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Z. Wahrscheinlichkeitstheorie verw. Gebiete 60, 171 - 184 (1982) Zeitschrift ftir Wahrscheinlichkeitstheorie und verwandte Gebiete Springer-Verlag 1982 Operators Generating Stable Measures on Banach Spaces Werner Linde Friedrich-Schiller-Universifiit Jena, Sektion Mathematik, Universitgtshochhaus, DDR-69 Jena, German Democratic Republic We investigate operators T from the dual E' of a Banach space E into Lp(f2, P), 0<p<2, for which the mapping a~exp (-IITalf) is the characteristic function of a (p-stable symmetric) Radon measure on E. Let us denote by Ap(E', Lp) the set of all operators from E' into L v having this property. Then the main results of this paper are the following: (1) Ap(E',L,) becomes a complete normed, resp. quasinormed space. From this we derive estimations for the r-th moments, 0<r<p, of p-stable symmetric measures. (2) Each operator TeAp(E',Lp) is decomposed by an E-valued random variable ~b with j'/l~llPdP< o% i.e. Ta-(O(.),a) for all aeE'. Q (3) A Banach space E is of stable type p iff there is a constant c>0 such that for all p-stable symmetric measures/~ the estimation sup tP#{Nx[]>t} <=c lim tP/~{l[x[I >t} t>0 t~o~ holds. (4) We construct two p-stable symmetric measures # and v on R 2 with #{[(x,a)l<l}<v{Kx, a)[<l}, a~R 2, such that ~(c) > v(C) for some absolutely convex closed subset C ~_ R 2. 1. Notation and Definitions By E we always denote a real Banach space with dual E'. L(E, F) denotes the set of all (continuous linear) operators from E into the quasi-normed space F. 0044- 3719/82/0060/0171/$02.80

Operators generating stable measures on banach spaces

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Page 1: Operators generating stable measures on banach spaces

Z. Wahrscheinlichkeitstheorie verw. Gebiete 60, 171 - 184 (1982)

Zeitschrift ftir

Wahrschein l ichkei t s theor ie und verwandte Gebiete

�9 Springer-Verlag 1982

Operators Generating Stable Measures on Banach Spaces

Werner Linde

Friedrich-Schiller-Universifiit Jena, Sektion Mathematik, Universitgtshochhaus, DDR-69 Jena, German Democratic Republic

We investigate operators T from the dual E' of a Banach space E into Lp(f2, P), 0 < p < 2 , for which the mapping

a ~ e x p ( - I I T a l f )

is the characteristic function of a (p-stable symmetric) Radon measure on E. Let us denote by Ap(E', Lp) the set of all operators from E' into L v having this property. Then the main results of this paper are the following:

(1) Ap(E',L,) becomes a complete normed, resp. quasinormed space. From this we derive estimations for the r-th moments, 0 < r < p , of p-stable symmetric measures.

(2) Each operator TeAp(E',Lp) is decomposed by an E-valued random variable ~b with j ' / l~l lPdP< o% i.e. Ta-(O( . ) ,a ) for all aeE'.

Q

(3) A Banach space E is of stable type p iff there is a constant c > 0 such that for all p-stable symmetric measures/~ the estimation

sup tP#{Nx[] >t} <=c lim tP/~{l[x[I >t} t > 0 t~o~

holds. (4) We construct two p-stable symmetric measures # and v on R 2 with

#{[(x,a)l<l}<v{Kx, a)[<l}, a~R 2,

such that

~(c) > v(C)

for some absolutely convex closed subset C ~_ R 2.

1. Notation and Definitions

By E we always denote a real Banach space with dual E'. L(E, F) denotes the set of all (continuous linear) operators from E into the quasi-normed space F.

0044- 3719/82/0060/0171/$02.80

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172 W. Linde

If # is a R a d o n measure or m o r e general a cylindrical measure, /2 is its characteris t ic funct ion (ch.f.), i.e.

fi(a) = ~ e i<~'"> d#(x), a~E'. E

An E-valued r a n d o m var iable (r.v.) is a (strongly) measurab le funct ion go f rom a measure space (f2, P), P(f2) = 1, into E. By Lp(f2, P, E) or Lp((2, P) if E is the field of real numbers , we denote the set of all E-valued r.v.'s with

Ilgollp = {~ ,go(co)ll" dP}l/p < ~ , 0 < p < oo, f~

resp. IlgoN~=esssup ]lgo(co)l[<oo, p = o o . If not otherwise stated p always de- notes a real n u m b e r with 0 < p < 2 . Then 01, 02, . . . is a sequence of inde- pendent real r.v.'s with

Oj(t) = j' exp (iO i t) dP = e-I,Ip, j = 1, 2, .. . , 12

i.e. 0j are s tandard p-stable r.v.'s. A Banach space E is said to be of stable type p ([201) if for one (each) r

with 0 < r < p there is a cons tant c > 0 such that for all x, . . . . . xneE the following es t imat ion is valid:

Let us men t ion that for 0 < p < l each Banach space is of stable type p (cf. [201).

