property that Recf>(w) = R e ( - ^ h " 1 ^ ) ) ) = I m f ( h - 1 ( w ) ) > 0 since Imh - 1 (u;) > 0. We may apply Herglotz theorem, Theorem 2.5.1, to find for w — h(z), z € H, that ""

pW + hfz)

/ where c = Im ((0)) G R and p is a bounded measure on (—n, TT]. Introducing new coordinates by e1^ = h(t), t £ R , and putting g(i/>) := h _ 1 (e"^), we find ° ° t y -i- 1 J

/

7

-g(p)(d<)-

With the notation a := —c, /? := g(/5)({+oo}) and a := g(p) it follows that 0°

/

j.

_j_ -I

——a(dt), •oo

(

-

zGH.

(3.268)

z

But since f(z) = f(z), this formula holds for z G C \ R. The uniqueness result in the theorem of Herglotz and the bijectivity of h imply the uniqueness of the representation (3.268). We prove now that supper C (—oo,0]. For this let q > p > 0 be two points not being atoms of a. For z = x + iy £ H we find i /

q />oo fpq f°° tz +1 Im i(x + iy) dx = (3y + / / Im cr(di) da;

Jr> — oo oo p •/J — q /•oo

Py+ I I Jp

t —Z J2,

ntZyJy„.i*(
J —C

/•oo oo

r
/ (t2 + 1)( arctan

Py+ J-oo

V

arctan y

) a(dt). y

)

196

Chapter 3 Fourier Analysis and Convolution Semigroups

We may use Fatou's lemma for taking the limit y —> 0, and it follows that lim

r / Im i(x + iy) dx JP

p-t

/

oo

(t2 + l) I arctan / -oo V V

arctan

] a(dt)

>fr + r + n \J — oo

Jq+ J

Jp

(t + 1) lim

= f\(t2

arctan

arctan

&(dt)

+ l)a{dt).

Jp Since the function (x, y) i-> Im f(a; + iy) is continuous and [p, q] is a compact subset of [0, oo), it follows that 0 < 7T / (t2 + 1) a(dt) < / lim Im f(a; + iy) dx = 0. (3.269) Jp Jp y-»0 The measure a is bounded, hence, it may have at most countably many atoms. Therefore, (3.269) holds for a dense subset of values p,q £ [0, oo), implying that <5l[o,oo) = 0- Now let a be the measure obtained from a by a reflection at the origin. Thus we find f

f(z) = a + (3z •

I

°°tz-l cr(di), t+z

=

f

(3.270)

z€C\R.

The boundedness of aJoimplies further that (3.270) holds also for z € (0, oo). Let x £ (0,1). Then it follows that |f(a;) -Px-a\

tx-1 a{dt) t +x

Jo

> >

Jo

J0

t+x

Ji

t + x a(dt>

A

t+x tx-1 a(dt). t+x

But for t > 1 and 0 < x < 1 we have tx-1 t+x

<

tx + 1 t+x

<

tx + t t+x

<

(t + x)(x + l) = x+1 t+x

3.9

Bernstein Functions and Subordination of Convolution Semigroups

197

implying t h a t Z"1- 1 -tx f00 \i(x) - (3x - a\ > / a(dt) - (x + 1) / Jo t + x Ji

a(dt).

For x —> 0 we get using Fatou's lemma |f(0) -a\>

/•!liminf / x-+0 JO Z -1> liminf 7o i->o

1 _ tT

r°° / a{dt) *+ Jl 1 - tx f°° ^ — ^ a(dt) - /

= £ ±*(dt)-J™ a(dt), thus we have /•OO /*00

/•OO -i

Jo -tcr(dt) + J^ a(dt)<

00.

Since f(0) > 0, it follows from (3.270) t h a t a > J0°° ± 0, hence 4. is proved. 4. =>• 5. 5. =>• 6.

This implication is trivial. We define ,oo

a:=a-

/

1

- aids) s

Jo

> 0, b :=/3 _> 0

and /i(dt) := m(t)\W(dt)

where m(i) = /

e

- s t ( s 2 + l ) cr(ds),

taking a, (3 and /x from the representation formula in 5. By definition /z is absolutely continuous with respect to the Lebesgue measure A^1) and its density is the Laplace transform of the measure r ( d s ) := (s 2 + l)cr(ds), T being a measure on [0,oo). Further we find rl —

/>oo

/ t/x(dt)+ / Jo+ Ji rl

fj.(dt) y-OO

rOO

= / / Jo Jo

te~st (s2 + 1) a(ds) dt+ Ji

/-OO

/ Jo

e~st (s 2 + l )
198

Chapter 3 Fourier Analysis and Convolution Semigroups /•OO

=

/

/*00

pOO

/•!

te~st (s 2 + 1) dt a{ds) +

/

= f(-< st+1)f ?) < (-5

/

e~st (s2 + l ) At a{ds

(S 2 + 1) (7(ds)

(s2 + 1) ff(da)

/•°° s-* 4- 1

On [1, oo) the function s >->• ^^-(1

— e s) is continuous with lim ^ - ^ ( 1 — e ~ s ) s—»oo

= 1, hence, this function is bounded by some constant c on [0, oo). But for 0 < s < 1 we find

and therefore /•l— /

i fi(dt) + /

Jo+

/>oo r^-~ i /*oo /i(dt) < 2 / -s a(ds) +c a(ds)

Ji

Jo

Ji

< oo.

It remains to prove t h a t f has a representation characterised by the triple (a, b, /x). By our assumptions we have i(x) =a + /3x+

(

f°° sx - 1 / o-(ds) Jo s+ x

r001

/•OO

= a + bx+

r°°x(s2 + i)

\

,-00

/

(e"8' -

e-

(s+a:)

M (s 2 + l )
/•OO

(1 - e - a : t ) (s 2 + l ) m ( t ) dt

= a + bx+ /•OO

(1 - e - * ' ) (s 2 + 1) /x(dt).

= a + bx+ JO

3.9 Bernstein Functions and Subordination of Convolution Semigroups

199

Now, for the measure r(ds) := (s 2 + l)cr(ds) we finally find /•i— 1

r°°

1

f i

-

« 2 4-1

< 2/

r°° s 2 -I- 1

- a(ds) + 2 /

a(ds) < oo.

In particular, T ( { 0 } ) = 0 and we may identify r with r|(o)00)6. => 1. From the properties of r it follows that

£ s (^ r ) ds+ r(^ T ) ds{x) :=b + ax +

(1 -

e

1 - x ) - j r(ds)

s /o+ 7o+ is a Bernstein function by Theorem 3.9.4. Furthermore, we have

x2C{){x) =x2 f°° (b + at+ f /•OO

= a;2 / Jo

(1 - e~st) -^ r(ds) J e~*x dt /"OO

be"'* dt + x2

ate-xt dt + Jo

+x2£(f+(l-e^)e-*°d?)±T(ds) = bx + a + x2 = a + bx+

( Jo+ \x

[ Jo+ \s /•OO

= a + bx+

) -, r(ds) s + xj s2 ) rids) s + xj

/-OO

/ ( 1 - e~tx) e~st r(ds) dt Jo Jo+ /•OO

= a + bx+

(1 - e _ t x )m(t) dt = f(x),

hence, f is a complete Bernstein function. 1. =>• 2. Let f(:r) = x2C(cj>)(x) for a Bernstein function . From (3.192) it follows that ^

a;

= xC(4>)(x) = 0(0) + £(')(*)•

Chapter 3

200

Fourier Analysis and Convolution Semigroups

Since <j> is a Bernstein function, the function $ is completely monotone, hence, by Bernstein's theorem, Theorem 3.8.13, it is the Laplace transform of a suitable measure p on [0, oo). Thus from Definition 3.8.17 we find f(rc) X

= <^uj -r ^ \P)(*) = yyv) -r

JQ_L_

S -f- X

On the other hand, the Bernstein function f has the representation /•OO

/-OO

( l - e~tx)

f(z) =a + bx+

/ s2e~st Jo+

Jo

p,(ds) dt

when <)> has the representation /•OO

( l - e~sx)

4>(x) =b + ax+

p,{ds).

Jo+ Since these representations are unique, we may introduce the measure such t h a t p(ds) = s~1p(ds) + aeo and find i(x)

a

X

X

,

[°°

±^ = -+b+

J0+

Furthermore, s~2p(ds) 1

r/ VO

1 S(S +

...

X)

p(ds).

satisfies (3.228). From this we find

r1-1

r°° i p(ds) + / VI

p(ds)

- p(ds) =a+

r°° 1 - p{ds) + /

=<,+ s w+

S

s

V0+

< OO.

Jl

- j p(ds) s

ir (» r(^)



2. =>• 3. By our assumptions f has the representation i(x) = a + bx+

r°°

x

—. r p ( d s ) , a; > 0. 7o+ s(s + x)

Choose a measure p such t h a t p = ^ j ^ and /

p(ds) + /

- p(ds) < 2 /

(3.271)

+ aeo(ds) and note t h a t p|(o,oo) = \Pi

- — p(d«) + 2 /

2f&)

+ s

— -

< 2f(1)<00 .

p(ds)

3.9 Bernstein Functions and Subordination of Convolution Semigroups

201

Foi z = x + iy e C\ {x E R | x < 0} it follows that 1 s{s + z)

x2 + y2 (s + x)2 +y2'

Let £ > 0 be fixed and take z from a compact subset K, K C C\ {a; G R 11<0 }, such that dist(if, (—oo, 0)) > £. For x > 0 and any y € R we have x2 + y2

1

- , 0 < s <2e, s y (s + a;)2 + y2 < s' x2 +y2 s\J(s + x)2+y2

<1s]j(s -J^±t = ^±t<4^^ s>2e, + x)2 s(s + x) s(s + x)

and for x < 0 and y ^ O w e find 1 / x2+y2 s V (s + x)2 + y2 x2 + y2 s]](s + x)2+y2

1 x2 + y2 s V y2

c'(K,e) s

„, .

< ls]!ts.\ , / »2a + ^2 <2y-Jt±J< 2 2 + y

s

s

+2/

s

K) ^ .,s>2\x\. ( s + |y|)

Since dist(z, (—oo,0)) > e, the integral J0°° s(s+z) p(ds) converges for z S K Moreover, since e > 0 was arbitrary, it follows that f extends to an analytic function on C \ {x £ R | x < 0}. Moreover, the functions hx(s) := ; x -, = i — 7Tx decrease to zero as x —> 0. Thus, by the monotone convergence theorem we get f(0) = lim f(z) = a > 0. From (3.271) it is clear that f(z) = f(z). Further, using the measure p, we see Imi(z) = blmz+ = b Im z + / Jo

Im ( -^-_ ) p(ds) s + z> \s + z\2 p{ds),

we finally find Im z • Im f(z) > 0, hence, 3. is proved.

202

Chapter 3

Fourier Analysis and Convolution Semigroups

Finally, we observe t h a t 7. and the identity f(x) = - § - imply 1. On the other hand, x M- -^

satisfies 3., hence is a complete Bernstein function.

R e m a r k 3 . 9 . 3 0 Obviously, above theorem is fulfilled if

F

r(dt)

the condition

•

on the measure r in part 6. of the

< oo

/o+ *(* + !)

holds which yields

Since for s > — 1 we have -r~- < 1 — e~s, we arrive at m(s) ds < oo. / W o+ 1 + s Jo With these remarks we can give an additional charactrisation of complete Bernstein functions which is due to F. Hirsch [134], p. 103: A function f is a complete Bernstein function if and only if f has a representation f(i) = a+

—— iU&t) l + tx

Jo

with a measure /x(di) satisfying

fQ

(3.272) j ^ . £i(dt) < oo.

Proof: Suppose t h a t f has a representation (3.272) and set a = f(0), b = /}({0}). It follows t h a t i(x) =a + bx+

f°° x / p,(dt) Jo+ l +tx

/•oo

= a + bx+

/-oo

/ Jo+ Jo+

p-s/t

(l-e-sx)—5-/i(dt)da *

/•oo

(1 - e-sx)h(s)

= a + bx+ Jo+ 10+ where /•OO

1

ds

3.9 Bernstein Functions and Subordination of Convolution Semigroups

203

Since JQ j ^ fidt) < oo we find further that /„ j4^h(s) ds < oo, which proves that by (3.272) a complete Bernstein function is given. The other direction of the new characterisation follows by the considerations just made above. D Lemma 3.9.31 Let f be a complete Bernstein function with representation /•OO

(1 - e~sx) /x(ds)

i(x) =a + bx+

(3.273)

Jo and /•oo

fi(ds)=m(s)\U(ds),

re~'r p{dr)

m(s)=x[Qi00)(s)-j

(3.274)

according to Theorem 3.9.29 with a measure p satisfying f1 f°° 1 / p(dr) + / -r p(dr) < oo. Jo Ji Then we have for any a > — 1 / sCT n(ds) < oo Jo

(3.275)

if and only if ,oo ji

1

^P(dr)
(3.276)

Proof: Using (3.273) and (3.274) we find /*1

pi

/«00

/ s° p,(ds) = I s" Jo Jo Jo

re~sr p(dr) ds

/>1

/>oo

= / Jo

/ s°re-sr Jo

= / J\

/ — e~T dr p(dr) + sare~sr Jo r Jo Jo

/•l

ds p(dr)

/-l

/»oo

sare~sr dsp{dr)+

= Jo Jo

I Ji

ds p(dr)

rr

/ T°e~T d r r'" Jo

p(dr).

204

Chapter 3

Fourier Analysis and Convolution Semigroups

Thus we find /-00

/•l

/

T°e-TdT

r~a p(dr)

/•l

rl

/-oo

poo

< / sa fi(ds)
r _ f f p(dr),

where ca = J0 s" ds < oo, which proves the lemma. • E x a m p l e 3 . 9 . 3 2 The following functions S

s" =

^±

r

7T

Jo

- ^ r ^

s+X

—— £\(dr),

Jo s + r r°° s I

log(1 + s) =

a € (0,1);

functions: (3.277)

s" +' ~r

f°° s

/

dr,

are complete Bernstein

(3.278)

A>0;

/ 0 7T7r x a.-) ( r ) d r ;

(3 279)

-

00

A Z" s 1 Viarctan^=/ - — — -/s J0 s + r 2y/r

x

E x a m p l e 3 . 9 . 3 3 Consider the Bernstein as in (3.277). Thus we have p(dr)

=

(r) dr, A > 0 . (°>A ) s M- sa with

function

(3.280) representation

sinew !^^r-lA(i)(dr) •K

and

Hence, Lemma 3.9.31 yields for 0 < a < 1 that POO

OO

(3.281)

/

r" p(dr) ~c + c' if and only if a > a!

r ^ - * ) " 1 d r < oo,

L e m m a 3 . 9 . 3 4 A. For every Bernstein

^ 4 < - for s > 0. f(s)

s

function

f we have

(3.282)

3.9 Bernstein Functions and Subordination of Convolution Semigroups

205

B. For every Bernstein function f and all c > 1 it follows that (3.283)

-f(s) < i(cs) < d(s). c C. If f is a complete Bernstein function, then <

f(*)(

k+ l

(3.284)

, «>0,

holds for all k e N . D. For the derivatives of any Bernstein function we have k\ s

|f(fc)00 < 4kf ( s )> s>0 andk € N0.

(3.285)

Proof: A. By the inequality 1 + sr < ers or, equivalently, rse we obtain by differentiating (3.273) under the integral sign yoo

< 1 — e~rs,

poo

rse~TS fi(dr)
si'(s) =bs+

rs

7o

(l - e - " ) fx(dr) = f(s). Jo

B. Since f > 0 the function f is increasing, and since c > 1 we have s < cs implying that f(s) < f(cs) < cf(cs), and the first inequality in (3.283) follows. Using f" < 0 and (3.282) we find that f ' ( 0 < f'(s) < — , s

0<s
Integrating this estimate, we get /

f'(r) dr<

f

^

dr = (c - l)f(s),

which proves the second inequality in (3.283). C. The complete Bernstein function f has the representation

^=7+6+r^(dr)

(3 286)

-

by Theorem 3.9.29 part 2. Differentiating (k + l)-times in (3.286) we get 1 f( f c +D( s )= / • ( - ! ) * " ( * + \k+2 (« + ' Jo+

)

^

206

Chapter 3 Fourier Analysis and Convolution Semigroups

which yields [°°

f (*+U( s )

(k + l)\r

f

k\r

k + l

'o+ (s + r) f c + 1 s + r

<

k + l s

J

k\ r

J0+

(s +

r)^1

k + l

p(dr) =

p{dr)

f(s)

Jo

and C. is proved. D. For complete Bernstein functions (3.285) follows from (3.284). T h e following proof is taken from W. Hoh [145], Proposition 3.6.4. Observe first t h a t I + j ^ < ex, x > 0, hence we have xke-x

<

*

•

'

(

for x > 0.

!

Thus it follows for f as in (3.273), but taking a = b = 0 since for s >->• a + bs the estimate is trivial, /•OO

f (*>(*)

(1 - e " " ) fx(dr

k

ds

-5r

/

r

ke-rs

fj,(dr)

(i_e_rs)M(dr)= f(s)

and the lemma is proved.

5

Jo

•

R e m a r k 3 . 9 . 3 5 The function S H I - e~@s, f3 > 0, is a Bernstein function that is not a complete Bernstein function. A direct computation shows that (3.284) does not hold for this function. Finally, we want to have a further look at the structure of the set of all Bernstein functions. For this we note first C o r o l l a r y 3 . 9 . 3 6 Let f and g be two Bernstein a Bernstein function.

functions.

Then f o g is also

Proof: Let ip e C N ( R n ) for n e N. It follows t h a t g o T/>, hence ( f o g ) o 0 e C N ( R n ) for all n G N and ip € CN(R"). Therefore, by Harzallah's theorem, Theorem 3.9.20, it follows t h a t f o g is a Bernstein function. • Since the identity m a p on [0, oo) is a Bernstein function, it follows t h a t the set of all Bernstein functions together with the operation of composition of

3.10

Some Function Spaces related to Continuous Negative Definite Functions 207

mapping forms a monoid, see [196], p.3, for a definition. This monoid is called the Bernstein monoid and was studied recently by N. Bouleau and O. Chateau [44]. Let us denote for a moment by 5o,i the set of all Bernstein functions f such t h a t f(0) = 0 and f(l) = 1. On Bo,i we introduce a partial order "-<". For f , g g Bo,i we write f -< g if and only if g = f o h for some h G -Bo.i- A subset of the partially ordered set (-Bo.i, -<) is called a branch if it is maximal totally ordered. E x a m p l e 3 . 9 . 3 7 A. For a G (0,1] let ia(s) = sa, s > 0. Then {fa | aG(0,1] } is a branch of (JBO,I> -<)• This branch is called the stable branch. B. For a > 0 set h a ( s ) = V f > s > 0. Then {h a | a £ (0, oo)} is called the homographic branch of (-Bo,i, -<). We refer to the paper [44] where the structure of (-Bo,i, ~"0 i s investigated further.

3.10

Some Function Spaces related to Continuous Negative Definite Functions

In this section we discuss a class of anisotropic function spaces B% p ( K " ) related to a continuous negative definite function ijj : R n —>• C. For the function f h-> i/>(£) = |^| 2 and p = 2 these spaces will coincide with the classical Sobolev spaces of fractional order which we will discuss in the next section. T h e spaces B!^p(M.n) are constructed like the spaces Bk,p(M.n) introduced by L. Hormander [150] and we will come back to this relation later on. In this section, unless it is stated otherwise, ip : M™ —• C will be a continuous negative definite function, s g l and p G [1, oo]. D e f i n i t i o n 3 . 1 0 . 1 The space B^p(R.n) u G S'(Rn) such that \u\\iP,s,p

(i + iv(.)ir/2u(.)

consists of all tempered

< oo.

distributions

(3.287)

LP(R")

R e m a r k 3 . 1 0 . 2 In later chapters we will write H^(Mn)

:= 5 J , 2 ( R n ) and \\*\\

:= | | u | | ^ > 2 .

(3.288)

208

Chapter 3 Fourier Analysis and Convolution Semigroups

T h e o r e m 3 . 1 0 . 3 The space BL „ ( R n ) is a Banach space and in the sense of continuous embeddings we have S(Rn)

-><S'(R n ).

<-> B.Tp,p\

(3.289)

Moreover, for 1 < p < oo the test functions B. 1p,p\ r). Proof:

Co°(R") form a dense subspace of

Let C^>StP be t h e Banach space of all measurable functions v such

t h a t ||(l + |V(.)l) a/2 v|lLp is finite. Since (1 + | G <S(R") a n d - + -r = 1 that /

f>vd£ <

(i + iv(.)i) s/2 v

(i + m.)\)-'/24 ,.

T LP

We prove t h a t M-(i

+ h/>(-)l):

11 LP

I LP'

is a continuous semi-norm on

Indeed, for <j> G 5 ( R " ) it follows that

(1 + | V ( . ) I ) ^


I»I

(1 + I-I)V<

LP''

which implies t h e assertion and yields «S(Rn) «-> C^s>p -»• 5 ' ( R " ) in t h e sense of continuous embeddings. Obviously, <S(Rn) is dense in £^tS,p for p < 00. Since t h e Fourier transform maps <S(Rn) and <S'(R n ) each continuously and bijective onto itself, we have (3.289), and it follows t h a t B^, (W1) is complete and for p < 00 we find that <S(Rn) is a dense subspace. Since Co°(R") is dense in <S(R n ), we finally have proved t h e theorem. • T h e o r e m 3 . 1 0 . 4 Let i>i,ip2 • functions such that

C be two continuous

(1 + 1^(01) < c ( i + |Vi(OI)

negative

definite

(3.290)

holds for all £ G R™. Then, for any s > 0 the embedding

^liP(R")^^2iP(R")

(3.291)

3.10 Some Function Spaces related to Continuous Negative Definite Functions 209 is continuous. Conversely, suppose that for some open set G C R n , G ^ 8 , and some s > 0 the embedding B$ l i P (R n ) n £'{G) ^ B^p(Rn)

(3.292)

is continuous. Then (3.290) holds for all ( £ R n . Proof:

The first part of the theorem is trivial by the definition of the norm o

II-IL s p- Now, let K C G be a compact set such that 0 ^ K and set H:=B^p(Rn)n£'(K). The space H is a closed subspace of B i (E n ) and the embedding i : H c-> B^ 2 p(]Rn) is a closed mapping. Hence, the closed graph theorem, Theorem 2.7.8, implies that ||u||^ i 3 i p < c 0 | | u | | ^ p

(3.293)

holds for all u € H. Take u e Cg°(#), u ^ 0, and put u ^ x ) = u{x)eixr>,

(3.294)

thus we have u,,(£) = u(£ — 77). For s > 0 it follows using a consequence of Peetre's inequality, Corollary 3.6.24, that

(1 + |Vi(£)l)s/2u(£ - v)\ <

r/\i+\MvW/2(i+\Mt-vW/2m-v)\

and (l + l V ' 2 ( O I ) s / 2 u « - r ? ) | > 2 - s / 2 ( l + |V 2 (r 7 )|r/ 2 (l + | V 2 ( ^ - ^ | ) - s / 2 | u ( ^ - r ? ) | which leads to KIU...P ^ c i(! + \Mv)\Y/2hhu.,P

(3-295)

IKH^,.,,, > c 2 (l + I ^ W D ' ^ H u l l ^ ^ ^ .

(3.296)

and

Hence, we get /

.I

I,

\ 2/«

(1+iv^wi) < c C ! ! 1

(i+IV'IWD,

V C2llUIU2,,,p/ implying the theorem. D Next we will characterise compact embeddings.

Chapter 3 Fourier Analysis and Convolution Semigroups

210

T h e o r e m 3 . 1 0 . 5 Let K c K™ be a compact set, K ^ 0, and s > 0. Further, let V'l, V'2 : Rn —* C fee two continuous negative definite functions such that

lim I«I-

i±^M = o

(3.297)

i + l^i (01

holds. Then the embedding t : B^up(Rn) f~l £'(K) -> B^ 2 p (K™) is compact. Conversely, if c: B^up(Rn)D£'(K) ->• B ^ , 2 p ( R n ) is compact for some compact set K cRn,

K ^®, then (3.297) /iotas.

Proof: A. Let (u f e ) f c 6 N , ufc G B^up{Rn) n £'(K), such t h a t ||u f c ||^ i ) S p < 1. We have t o prove t h a t a subsequence of (ufc)fceN converges in 5 1 p ( K n ) . Let G Cg°(R n ) satisfying 4>\K — 1. It follows t h a t Ufc = Ufc and t h a t Ufc(0 = (27T)-"/ 2 /

^-T^U^dr?.

(3.298)

Once again we apply Lemma 3.6.23 t o find with (3.298)

(i + IVi(0l)s/2M0l (2TT

.-nil

\M0\y/2kt-v)Mv)dv

[ (i + n

JlH


(i + ivi(.)irv

LP

where we used Holder's inequality, I + i = 1, a n d H u ^ H ^ ^ < 1. Further, for a G No we find D?ufc(0 = (2TT)-"/2 /

D £ ^ ( £ - 77)^(7?) dry

JRn

= (27r)-"/2/

fe)

ufc(r7)d77,

which gives

(i + ivi(oi)'/2pafijb(oi <e (i + iv-i(.)ir/2Da^| .

3.10 Some Function Spaces related to Continuous Negative Definite Functions 211 It follows that the sequence (ufc)fceN is uniformly bounded and on any compact set K' C R n uniformly continuous. Therefore, by the theorem of Arzela and Ascoli there exists a subsequence converging uniformly on compact subsets. We denote this subsequence once again by (ufc)fcgN. Now, let e > 0 and p > 0 such that

This yields that I K ~U'II,/-2,S,P

= (jf n ((i + l^(0l) a/2 M O - u((0l)P dc)"

= (([

+ / ) ((i + i^(oir/2iufc(o-uKoi)p^V

^^k-m\\^,a,P

+ o(jg

Q

((i + |V(0ir / 2 K(0-u/(0l) P dCj

The second term converges for k, I —• oo to zero, while for the first term we find that ||ufe — u/|| . ps < 2. Hence, (ufc)fceN is a Cauchy sequence in B^2 (Mn) and therefore convergent. B. Now let K C R n be a compact set with nonempty interior such that i: 5 ^ p ( R n ) n £'(K) ->• B^2p(Rn), s > 0, is compact. We will prove that for any sequence (£fc)fceN> £fc S R n , such that |£j,| —> oo it follows that , , ' —• 0 as fc —>• oo. 1 + 1^1(^)1 For this let u € Co°(.fO, u ^ 0, and consider Ufc(x) :=u(x)-(l + hM&)l)' /2 ' Using (3.295) and (3.296) we find now that

IKIUa,,,p
(3.299)

Chapter 3

212

Fourier Analysis and Convolution Semigroups

and \\nk\

>C2a±j*iiiiellull

(3.300)

H2,s,p - ^ (1 + | ^ ( ^ ) | ) V 2 " " * I . — .P-

From (3.299) we conclude t h a t (ufc)fc€N is bounded in B i _ ( R n ) , hence conditionally compact in B^ (R™) by our assumptions. For 4> € <S(Rn) we find now / nk(x)(p(x) VR"

dx = / u(x)(j){x) . 7K" (l + |Vi(£fc.

dx 1

- WW)"(-^> (1 + | , l ( f a ) | ) . ^ Since (u) oo t h a t /

Ufc(:r)(/>(a;) da; —>• 0 as k —>• oo

implying that uj; —> 0 in S ' ( R " ) . Since the t h a n the topology on B%,2tP(Rn), the only limit could be the function identically equal to zero. of (u.k)keN in 5 ^ 2 ] P ( R n ) implies t h a t llufcll^^p existence of a limit. But now (3.300) yields the

topology of <S'(R n ) is weaker point of (u f c ) f c e N in B^2p(Rn) T h e conditional compactness —> 0 as A; —>• oo, hence, t h e theorem. •

R e m a r k 3 . 1 0 . 6 Since we may take in Theorem 3.10.5 as function V'2 the continuous negative definite function £ i-» 1, it follows that the embedding : B:i,!,2\

n £'{K) -* Bs12{Rn) = L2(Rn)

is compact if and only if lim |V>i(£)| = oo|£|-»0O

D e f i n i t i o n 3 . 1 0 . 7 Let V'i;V'2 '• Rn —> C 6e iwo continuous negative definite functions. We define the norm ||.|L ^, s / o r s g R and p G [1, oo] by

(l + |Vi(.)|)' /2 (l + |^(-)l),/2
lxpl,1p2,S,P ' "

and i/te space 5 ^ ^ B

k^,P(^n)

p(K

••= { u

n

(3.301) LP

) is defined by 6 5

'( K ")

HUIUI,*,,.,P < °° }•

(3.302)

3.10

Some Function Spaces related to Continuous Negative Definite Functions 213

Obviously, Theorem 3.10.3 holds also for the space

B^aiP(Rn).

T h e o r e m 3 . 1 0 . 8 Let ipi,ip2 : R n —• C be two continuous negative definite functions. Further, fors>0 letm G £ ^ ] P ( R n ) n 5 ' ( R " ) andu2 G £ ^ 2 i 0 0 ( R n ) . n Then u i * u 2 G % i V , 2 , p ( R ) and

IK * u2lU,^,s,P ^ (2^n/2lluilU1,.,PHu2||^,a,o0

(3-303)

holds. Proof:

Since (m * u 2 ) A ( 0 = ( 2 T T ) " / 2 U I ( £ ) U 2 ( 0 we find

||u 1 *u 2 || v , l j V , 2 ) S ) P

= (jfB ( a + ivi(0i) s/2 (i + i^2(oi)s/2 K^i * U2 ) A (e)|) p dc) =

(2TT)"/2 Q T B

((i + ivi(oi) a/2 (i + !v>2(oir /2 Moi iu 2 (oi) P ^ )

< (27r)"/ 2 ||u 1 || v , i ] S i J ) ||u 2 || v , 2 i S i 0 0 .

•

T h e next theorem shows t h a t the space B^, ( R n ) has many multipliers. T h e o r e m 3 . 1 0 . 9 Let ip : R™ —> C be a continuous and G <S(R"). Then cpu G £^ >p (R™) and

negative definite

function

(3.304)

&u|L, „„ < c | | ^ | | , l o l | | u | | n »S ZioZds / o r any u G B^, D(R ). V'IP

Proof:

First suppose t h a t e Co°(E n ) and put v : = <j>u. It follows t h a t

v ( 0 := (2TT)-"/2 /

4>(£-T,)vi(T,)dr,

and using Lemma 3.6.23 we get

(i + KK0l)s/2l*(0l
(1+m

-vW'^lkt-v)

(l + | ^ ) i r / 2 | u ( r / ) | d r 7

= c ( ( a + hK.)ir /2 |^|) * ( a + IV(.)D S / 2 N)) (o-

Chapter 3 Fourier Analysis and Convolution Semigroups

214

From Young's inequality, Lemma 2.3.15.A, we deduce

(i + WOI) •/2<

< C

L"

(l + \iP(.)\y/24>

T1

(l + |V-(.)l)' /2 uL .(3-305)

I L 1 II n

I U> n

or equivalent^, for <j> G <S(R ) such that j> G Cg°(R )

Since Cg°(R") is dense in <S(R"), it follows that (3.305), and therefore (3.304) holds for all G S(Rn). D Remark 3.10.10 Clearly, from (3.304) and the density ofS(Rn) in B ^ ^ R " ) we conclude that also all elements v G Bl 1 (R n ) are multipliers for the space 5;,p(Rn). We will need a characterisation of the dual space of B5, „(Rn). Theorem 3.10.11 The dual space of Bl (Rn), p < oo, is the space #0,p'( Rn )> £ + j? = 1; in the sense that for any I G [B^p(Rn)}* there exists v G BTS ,(Rn) such that we may interpret v • u as an element of L : (R") with the property that *(u)= /

JRn

v ( 0 u « ) d f = v~(u)

holds. The norm of I is given by | | v | | ^ _ s p , . Conversely, for any v G an element of [B^p(Rn)]* is defined 'by (3.306).

(3.306)

B^pl(Rn)

Proof: First we prove that for any v G B7S ,(Rn) we can define a continuous linear functional on B%,p(Rn) by (3.306). We know that v G B^p,(Rn) if and only if (1 + |V>(.)|)-s/2v(.) G L P ' ( R " ) . NOW, for u G B^p(Rn) it follows that (1 + |-0(.)|) s/2 u(.) G L p (R n ), implying that v • u G L J (R")- Hence we may consider the integral in (3.306) to find

/ v(0u(0^ < / (i + M0l)-j/V(0l(i + W0l)' /2 K0l^ JR"

JRn

< llVIU,-S,p'HUllv,s,P' thus, any v G B^sp,(Rn) B^p(Rn), i.e. B^,(Rn)

defines by (3.306) a continuous linear functional on C [Bs^p(Rn)}*. Let I G [B^p(Rn)]\ Any linear form

3.10 Some Function Spaces related to Continuous Negative Definite Functions 215 On <S(Rn) we may introduce the

I G [Bs, (R n )]* is determined on norm

IIMII:= (i + m.)\y/24>

LP

Any continuous linear functional I on (5(R n ), |||.|||) has a representation by a function w such that (1 + |V>(-)|)~V2w e l / ( R " ) and

(i + hKOI)

-s/2

W

LP' x

In particular, we have w G L (R n ) and l{<j>) = JRn w(£)(£) d£. The function (l + \ip(.)\)3/2 is polynomially bounded, hence, for any f G L p (R n ) the function (1 + |V'(.)|)s/2f belongs to S'(Rn), see Lemma 2.6.7. Therefore we find that w G S'(Rn) and hence there exists v G S'(Rn) such that v = w, i.e. (1 + |V>(.)l)_s/2v G l / ( R n ) . But from the very definition of B^p,(Rn) we deduce now that v G B,ip,p > , i-e. r

m = / Horn <%• Since every I G [Bl (R n )]* induces a linear continuous functional on (<S(Rn), |||.|||), the theorem is almost proved. To get rid of the Fourier transform in 3.306 observe that for u G iS(Rn) this reads as /

v ( 0 u ( 0 d£ = (v,u) =
(3.307)

D

The next theorem generalises the classical Sobolev embedding result. Theorem 3.10.12 Letip : R™ —> C be a continuous negative definite function and suppose that for some m G No we have

(1 + | . | 2 ) W 2 s/2

1

GL p

(1 + MOI)

Then it follows that B^p(Rn)

p

^

- — 1 and 1 < p < oo. p

(3.308)

C C£(R n ) and the estimate

sup I D ^ u ^ l < c||u||^ i p , |a| < m,

(3.309)

x6R n

holds for all u G B^ (Rn). G ^ 0, the inclusion B^p(Rn)

Conversely, when for some open set G C Rn, n£'(G) C C™(Rn) holds, then (3.308) follows.

216

Chapter 3 Fourier Analysis and Convolution Semigroups

Proof:

A. Let u e B^p(Rn)

and a G Nft, \a\ < m. Since

pa

Tu(0

(1 + | ^ ) | ) V 2 (l

+ |V(0l)5/u(0,

(3.310)

we find using Holder's inequality and (3.308) t h a t £ h-> £ Q u(£) is an element in L ^ K " ) . Hence, we get u(z) = ( 2 T T ) - " / 2 /

eix«u(£)d£

(3.311)

and differentiation in (3.311) gives t h a t u G C m ( R n ) . Furthermore, for \a\ < m it follows t h a t |Dau(z)| =

(2n)-n/2 JR"

in a/2 (i < (27r)-"/ 2 f (i + WOI) Jw. (i + i.i 2 ) m / 2 n/2 < (27r)-

(l + |^(.)|)'/2

+ i^oir/2Koid^ I V>,s,p*

LP'

For every u e B J p ( R " ) we find a sequence (^fc) fe6N , fc G Co°(R n ), such t h a t ^fc —>• u in B% p ( K " ) . Using this sequence we see t h a t for \a\ <m sup | D a u ( a ; ) - D a 0 i f e ( x ) | < c | | u - ^ | | ^

,

implying D Q u G C 0 0 ( R n ) , hence u G C™(R n ). B. Without loss of generality we may assume t h a t 0 G G. Further let G Cg°(G) such t h a t {x) = 1 in a neighbourhood of 0. For v G B^p(Rn) n it follows t h a t v G l?^, ( R ) fl £'(G), and by our assumptions we have v G C m ( R " ) . As in the proof of Theorem 3.10.4 an application of t h e closed graph theorem, Theorem 2.7.8, yields for the closed mapping <j) M- v t h a t

£

sup |D"(
\a\<m

x£W

For \a\ < m and v G <S(R") we get further

/ rv(o d£

|D%(o)|
LP

3.10

Some Function Spaces related to Continuous Negative Definite Functions 217

which implies by Theorem 3.10.11 that ( £ a ) A £ B^pl(Rn),

^R"

d£

(i + WOI)' /2

thus


for all | a | < m, proving the theorem. D We will prove two approximation results for the spaces B%, „{Rn). T h e o r e m 3 . 1 0 . 1 3 Let e Cg°(R n ), 0(0) = 1 and <j>s(x) = {ex). and p < oo it follows that

ll^u-ull^,P—>°

as e

—>0

Fors>0

(3.312)

forallu£B^p{Rn). Let u € B^p(Rn)

Proof:

and e G (0,1). It follows t h a t

II^ U IU,5,P^ C II^IU,.,III U IU,«,P by Theorem 3.10.9, and since s > 0 we find

ll^lk.,i= / (l + IV-(eOI)'/20(O dC


(l + e*\tfy/2fo) d£

n

JR

i.e.

He

b>s,p

< c^ ( ^

(1 + |£| 2 ) S / 2 |<^)| d ^ ||u||^ iiip .

Hence, there exists a constant c = c$^ independent of e such t h a t

ll<M 1p,S,p

— C\\U\\lp,

(3.313)

s,p

holds. For this reason it is sufficient to prove (3.312) only for all u S Co°(R n ). But for u G Cg°(]R n ) we find because of (&u

- u)A(0

= (2TT)-"/2 f

4>(T)U(£ - ST) d r

-

u(0

and (27r)~"/ 2 / R „ 0 ( r ) d r = F _ 1 ^ V o ) = 0(0) = 1 the convergence of 0 £ u to u in

B

^,P\

as e

0. •

218

Chapter 3 Fourier Analysis and Convolution Semigroups

Theorem 3.10.14 Let 4> G C§°(Rn) such that JRn (j>{x) Ax = 1. Further set 4>£(x) = £~n(|). For u G B^p(Rn), p < oo, the function u * 0 we have llu*<^-u||^ip-^0.

(3.314)

Proof: We know already that u * <j>e G C°°(R n ). Further, we have <j>e(e£) —>• (0) = 1 uniformly on compact sets as e —> 0, and since ( u * ^ ) A ( 0 = (27rr/ 2 u(0^(eO the theorem is proved. • Remark 3.10.15 The function spaces Bl p (R n ) are defined analogously to the spaces -BfciP(Rn) in [150]. Here k : R n —>• R + is a function satisfying with two constants c, m > 0 k(£ + 77)<(l + c|£|rk(77)

(3-315)

and BktP(Rn) := {u G S'(Rn) | k(.)u(.) G L P (R") }. Taking j> : R n -> C to 6e a continuous negative definite function we know by Lemma 3.6.25 that

i + ms+v)\ < (i + Vmj\)\i+Mv)\)-

(3.316)

It is possible to show that because of (3.316) the modulus of a continuous negative definite function share many properties of the class K of all functions satisfying (3.315). We do not carry out these results further since the main application of the class SfciP(Rn) lies in regularity results for solutions of partial differential equations. Now we will prove an interpolation result for the spaces Bfj, (M.n). Theorem 3.10.16 Letip : R™ -> C be a continuous negative definite function, p G (l,oo) and si < s2. Further let 9 G [0,1] and s := (1 — 9)si + 9s2- Then we have

[ % ( « " ) , B^(R")] = B^p(Rn).

(3.317)

3.10 Some Function Spaces related to Continuous Negative Definite Functions 219 Proof: We use the notation of Section 2.8. We set Be:=\B£p(p.n),B%p(Rn) Let u € -Be and £ > 0. Then there exists f G u(.Bs^p{Rn),Bsjp{Rn)\ such that i{6) — u and ||f||^ < ||u||e + £, where \\.\\e denotes the norm in Bg according to (2.213). Consider the mapping 9 >->• g(0) = (1 + \ip(.)\)Sl/2(l + A-2—1) \ip(-)\) 2 F(f(0)) where F denotes as usual the Fourier transform. Recall that t(z) € B£p(Rn) C S'(R n ). For x = Re z, 0 < x < 1 it follows that g(z) G ^ " ^;(S2R (R"n ) for each z in the strip { z e C | 0 < Rez < 1}. Now let n v G S(R ) and put h(z) = /

g ( z ) ( 0 v « ) d£.

It follows that z M- h(z) is continuous and bounded in the strip {z G C | 0 < Re z < 1} and analytic in its interior. Moreover, we have for ^ + -h = 1 for 6 = Re z = 0 |h(iy)|

/

g ( i y ) ( 0 v ( 0 ^ < Hf(i2/)IU,,1)PHv||Lp'

and for # = Re z = 1 |h(l + zy)| =

<\m + iv)\ ^,S2, W \\hP' V

P

Thus, by Lemma 2.8.4 we have IM*)I
0
In particular, for z — 6 we get /

g(«)(0v(0^

< ( H e + e ) | | v | LP'

implying that HSWIILP <

Hie-

But Hg(«)llLP= (l + I W I ) i l = f i * ± f a f i | LP

•

220

Chapter 3 Fourier Analysis and Convolution Semigroups

which gives

IHU s , p = l|gWII LP
Now let u G B^p{Rn)

and put

\if>(.)\f2~'^"~*)Fu,

+

thus we have f(#) = u. Further, it follows that (1 + |V(.)ir i / 2 |Ff(x + iy)\ = (1 + |V(-)l) Sl/2 (l + MXf'"*1"*

\F*\

s/2

< ( l + |V(.)l) |Fu|, implying that f(z) G B£p{Rn) \\Kz)\\i,,sUP ^

for 0 < x = Re z < 1 and

\\uh,s,P-

Hence, f is a continuous and bounded function from the strip {z G C | 0 < Rez < 1} into the space i?! 1 (R n ) which is analytic in the interior of the strip. Moreover, we have (1 + |V(.)I) S2/2 |F f(l + iy)\ = (1 + |V>(.)I)S2/2(1 + |V(.)l) ( ' 2 "'' 2 " 8 " 1) |Ful = (l + |^(.)ir / 2 |Fu|. This shows that f G H(B^p(Rn),B£p(m.n)^

and

llfllw < \Hn,,s,PThus u G Be and ||u|| e < ||u|| . s , proving the theorem. • Let ip : R™ -> R be a continuous negative definite function and consider the space i^> s (R n ) = 5 ^ 2 ( R " ) for 0 < s < 1. In this case llullv>,*,2

/ (i+v(orKoi2d^,

and it is clear that H^'s(Rn) product u,v)^=/

is a Hilbert space for any s G R with scalar

(1 + V ( 0 ) s u ( £ ) v ( 0 d e

(3.318)

3.10 Some Function Spaces related to Continuous Negative Definite Functions 221 Since for 0 < s < 1 the function £ \-> V;S(0 is a l s o a continuous negative definite function we may consider on H^'2(Rn), 0 < s < 1, the equivalent norm ||.|L, x, and therefore we may reduce the following consideration to the case s = 1. Thus we look at the scalar product

/ (i+v(0)u(o^(i)^= / u(x)y(x)dx+ [ mmW)
7R™

JRn

where we assume the functions u and v to be real-valued. Now, ip has accordingly to (3.186) a Levy-Khinchin representation V-(0=c + q ( 0 + /

(l-cos(x-O)^(dx),

(3.319)

J13Ln\{0}

c > 0, q(£) = 5^™,-=1 qij£i£j > 0 and qij = q^ G R, and the measure v is given on R" \ {0} by v{dx) = 1tj|cJ fi(dx), /x being a bounded symmetric measure on R™ \ {0}. Let u, v e <S(R"; R) be real valued and consider

/ mHow)dt=c[ u(xMx)dx+[ f ^ y y ^ + /

/

(1 - cos y • £)u(0WJ

Hdy) d£-

JRn\{0}

JUL"

To handle the last integral on the right hand side, note that for f, g E S(R n ;R) f(t)g(-*) dt = (27r)- n /

/

/

,/R™

e^+^f(^)g(y)d^dydz.

/ n

JR" JR

n

JR

Hence, we find / JRn

/

(l-cos(y-0)Fu(OFv^)i>(dy)de

JRn

= \ l

I

^ JRn

=

(e^-l)(e-^-l)Fu(0Fv(-0d^(dy)

JRn

( 2 ! r p /• r ^

x e
= \ [ z

/• /

JR" JR« JR"

/

cto.,(e^_l)pu(a)

JR"

1)FV(T)

dcr dr i/(dy) dec

( u ( a; + 2/) - u(x))(v(x + y) - v(x)) v(dy) dx

JRn

[

VR" JR"

K * ) - u(y))(v(x) - v(y)) J(da:, dy),

222

Chapter 3 Fourier Analysis and Convolution Semigroups

where J(dx,dy) is a symmetric measure on Rn x R™ with no mass on the diagonal. Thus we have proved Theorem 3.10.17 Let 4> '• Rn ->K ie a real-valued continuous negative definite function with Levy-Khinchin representation (3.319). Then on i J ^ ' ^ R " ) an equivalent scalar product is given by I

X>^^>d* + (c + l ) / +\ l

i

«x)

u(,)v(,)d,

~ u 0/)) - ( v (z) - v(2/)) J(dx,

dy)

(3-320)

where c, qij and J(dx, dy) have the properties mentioned above. Clearly, the spaces H^'s(M.n) are modelled after the classical Sobolev spaces H (Rn), note that Hs(Rn) = H^2's(Rn). These spaces and their generalisations are considered in the next section. However, let us mention the following result which is proved in [298]. s

Theorem 3.10.18 Let u G £'(R n ). Then there exists s > 0 such that u G H~s{Rn). For example, we have s0 G H-?+r>(Rn) for every rj > 0. Corollary 3.10.19 Suppose that the continuous negative definite function tp '• R™ —>• C satisfies the estimate

wo i > <*Kr for some r0 > 0 and c0 > 0 and all £ G R™, |£| > p. Then for every u G £'(Rn) there exists some s > 0 swc/i that u G i!^'~ s (R n ). Finally, we introduce the spaces H?0'cs(Rn) by defining u G H?0>cs(Rn) if and only if for every G Cg°(R") the function 4>u belongs to H^'s{Rn).

3.11

Besov Spaces and Triebel-Lizorkin Spaces

In order to measure the smoothness of functions or distributions two scales of function spaces are at our disposal, the scale of Besov spaces B* (R n ) and

223

3.11 Besov Spaces and Triebel-Lizorkin Spaces

the scale of Triebel-Lizorkin spaces F*(M.n). Most of the known (isotropic) function spaces are covered by these spaces. Since we will need more often Besov spaces later on, we present a part of their theory more detailed and give not always the analogous result for the spaces F* (M.n). However, since detailed proofs of many results for the spaces Bpq(Rn) and i ^ (R n ) require a larger machinery and all these proofs are available in well written monographs, we will omit several proofs in this section. As standard reference for function spaces we mention the books [299]-[301] of H. Triebel. To construct these function spaces we introduce a special class of smooth dyadic partitions of unity. Fix 4>Q G Co°(Rn) with the properties supp<^oCB 2 (0) and ^ols^o) = 1-

(3.321)

Further we set for k G N

MO •= M 2 "**) - 4>o(2~k+1t).

(3.322)

It follows that supp 4>k C B2k+i (0) \ B2*-i (0),

(3.323)

hence supp k+2 = 0,

(3.324)

and for a G NQ we find that |Da<M£)l ^ c « 2 - f c H , k G No.

(3.325)

Finally, we have for all ( e l " OO

53^(0 = 1-

(3-326)

fc=o From (3.324) we conclude that for any a G NQ and m G N OO

a

d

E

OO

MZ)=

E

fc=m+l

OO

d<X

V

fc=ro+l

k=m+l

M0 and lim

d"^)

= 0.

For any u G <S'(Rn) we may define (^(Dju-F-^fcFu).

(3.327)

224

Chapter 3 Fourier Analysis and Convolution Semigroups

Note that fc(D) is a pseudo differential operator, see Introduction, and from the Paley-Wiener-Schwartz theorem, Theorem 3.4.6, it follows that <^(D)u extends to an entire analytic function. Let a,/3 G NQ and u G <S(Rn), then we find OO

CO

x?da J^ F(^F" 1 u)=^ J2 F(
fc=m+l

and further, with some ( € C CO

x? J2 F(
OO

•/Rn

k=m+l

-

oo

C/ (afe-te-«) Yl MOF-^uXOde •^ R "

fc=m+l

c / e-*** Y, PV ( E MO) d^F-\daum de w ^

^

vfc=t+i

/

Hence, we find sup xeEn

^ W £ F^feF-1!!) Vfc=m-fl

^E/

di £ MO |^-7 F -i(a« u )(^)|de k=m+l

Now, as m —>• oo the integrand | < 9 7 £ £ l m + 1 < M 0 | | 9 / 3 - 7 F - 1 ( a a u ) ( ^ ) | tends to zero, and from (3.325) it follows that

37 E Mi) |^-TF-1(aau)(o| < c7|a^-7F-1(aau)(o|. Thus the dominated convergence theorem implies lim sup xPd*l

771—>00 x(zM.n

Y

\k=m+l

FfaF-^ix)

225

3.11 Besov Spaces and Triebel-Lizorkin Spaces i.e. we have oo

u = ^F(0fcF-1u)1

(3.328)

fc=0

where the series converges in <S(Mn). From this we derive Lemma 3.11.1 For every u € (S'(Rn) we have u = ^ ^ ( D ) u = ^F-1(0fcu),

(3.329)

fc=0 fc=0

where the series converge in

S'(Rn).

Proof: For v € S(Rn) and m G N we find ( u - £ > f c ( D ) u , v \ = (u, \

fc=0

/

F(^F_1v)),

Yl

\

ro=fc+l

/

and from (3.328) the lemma follows. D Definition 3.11.2 Let 0 < p < oo, 0 < q < oo, s e t before. The Besov space Bpq(Rn) is defined by B°P:q(Rn) := {u e 5'(R") | ||u|| B . q < oo } ,

and (fc)fc€No be as

(3.330)

where for q ^= oo

\*;=0

/

and for q = oo ||u|| B .

:=suP2fcs||^(D)u||LP.

(3.332)

Definition 3.11.3 Let 0 < p < oo, 0 < q < oo, s £ R and (fc)fceNo ^e above. The Triebel-Lizorkin-space F£ (M.n) is defined by F;,q(^n)

:= {u e <S'(R")

\\VL\\F. q

< oo } ,

as

(3.333)

Chapter 3 Fourier Analysis and Convolution Semigroups

226 where for q < oo

1/9

fc

£ 2 ^(D)u|'

(3.334)

^fc=o

LP

and for q = oo sup2 fcs |«/> fc (D)u| fc€N0

(3.335) LP

R e m a r k 3 . 1 1 . 4 A. Note that for p < 1 we have to handle quasi-norms, see Section 2.7. In fact our applications to path properties of certain stochastic processes make it necessary to include this case. B. Formally, the norms \\.\\Bs and

are depending on the partition

of unity.

However, it can be shown

that any other smooth dyadic partition of unity having the properties (3.321), (3.323), (3.325) and (3.326) give rise to an equivalent (quasi-)norm. C. For the problems concerning the space F^ (M.n) we refer to [301], p.29-30. In the following lq, 1 < q < oo, denotes the sequence space lq := {{ak)keNo

ak e C and ||(a fc )||, g < oo | ,

where

ii(«*)ii?,= X > * i

1 < q < oo,

\k=0

and

IIK)IL = SU P M fc€N0

T h e o r e m 3 . 1 1 . 5 In the sense of continuous embeddings we have R n ) for 0 < q0 < qx < oo, 0 < p < oo, s £ R. 1. B: B P,qi < p,go^

p,qo v

^F:

2 B

- ££(

P,1iy

^•B*

and e > 0.

s Fa+e CRn) i M- FP,Q p,go \ and £ > 0.

ln) for 0 < q0 < qi < oo, 0 < p < oo, s e R. R n ) for 0 < go < oo, 0 < qx < oo, 0 < p < oo, s € R n ) for 0 < go < oo, 0 < qi < oo, 0 < p < oo, s €

227

3.11 Besov Spaces and Triebel-Lizorkin Spaces

Proof: We give here only a sketch of the proof and refer to [300], p.46-47. Let (afc)fcgN be s o m e sequence of complex numbers. For 0 < go < <7i < °° w e have ||K)|| / g i <||(a f c )|| Z T O .

(3.336)

But (3.336) implies immediately 1. To prove 2. note that for any sequence (a/OfceN °f c o m p l e x numbers and e > 0 we have jr2skqi\ak\qA

< sup (2<-s+£^\ak\)

^ " ^

< c sup (2( s+£ ) fc |a fc |) , fc€N0

which gives together with part 1. and the usual modifications for gi = oo the assertion 2. • It is convenient to introduce the notation

ll(ffc)llL,(M == (jRn ( f > ( * ) N ' dx)

(3-337)

and H(fc)ll,,(Lp) ••= (jt

(jf n \{k(x)\Pdxy\

"

(3.338)

for a sequence (ffe)fe€No of complex-valued measurable functions on R". In particular, taking ffc(:r) = 2sfc(F_1fcFu)(a;), we find \HBitq

= IKffc)IL,(Lp)

(3-339)

H\Flq

= ll(f*)llLp(j,)-

(3-340)

and

Now we have Proposition 3.11.6 Let 0 < q < oo, 0 < p < oo and s e R , then we have BsP,pAq(Rn) ^ F£iq(Rn) - » B;ipVq(Rn).

(3.341)

228

Chapter 3 Fourier Analysis and Convolution Semigroups

Proof: Once again, we sketch only the proof and refer to [300], p.46-47. For p> q we have

£N<

II (fit) IIl P (LP) <

dx

k=0

= ll(f*)llL,(Z,)

< (£(7

|ffc|Pda;)

ll(ffc)||,

(LP)'

^fc=0

and for 0 < p < q < oo we have always

ii^)ii^(L P) -ff:(/ K jf fc i p d^ ^fc=0

£|f*

<

da;

fc=0

= ll(/fc)llLP(l,) < 11(^)111,(1.-)' proving the proposition. • Corollary 3.11.7 For 0 < p < oo and s £ R we /iai/e 5.p.pV.

p,pV

r).

(3.342)

Theorem 3.11.8 A. The space B^q(Rn), s e R, 0 < p < oo and 0 < q < oo, is a quasi-Banach space, for 1 < p < oo and 1 < q < oo it is a Banach space. Further, we have in the sense of continuous embeddings <-» Bspg(M.n)

<-+ <S'(R n ).

(3.343)

For s £ K, 0 < p < oo and 0 < q < oo the space <S(Rn) is dense in Bpq(E.n). B. The space .F^ (R n ), s 6 i , 0 < p < oo and 0 < q < oo is a quasi-Banach space and a Banach space for 1 < p < oo and 1 < q < oo. Furthermore, ^

«S'(R n )

(3.344)

holds in the sense of continuous embeddings. For s £ 1 , 0 < p < oo and 0 < q < oo the Schwartz space <S(Rn) is dense in Fp (R n ).

229

3.11 Besov Spaces and Triebel-Lizorkin Spaces

This theorem is proved in detail in the monograph [300], p.48, of H. Triebel and requires some auxiliary results. For this reason we will omit its lengthy proof. Now, we will establish some relations to classical Sobolev spaces. Definition 3.11.9 Let 1 < p < oo and s £ l . H°(Rn) are defined by H;(Rn)

The Bessel potential spaces

:= {u G <S'(R") I F - 1 ((1 + |.| 2 ) s / 2 u) G L p (R n ) }

(3.345)

These spaces are Banach spaces with respect to the norm (3.346)

H a H ^ ' ( ( i + l-rT"*)!,,. In particular, we have H°(Rn) = L P (R"). Theorem 3.11.10 For m G No and p = 2 we have H?(Rn)

= {u G L 2 (R") | dau G L 2 (R n ) for \a\ < m}

(3.347)

and the norm ||.||#m is equivalent to the norm (3.348) \a\<m

Proof: Let u G H?(Rn). rem 3.2.17, that

It follows from Plancherel's theorem, Theo-

(i^iT^in^1^1-12)7"72*)

< oo,

but

|(I+I.IT/2»!=/

(i+iermfdt

> ^ E / £2a|u(OI2d£ = c ]T / |F(a«u)(0l2d^ J2 f \dau(x)\2 dx. | a | < m *•

230

Chapter 3 Fourier Analysis and Convolution Semigroups

On the other hand, we have J2 [ \dau(x)\2dx= |a|<"»

£ f l"l
|F(d«u)(0| 2 d£

l

i/2 - "

>c'

2

D

(1 + | . | T u • II

'

110

In general we have Theorem 3.11.11 For s e M and 1 < p < 00 we have #*(R") = Fps>2(Rn).

(3.349)

For a proof of this result we refer to [300], p.88. Let 0 < s < 1 and p = 2. It follows from Plancherel's theorem that .ff|(R n ) = -B|.|2Si2(^n)> where B},2, 2(^™) i s t n e space corresponding to the negative definite function £ t-> \£\2s- Using Theorem 3.10.17 we know that the scalar product in if|(]R n ;R) can be represented as in (3.319). It is not difficult recalling the calculation leading to (3.319) and using Example 3.3.8 that in the case of i72(R™;R), 0 < s < 1, to see that we have as an equivalent scalar product J

J (u(X)-u(y)){^)-v{y))dxdy+r

h« 7R«

F - y\2s+

u{xHx)dXj

(3.350)

JR«

whereas for s = 1 Theorem 3.11.10 gives the equivalent scalar product /

Vu(x) • Vv(z) dx+

[ u(i)v(x) dx.

(3.351)

Next we will give some embedding and interpolation results for the spaces 5 £ , ( R " ) and i £ , ( R " ) . Theorem 3.11.12 A. In the sense of continuous embeddings we have for 0 < Po < Pi < 00, 0 < q < 00 and —00 < s\ < So < °° 72

72

2J£,(R") C £ £ , 9 ( R " ) if s0 - - = Sl - - .

(3.352)

231

3.11 Besov Spaces and Triebel-Lizorkin Spaces

B. Let 0 < pQ < pi < oo, 0 < q,r < oo and —oo < si < s0 < oo. Then we have in the sense of continuous embeddings F^q(Rn)

Ti

c i £ , r ( R " ) if 80 - -

H

= Sl - - .

(3.353)

This theorem as well as the following corollary is proved in [300], p.129-131. Corollary 3.11.13 A. Let s > 0 be not an integer, s = [s] + {s}, {s} € (0,1], 0 < p < oo and 0 < q < oo. Then we have Bp£'(R n )cCM , < s >(R n ).

(3.354)

B. Let 0 < p < oo, 0 < q < 1 and m £ No- Then we have BPiqp(Rn)cCm(Rn).

(3.355)

C. Let Q < po < pi < oo, —oo < si < so < oo, 0 < q < oo and further so — — = si ——. Then we have F;iq(Rn)cB;ipo(Rn).

(3.356)

D. For 1 < po < pi < oo and — oo < s\ < SQ < oo we have H;°(Rn) c B;iP0(Rn)

if so - _ =

ai

- -.

(3.357)

Once again (3.354)-(3.357) is meant in the sense of continuous embeddings. Note, that (3.357) combined with (3.354) gives for s 0 - ^ € l + \ N 0 i^°(Rn)cc[s°-?]'{s°-?}(Rn),

0
(3.358)

Theorem 3.11.14 Let s0 € R, si £ R, 1 < q0 < oo, 1 < q\ < oo, 1 < p0 < oo and 1 < p1 < oo. For 6 G (0,1) set 1 1 - 9 - = P Po

0

1

1 1 - 0 9 , , „N , - = 1 , and s = (1 - 9)so + #si. Pi q qo qi

Then the following complex interpolation formulas hold [Bsp°,qo(Rn),Bplqi(Rn)]e

= B;jRn)

(3.359)

= F;iq(Rn).

(3.360)

and [F£iqo (Rn), F£iqi (Rn)}

9

232

Chapter 3 Fourier Analysis and Convolution Semigroups

This result is taken from [301], p.45, where even more interpolation results are discussed. For our later applications we need an atomic decomposition of elements in Besov spaces. These results are essentially due to M. Frazier and B. Jawerth

[96]. Let Qjk C R n , j £ No, k £ Z™, be an afflne copy of the unit cube with side length 2 _ J and centre 2~^k. By rQjk, r > 0, we denote the dilated cube (with respect to its centre). D e f i n i t i o n 3 . 1 1 . 1 5 Let 0 < p < o o , s e R and K,L K > ([s] + 1 ) + and L > n\ - - 1 | P A function

m : R " - > C i s called s-atom

supp m C 5Q0k

-

8

eZ

such that

V(-l).

(3.361)

if

for some k £ IT

(3.362)

and | D a m ( x ) | < 1 for alia £ NJ, \a\ < K. Moreover, the function to K and L if

(3.363)

m is called (Qjk,s,p)-atom,

j £ No, k £ IS1,

supp m C 5Qjk,

relative

(3.364)

|D a m(a;)| < {2-in)~i+-~^

for alla£

NQ1, | a | < K

(3.365)

and

/

xpm(x)

dx = 0 for all (3 £ NQ\ |/3| < L.

(Note that for L — —1, condition (3.366) is

(3.366)

empty.)

T h e o r e m 3 . 1 1 . 1 6 Let 0 < p,q < oo, s £ M. and K,L £ Z be as in Definition 3.11.15. Further, let u £ S'(Rn). Then u belongs to the Besov space BsP:q(Rn) if and only if /

U=

CO

S m

X

\

S

m

x

XI I k k( ) + X ] J,k 3,k( ) feez \

j=o

(3.367) I

3.12

233

Fourier Multiplier Theorems

holds in <S'(R n ), where rrifc is an s-atom atom, and if for the coefficient Sk, Sj^ 1/P \k&n

/

/oo

relative to Qo,k, mj,k is

(Qjk,s,p)~

y/P\1/q

/

\j=o VfceZ"

a

/

J

holds. This theorem is proved essentially in [96], p.795, see also [301]. R e m a r k 3 . 1 1 . 1 7 We will encounter some applications of Theorem 3.11.16 where we have to reduce the smoothness assumption on m^ and rn^fc. In our case, we will work only with Lipschitz functions, see R.L. Schilling [266]. In the paper [302] of H. Triebel and H. Winkelvofi atoms with Holder regularity are systematically used. R e m a r k 3 . 1 1 . 1 8 Later we also have to use the theory of weighted Besov and Triebel-Lizorkin spaces. This theory was recently developed by D. Haroske and H. Triebel [114]-[115] and applied to the eigenvalue distributions of some degenerate pseudodifferential operators. We will use their theory to discuss the results of R.L. Schilling [270] about global properties of paths of some classes of Feller processes. It seems to us more reasonable to present the theory of weighted spaces when discussing their applications.

3.12

Fourier Multiplier Theorems

Let m : R n —> C be a bounded measurable function. theorem we find t h a t T m ( u ) := F - ^ m F u ) = F ^ m u ) ,

Using Plancherel's

(3.369)

defines a bounded linear operator on L 2 ( R " ) . Thus m £ L°°(R n ) is an L 2 Fourier multiplier in the following sense. D e f i n i t i o n 3 . 1 2 . 1 Let m £ L°°(R"). 1 < p < oo, if the operator u M. Tm(u)

:= F - ' H

is a bounded operator on L p ( R n ) .

We call m an L p -Fourier multiplier,

(3.370)

Chapter 3 Fourier Analysis and Convolution Semigroups

234

A word to the definition of (3.370). For u € <S(Rn) it follows t h a t u £ <S(R"), implying t h a t mu e L ^ R " ) D L ° ° ( R n ) , provided m € L ° ° ( R n ) . Hence, on «S(Rn) the operator T m is well defined. The question whether m is an L p Fourier multiplier is therefore the question whether it is possible to extend T m from «S(Rn) continuously to a n operator from L P ( R " ) into itself, 1 < p < oo. In this section we will prove a variant of the Michlin-Hormander Fourier multiplier theorem. For this we need as an auxiliary result one of the cornerstones of real-variable theory, the Calderon-Zygmund decomposition lemma, [53]. In our presentation we follow essentially the monograph [284] of Chr. Sogge. L e m m a 3 . 1 2 . 2 ( C a l d e r o n - Z y g m u n d ) Let f e L ^ R " ) and a > 0. we may decompose f as

Then

oo

3=1

where oo

llg|| L 1 +Ell b A>< 3 ll f ll L 1 >

( 3 - 372 )

|g(a;)| < 2na

(3.373)

almost everywhere inM71,

and for certain non-overlapping axes we have bj(x) = 0 if x^Qj

and /

cubes Qj with axes parallel to the

bj(x) dx = 0,

coordinate

(3.374)

and oo

1

X;A ( n ) (Q,)<-||f|| L 1 .

(3-375)

j=i

Proof: First let us divide R™ into a lattice of cubes with axes parallel to the coordinate axes and of volume larger t h a n ^ | | f | | L i - For such a cube we have

3,4;/l«MI<•*<••

(»•"»)

235

3.12 Fourier Multiplier Theorems

Next we divide each cube into 2 n equal non-overlapping cubes. We denote by Q n , Q12, • • • these new cubes for which (3.376) does hold, i.e.

i4)jLMd,sa

(3377)

From (3.376) and since 2nX^n\Qxk) oc\{n\Qik)<

I

= \^n\Q)

it follows that

|f(x)|dx<2"aA(")(Q l f c ).

(3.378)

JQik

Now define on Qik the functions gifcfr) ==

Un)jn

x/

|f(l/)|dy, xGQlk,

(3.379)

and on Kn we define M x ) =

|fW-gW;:eQ;:

(3380)

Further, let us consider all the cubes not belonging to the family {Qik}- These cubes satisfy (3.376). We divide them as before into 2™ cubes and denote by Q2I1Q22J • • • those new cubes for which

A«k)/ a J t W ! d * S Q

(3381>

'

holds. Now we extend (3.379) and (3.380) for these cubes to get functions g2fc and h-2k- Continuing this procedure, we obtain non-overlapping cubes Qik and functions gik, hik- Since these families are countable, we may rearrange them to sequences (Qj). N and (gj). N, (bj). N . We consider now the function 00

g(z):=£g,(z)xQ.(z),

x€Rn.

It follows that f = g + Yl'jLi ^ji i-e- (3.371) holds, and since / JQJ

|g(a;)| dx < f JQJ

\i(x)\ dx

236

Chapter 3 Fourier Analysis and Convolution Semigroups

it follows, see (3.380), t h a t /

(|g(*)| + | b i ( s ) | ) d a : < 3 / '

|f(x)|dx.

Since the cubes Q , are non-overlapping and g = f in ( U j l i Q j )

>w e Set

(3.372). For x e \J™=1 Qj it follows t h a t (3.373) holds, but for x e ( U ° l i we find a small cube Q containing x such t h a t TJ^SIQ

x

ft( )\

x

^

QjX

a

< > implying

(3.373) for almost all x £ R n . Condition (3.374) is clear from the definition of the functions b j . Finally, (3.375) follows since (3.375) holds for all Qj and the lemma is proved.

•

Now we are in a position to prove the following variant of the Hormander multiplier theorem which we take from [284].

Michlin-

T h e o r e m 3 . 1 2 . 3 Let m <E L°°(R n ) and T m be defined by (3.370). Suppose that for some k G N, k > | , we have for all 4> € Co°(R™ \ {0}) the estimate

/^"(X^H

2_] sup/i 0<|a| °

0

2

< CO.

(3.382)

Then for all p, 1 < p < oo, we find (3.383)

||T m u|| L P < c p | | u | | L P , and further,

for a > 0, we have

A<") {x£Rn

\ |T m (u)(a;)| >a}<

i.e. T m is of weak-type

4||u||Ll, a

(3.384)

(1,1).

Proof: We know already t h a t T m is a bounded operator from L 2 ( R n ) into L 2 ( R n ) . Suppose we have proved (3.384), i.e. we have shown t h a t T m is of weak-type (1,1). T h e n the interpolation theorem of Marcinkiewicz implies t h a t T m is a bounded operator from L P (R") to L P (R") for 1 < p < 2. T h e function m satisfies the same hypothesis as m, thus T ^ : L p ( R n ) —> L P ( R " ) , 1 < P < 2, is also a bounded operator. Since for f , g £ <S(Rn) we have / JR"

(Tmf)gdz=

/ V/R"

i(Tmg)dx,

237

3.12 Fourier Multiplier Theorems

it follows that T m as adjoint operator of the bounded operator Tm : L p (R n ) —»• L p (R n ), 1 < p < 2, is a bounded operator from l / ( R n ) to l / ( R " ) , i + ± = 1, 1 < p < 2, i.e. 2 < p' < oo. Thus it remains to prove the weak-type estimate (3.384). To do so we use a modified dyadic partition of unity, compare Section 3.11. Let 0 £ Cg°(R n ) such that (3.321) holds and define the function <j)by


(3-385) c

The function <j> belongs to Co°(E"), 0 G (supp ) and we have oo

J2 H2~jt) = 1. ^

°-

(3-386)

j=—oo

Note that (3.323)-(3.325) still hold for ,• (£) = <j>(2~^). The main point in using this dyadic partition of unity is that it is now easier to use scaling properties. We set

m„(0 = #0m(A*0. M>0,

(3.387)

to find

= l|^lE(^-|"l^^-^)(f)(^)(,) d£ HndV

=/

" | a | E ( f l ) " " | a | + " " ( ^ ) W 8 ? ( m N ) / ' " l / " »n*n

= f ^|a«(^)m( W ))| 2 d,, = /

/x"|9QmM(0|2de

Chapter 3 Fourier Analysis and Convolution Semigroups

238

Since for some constant c = Ck>n we have

(id - A)*/2mM(0 2 d£ < M" J2 I \da^(0\2 <%

of f

a
it follows from (3.382) that (id-A)fc/2mM(0

f n

d£
(3.388)

JR

with a constant independent of fi. Define KM by KM = m^. We find that / JR

n

|K M (a;)| 2 (l + |a;| 2 ) f c da;
(3.389)

with c independent of fi. Since k > { i £ M" | |ajj| < R}, we find further /

\Kti(x)\dx

C Q A , QR

:=

x-fc/2

|KM(a;)|(l1 + l a,2\fe/2 : ! 2 ) " " ^ + l^l2 2 )"*" dx

= f

•'Oft

^ and B#(0)

•'Of. 1/2


2

2 fc

( 7 |K^)| (i + N ) dx) \JK"

1/2

/

f/

v 51,(0)

2 fc

(i + N )- dx')

y

The substitution a; = Ry yields (l + \x\2ykdx

f

= Rn[

JB°R(O)

(l+JR2|y|2)-fcdj/, JB°(O)

and since (l + R2\y\2)~k

< (R2\y\2)~h,

we find

(l + \x\2ykdx
f JB%(O)

\y\-2kdy, JB$(O)

implying by (3.389) that \Kli(x)\dx
/

(3.390)

C

JQ „

Next observe that £,-mM(£) = (£j<£(£))m(M£) satisfies the same estimate as m^ since we may substitute the function <j>o by £ — i >• £J^>O(£)J n ° t e that so

239

3.12 Fourier Multiplier Theorems

far we used only <j> G Cg°(Rn \ {0}) C Cg°(R"), which is not changed when considering £ H-» £j(j>o{£) instead of 4>Q. In particular, since K^ = m^ it follows that f

|VK M (x)| dx < c,

(3.391)

where we have to use (3.389) for £j i-> £jmM(£) and (3.390). The identity f1 d I —Kfi(x + ty)dt

Kp{x + y)-Kp{x)=

= / (tj/)-(VK M )(a; + ty)dt Jo leads now to |KM(a: + y)-K / J (a:)|da:
/

(3.392)

Now we want to use the Calderon-Zygmund decomposition for u = g + X ^ l i Dj with cubes (Qj)jeN as in Lemma 3.12.2 to prove (3.384). For this let K = m, i.e. |T m u| = |K * u| for suitable functions u, and denote by Cfc the cube 2Qj. We will prove /

\Tmbj\dx=

Jx<£Q*

\K*bj\dx
\bj\dx.

(3.393)

JMn

Jx£Q'

Suppose that (3.393) is shown. From (3.373) we deduce |g0r)|2dz<2"a/

/ n

JR

|g(z)| dz < 3 • 2"a||u|| L1 ,

JR"

where we used in the last step (3.372). The L 2 -boundedness of T m and Chebyshev's inequality (2.217) yields \^({x

e R" | |T m g(x)| > f } ) < c^||g|| 0 2 < c'l||u|| L 1 .

(3.394)

For Q* := {f°=l Q* we find by (3.375) oo A (n) ( g * } =

J2\^(2Qj)

oo

1

= 2» £>(">(<&) < 2n-||u||L1.

(3.395)

240

Chapter 3 Fourier Analysis and Convolution Semigroups

On the other hand, (3.383) implies for b := Y27Li ^>j that

A(n)({ze(QT

\Tmb(x)\>^\)<-J2

\(K*bj)(x)\dx j=l

<

2

C

Jx

?Q'j

°° r

a £ /. " ^ j=l

da;,

•'K"

and using (3.372) we get A(n) ( { z G (Q*)C | T m b(:r) > § } ) < c||u||L1.

(3.396)

Since {xeRn\

|Tmu(aO| > a } c { x e r | |T m g(z)| > | } U

{i 6

r m b(a;)|>|}

it follows from (3.394)-(3.396) that (3.384) holds. Thus it remains to prove (3.383). Without loss of generality we may assume Qj = QR. The Fourier transform of x i-» finKfj,(fix) is the function £ i-> mM( M , see Lemma 3.1.9.5, and using (3.386) we have

K(a:)= J2 2" j K 2 ,( 2 ^), j=-oo

where the series is convergent in <S'(R™) follows from (3.390) that

/

Since bj vanishes outside Qj, it

\(^Kfl(n.)*bj)\(x)dx = [

f

IfK^x-yVWbjMldxdy \Kll(x)\dx
<\\bj\\L1[

(3.397)

J

QR

Since k > ^ this estimate is useful for ^ < R. To handle the other case, we note first that JR„ bk(x) dx = 0 leads to KM(/i .) *bj=

/

(KM(/x(z - y)) - K^x))bj(y)

dy,

241

3.13 Notes to Chapter 3 and using (3.392) we find

< f

f

/in|KM(/x(a: - y)) - K„(Az)| |b,(2/)| dx dy


(3.398)

Combining (3.397) and (3.398) we arrive at

[ jcK^b^^idx
Jx

£Q'j

\2iR>\

2JR

)

2iR<\

J

hence (3.393) is proved. • Corollary 3.12.4 For m e C°°(R" \ {0}) D L°°(]Rn) assume | D a m ( 0 | < c Q |£|-H

(3.399)

for all a £ NQ (or a € NQ ; |a| < k). Then m satisfies the assumptions of Theorem 3.12.3. Note that (3.399) is always fulfilled for a function m 6 C°°(E n \ {0}) being homogeneous of degree zero. Finally, let us state without proof a multiplier result due to P.J. Lizorkin [206]. Theorem 3.12.5 Let m 6 L°°(R n ) be a function such that sup |(£-d) a m(£)| < c for all a € Nff, a, € {0,1}, j = 1 , . . . , n, and (£-d)a = {^d^1 Then m is an L p -Fourier-multiplier for 1 < p < oo.

3.13

(3.400) •... • {£ndn)a".

Notes to Chapter 3

In 3.1-3.4 we used mainly our book [165], but in writing that book we borrowed much from the monographs of F.G. Friedlander [97], L. Hormander [152] and

242

Chapter 3 Fourier Analysis and Convolution Semigroups

W. Rudin [255]. Further, in 3.2 when discussing L p -derivatives we followed closely E.M. Stein and G. Weiss [289]. For handling the Wiener algebra and related questions we used mainly P. Malliavin's book [217]. In 3.3, besides the books already mentioned, we often worked with C. Zuily's guide to distributions [317]. In 3.4 we used W. Rudin's book [255] for Theorem 3.4.2, but in proving Theorem 3.4.5 we followed F.G. Friedlander [97]. In both cases, however, the influence is indirect since we mainly looked into our book [165]. For a comprehensive treatment of abstract harmonic analysis we refer to the monograph [92] of G.B. Folland and that of W. Rudin [254]. When working on 3.5-3.9, often the book of Chr. Berg and G. Forst [25] was our source which we followed quite closely. However, we also proved various results that are quoted but not proved in their book. As a remarkable survey for these topics we would like to mention Chapter 1 of V.P. Guraii [112]. A readable treatment of the theory of distributions and Fourier transforms is the monograph [74] of W.F. Donoghue, where aspects related to positivity of distributions are also discussed. In 3.5 many results are worked out by using the book [18] of H. Bauer. The proof of Bochner's theorem is essentially taken from R. Strichartz [293], we refer to the paper of M. Bingham and K. Parthasarathy [32], where a probabilistic proof of Bochner's theorem is given. The original proof was given by S. Bochner in [38]. Theorem 3.5.24 is taken from T. Kawata [181]. For the other results in 3.5 we used the book [25] of Chr. Berg and G. Forst. Many results on positive definite functions and distributions are presented in the lecture notes of C. Herz [125]. Of course, in probability theory positive definite functions are very much used, but under the name characteristic functions. The books [208]-[210] of E. Lukacs should be regarded as the standard reference for (one—dimensional) characteristic functions, but one should also consult the monographs [181] of T. Kawata and [202] of Yu.V. Linnik and Z.V. Ostrowskii. More recently K. Sato published lecture notes [260] where also many results on characteristic functions are discussed. Generalisations of positive definite functions as well as positive definitizable functions are treated by Z. Sasvari [258]. We also want to mention the historical survey of J. Stewart [290] on positive definite (and negative definite) functions. Further it should be stressed that many results in 3.5 and the following sections were discussed for the first time in the monograph [40] of S. Bochner. In 3.6 we followed once again Chr. Berg and G. Forst [25], but we rearranged partly their presentation. Formula (3.131) can be found in L. Hormander [152],

3.13

Notes to Chapter 3

243

T h e validity of Lemma 3.6.23, Peetre's inequality, was pointed out to us by W. Hoh who used it in [143] to improve earlier results of the author's [159]. Lemma 3.6.25 was communicated to us by R.L. Schilling [267]. We also refer to the historical important paper [175] of J . - P . Kahane and to the paper [123] of C. Herz where a very general theory for negative definite functions is outlined. T h e definition of negative definite functions is due to I.J.Schoenberg [275]. It has been introduced in connection with isometric embeddings of metric spaces into Hilbert spaces, see also K. Ito [157], Having his results in mind, the validity of an analogue to Peetre's inequality is not surprising. This aspect of negative definite functions had been taken up more recently by A.L. Koldobsky [186], we refer also to P. Levy [200], p.63, and C. Herz [123]. We also want to mention the work of J. Christensen and P. Ressel [60] where a probabilistic characterisation of negative definite functions (on Abelian semigroups) is given. T h e proof of the Levy-Khinchin formula we give in 3.7 is taken from our joint paper [172] with R.L. Schilling. Although there is many a textbook or monograph t h a t gives a probabilistic proof of the Levy-Khinchin formula in one space dimension, see for example the text of L. Breiman [49], it is rather difficult to find a proper reference for the higher dimensional case. Exceptions are the book of R. Cuppens [64] and the work [259], [260] of K. Sato. In the sixties several purely analytic proofs of the Levy-Khinchin formula were given'. Ph. Courrege [61] used quite hard analytic results on certain integro-diiferential operators, in [250] M. Rogalski introduced the radial limit methbd which he attributes to A. Beurling. There are other proofs which employ extreme-point methods and Choquet theory, see S. Johansen [174] or D.G. Kendall [182]. Further proofs are given by K. Harzallah [117] and G. Forst [93]. In addition, using Courrege's idea and representation results for linear functionals satisfying the positive maximum principle, see Section 4.5. F. Hirsch gave an unpublished proof of the Levy-Khinchin formula which is close to our approach, see F. Hirsch [137], p. 115. In addition, following ideas of J . - P . R o t h [253], in [136] F . Hirsch supplied a further analytic proof of the Levy-Khinchin formula using only analytic methods. In the monograph [127] of H. Heyer results related to the Levy-Khinchin formula are discussed rather comprehensively. Proposition 3.7.12 is taken from Chr. Berg and G. Forst [25]. Theorem 3.7.13 is due to W. Hoh [145], it will become rather important later on for the construction of Feller semigroups. Independently, P. Glowacki [105] made similar observations when introducing a type of Weyl calculus for certain pseudo-differential operators with non-smooth symbols.

244

Chapter 3

Fourier Analysis and Convolution Semigroups

Introducing the Laplace transform in 3.8 we followed partly the book of S. Mizohata [225] and t h a t of D.V. Widder [311]. We did not discuss the Laplace transform for distributions. This is done in the monograph [237] of B. Petersen, and of course by L. Schwartz in [280]. T h e discussion of completely monotone functions follows essentially the book of Chr. Berg and G. Forst [25], the proof of Bernstein's theorem, Theorem 3.8.13, is taken from D.V. Widder [313]. In discussing the Stieltjes transform we used besides the book of Chr. Berg and G. Forst the papers [132] and [133] of F. Hirsch. A nice introduction to the Laplace and Stieltjes transform is given by J.J. Hirschman and D.V. Widder [140]. Further results on Stieltjes transforms are given in the papers [21]-[22] of Chr. Berg. A readable paper on the representation and inversion of Laplace transforms is the paper [120] of B. Hennig and F. Neubrander. In the beginning of 3.9 we have taken once again larger parts from Chr. Berg and G. Forst [25]. As mentioned in the text, Theorem 3.9.20 is due to K. Harzallah [116], we followed the proof given by R.L. Schilling in [263]. This enables us also to give a short proof of Theorem 3.9.25 which is due to I.J. Schoenberg [275], see also J.-P. Kahane [176]. Our proof follows R.L. Schilling [265], another proof is given by P. Ressel [248], we refer also to [26]. T h e discussion of complete Bernstein functions is oriented at the work [263] of R.L. Schilling, in particular Theorem 3.9.29 is taken from there, we refer also to the work [22] of Chr. Berg and t h a t of F. Hirsch [134]. Complete Bernstein functions had been considered by K. Lowner [207] and E. Heinz [119] as operator monotone functions. They were also used very much by Chr. Berg, K. Boyadzhiev and R. de Laubenfels [23] and J. Priiss [244] in connection with operational calculi for generators of semigroups. We will come back to this point in the notes to the next chapter. Lemma 3.9.31 and Lemma 3.9.34 is taken from our paper [171] jointly written with R.L. Schilling. T h e Bernstein monoid was introduced by N. Bouleau and O. Chateau in [44], some results are also given in the monograph [48] by N. Bouleau and D. Lepingle. Recently, Th. Simon [283] proposed an interesting generalisation of Bochner's notation of subordination. In Volume 3 we will discuss the probabilistic counterpart of subordination. However, we refer already now to the monograph [27] and the survey [28] of J. Bertoin concerning these probabilistic aspects. Further we want to mention the monograph [33] of N.H. Bingham et.al. where connections to regular variations of functions are discussed.

3.13 Notes to Chapter 3

245

In 3.10 we followed our lecture notes [167], but of course, these results are essentially an exercise to what is worked out in the book [150] of L. Hormander, see also his book [153], The interpolation result, Theorem 3.10.16, is derived by using the presentation [262] of M. Schechter, see also his paper [261]. The calculation leading to Theorem 3.10.17 is borrowed from our joint paper [148] with W. Hoh. In 3.11 our source were the books [299]-[301] of H. Triebel. We refer to Chapter 3 in [1] were many historical comments to the theory of function spaces are given. The proof of the Calderon-Zygmund decomposition lemma and the proof of the Michlin-Hormander Fourier-multiplier theorem is taken from Chr. Sogge [284]. Note that in case of Fourier series the analogue was proved before by J. Marcinkiewicz [219]. Fourier-multipliers in Besov and Triebel-Lizorkin spaces are discussed in the monograph [300] of H. Triebel.

Chapter 4

One P a r a m e t e r Semigroups In this chapter we will discuss the general theory of one parameter semigroups of operators in Banach spaces. Our special interest is to present those parts of the theory being related to sub-Markovian semigroups on Coo(R n ;E) and on LP(R™;R), 1 < p < oo. After an introductionary section on the general theory and after having handled analytic semigroups we discuss subordination in the sense of Bochner in the context of operator semigroups. Then we present some important perturbation results. Section 4.5 to Section 4.7 are devoted to certain positivity preserving semigroups. More precisely, we determine the structure of generators of Feller semigroups and discuss sub-Markovian semigroups in V, 1 < p < oo. For p = 2 Hilbert space methods are available leading us for a first time to the theory of Dirichlet forms. In a last section we will treat the problem to extend certain semigroups and their generators to larger spaces.

4.1

Strongly Continuous Operator Semigroups

Let (X, \\.\\x) be a real or complex Banach space. Definition 4.1.1 A. A one parameter family (Tt) t > 0 of bounded linear operators Tt : X —> X is called a (one parameter) semigroup of operators, if T 0 = id and Ts+t = T s o T t hold for all s,t>0. B. We call ( T t ) t > 0 strongly continuous if lim||Ttu-u||x = 0 t-s-0

(4.1)

248

Chapter 4 One Parameter Semigroups

for all u £ l . C. The semigroup (Tt) t > 0 is called a contraction semigroup, if for allt>0 IITtll < 1

(4.2)

holds, i.e. if each of the operators T< is a contraction. As usual, ||Tt|| denotes the operator norm W^tWxX' Example 4.1.2 Let A : X —> X be a bounded linear operator and define °° 1 T t u := e t A := ^ - * f e A f c , t > 0.

(4.3)

First we find for t > 0 that

fc=0

'

k=0

and as in the finite dimensional case we find now e(-+t)A =

e ,A e tA

and

e 0A = i d _

Furthermore, we have the uniform continuity of the family (Tt) t > 0 as t tends to zero, i.e. lim ||e t A - idll = 0, t-x> implying that lim e*Au — u | | x = 0. Hence, (etA)t>0 t->o one parameter semigroup on (X, \\.\\x)-

is a strongly continuous

Example 4.1.3 Let (fit)t>o be a convolution semigroup on R™, see Definition 3.6.1. On the Banach space (C 0 0 (R n ), H-H^) we define the operator T 4 u(x) := /

u(x - y) fit(dy),

(4.4)

see (3.153). We claim that (Tt)t>0 is a strongly continuous contraction semigroup. First, since u € Coo(Rn) is bounded we find \Tt(x)\ < f

\u(x - y)\ inidy) < IMI^/xtCR").

4.1

249

Strongly Continuous Operator Semigroups

But fit(M.n) < lj which

implies

sup | T t u ( z ) | < H ^ ,

(4.5)

x€E"

i.e. Tt is defined on C 0 O (R") and T*u is a bounded function. But now it is n easy to see that T t u e Coo(R ). In fact, for u G <S(R") we / m d usin<7 the convolution theorem and Theorem 3.6.16 that ( T t u ) A ( 0 = (27r)"/ 2 u(OAt(0 = u ( O e - t l / , ( ? ) ,

(4.6)

where ip : R " —>• C is a continuous negative definite function. But (4.6) implies that (Ttu) e L 1 ( R n ) / o r u € <S(R n ), and the Riemann-Lebesgue lemma, Theorem 3.2.1, implies T*u € C 0 0 ( R n ) . Thus, by (4.5) we find using the density o/«S(R n ) in C 0 o(R Tl ) that Tt is a contraction on C00(Wn^. From the definition of the convolution of measures, Definition 2.3.5, we find Ts o T t u ( z ) =

/

I

u(x - z - y) fit(dy)

=

/

\i(x-

z) (fit *

=

/

u(x - z)

=

\

fis(dz

fis)(dz)

fit+s(dz)

Ts+tu(x).

Since fio = EQ, we have immediately To = id. Finally, we prove that ( T t ) t > 0 is strongly continuous for t — • 0. For this note that any function in Coo(R") is uniformly continuous. Hence, for s > 0 there exists 6 > 0 such that \u(x) — u(x — y)\ < e for \y\ < S. The continuity

of (fit)t>o in the Bernoulli

\im fit(B5(0)) t-»o

= £0(Bs(0))

topology implies

that

= 1,

which gives fit(Bcs(0))

<e

and 1 - fit(Rn)

< e

(4.7)

250

Chapter 4 One Parameter Semigroups

for 0 < t < to. Now we find |T«u(a;)-u(a:)| < /

{u(x - y) - u(x)} fxt{dy) + | u ( x ) | ( l - ^ ( R n ) )

< /

\\i(x - y) - u(x)\ fit(dy)

JBS(O)

+ f

\u(x -y)-

u(x)\ ^(dj,) + 1^11^(1 -

Mt (R"))

JBS(O) -1(0)

< e + 2 e | | u | | 0 O + e | H | o o = e(l + 3 | | u | U , implying that (Tt)t>0 is strongly continuous as t —> 0. Note that Tt, t > 0, is positivity preserving, i.e. u > 0 yields Ttu > 0. Definition 4.1.4 Let (Tt) t > 0 be a strongly continuous contraction semigroup on (C 0 0 (R";R), H-H^) which is positivity preserving. Then ( T t ) t > 0 is called a Feller semigroup. Since for u £ C 0o (]R n ;R) the function denned by (4.4) is real-valued, it follows that all semigroups from Example 4.1.3 are Feller semigroups. Unfortunately, in the literature there are many different notions of a Feller semigroup, see [251], p.241, for related remarks. Example 4.1.5 Let (^t)t>o be as in Example 4.1.3 a convolution semigroup on R n . For u € <S(R") we define as before Ttu(x) = /

u(x - y) fitidy)

(4.8)

and find with a continuous negative definite function tp : R™ —>• C that (T t u) A (0 = e - ^ K ) i i ( 0 -

(4-9)

Now, Plancherel's theorem, see Corollary 3.2.17, implies ||T t u|| 0 = ||(T t u) A || 0 <||u|| 0 ,

(4.10)

where ||.||0 denotes the norm in L 2 (R n ). Since <S(Rn) is dense in L 2 (R"), it follows that each of the operators Tt has an extension to L 2 (R") and that this

4.1 Strongly Continuous Operator Semigroups

251

extension is a contraction. We denote this extension once again by Tt, t > 0. Moreover, we find f

l|2

l|Ttu • u

llo = /

e-^«>u(0-fi(0' 2 d£ e-tm

_ i u(0! 2 ^,

-j

implying the strong continuity of (Tt) t > 0 as t tends to 0. From (4.9) it is obvious that (Tt)t>o *s a semigroup. Hence, (T() t > 0 gives a strongly continuous contraction semigroup onL2(Rn). Letu G L 2 (R n )~nL°°(R"). Then (4.8) makes sense as a Lebesgue integral and we find that for 0 < u < 1 almost everywhere it follows that 0 < Ttu < 1 almost everywhere. As before, let us note that by (4.8) the operator Tt maps real-valued functions onto real-valued functions. Definition 4.1.6 A. Let (Tt) t > 0 be a strongly continuous contraction semigroup on L P (R"; R), 1 < p < oo. We call (Tt) t > 0 a sub-Markovian semigroup on IP, 1 < p < oo, if for u € L p (R n ;R) such that 0 < u < 1 almost everywhere it follows that 0 < Ttu < 1 almost everywhere. B. Let (Tt) t > 0 be a strongly continuous contraction semigroup on L P (R"), 1 < p < oo, or on C 0 0 (R"). We call (T t ) t > 0 symmetric if for all u,v <E L P (R") n L 2 (R") or u,v e Coo(R") n L 2 ( R " ) , respectively, we have (T t u,v) 0 = (u,Ttv) 0 .

(4.11)

Note that by the very definition a sub-Markovian semigroup on L p is always strongly continuous. All examples given in 4.1.5 are sub-Markovian semigroups on L 2 (R n ;R). Let ip : R n —> R be a real-valued continuous negative definite function. Further, let {nt)t>o be the corresponding convolution semigroup on R™ such that At(0 = (27r)-"/ 2 e-^(«. From (4.9) and Plancherel's theorem we deduce that in this case the semigroup denned by (4.8) is symmetric on L 2 (R"). In fact we have for u, v e L 2 (R") (Ttu,v)0= /

JRn

= I

e

- ^ > u ( 0 v ( 0 d£ u(0(e-'V>(Ov(0)d£=(u,T t v) 0 .

252

Chapter 4 One Parameter Semigroups

But t h e same calculation gives t h a t for a symmetric semigroup on L 2 ( K n ) given by (4.8) t h e function ip must be real-valued. Now let us return to t h e general theory of semigroups. L e m m a 4 . 1 . 7 Let ( T t ) t > 0 be a strongly continuous semigroup Then there exist constants w > 0 and Mw > 1 such that

on

(X,\\.\\x).

||T t || < Mwewt,

(4.12)

where once again ||.|| denotes the operator norm Proof: ||Tt|| < t h a t tn in form

\\-\\xX'

First, note t h a t there exists a constant M > 1 and to > 0 such t h a t M for all t € [0,£o]- Otherwise we could find a sequence ( i „ ) n e N such —> 0 and ||Tt n || —> oo. However, t h e uniform boundedness principle of Corollary 2.7.6 then yields t h a t sup | | T t „ u | | x = oo for some u e X. nGN

But since (T t )t>o is strongly continuous, it follows t h a t {||T t || |0 < t < t0} is bounded. Now take w := t^1 I n M and t > 0. T h e n there exists k G N a n d s, 0 < s < to, such t h a t t = kto + s and we get ||T t || = ||T s T t fc 0 || < MMk

< MMt/t0

= Mewt.

D e f i n i t i o n 4 . 1 . 8 Let ( T t ) t > 0 be a strongly continuous (or the growth bound) of ( T t ) t > 0 is defined by

D

semigroup.

w0 : = wo ( ( T t ) t > 0 ) : = inf {w e R | ||T t || < Mwewt} where \\Tt\\ < Mwewt

The type

,

is to hold for some Mw > 1 and all

t>0.

C o r o l l a r y 4 . 1 . 9 Let ( T t ) t > 0 be a strongly continuous semigroup on (X, For any u G X the mapping t H-^ T t u is continuous from [0, oo) to X. Proof:

For u € l , t > 0 and h > 0 we find using Lemma 4.1.7 t h a t

||T t + f c u - Ttulljr = ||T t (T f c u - u)\\x < Mewt\\Thu

-

u\\x

and for 0 < h < t it follows t h a t ||T t _ f c u - T t u | | x = ||T t _ f c (T h u - u ) | | x < Me^HTfcU - u | | x implying t h e corollary. D

\\.\\x)•

4.1

253

Strongly Continuous Operator Semigroups

R e m a r k 4 . 1 . 1 0 Let ( T t ) t > 0 be a strongly continuous such that (4.12) holds. Define ( S t ) t > 0 by Stu = e - u r t T t ,

i>0.

\\.\\x)

(4.13)

Since S t + S u = e-w(t+s^Tt+su = e-wtTt(e-wsTsu) semigroup on X which is strongly continuous and ||St||<M

semigroup on (X,

= S t o S s u, ( S t ) t > 0 is a satisfies

fort>0.

In particular, for M = 1, the semigroup ( S t ) t > 0 is a strongly continuous traction semigroup on X.

con-

A central notion in the theory of one parameter semigroups of operators is t h a t of the generator. D e f i n i t i o n 4 . 1 . 1 1 Let ( T t ) t > 0 be a strongly continuous semigroup of operators on a Banach space (X, \\.\\x). The generator A of ( T t ) t > 0 is defined by A.u := lim t->o with

(strong limit)

(4-14)

t

domain

D(A):=<JueX

lim t-+o

t

exists as strong limit > . J

(4-15)

Obviously, D(A) is a linear subspace of X. Note t h a t it is important to have the strong limit (4.14) and not the uniform limit. In the latter case it would follow t h a t A is a bounded operator. A proof of this result can be found in A. Pazy [235], p.2. E x a m p l e 4 . 1 . 1 2 Let tp : K™ —» C be a continuous negative definite function with corresponding convolution semigroup (fJ,t)t>o, i-e. jlt(0—('^7T)~n^2e~t^'^• Moreover, let ( T t ) t > 0 be the (Feller) semigroup defined by (4.4). Let u £ «S(R n ), note S(Rn) c Coo(]R n ). It follows that Ttu-u /n_,_n/2 f = (27T)-/2 / t

.fa.fe-**«)-lA e»-« i f i ( 0 dC

254

Chapter 4 One Parameter Semigroups

Since u G <S(R") onrf 6?/ lemma 3.6.22 |^(0I
e ^ t y t f W O d£.

(4.16)

We c/aim i/iaf <S(R") c D(A(°°>) and A<°°)u = - ^ ( D ) u , where A(°°) is i/ie generator of the (Feller) semigroup ( T t ) t > 0 . E/sm

which gives Ttu-u + V»(D)u t


(l + |£| 2 ) 2 |u(£)|d£,

which implies Tfu-u -V»(D)u. t t-yO In general it is not possible to characterise D(A°°) completely in terms of function spaces. However, for the (sub-Markovian) semigroup on L 2 (R n ) related to (/zt)t>o this is sometimes possible. lim

Example 4.1.13 Let (/xt)t>o and ip : R n ->• C be as in Example 4.1.12. The operator semigroup (Tt) t > 0 related to (/it)t>o by (4.4) is now considered as a (sub-Markovian) semigroup on L 2 (R n ), see Example 4.1.5. Using Plancherel's theorem, Corollary 3.2.17, we find for u G <S(R")

Ttu-u t

e-H(i)

+ ip(D)u 0

JR*

< t24

/

JRn

_ i

\mf<%

(1 + |C| 2 ) 4 |u(0| 2 d£ = t 2 c 2 | | u | | ^ (4.17)

which implies that lim

Tt U

7 +^(D)u

t->0

for all u G «S(Rn). Denoting the generator of the (sub-Markovian) semigroup in L 2 (R") by A^, we have shown that <S(R") C D(A^2)) and on <S(Rn) we have A(2)u = -V'(D)u.

255

4.1 Strongly Continuous Operator Semigroups

We will prove in Example 4.1.16 that D(A<2)) = fl^^R") holds. However, to do so we need further results from the general theory of semigroups. In the following we use frequently Lemma 2.3.24. Lemma 4.1.14 Let (Tt)t>o be a strongly continuous semigroup on the Banach space (X, \\.\\x) and denote by A its generator with domain D(A) C X. A. For any u G X and t > 0 it follows that fQ T 5 u ds G D(A) and T t u - u = A [ Tsuds. (4.18) Jo B. For u G D(A) and t > 0 we have T t u G D(A), i.e. D(A) is invariant under Tt, and ^ T t u = AT t u = T t Au. at C. For u G D(A) and t > 0 we always get

(4.19)

T t u - u = / AT 5 u ds= J T s Au ds. Jo Jo Proof: A. Let h > 0. It follows that

(4.20)

^ (T h - id) f T s u ds - i f ( T , + h u - T s u) ds

(4.21)

=s{ J T T - uds ~ J (' T ' uds } i

= -/

pt+h

-i

Tsuds--/

rh

Tsuds.

For h —> 0, the right hand side in (4.21) converges in X to T 4 u — u, which yields (4.18). B. For h > 0 let Ah := ±(Th - id). It follows that ^(T t + f c u - T t u) = A ft T t u = T t A h u, hence, T t u G D(A) for u G D(A). Furthermore, we have ^ T t u = AT t u = T t Au,

(4.22)

256

Chapter 4 One Parameter Semigroups

where ^ - denotes the derivative from the right hand side. But for 0 < h < t we find - i ( T t _ f c u - T t u) - T t Au = Tt_fc(Afc - A)u + (T t _ fc - T t )Au,

(4.23)

which gives for h —> 0 that ^ T t u = T t Au, where ^ - denotes the derivative from the left hand side, hence B is proved. C. Equality (4.20) follows immediately from part B and Lemma 2.3.24.5.

• Corollary 4.1.15 Let A be the generator of a strongly continuous (T()t>o on the Banach space (X,\\.\\x). Then D(A) C X is a space and A is a closed operator. Moreover, (Tt)t>o is a strongly semigroup on D(A) when D(A) is equipped with the graph norm ||Au||^ + ||u|| x .

semigroup dense subcontinuous \\u-\\AX =

Proof: Since rt

lim i r im - lJo/ T s u ds = u 14.0 tio

tj0

for all u e -X", it follows from Lemma 4.1.14.^1 that D(A) is a dense subspace of X. We prove the closedness of A using Lemma 2.3.24. Hence, for u„ G D(A) suppose that u„ —> u and Au„ —> g in X. It follows from Lemma 4.1.14. C that T t u „ - "U„ == / T s A Uj/ ds Jo for all t > 0,, and for v —> oc• weget Ttu- u =

f Tsgds,

or 7 ( T t u - u ) = j / T 5 gd S . t t J0

(4.24)

4.1

257

Strongly Continuous Operator Semigroups

But as t —>• 0 we find t h a t the limit on the right hand side exists and equals g, thus u G D(A) and Au = g, implying t h a t A is closed. Next we prove t h a t m(A),||.||AjAJ.

(T t )t>o is a strongly continuous semigroup on

Obviously, ( T t | D ( A ) ) t > 0 has the semigroup property since

T t D ( A ) C D(A). To prove t h a t ( T t ) t > 0 is strongly continuous with respect to

II -IIA x n °te that ||T t u - u | | A j X = ||AT t u - Axx\\x + ||T t u -

u\\x

= ||T4Au-Au||x + ||Ttu-u||x, which implies the assertion.

•

Now we determine the domain D(A^ 2 )) of the generator of the ( s u b Markovian) semigroup (T t )t>o on L 2 ( R n ) associated with a convolution semigroup (nt)t>o on R n which is characterised by the continuous negative definite function ip : K™ -> C. E x a m p l e 4 . 1 . 1 6 We claim that the operator —ip(D) defined on <S(R") by (4.16) is closable on L 2 ( R " ) . For this let us consider a sequence ( u i / ) I / 6 N , u„ G <S(R"), converging in L 2 ( R n ) to zero. Moreover, assume that (— , i/'(D)ut / ) l/eN converges in L 2 ( R n ) to some element —g G L 2 ( R " ) . We have to show that —g = 0. For this take 6 G <S(R") and note that /

V(D)u„da: = /

u„ip(D)
which gives (g, G iS(R n ), thus —ip(D) is closable. Next we show that the norm ||.|| , x on <S(Rn) is equivalent to the graph norm ||u|L/D-> L 2 = ||u|| 0 + ||V>(D)u||0. Recall that

\Hl,i=

[ (l + |V(OI)2|
(4.25)

Thus we have

which clearly implies that ||.||^ : is equivalent to ||.|L( D \ L 2 - By Theorem 2.7.1 it follows that the domain of the closure A of —ip(D) is given by D(A) = 5(R J 0 I I '"* ( D ) - L a = 5 ( I S ) I M I , M =

H^\Rn).

Chapter 4 One Parameter Semigroups

258

Besides the generator A we may associate a further object, the resolvent, with a strongly continuous contraction semigroup. The resolvent set p(A) for a closed operator was defined in Definition 2.7.24. Definition 4.1.17 Let A be a closed operator on the Banach space (X, \\.\\x) with domain D(A) C X. The resolvent of A is the family (RA) A € „(AV ^ : = (A — A ) - 1 . The operator RA is called the resolvent of A at A. Lemma 4.1.18 Let (Tt)t>o be a strongly continuous contraction semigroup on the Banach space (X, \\.\\x) w^h generator (A, D (A)). Then {AGC | Re A > 0} C p(A) and we have /»oo

R A u = (A - A)- X u = / Jo for all u G X and Re A > 0.

e~ At T t u di

(4.26)

Proof: Let Re A > 0 and denote by UA the operator U A g:= / e-xtTtgdt. Jo

(4.27)

Since f°° 1 ||UAg||x 0 w e find further 1

1 r°°

-(Th - id)UAg = -J

e-xt(Tt+hg

„Xh _ 1

/-oo

- Ttg) di Xh rh

which implies for h —> 0 that UAg G D(A) and AUAg = AUAg — g. Hence, for all g G X we get (A - A)UAg = g.

(4.29)

Furthermore, for g G D(A) it follows by Lemma 2.3.24 that /•oo

UAAg= / Jo

/«oo

e-xtTtAgdt= xt

Jo

e- Ttgdt

Jo = AXJxg,

A(e-xtTtg)dt

4.1 Strongly Continuous Operator Semigroups

259

and by (4.29) we find (4.30)

U A ( A - A ) g = g.

But (4.30) says that A—A is injective and from (4.29) it follows that R(A—A) = X. Therefore, we have UA = (A - A)" 1 and A e p{k). • Remark 4.1.19 Note that it is possible to define the resolvent (R-A)^>O (or A 6 p{A)) for a (strongly continuous) contraction semigroup directly by /*oo

RAu= / Jo

e- A t T t udt,

(4.31)

i.e. without using its generator. For u £ D(A) we find now for A > 0 that |Au-

A2R^U +

Aul

x

r

e - A t A 2 ( u - T t u ) d t + Au X

U-T4LU

ds + Au

— Tiu

e

jf "-'(Au + u -— T i i

Jo

ds

Au+

e

ds,

s

x

which implies that in the sense of the strong limit we have for all u £ D(A) lim (-Au + A2RAu) = Au. A—•oo

Lemma 4.1.20 Let A be a closed operator. For X,p£ equation RARM =

R M RA

= (A -

M)_1(R^

-

RA)

p(A) the resolvent

(4.32)

holds. Proof: Since (A - A)(/x - A) = (p - A) (A - A) we find for A, p e p(A) that (p - A)-\X

- A ) - 1 = (A - A ) " 1 ^ - A)" 1 .

260

Chapter 4 One Parameter Semigroups

Furthermore, we have Rx - RM = Rx(v - A)RM - RA(A - A)RM = -R A AR M + fiRxR,, + RAARM - ARAR^ = {n - A)RARM, which yields (4.32). • Now, (4.32) implies that RA = (id -fa-

A)RA)RM

and for \/i — A| ||RA|| < 1 we get

RAi = ( i d - ( / i - A ) R A ) - 1 R A or oo

^ = E^- A ) J R i + 1 -

(4-33)

3=0

The series on the right hand side of (4.33) is called Neumann series. From (4.33) we deduce immediately Corollary 4.1.21 The resolvent set p(A) C R is an open set. Example 4.1.22 Let (/Xt)t>o be a convolution semigroup on R™ with associated continuous negative definite function ip : R™ -4- C, i.e. (H(0 = (27r) - ' 1 ' 2 2 e-tip(Z)_ £e£ (Tt)t>o denote the (sub-Markovian) semigroup on L (R") associated with (fJ.t)t>o- Thus Ttu(x) = f

u(x - y) 0t(dy) = (2TT)-"/ 2 /

e ^ e - ' ^ u t f ) d£

for u G <S(R"). For A > 0 and u e <S(Rn) we find POO

RAU(X)

= /

e-xtTtu{x)

di

= (2TT)-™/2 f°° e~xt [

e fa «e-**^u(^) d<£ di.

4.1

261

Strongly Continuous Operator Semigroups H* e - ( A + ^ ) ) ' u ( £ ) e " £ is in ^ ( ( O . o o ) x Rn)

Since (t,£) theorem Rxu(x)

= (27T)-™/2 f

i

r

we have by

e - ( A + ^ « » ' dt j e " « u ( 0 d£

= W~n/ L^XTW)^)^ We claim the

Fubini's

(4-34)

estimate

IIRAUII^J < c||u|| 0 .

(4.35)

For u G <S(Rn) we / m d since Re -0(0 > 0 f/iaf I|RAU||^=

/

(i + |^(OI) 2 |(RAu) A (orde

=1 (1+ ^ l)J raF |fi(0|J * - K 1+ *0 l|u|l°' implying (4.35). Thus we may consider (4.34) by an appropriate interpretation as an expression for R\ on L 2 ( R " ) . In particular, we have by the convolution theorem

R

^ = F'ixT^rf) = ^-n/^^'

where i\

:= F

1

But

by

Corollary 3.6.13 x+\, ^ is a continuous positive definite function, implying Bochner's theorem, Theorem 3.5.7, that ix € iS'(R n ) is already a measure Thus we have

by p\.

Rxu(x)

( x+\/

= (2n)-n/2

\ ) which makes

f

sense at least in <S'(R").

u(x - y) px(dy).

(4.36)

Moreover, when we consider the Feller semigroup associated with (nt)t>o> for any u G <S(Rn) formula (4.34) remains valid and we get also (4.36). Now, we come to a central notion in the theory of operator semigroups. D e f i n i t i o n 4 . 1 . 2 3 A linear operator A : D(A) -» X, D(A) c X, dissipative, more precisely X-dissipative, if

IIAu-AuH* >A||u|| x holds for all A > 0 and u G D(A).

is called

(4.37)

262

Chapter 4 One Parameter Semigroups

Example 4.1.24 Let q^ = q,-j e C ^ G ) , G C R n , be real-valued functions such that for all x e G and ( g R n n

Yl tbjW&tj ^ °

(4-38)

holds. Further let L

(^D)=E^:(%W^-)

(4-39)

he a second order differential operator. We claim that on L 2 (G) with domain CQ°(G) this operator is dissipative. For this let A > 0 and u € C Q ° ( G ) . It follows that ||Au - L(x,D)u||g = A2||u||* - 2A(u,L( a ;,D)u) 0 + HL^.DJuHg

> A 2 IIUJ!;, Zience rue have ||Au-L(:r,D)u|| 0 >A||u|| 0 . Example 4.1.25 Let tp : R" —> C be a continuous negative definite function. On S(R n ) C L 2 (R") we define the operator ^(D)u(a:) =

(2TT)-"/ 2

f

e^tPdMW,

(4.40)

and claim that —ip(D) is L2(R")-dissipative on <S(Rn). Indeed, for A > 0 and u € <S(R") we ftaue ||Au + V(D)u||2 = A2||u||^ + 2A / >A 2 ||u||^ implying the L2 -dissipativity of —ip(D).

Re V ( 0 | u ( 0 | 2 ^ + ||V(D)u|

4.1

263

Strongly Continuous Operator Semigroups

L e m m a 4 . 1 . 2 6 Let (A, D (A)) be a dissipative operator on X and A > 0. The operator A is closed if and only if the range R(A — A) is closed. Proof:

Suppose t h a t A is closed. For ( u „ ) „ 6 N , u„ G D(A) such t h a t lim (A u—foo

A)u„ = h it follows by dissipativity, see (4.37), t h a t (u [ / ) 1 / € N is a Cauchy sequence in X. Hence, there exists u G X such t h a t u„ —> u as v —> oo, which implies Au„ —> Au — h as v —> oo. Since A is closed, it follows t h a t u £ D(A) and h = (A - A)u, i.e. R(A — A) is closed. Now assume t h a t R(A — A) is closed and take a sequence ( u i / ) ^ € N , u„ G D(A) such t h a t u„ —> u and Au,/ — • g as u —> oo. We find t h a t (A — A)u„ —> Au — g and the closedness of R(A — A) implies t h a t Au — g = (A — A)uo for some uo G D(A). T h e dissipativity of A gives now ||A(u„ - u 0 ) - A(u„ - u 0 ) | | x > A||u„ -

u0\\x,

hence, uu —> uo as n —> oo. But now we deduce t h a t u = uo G D(A) and Au = g, i.e. A is closed. • Furthermore, we have L e m m a 4 . 1 . 2 7 Let (A, D ( A ) ) be a dissipative and closed linear operator on the Banach space (X, \\.\\x). Further, set p+(A) = p ( A ) n ( 0 , oo). If p+(A) ^ 0, then p+{A) = (0, oo). R e m a r k 4 . 1 . 2 8 Recall that A G p(A) if and only if X — A is injective, R(A — A) = X and (A — A ) - 1 is bounded. In particular, A e p(A) implies that the equation Au - Au = f

(4.41)

is uniquely solvable for all f € X. Now, Lemma 4.1.27 says that if for a closed and dissipative operator A and one A > 0 equation (4.41) is uniquely solvable for all f G X, then for all A > 0 it is uniquely solvable for all f G X. P r o o f of L e m m a 4.1.27: We show t h a t p+(A) is an open and closed subset of (0, oo) which will imply the assertion. Since p(A) is always an open set in R„ see Corollary 4.1.21, it follows immediately t h a t p+(A) is open in (0, oo). Now, let (A^)^ 6 N , A„ G p+(A), be a sequence such t h a t A„ — • A, A > 0, and take g G X. P u t t i n g g„ := (A - A)(A„ - A ) _ 1 g , i.e. g„ G R(A - A), we have g„ - g = (A - A)(A„ - A ) " ^ - (A, - A)(A„ - A ) " ^ = (A-A,)(A,-A)-1g,

264

Chapter 4

One Parameter Semigroups

leading to

lis, - e\\x = IKA - A *)( A - - A ) _ 1 g | L ^ lA - A -l IKA- - A ) " x g | L T h e dissipativity of A gives further

\\(\„-A)-h\\x
l|g, - g|| x < ^ ^ l l g l l x —> 0 as A, - » A. Thus R(A — A) is dense in X. But since A is closed and dissipative, by Lemma 4.1.26, it follows that R ( A - A ) is closed, hence R(A —A) = X. Furthermore, the dissipativity of A implies t h a t A — A is injective and II (A — A ) - 1 II < A - 1 . For this reason A G p+(A) and therefore p+(A) is closed in (0, oo), implying the lemma. • Our aim is to construct a strongly continuous contraction semigroup (Tt)t>o from a given operator A which should t u r n out to be the generator of (Tt)t>o- For bounded operators we may take just the exponential function of A, see Example 4.1.2. T h e idea is to approximate a more general operator by a family of bounded operators and pass to the limit of the semigroups generated by these operators. For this reason we introduce the Yosida approximation of A. T h e o r e m 4 . 1 . 2 9 Let A be a closed and dissipative operator which is densely defined on a Banach space (X,\\.\\x). We assume that (0, oo) C /o(A). The Yosida approximation of A is defined for A > 0 by A A = AA(A - A ) " 1 = AAR A . It has the following

properties

1. For all A > 0 the operator A\ is bounded on X and the (etAj>l) is a strongly continuous contraction semigroup. 2. For all X,fi>

(4.42)

0 we have

A A A^ — A^A A .

semigroup

265

4.1 Strongly Continuous Operator Semigroups 3. For u e D(A) it follows that lim | | A A u - A u | | A . = 0 . A—>oo

Proof: Let A > 0. Since (A - A)RA = id on X, i.e. ARA = ARA - id and RA(A - A) = id on D(A), i.e. XR\ - id = RAA, we get AA = A2RA - Aid on X

(4.43)

AA = ARA A onD(A).

(4.44)

and

From (4.43) we deduce the boundedness of AA, A > 0, and „tA A

,-*A

,*A 2 R A

< e -'V A2 ll R *H < 1,

(4.45)

for all t > 0 since the dissipativity of A implies ||RA|| < j for all A > 0. To prove 2. we use (4.43), RAR^ = R^RA and the resolvent equation (4.32) AAAM = (A2RA - A id) (/x2RM - /x id) = A V R A R M - A/i2RM - /iA2RA + A/x id = A V R ^ R A - /iA2RA - A^2RM + Xfj. id = (M 2 R M -Mid)(A 2 R A -Aid) = AMAA. To prove 3. we show first lim | | A R A U - U | | X = 0 A-»0

for all u e

(4.46)

X.

Since ARA - id = ARA - RA(A - A) = RAA we find for u € D(A) ||ARAu - u | | x = ||R A AU|| X < A"11| Au|| x —> 0 as A —• oo. Now, let u € X be arbitrary. Then there exists a sequence (u^) J/£N , u„ G D(A), such that Uj, —>• u as u —> oo. Furthermore we have ARAU - u = ARA(U - u„) + ARAU„ - (u - u„) -

u^,

266

Chapter 4 One Parameter Semigroups

which implies ||ARAu-u||x < ||(ARA-id)(u-ul,)||x + ||ARAuI/-u^||x < ||(ARA - id)|| x ||(u - u „ ) | | x + ||ARAu„ - u„\\x < 2||(u - \i„)\\x + ||ARAu„ - u „ | | x , where we used ||ARA - id|| < A||RA|| + ||id|| < A^ + 1. Now, we get (4.46) when we first let A tend to oo and then v to oo. But (4.46) and (4.44) imply 3. D Example 4.1.30 Let (/xt)t>o be a convolution semigroup on R™ with associated continuous negative definite function ip • R n —> C and consider the strongly continuous semigroup (T t ) t >o induced by (/it)t>o on L 2 (R n ). Combining Example 4.1.13 with Example 4.1.16, we know that the generator A of (Tt)t>o is the operator —ip(D) with domain H'^'1(Mn). Moreover, from Example 4.1.22 we find that the resolvent (R\)\>0 of (Tt)t>o consists of operators R\ given by (4.34), thus we have Au(z) = -V(D)u(x) = -(27r)-"/ 2 /

e i x ^ ( 0 u ( £ ) d£

( 4 - 47 )

and

RAu(x) = Wn/2l

** A T W " ( 0 *'

(4 48)

-

where at least on <S(Rn) the integrals are well defined Lebesgue integrals. The Yosida approximation of A = —ip(D) is given by A\ = AARA and for u 6 <S(R") we find

= -(27r)-"/ 2 f

e^MtMQ

<&

which defines of course by Plancherel's theorem a bounded operator on L 2 (R"). By Corollary 3.6.14, the function

A + <M0

4.1

267

Strongly Continuous Operator Semigroups

is once again a continuous negative definite function and therefore the semigroup ( T ^ ) , > 0 on L 2 ( R " ) generated by the bounded operator A\ is associated with a convolution semigroup (/•**) t>0 on ^-n- Clearly, this is nothing but the convolution semigroup obtained from (/it)t>o by subordinating with respect to the Bernstein function i\(x) = -£n£, see Example 3.9.23. For A —> oo we have t})\(£) —>• V'CO- Therefore, it is reasonable to ask whether in some sense TtA —>T t as X — • oo. We need L e m m a 4 . 1 . 3 1 Let A, B be bounded, linear operators (X,\\.\\x) such that AB = BA, | | e t A | | < 1 and \\etB\\ Then we have for all u € X and t>0 that

on a Banach space < 1 for all t > 0.

||e t A u - e t B u | | x < t||Au - B u | | x . Proof:

(4.49)

For u € X and t > 0 we have

e t A u - e t B u=

rt

/ Afe'V^juds

= /

(4.50)

esA(A-B)e(t-^Buds

= / esV*-s)B(A-B)uds, Jo where we need only in the last step t h a t AB = BA. But now (4.49) follows immediately. • R e m a r k 4 . 1 . 3 2 Using Lemma 4.1.14 we obtain from the calculation in the proof of Lemma 4.1.31 that whenever A andli are generators of strongly continuous contraction semigroups ( T A ) t > and ( T B ) , respectively, on (X, | | - | | ^ ) , the equality T A u - T f u = / T A o (A - B) o T ? _ s u ds Jo holds for all u € D(A) fl D(B). In addition, on X, then (4.51) holds for all u £ X.

when A — B is a bounded

(4.51) operator

268

Chapter 4 One Parameter Semigroups

Now, we can characterise all generators of strongly continuous contraction semigroups. Theorem 4.1.33 (Hille and Yosida) A linear operator (A, D (A)) on a Banach space (X, \\.\\x) is the generator of a strongly continuous contraction semigroup (Tt)t>o if o,nd only if the following three conditions hold 1. D(A) C X is dense; 2. A is a dissipative operator; 3. R(A - A) = X for some A > 0. Proof: We prove first that 1.-3. are necessary conditions. For this let A be a generator of a strongly continuous contraction semigroup. From Corollary 4.1.15 it follows that D(A) is dense in X. Furthermore, using Lemma 4.1.18 we find for A > 0 and u G D(A) that ||u|| x = | | ( A - A ) - 1 ( A - A ) u | | x =

/

e - A t T t ( A - A ) u dt

Jo

r°° < Ja

i e-xt\\(X - A)u|| x dt = -||(A - A)u|| X)

implying that A is dissipative. Finally, we have A G p(A) for any A > 0, once again by Lemma 4.1.18, hence 3. is proved. Now we prove that 1.-3. are sufficient that A is the generator of a strongly continuous contraction semigroup. From Lemma 4.1.26 we know that A is a closed operator and since p(A)(l(0, oo) ^ 0 by 3., it follows from Lemma 4.1.27 that (0,oo) c p{A). Denote by A\ = AA(A - A ) - 1 the Yosida approximation of A and let (T^) > 0 , TA = etAx, be the strongly continuous contraction semigroup on (X, ||.|[ x ) generated by A^. Due to Theorem 4.1.29 part 2. we may apply Lemma 4.1.31 to find ||TtAu - T ? u | | x < *||AAU - A M u|| x

(4.52)

for all u £ X, t > 0 and A, /J. > 0. Furthermore, part 3. of Theorem 4.1.29 implies that lim T^u exists (as a strong limit) for all t > 0 and this limit is A—>oo

uniform with respect to t on compact intervals for all u G D(A), hence for all

4.1 Strongly Continuous Operator Semigroups

269

u € D(A) = X this limit exists. For u E l w e denote this limit by T t u, t > 0. Since T s + t u - T , T t u = T s + t u - T* +t u+T*(T t A u - T t u)+(T A - T,)T t u, (4.53) it follows by passing to the limit A —> oo that (T t )t>o is a semigroup. Moreover, the equality T t u - u = T t u - T A u + TtAu - u implies that (T t )t>o is strongly continuous, hence, since ||T A || < 1, (T t )t>o is a strongly continuous contraction semigroup. It remains to prove that A is a generator of (Tt)t>o- By Lemma 4.1.14.Cwe have for all u € X, t > 0 and A>0 />OC L

t

!iu-u = / Jo

T A A A uds,

(4.54)

and for u € D(A) and t > 0 we find T^A A u - T s Au = T^(A A u - Au) + (TA - T s )Au.

(4.55)

Thus, Theorem 4.1.29 part 3. gives that ||T*AAu - T s A u | | x —• 0 as A —> oo uniformly for 0 < s < t, and (4.54) implies for A —> oo that Ttu-u=

/ T s Auds Jo

(4.56)

for all u G D(A) and t > 0. Hence, the generator A of (T t )t>o must be an extension of A. For any A > 0 the operator A — A is injective since A must be dissipative by 2. Moreover, R(A - A) = X since A e p(A), and it follows that A = A proving the theorem. • We stated the theorem in a form due to R.S. Phillips [241]. The original version of the Hille-Yosida theorem uses direct conditions on the resolvent of A, namely that (0, oo) C p(A) and that ||RA|| < j for all A > 0. Moreover, this version of the Hille-Yosida theorem generalises to non-contractive, strongly continuous semigroups in the following manner. Suppose that (A, D(A)) is a closed densely defined operator in the Banach space (X, \\.\\x) such that (0, oo) C /9(A) and ||(R A )"|| < # for A > 0 and v e N. Then A is the generator

270

Chapter 4 One Parameter Semigroups

of a strongly continuous semigroup satisfying ||T t || < M. Conversely, the generator of a strongly continuous semigroup with ||Tt|| < M is a densely defined, closed operator (A, D (A)) with (0,oo) c p(A) and it satisfies || (R-A)1' |[ < jffor A > 0. Since we are mainly concerned with contraction semigroups we omit the proof of this result. It can be found in the book of A. Pazy [235], p. 19. Note that from the closed range theorem, Theorem 2.7.21, it follows that if A* is injective, then R(A — A) = X, hence, the injectivity of A* implies for a densely defined closed and dissipative operator that it is the generator of a strongly continuous contraction semigroup. In particular, when A* is dissipative too, A, being closed and dissipative as before, generates a strongly continuous contraction semigroup. In order to prove that a strongly continuous contraction semigroup is uniquely determined by its generator we need Lemma 4.1.34 Let A be a linear dissipative operator on a Banach space (X,\\.\\x). Moreover, let u € C([0,oo),X) such that for all t > 0 we have u(t) € D(A), Au € C([0, oo), X) and for allt>£>0

[M

u(t) = u(e) + / Au(s) ds.

(4.57)

Then we have the estimate \\u(t)\\x < 1^(0)11^- for all t > 0. Proof: Let 0 < e = t0 < h < ... < tn = t. It follows that n

HuWII* = Me)\\x + X) (MWWx - K
= He)\\x + X (HWWx - Htj) - (*i - *i-i)Au(tj)||x) n

+ X(l|ufe)-(i J -^-i)Au(i j )|| x 3=1

-HtiJ-U^J-U^--!)^). Since A is dissipative we have for all A > 0 and v € D(A) that \\x

v - -Av A

X

4.1

271

Strongly Continuous Operator Semigroups

hence n

E (llufe)Hx - Hufe) - & - *;-i)Aufo)llx) < 0. 3= 1

Using (4.57) we get u(tj) — u ( t j _ i ) = /

Au(s) ds,

which implies ||u(t)||^. <

+£

\\u(e)\\x

ll u (*i)-&-^-i) Au &)llx

u(tj) -

/

Au(s) d s

Using (tj - i j _ i ) A u ( t j ) = /

Au(tj) ds

and the converse triangle inequality \\u\\x — \\v\\x < ||u + v\\x at l|u(t)||x < K ^ l l x + E

r

we finally arrive

l|Au(i,)-Au(S)||xds.

Now by our assumptions we may take first the limit m a x \tj — t / - i | —> 0 and then we let e tend to zero to get the estimate | | u ( i ) | | x < | | u ( 0 ) | | x . D C o r o l l a r y 4 . 1 . 3 5 Let ( T ^ ) > Q and ( T f ) > 0 be two strongly continuous contraction semigroups with generator A and B respectively. If A = B (as operators) it follows that T f = T f for allt>0. Proof: For t G [0, oo) and u G X we consider the mapping t h-> T ^ u — T ^ u . Since this mapping fulfils the assumptions of Lemma 4.1.34, we get | | T f u - T t B u | | x < | | T £ u - T*u\\x

= ||u - u\\x = 0,

implying t h a t T f u = T f u for all u G X. D

272

Chapter 4

One Parameter Semigroups

It turns out t h a t often the theorem of Hille and Yosida is not applicable in concrete situations since the operators envolved in concrete problems are often not closed, but closable. Therefore, we will give a version of Theorem 4.1.33 for closable operators. For this we need some preparations. Recall t h a t by Lemma 2.7.12 a linear operator A : D(A) -*• X, D(A) C X, (X,\\.\\x) a Banach space, is closable if and only if for any sequence (u i / ) ( / € N , u„ € D(A), converging to 0 and lim Au„ = g, it follows t h a t g = 0. In the following V—KX>

A always denotes the closure, i.e. the minimal closed extension of a closable operator. L e m m a 4 . 1 . 3 6 Let A be a densely defined linear dissipative operator (X, \\.\\x). Then A is closable and R(A - A) = R(A - A) for all A > 0.

in

Proof: Using the criterion from Lemma 2.7.12 just mentioned above, we have to prove for a sequence (u i / ) J / € N , u„ G D(A), such t h a t lim u„ = 0 and v —loo

lim A u y = g in X it follows t h a t g = 0. For this take a sequence (g^)

gN,

v—>oo

gfi £ D(A), such t h a t lim gM = g in X. T h e dissipativity of A implies t h a t ||(A - A)g„ - Xg\\x = lim ||(A - A)(g M + \uv)\\x

(4.58)

v —>oo

> lim AHg^ + AuJ^ > A||gJ x v—»oo

for all A > 0 and /i G N, where we used t h a t Hu^H^ —> 0. Now we find using (4.58)

hiM-e\\x +

xAg-

> g/i - g - ^Ag M X

= -(A-A)gM-g

> hAx> X

for A —> oo we have llgfi - g|| X ^ \sJx and for /i —> oo it follows t h a t g = 0. Now, let A > 0. We have always t h a t R(A — A) C R(A — A). Hence, we have to prove t h a t R(A - A) c R(A - A ) . By definition of A we have R(A - A) C R ( A - A ) C R ( A - A ) . But R(A - J) is closed by Lemma 4.1.26, implying R(A — A) = R(A — A ) . •

4.1 Strongly Continuous Operator Semigroups

273

Now we have Theorem 4.1.37 A linear operator on a Banach space (X, \\.\\x) is closable and its closure A is the generator of a strongly continuous semigroup on X if and only if the following three conditions are satisfied 1. D(A) C X is dense; 2. A is a dissipative operator; 3. R(A — A) is dense in X for some A > 0. Proof: From Lemma 4.1.36 it follows that A satisfies the conditions 1.-3. if and only if A is closable and A satisfies 1.-3. from Theorem 4.1.33. • For later purposes it is helpful to characterise dissipative operators using the dual space. For this let (X, \\-\\x) be a Banach space with dual space (X*, \\-\\x.) and for u G X define J(u) := {x* G X* | <x*,u) = ||u|| x = ||x*|| 2 x . } •

(4-59)

Clearly, by the Hahn-Banach theorem, Theorem 2.7.4, J(u) =fi 0 for any u G X. Lemma 4.1.38 Let u,v e X.

Then

| | u | | x < | | u + Av|| x

(4.60)

holds for all A > 0 if and only if there exists x* E J(u) such that Re(x*,v)>0

(4.61)

holds. Proof: For u = 0 G X nothing is to prove, hence suppose that u / 0. Let x* £ J(u) such that Re (z*,v) > 0. For A > 0 we find ||u|| x = ( x * , u ) = R e (x*,u) < R e (z*,u + Av) < | | u + A v | | x | | * l x . = ||u + A v | y | u | | x . Thus (4.61) implies (4.60). Suppose now that (4.60) holds for all A > 0. For x*x G J(u + Av) we put z\ := -n—^—, i.e. ||z£|| x . = 1 and "Mix* u

H\x ^ ll +

Av

llx = (*A,u + Av)

= Re (zx,u) + ARe (z£,v) < ||u|| x + ARe (z*x,v),

274

Chapter 4 One Parameter Semigroups

which implies Re(^,u)>||u||x-A||v||x

(4.62)

Re(zl,v)>0.

(4.63)

and

By the Banach-Alaoglu theorem, Theorem 2.2.2, the unit ball in X* is weak*-compact. Therefore there exists z* £ X* with ||z*|| x . < 1 such that for any neighbourhood V* of z* in the weak-*-topology, and for any A > 0, we find some a £ (0, A) such that z* £ V*. In particular, for any e > 0 and any A > 0 there exists an a £ (0, A) satisfying |(z* — z*,u)| < e. Now, (4.62) implies that Re (z*,u) = R e ( z * , u ) + R e (z* - z*a,u) > \\u\\x - a\\v\\x - e >l|u||x-A||v||x-£, which leads to Re (z*,u) > ||u|| x . Moreover, from (4.63) it follows that Re (.z*,v) > 0. On the other hand we have (z*,u) = HuH^ since Re {z*,u) < l( z *i u )l ^ ll u llx r Therefore, taking x* = ||u||^z*, we have x* £ J(u) and Re (a;*,v) > 0. • Now we can prove Theorem 4.1.39 A. Let (A, D(A)) be a linear operator in a Banach space (X,\\.\\x). The following conditions are equivalent: 1. A is dissipative; 2. for any u £ D(A) there exists x* £ J(u) such that Re (z'.Au) < 0 ;

(4.64)

3. for all u £ D(A) and all A £ C, Re A > 0, it holds ||(A-A)u||x>ReA||u||x. B. Let (A, D(A)) be a closed, densely defined dissipative operator. Re {x*, Au) < 0 for all u £ D(A) and all x* £ J(u).

(4.65) Then

4.1 Strongly Continuous Operator Semigroups

275

Proof: A. Suppose that A is dissipative, i.e. we have for all A > 0 the estimate ||u|| x < ||u — AAu|| x . By Lemma 4.1.38 there exists x* G J(u) such that Re {x*, Au) < 0, i.e. it follows 2. Now suppose that 2. holds. Let u G D(A) and A G C, Re A > 0. Further take x* G J(u) satisfying Re (a;*,Au) < 0. It follows that Re (z*,(A-A)u) = R e (x*, Au) - Re A (a;*, u) < -ReA||u|&, implying that ||(A-A)u|| x ||u|| J C = | | ( A - A ) u | | x | | a : * | | x . > - R e (x*, (A - A)u) > Re A||u||^, which leads to ||(A — A)u|| x > Re A||u||^. Since obviously 3. implies 1., part A is proved. B. For all A > 0 we have (id - AA)" 1 = A"1 (A - 1 - A ) " 1 and by part A it follows that 11(id - AA) _1 |j < 1. Thus for u G D(A) it follows that ||(id - A A ) " ^ - u ^ = ||A(id - AA)" 1 Au|| x < A||Au|| x —> 0 as A —> 0. Since D(A) is dense (id — AA) _1 u —>• u strongly for all u G X. Now, for u G D(A) and x* G J(u) we find Re (x*, (id - A A ^ A u ) = A- x Re (x*, (id - AA) _1 u - u)

For A —> 0 we now obtain Re (a;*, Au) < 0, proving part B. • Theorem 4.1.40 Let (A, D (A)) be the generator of a strongly continuous contraction semigroup on the Banach space (X, ||.||x)- Further let (A, D (A) J be a dissipative operator on (X,\\.\\x) extending (A,D(A)), i.e. D(A) C D(A) and A|D(A) = A. Then we have D(A) — D(A) and therefore A = A. Proof: Take u G D(A). Since for A > 0 we have A G p(A), there exists h G D(A) such that Au - Au = Ah - Ah,

276

Chapter 4

One Parameter Semigroups

or, since Ah = Ah for h G D(A) A(h - u) - A(h - u) = 0. By our assumptions A is dissipative, implying A||h-u||x<|(A-A)(h-u)

*=°'

thus h = u G D(A). D We want to investigate the dual semigroups (TjT ) t > 0 of a strongly continuous contraction semigroup (T t ) t >o. In our presentation we follow closely the monograph [52] of H. Berens and P. Butzer. T h e o r e m 4 . 1 . 4 1 Let (Tt)t>o be a strongly continuous contraction semigroup on a Banach space X. Then the family {Tt)t>o °f adjoints has the following properties Tt*+s=Tt*oT:

and T*Q = id,

(4.66)

l|Ttl
(4.67)

and w * - l i m T > = u for all u G X*, t-»o where iu*-lim denotes the limit in the

(4.68) weak-*-topology.

Proof: Since Tt is bounded T^ is a bounded linear operator on X* with the same operator norm as T t , which implies (4.67). Moreover, by the very definition of T* we have for all u G X* and v G X, which yields (4.66). Moreover, for u G X* and v G X it follows t h a t lim (Tt*u - u, v) = lim (u, T t v - v) = 0 t->0

t-*0

by the strong continuity of (T t )t>o, hence (4.68) holds. •

4.1

277

Strongly Continuous Operator Semigroups

In general the adjoint (A*, D (A*)) of the generator (A, D (A)) of a strongly continuous contraction semigroup (T t ) t >o on a Banach space X is not the generator of the semigroup ( T j ) t > 0 . However, we have T h e o r e m 4 . 1 . 4 2 Let (T t ) t >o be a strongly continuous contraction semigroup on a Banach space X. A. The operator (A*, D (A*)) is a weak-*-closed operator and D(A*) is weak~*-dense in X*. B. For u G D(A*) it follows that Tt*u G D(A*) and A * T > = Tt*A*u. Moreover, we have
(4.69)

for all v G X and t > 0. C. An element u G X* belongs to D(A*) if and only if lim/T*U~U-A*u,v\ =0 t-+o \ * /

(4.70)

for all v G X. Proof: A. Since (A, D ( A ) ) is a closed, densely defined operator in X, the assertion of part A follows directly from Proposition 2.7.20. B. Let u G D(A*) and v G D(A). It follows from the definition of A* and by Lemma 4.1.14.5 that (T t *A*u,v) = , A v ) , which proves t h a t T*u G D(A*) and A*T£ = Tt*A*. (Tj A*u,v) = (A*u,T t v) is continuous which yields

T h e function t M.

/ ( T : A * u , v ) d s = f (A*u,Tsv)ds Jo Jo A*u, f Tsvds\

= /U,A([

(u,Ttv-v) = (Tt*u-u,v), where we used Lemma 4.1.14. Hence, part 5 is proved.

Tsvds)

278

Chapter 4 One Parameter Semigroups

C. Suppose for u G X* that *" converges in the weak-*-topology to some f G X* as t tends to zero. For all v G D(A) we find

= l i m p p , v = lim ( u, -±— t->o \ t

) = (u, Av), J

which implies that u G D(A*) and A*u = f. Conversely, for u G D(A*) we find using (4.69)

T?u _

u

= i
( A * u , T » ds = / A * U , - t

1svds

for all v G X. Since | JQ Tsv ds —> v strongly as t —> 0, it follows that the limit lim( t-x> \

,v t

exists and equals (A*u, v) and the theorem is proved. D The weak-*-generator of (T*) t > 0 is the operator (A*, D(A*)), D(A*) C X* with D(A*) := {u G X*

lim / T * U ~ ~ U , v \ exists for all v G X \

and A*u,v\ = lim / T * U ~ U , v \

for all v G X,

we have proved Corollary 4.1.43 The operator (A*,D(A*)) is the weak-*-generator

of

\*-t)t>oNext we want to determine the subspace XQ C X* where t \-> T*u is strongly continuous as t —• 0.

4.1

Strongly Continuous Operator Semigroups

279

P r o p o s i t i o n 4 . 1 . 4 4 Let (Tt)t>o be a strongly continuous contraction semigroup on the Banach space X. Then XQ is a strongly closed subspace of X*, hence (XQ, | | . | | X . ) is a Banach space, and it is invariant under T£, i.e. for all t > 0 we have T$XQ C XQ. Moreover, we have D(A*) C XQ and ||Tt*u-u||x,
(4.71)

holds for all u G D(A*). Proof: Obviously, XQ C X* is non-empty and a linear subspace. Let ( u l / ) ! / g N , u„ G XQ, be a sequence converging in X" to u. Hence, for each e > 0 there exists vo(e) such t h a t ||u„ — u\\x, < | for v > vo(e). It follows t h a t ||Tt*u - u\\x. < | | T > - T ? u „ l l x . + ||T?u„ - uv\\x.

+ ||u„ - u | | x .

< 2||u„ - u | | x . + ||T*u„ - u „ | | x . Since u„ G XQ we have lim ||TjU„ — u „ | | x , = 0, hence there is a S(e, u) > 0 t->-o

such t h a t ||T£u v — u „ | | x . < § for all 0 < t < S, which implies ||T£u - u\\x. < £ and therefore XQ is closed. T h e semigroup property of ( T * ) t > 0 implies t h a t u G XQ if a n d only if t h e function t t-> T £ u is strongly continuous, see t h e proof of Corollary 4.1.9, implying t h a t XQ is invariant under T£, t > 0. Now, for u G D(A*) we find using (4.69) t h a t |-u,v)|<

f w i A ' u ^ l d s K t J0

sup | | T : | | | | A * U | | X . | | V | | X I 0
and since T£ is a contraction, we arrive at (4.71) a n d consequently D(A*) C •

XQ-.

Let us denote by T5 i t t h e restriction of T£ onto XQ. Clearly, (Tj$ t )t>o is a strongly continuous contraction semigroup on the Banach space (XQ, \\.\\X.) having t h e generator (Ajj,D(Ao)), D ( A Q ) C XQ. T h e o r e m 4 . 1 . 4 5 Let (T t )t>o and (To it )t>o be as above. A. The space XQ is equal to the strong closure of D(A*) and D ( A Q ) is weak-*~dense in X*. Moreover, we have D(AS)cD(A*)cX0*. B. The operator Ag is the largest restriction XQ , while A* is the weak-* -closure of AQ .

(4.72) of A* with domain and range in

280

Chapter 4 One Parameter Semigroups

Proof: A. We know already that D(A*) C X0* by Proposition 4.1.44. If u G D ( A Q ) , it follows that lim «"~u — AQU in the strong sense, hence in i-s-0

the weak-*-topology too, which by Theorem 4.1.42 implies u G D(A*), i.e. D(A^) C D(A*) and (4.72) is proved. Since D(AQ) as the domain of a generator of a strongly continuous contraction semigroup on XQ is dense, the same must hold true for D(A*) by (4.72). Furthermore, Theorem 4.1A2.A yields that D(AQ) is weak-*-dense in X*. B. We prove next that for u G D(A*) with A*u G XQ it follows that u G D ( A Q ) , which will imply the first part of the assertion. Thus, let A*u G XQ. It follows that s H-» T*A*u is strongly continuous and from (4.69) we deduce that < T S i t u - u , v ) = jf (T5, s A*u,v)d S = ^

T5, s A*ud S ,v^

for all v G X, and therefore we get TS,tu-u= / T^A'uds Jo which implies that u G D(Ajjj) and A^u = A*u. The second statement of part B is that the graph of A* is the weak-*-closure of the graph of AS in X* x X*. Since the operator is weak-*-closed by Theorem 4.1.42..A, it suffices to prove that r(AJ) = {(ASu, u) G X* x X* | u G D(A£)} is weak-*-dense in T(A*). For this we prove that any weak-*-continuous linear functional on X* x X* that vanishes an T(Ag) also vanishes on T(A*). The space of all weak-*-continuous functionals on X* x X* is given by X x X. Let ( v i , v 2 ) e l x l such that (ASu,v1>-(u,v2>=0

(4.73)

for all u G D(A£). Using T ^ u - u = /„* TS,sA*u ds and (4.73), we find
Vl>

- (TS it u - u , V l ) = /J

T^uds.v^

= / ( A o T S , s u , V l ) d S = / (T£, s u,v 2 )ds Jo Jo = f (u,T a v 2 ) ds = / u , f Tsv2 ds

4.1

281

Strongly Continuous Operator Semigroups

Since D ( A Q ) is weak-*-dense in X* we arrive at = /u, f

Tsv2ds\

for all u G X*. Hence, it follows t h a t vi G D(A) and Avi = v 2 , implying t h a t the linear functional defined by (4.73) vanishes for all u G D(A*) which finally proves the theorem. • C o r o l l a r y 4 . 1 . 4 6 In addition to the assumptions of Theorem 4.1.45 suppose that X is reflexive. Then we have X$ = X* and AQ = A* as closed operators. Proof: From Proposition 2.7.20 we conclude t h a t D(A*) is strongly dense in X* and therefore the corollary follows from Theorem 4.1.45. • In Section 4.6 we will discuss the dual semigroup of sub-Markovian semigroups more detailed. Let us close this section with a collection of formulas telling how to obtain (Tt)t>o when the generator (A, D ( A ) ) or the resolvent (R-A),\>O a r e given. These formulas are taken from the monograph [131], p.354, of E. Hille and R.S. Phillips. However, some of these formulas had been proved in this section. We use the notation (ATt)u:=i(Ttu-u),

t > 0,

and

fc=o

^ '

T h e following formulas do hold: T t u = lim exp ( t A T , ) u ;

(4.74)

»)->0

T

T

t

*

u = l i m V ^ A j T K "->°fc=o u =

E|

A f c u

s

+(^I)T/

u ,

t 0

s

>0;

(t-s)"

(4.75)

_ l T

^

A n u d s

'

uGD(A);

(4.76)

282

Chapter 4 One Parameter Semigroups

;™ h (Ttu - £ ^Aku)= \

fc=0

'

^A"u'u G n D ^ A "); /

T t u = lim exp (t(\2Rx

'

(4-7?)

neN

- A))u;

(4.78)

A—>oo

T t u = lim fyRfc/t) u fc—>oo \ I

(4.79)

/

and T t u = lim ( i d - - A )

4.2

u.

(4.80)

Analytic Semigroups

Let (Tt) t >o be a strongly continuous contraction semigroup in the Banach space (X, \\.\\x) "with generator (A, D (A)) and resolvent ( R A ) A > 0 . The relation between (T t )t>o and ( R A ) A > 0 i s given by

-F

R A u = / e _ A £ T t udi. (4.81) Jo Thus we mayJolook at R^u as the Laplace transform of T^u at A, A > 0. Recalling Theorem 3.8.7 we may try to invert (4.81) in order to express Ttu with the help of (RA)A>O- For ^ n ' s however, it is necessary to extend A H* RAU to some sector in the complex plane. In addition, it will be of importance whether or whether not t M- T t u has an analytic extension to some sector inC. Let w e R and 6 € (f, TT). Then the sector SgtU C C is defined by Se,uj := {A £ C | A ^ co and |arg(A -w)\<0},

(4.82)

where arg z £ (—ir, ir] is the argument of the complex number z. Definition 4.2.1 Let A : D(A) —*• X, D(A) C X, be a densely defined linear operator in the complex Banach space (X, ||.||^)- We call A sectorial if there exist constants u> £ R, 9 £ (5, n) and M > 0 such that So*, C p(A)

(4.83)

4.2

283

Analytic Semigroups

and

M

HRAII^TT^, \A — LJ\

XeSe,u,

(4.84)

hold. R e m a r k 4 . 2 . 2 A. Any sectorial operator is closed. B. Let (X, \\.\\x) Hilbert space (H,(.,.)H) and ( A , D ( A ) ) be a sectorial operator on H. —A is form sectorial with corresponding sector Se-^,-u-

be a Then

Suppose t h a t ( A , D ( A ) ) is sectorial with w G M, 9 G ( f ,7r) and M > 0 as in Definition 4.2.1. Then the operator A " := A — u) is sectorial now with w = 0, but 9 and M the same as for A. For a sectorial operator A with w — 0 and M = 1 it follows t h a t (0, oo) C /o(A), hence for all A > 0 the operator A — A is surjective, i.e. for all f G X the equation Au — Au = f has a solution. Moreover, since M = 1, we find for u G D(A) and A > 0 t h a t ||u||x =

||RA(A-A)u||x<^||(A-A)u||x,

i.e. ( A , D ( A ) ) is a dissipative operator. Hence, by the Hille-Yosida theorem, Theorem 4.1.33, we have C o r o l l a r y 4 . 2 . 3 Let (A, D ( A ) ) be a sectorial operator in the Banach space (X, \\.\\x) such that (4.84) holds with M = 1. Taking LJ G R as in (4.83), it follows that the operator A " := A — w with domain D(A) is the generator of a strongly continuous contraction semigroup (Tt)t>o on X. Moreover, the operator ( A , D ( A ) ) is the generator of the strongly continuous semigroup (cwtTt)t>0. Now suppose t h a t A fulfils (4.83) and (4.84). Take r > 0, 77 G ( § , 9 ) and consider the curve 7r,T7 := {A G C I |arg A| = rj and |A| > r } U {A G C I |argA| < n and |A| = r} , which we regard to be oriented counterclockwise. Then the curve u + j r i V is contained in S$)Uf and we may consider the operators U t u := —

/

e a R A u dA, t > 0, and U 0 u = u,

(4.85)

Chapter 4

284

One Parameter Semigroups

where the integral is to be understood as a Dunford integral, see Section 2.7. T h e function A i-> etAR.AU is analytic in SQIUJI hence U^u is independent of the special choice of r and rj. P r o p o s i t i o n 4 . 2 . 4 Let A be a sectorial operator with sector SgtUJ and let ( U t ) t > 0 be defined by (4.85). Then we have for all t, s > 0 and u G X Ut o U s u = U t + S u ,

(4.86)

and lim U t u = u t-+o as a strong limit in (X,

(4.87) \\-\\x)-

Proof: Let us define the operator Aw by A w u := Au — urn. It follows t h a t the resolvent set of A w contains Sgto and t h a t R ^ = R ^ + w , where R ^ denotes the resolvent of A at A. Moreover, we have VtU

:= —

/

e u R f u dA = —

/

e A t e - ^ R £ u dA

= e-"*U t u, hence we may consider the case w = 0. For t, s > 0 and r > 0, ^ < rj' < rj < IT we find using the resolvent equation, Lemma 4.1.32, t h a t 2

(V, o V s )u = (±-\

\2ni) =

etARf J

J

U J ^ ,

M-A

etARfudA/"

(J_V/ — ^

e*"Rf u d/i dA

/

e^Rfud/if

-^-d

M

-^-—dA.

Now, the analyticity of the exponential function and the special choice of 7 r ^ and 72r,T7', respectively, imply by Cauchy's integral formula t h a t

1 'W

du = 2-7ri e * -

X

for A € 7 r „,

4.2

285

Analytic Semigroups

and Ptx

r

—

M

J -vIT

^

dX = 0 for // £ 72r,77' •

Thus we get (V t o V s ) u = - L /

e f + ^ R f u dA = V t + S u ,

and (4.86) follows. Next we will prove (4.87). For A G p(A) we have AR A = A(A - A ) " 1 = (A + A - A)(A - A ) " 1 = ARA - id, implying t h a t AUtu = —

e t A AR A udA = —

[

f

e t A AR A udA,

(4.88)

where we used also the fact t h a t — /

e t A udA = 0.

(4.89)

Hence, for any u e X we have U t u € D(A). Further, differentiating in (4.85) with respect to t gives l u dt

t

u = - ^ / 2m

e t A AR A udA, Ju+^

implying together with (4.88) t h a t - ^ U t u = AU t u. at In particular, we find t h a t ( U t ) t > 0 is strongly continuous and t h a t d

T T

I

(4.90)

A

-T7Utu|t=o = Au, and the proposition is proved.

•

T h e proof of Proposition 4.2.4 implies t h a t (A, D (A)) is the generator of the strongly continuous semigroup ( U t ) t > 0 and we write in the following (Tt)t>o instead of ( U t ) t > 0 . From Lemma 4.1.8 we find t h a t ||T t || < M 0 e " " , but we may strengthen some of the results just derived.

(4.91)

286

Chapter 4 One Parameter Semigroups

Proposition 4.2.5 Suppose that A is a sectorial operator with sector Se,u and denote by (Tf)t>o the strongly continuous semigroup generated by A. It follows that for any u G X, k G N and t > 0 we have T t u G D(Afc), hence T t u G f| D(Afc), fceN

(4.92)

and we have for u G D(Afe) A fc T t u = TtAfcu, t > 0 ,

(4.93)

and for suitable constants Mk, k GN, we find for t > 0 that ||i f c (A-u;id) f c T t || <Mke"\

(4.94)

Moreover, the function t H-> Tt is arbitrarily often differentiable and satisfies ^ T t u = A fc T t u

(4.95)

for all u G X. Finally, the mapping t — i >• Tt has an analytic extension to the sector u 4- S, where S is given by S := {A G C \ {0} | |arg A| < 6 — ^ }. Proof: We may use the representation (4.85) for Tt, and moreover, without loss of generality we may assume w = 0. We know already that T t u belongs to D(A) for any u G X and that AT t u =—

AetARAu dA.

f

(4.96)

Iterating the arguments leading to (4.88), in particular the equation ARA = ARA - id and (4.91), we find T t u G D(Afc) and AfeTt= u = - -^ / / Afce'ARAudA, (4.97) 1-KiL " thus Ttu G D(Afc) for all u G X, k G N and t > 0. From the representation of T t we get AT t u = T t Au for u G D(A), since ARAu — RAAu for u G D(A), and we have proved (4.92) and (4.93). Next note that T t u G D(A) and therefore we have IT,7}

AeAtRAudA=-^/

ATtu=-^-/

°

jr,r)

r

W

^

287

4.2 Analytic Semigroups Thus taking into account the special definition of 7r>7,, we deduce that

||AT,u|| < ^

UpeP^&p

+ rp

ercos* dA ||u|| x ,

which leads to (4.94) for k = 1 (and as assumed v = 0). Since AfcTt = TtAfc and A fc T t u = (AT t/fc ) u for u G D(Afc), we conclude that

||A^u|L<(^)fc<(Mie)^!^ implying (4.94). Moreover, differentiating in (4.85) with respect to t and taking into account (4.92), we get (4.95). Now let 0 < e < 6 — | and take 77 = 6 — e. It follows that the function

z^Tz:=^-[

ezARAdA

is well defined and analytic in the sector S£:={zG0 Se, it follows that z H* T Z is analytic in S and the semigroup property for Zi,Z2 £ S is proved as in Proposition 4.2.4. D Definition 4.2.6 A strongly continuous semigroup (Tt)t>o is called an analytic semigroup of angle 8, if the mapping t H-» T J has an analytic extension to the sector S := {z GC\{0} | |arg z\ < 9 - § }. Thus we have proved that a sectorial operator generates an analytic semigroup. We have the following characterisation for analytic semigroups. Theorem 4.2.7 Let (Tt)t>o be a strongly continuous semigroup on a Banach space (X, \\.\\x) such that ||T t || < M for all t > 0 and some M > 0. Further, let (A, D (A)) be its generator and assume that 0 € p(A). Then the following conditions are equivalent: 1. The semigroup (Tt)t>o has an analytic extension to a sector Sg-a.^, 9 G [~,7r) and ||T 2 || is uniformly bounded in every closed subsector Sg>_ » $, 6' G (§,7r) and0' < 9.

288

Chapter 4 One Parameter Semigroups 2. There exists a constant c such that for every a > 0 and r ^ 0

(4-98)

IIR^+trll < A

M holds. 3. There exists 5 € (0, §) and M > 0 such that £ := JA e C I |arg z\ < | + 6 } U {0} C p(A)

(4.99)

and ||RA||

< ^

/orAGE\{0}.

(4.100)

4- The mapping t t-j- Tj is differentiable in (0, oo) and wzi/i some constant c' we have ||ATt|| < j , t > 0.

(4.101)

Proof: 1. =» 2.: Let 0 < 6' - § < 6 - § be such that ||T Z || < cx for z G Se'-^-fi. For u € l and cr > 0 we have /•OO

R<7 + 7 r u =

/ e-^+^Ttudi. (4.102) Jo Since 2 1—>• T z is analytic in Se-^-,o and uniformly bounded in Sg>-z.:o, we may shift the path of the integration in (4.102) from the positive real axis to any ray p i->- peir>, p G (0, 00) and \rj\ < 9' — ^. Taking r > 0 we get when changing the path of integration t o / ) r t pe%\ ~*> /•OO

||IWu|lx< / ./o

e-H—(«'-i)+-'«K-f))Cl||u||xdp

Clllullv - t7cos(6>'-f)+Tsin(fl'-f)

-

C .. ,, T " IIX'

Analogously, when r < 0 we shall shift the path of integration to p H pe-K6'-^) to get (4.98). 2. => 3. For Re A > 0 we have by (4.28) that ||RA|| < ^ \ . From 2. we know for Re A > 0 that ||RA|| < TJ^JJ and therefore we have ||RA|| < pjr for Re A > 0. The Taylor expansion of ~R,\ around A = a + ir, 0, yields 00

RA = £ R ^ > fc=0

+ ir-A)fc,

(4.103)

4.2

289

Analytic Semigroups

compare (4.33). Further, we know t h a t this series converges in \a + ir — A| < 6 < 1. Choosing A' = Re A + ir in (4.103) and using (4.98) it follows t h a t the series converges uniformly for \cr — Re A| < - | ^ , since a > 0 and 5 < 1 are arbitrary, it follows t h a t p(A) contains < A e C

Re A < 0 and Ljj A < £ >. In

particular, we have {AeC

(4.104)

||argA|<|+<5}cp(A),

where 5 = J a r c t a n (A), 0 < 6 < 1. Moreover, in this sector we have

IIRAII <

1

i-*M

Vc^+l

1

Af

(4.105)

(!-l) W W

By our assumptions 0 e p(A), it follows (4.99), hence 3. is proved. 3. => 4. From Proposition 4.2.4 we deduce from 3. t h a t

= - / <e A t R A dA,

(4.106)

where 7 is the p a t h composed by the two rays pel9 and p 7 oriented counterclockwise. Since

l0

, 0 < 9 < 00, and

y 7T COS 9 J t

7T ,/ 0

we may differentiate in (4.106) with respect to t and we find

= 5 ? / * " * dA

(4.107)

and d

T

<

M

1

v7TCOSP J t

(4.108)

Thus (T t ) t >o is differentiable for t > 0 and I AT, II =

d

T

c' < - for t > 0.

(4.109)

Chapter 4

290

One Parameter Semigroups

4. => 1. We will show in Lemma 4.2.8 t h a t for a differentiable semigroup (T*)t>o we have d" dtn

d

T

<

d

T

(4.110)

Using (4.109) and the estimate n! en > nn we find l_ dtn

<

ce

(4.111)

T

Now, let us consider the power series n T< + £ : ! Vdi" Tt)(z-t) . n=l

(4.112)

Because of (4.111) this series converges for \z — t\ < -^r, for every 5 < 1, implying t h a t z t-> Tz is analytic in {z G C | |arg z\ < arctan ( ^ ) } and for t £ R n { z e C | |arg z\ < arctan ( ^ ) } it follows t h a t T t = Tz, i.e. z H-> Tz is an analytic extension o f t H+ T t . Further, from (4.112) it follows t h a t ( T z ) * e { 2 e c | | a r g 2 | < a r c t a n ( ^ ) } i s a semigroup and Tzu —»• u for z — • 0, z £ J z £ C I |arg z| < arctan (JTJ) }• The uniform boundedness on closed subsectors {z s C | | a r g z | < arctan (JTJ) — e } , £ > 0, is now obvious. Thus the theorem is proved. • L e m m a 4.2.8 Let (Tj)t>o be a strongly continuous semigroup on the Banach space (X, \\.\\x) with generator ( A , D ( A ) ) . Suppose that t t-± Tt is differentiable. Then —Tt

= (ATt/„)n =

(-Tt/r

(4.113)

holds for t > 0 and n G Proof: Since t >->• T t u is differentiable, it follows t h a t T t u G D(A), t > 0, and the proof of Lemma 4.1.14.S yields (4.113) for n = 1. Now suppose t h a t (4.113) holds for n G N fixed and t > s. It follows t h a t

dtn

Tt = (ATt/„)n = Tt_s(ATs/n)".

(4.114)

4.2

Analytic Semigroups

291

Differentiating (4.114) with respect to t we get | ^ r T « = ATt_.(ATj/B),\

(4.115)

substituting s = -^TT, we obtain (4.113) for n+ 1, and the lemma is proved. D R e m a r k 4 . 2 . 9 In proving [235] of A. Pazy.

Theorem 4.2.7, we follow closely the

monograph

When dealing with analytic semigroups it is helpful to work in complex Banach spaces. On the other hand, when working with positivity preserving semigroups, it is reasonable to restrict ourselves to real Banach spaces. T h e following proposition shows t h a t we may switch from the real to the complex situation and vice versa without any problem. Recall t h a t for a real Banach space X and a linear operator A : D(A) —> X, D C X, we define the complexification by Xc := X + iX, D C (A) := D(A) + zD(A) and A c : D C (A) -> Xc (u + iv)

M-

Au + iAv.

Here we define in Xc for ui + iu2 and vi + iv2 the addition by (ui + 1112) + (vi +ZV2) = (ui + v i ) +z(u2 + V2) and for Ai +i\2 £ C the scalar multiplication is defined by (Ai + iA 2 )(ui + iu 2 ) = Aiui - A 2 u 2 + i(Aiu 2 + A 2 ui). P r o p o s i t i o n 4 . 2 . 1 0 Let (X,\\.\\x) be a real Banach space and ( A , D ( A ) ) a linear, densely defined operator on (X,\\.\\x). If the operator (Ac,D(Ao generated by Ac leaves X invariant, i.e. TtX c X. Proof: We may use the Yosida approximation Ac,A of Ac to consider the semigroup ptA c ,A _

V^—A

f c

fc=0

Since Ac,A leaves X invariant, it follows t h a t e tAc -* leaves X invariant. Since Ttu=

lim e* Ac '*u, A—¥ OO

(Tt)t>o leaves X invariant.

•

292

Chapter 4 One Parameter Semigroups

Example 4.2.11 Let ip '• M.n -+ C be a continuous negative definite function such that |ImV(£)l
(4.116)

holds for all { e R " . Then the operator Au(i) = -ip(D)u(x) = -(27r)-"/ 2 f

e " ^ ( O u ( 0 d£

(4.117)

with domain i J ^ ' ^ R " ) generates an analytic semigroup on L2(R™). For u € L 2 (R n ) we find T t u(z) = (27r)-"/ 2 /

e ^ e - ' ^ u O O d£,

hence we have AT t u(z) = -(27r)-"/ 2 /

e i x -^(Oe"*' / ' ( 5 ) u(0 d£,

JMV-

where both formulas hold at least in <S(Rn). Now, (4.116) implies

t^(Oe-^ ( 0 < tyjl + tf Re V(Oe"*Re * ( 0 and for s > 0 we have se~s < 1, implying that

< V±±4 t Thus i i-> e i x «V(O e " t V ' ( e ) u(0 fce^on^s to L 2 (R n ) and we /iave that t ^ Tt is differentiate. Moreover,

l|ATtu||0 < ^ ± 1 /loMs /or £ > 0. Thus the proof of Theorem 4.2.7, in particular the implication 4. =>• 1. j/ieZds i/iai (Tt)t>o has an analytic extension. Finally we want to prove a result due to E.M. Stein [288] giving a lot of important examples of analytic semigroups.

4.2 Analytic Semigroups

293

Theorem 4.2.12 Let (T t ) t >o be a family of operators such that T s + t u = (Ts o T t )u, t, s > 0, whenever both sides are defined for a function u : R™ —> C In addition assume that (Tt)t>o is strongly continuous on L 2 (R n ), and that for each 1 < p < co l|T t u|| LP < ||u||LP

(4.118)

holds. Further suppose that each of the operators T( is selfadjoint on L 2 (R"). Then for 1 < p < oo the mapping t >-> Tt has an analytic extension from the sector Sp := iz G C |arg z| < f ( l - | - l h | to LP(Rn) which is continuous on Sp. For these analytic extensions we still have the semigroup property, i.e. TZl+Z2 = TZl o T 22 for z\, Z2 £ Sp, and the operator u i->- Tzu, z G Sp and u G L p (R n ) ; is bounded on L p (R n ), 1 < p < oo. Proof: Note that for p = 1 and p — oo we have Si — M.+ and 5"^ = R+, respectively, thus nothing is to prove in these cases. Next we prove the result for p = 2 and then we will apply an interpolation result to obtain the general case. Now, let p = 2. The operator Ti is bounded and selfadjoint, hence by the spectral theorem, Theorem 2.7.30, we have

M/:

AdE A u,

where (E^) is the corresponding resolution of identity on L 2 (R"). By our assumptions we have Ti = T j / 2 ° T i / 2 = Tj ,2 o Ti/2, where Tj ,2 is just the adjoint of Ti/ 2 - Hence, it follows that E\ = 0 for A < 0 and we find T1vL=(f

AdEAU

(4.119)

for all u G L 2 (R"). From (4.119) it follows immediately that

T~.u=(Y A3^dEAJu for m, k G N. Since by assumption (T t )t>o is strongly continuous on L 2 (R") we find for t > 0

Ttu=r/

A*dEAV.

(4.120)

294

Chapter 4 One Parameter Semigroups

Now we define T z for z € S2 = {z € C | |arg z | < f } = { z € C | R e z > 0 } by

-a

1

A*dE A )u.

(4.121)

Clearly, z >-> T z is analytic from S2 into the space of linear bounded operators on L 2 (R n ), and (T z ) z € i S has the semigroup property. Moreover, we find ||T Z || < 1 for all 2 £ C, Re z > 0. Thus the case p = 2 is proved and we know ||T z u|| 0 < ||u|| 0 , Re z > 0,

(4.122)

||T t u|| L1 < ||u|| L1 , teR+,

(4.123)

and

moreover, the mapping z H-> fRn (T z u)v da; is analytic on S2. We want to interpolate between (4.122) and (4.123) using Stein's interpolation result, Theorem 2.8.3. For this let 77 > 0 and - f < 6 < § and define U(z)u := T ^ i ^ u , 0 < Re

z<\.

The family (U(z)) is an analytic family of operators in the sense of Definition 2.8.2. From (4.122) we find that ||U(z)u|| 0 < ||u||0 for Re z = 1 and (4.123) gives ||U(z)u|| L1 < ||u|| L i for Re z = 0. Thus Theorem 2.8.3 implies that for 1 < p < 2 I LP

< IMILp

(4.124)

where z = rjeie(2 D, and (4.124) holds for z € Sp, 1 < p < 2. In order to obtain (4.124) for 2 < p < 00, we have to interpolate (4.122) with the estimate

IITtuI^ < H L , t>0.

(4.125)

Finally we have to prove that for 1 < p < 00 Z M-

/

(T z u)vd:r

(4.126)

is analytic in Sp and continuous in Sp for u € L P (R") and v G L p (R™), - + ^7 = 1. For this we approximate u and v by sequences (u/t)fceN and (vfc)fcgN,

4.3

Subordination in the Sense of Bochner for Operator Semigroups

295

respectively, such t h a t ufc e L 2 ( R n ) n L " ( R 0 ) and vfc e L 2 ( R n ) n L " ' ( R n ) . It follows t h a t zsr+ \

(T0ufc)vfc da;

is analytic for each k £ N and we find f

(T2u)vdz- f n

jR

{Tzuk)vkdx n

JR | T z u - T z u f c | |v f c |da;+ / | T z u | |vfc - v| da; 7K" JRn < ||Tzu-Tzufc||LP||vfe||LP' +||T2u||LP||vfc-v||LP',

<

hence z H-> fRn ( T z u ) v da; is the uniform limit of analytic functions in Sp which are continuous on Sp, and the theorem is proved. • A particular improvement of the sector Sp determined in Theorem 4.2.12 was given by D. Bakry in [14]. For the general case V.A. Liskevich and M.A. Perelmuter [204] improved the sector Sp to the sector Ep = L

GC

|arg z \ < \ - arctan A = j = } •

(4-127)

In [308] J. Voigt gave an example showing t h a t the condition (4.127) is sharp. In Section 4.6 we will apply Theorem 4.2.12 to symmetric Feller semigroups.

4.3

Subordination in the Sense of Bochner for Operator Semigroups

Let f : (0, oo) —• R be a Bernstein function and (?7t) t>0 the associated convolution semigroup on R supported by [0,oo), i.e. we have by Theorem 3.9.7 C{r]t)(x) = e - * f ( x ) , x > 0 and t > 0.

(4.128)

Furthermore, let (T t ) t >o be a strongly continuous contraction semigroup on the Banach space (X, \\.\\x) with generator (A, D ( A ) ) .

Chapter 4

296

One Parameter Semigroups

T h e o r e m 4 . 3 . 1 Let (T t ) t >o and (j?t) t > 0 with corresponding Bernstein tion be as above and define Tfu for u € X by the Bochner integral

func-

/•OO

Tfu=

/ Jo

T s u77 t (ds).

(4.129)

Then the integral is well defined and (TQ tion semigroup on X.

>Q

is a strongly continuous

contrac-

Proof: Since (Tt)t>o is a contraction semigroup on X and each of the measures r)t is a sub-probability, it follows that the integral in (4.129) is well defined and moreover we have /•OO

|Tfu||x<

/ Jo

||Tau||x7?t(ds)<77t([0,oo))||u| x <

Mx-

Hence, the operators T\, t > 0, are contractions. Moreover, using the semigroup properties of (T t ) t >o and (»?t)t>0, respectively, we find /•oo

T ^ u =

/ Jo /•oo

= / Jo /•OO

= / Jo

/*oo

TruVt+s(dr)

=

Tru(7?t*r?s)(dr) Jo

/»oo

/ Jo

Tr+qur]t(dr)r)s(dq)

/-OO

/ Jo

TroT9u7?s(dg)r?t(dr)

/•OO

= / T*(T,uH(d9) Jo ^TfoT^u, where we used also Lemma 2.3.24. C for the operator T£. Thus (Tj) > is a contraction semigroup on X. Finally, we show t h a t (T£) > 0 is strongly continuous on X. For this note t h a t the function r \-> | | T r u — u||_y is continuous and bounded on [0, oo) and lim | | T r u — u\\x = 0. Furthermore, the convolur-»-0

tion semigroup {vt)t>o tends to eo in the Bernoulli topology as t —> 0, see Lemma 3.6.3. Since /•OO

||Tfu-u||x<

/ Jo

||Tru-u||*»fc(dr) + (l-Tft(R+))||u||x,

4.3

Subordination in the Sense of Bochner for Operator Semigroups

297

it is sufficient t o prove for any function v G Cj,(R + ), v(0) = 0 t h a t /•OO

lim / t->oJo

v(r) r]t(dr) — 0,

which is however obvious by t h e definition of the Bernoulli topology a n d t h e definition of EQ. Thus t h e theorem is proved. • D e f i n i t i o n 4 . 3 . 2 Let (T t )t>o and (vt)t>o wHh corresponding Bernstein function f be as in Theorem 4.3.1. The semigroup (T*) > 0 is called subordinate (in the sense of Bochner) to (Tt)t>o with respect to (r)t)t>0 or equivalently with respect to f. E x a m p l e 4 . 3 . 3 Let (/xt)t>o be a convolution the continuous e-tip(()

negative definite function

Further,

ing convolution

semigroup

let f : (0, oo) —>• R be a Benstein semigroup

(/•*t)t>o a. Feller semigroup and sub-Markovian

(ijt)t>o> (T\°°'

semigroups

su

function

with

correspond-

PP'7t C [0, oo). We can associate

with

n

J (T[

on R™ associated to

tp : R™ —• C, i.e. (it(Q = (27r)~ n ' 2

on the Banach space (Coo(R ; R), H-H^) )

on the spaces L p ( R n ; R ) , 1 < p < oo

(for p y£ 2 see Example 4.6.29). On <S(R n ;R) we have always

T t u(z) = /

u(x- y) K(dy) =

(2TT)-"/ 2

/

e ^ e " ' ^ ^ ) d£

where T t is the restriction of any of the operators T j , 1 < p < oo, to <S(R n ). Now, for Tft, t > 0, we find by (3.241) that Ttu(rr) = /

u(x - y) £{dy)

= (2TT)-"/2/

e * s € e - t f ( * ( c » u ( C ) <^,

(4.130)

where (A t t) t > 0 *s the convolution semigroup on R™ associated with the continuous negative definite function ioip. Clearly, (4.130) gives a complete description of the subordinate semigroups to ( T j I , 1 < p < oo, by means of f. Moreover, it is obvious that ( T j '' 1 (Tj

' J

is once again a Feller semigroup and

is for 1 < p < oo a sub-Markovian

latter result is of general

character.

semigroup

on L P ( R " ; R ) . The

298

Chapter 4 One Parameter Semigroups

Corollary 4.3.4 Let (T t )t>o be either a Feller semigroup on C00(R™;R) or a sub-Markovian semigroup on L p (R n ;R), 1 < p < oo. Further let i be a Bernstein function. Then (T*) > is also a Feller semigroup on C00(R™;R) or a sub-Markovian semigroup onXp(R™;R), 1