I f q0 is an E-valued r.v. we denote by

the dis t r ibut ion of go, i.e.

# = dist (go)

#(B)= P { go(co)~B}

for each Borel set BGE. Then # is a R a d o n measure on E. A symmetr ic (#(B} = # ( - B ) ) R a d o n measure # is said to be stable if for given g, f i > 0 there exists a ? > 0 such that

dist (a~ + fl~p) = dist (?~)

where dist(go)= dist ( 6 ) = # and go and 6 are independent r.v.'s. It is k n o w n (cf. [51) that there exists a real n u m b e r p ~ ( 0 , 2 J such that ~P+f lP=?P for all a, f l>0 . In this case we say # is p-stable. By Rp(E) we denote the set of all p- stable symmet r ic R a d o n measures on E.

Final ly we want to men t i on that C[0, 11 denotes the Banach space of all cont inuous functions on the unit interval.

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Operators Generating Stable Measures on Banach Spaces 173

2, The Space AI,(E' , Lv)

Let us recall that an operator S from L2(~2, P ) into E is said to be 7- Radonifying if S(72) extends to a Radon measure on E where 72 denotes the cylindrical measure on L 2 with ~?2(f)=exp(-ILfll 2) (el. [11]). If 1 < p < 2 one may extend this definition to operators S from Lp,, lip + lip'= 1, into E. More precisely, in [10] we investigated operators S from Lv, into E for which S(Tp) has a Radon extension where ~p(f) = exp ( - Ik f Nv), feLv.

To include the case 0 < p < 1 it is more convenient to investigate the duals of such operators.

If Lp=Lv(g2, P), 0 < p < 2 , we put

Ap(E', Lp)= {TeL(E, Lp); a ~ e x p (-]h Tall p) is a ch.f. of a Radon measure on E}.

If TeAv(E' ,LP) we denote by #T the corresponding Radon measure on E, i.e. fir(a)=exp(-IlTallP). Of course #reRp(E). The first theorem shows that all #eRp(E) can be written in this way.

Theorem 1. For each #eRp(E ) there exists an operator TeAp(E', Lv) such that # ~#T"

Proof. Let # be in Rp(E) and let a be the spectral measure of # on the unit sphere 3U of E (cf. [25]). Then we !define

P(B)=a(B)/a(OU), ~B~ ~U

measurable, in the case a:#O. TeAp(E', Lp(~U, P)) is defined by

Ta = ( . , a) a(O U) l/v, acE'.

If a (3U)=0, i.e. #=go , we choose P arbitrary and put T=0 .

Remark 1. If "cc(E', E) denotes the topology of uniform convergence on compact subsets of E then each TeAp(E',Lp) is zc(E',E)-continuous. Consequently, if l < p < 2 , TeAp(E',Lp) iff T=S', SeL(Lv,,E) and S(7p) extends to a Radon measure on E ( = # r ).

Remark 2. The operator constructed in the proof of Theorem 1 has an ad- ditional property, namely

Ta=(O(.),a), acE',

where O(x)=x-a(~U) l/p, i.e. r i s decomposed by a r.v. OeL~(OU, P,, E). We want to prove now that Ap(E', Lv) is a linear space. To do so we need a

result about embeddings of Lp into L,, 0 < r < p (cf. [4] or [14]).

Lemma 1. Suppose 0 < r < p . Then there is an operator cl)from L v into Lr(f2',P' ) such that for each f e L v the function q)(f) is a p-stable r.v. with

~ ( f ) (t) = exp ( - II f lip ItlP) �9

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174 W. Linde

Consequently,

where II qb(/) L =crp II f IPp

% = (El01 r} l/r, 0 ( t ) = exp (-It[P).

L e m m a 2. An operator T~L(E', Lp) belongs to Ap(E', Lp) iff ~ T is decomposed by an E-valued r.w (p.

Moreover, #T = dist(q)).

Proof The opera tor ~ T f r o m E' into Lr(f2', P') generates a cylindrical measure v on E with

f ( a ) = ( ~ a ) ( 1 ) (cf. [19] p. 350).

It is known (cf. [19-1 p. 351) that v extends to a Radon measure on E iff 4~Tis decomposed by an E-valued r.v. ~o. Moreover , in this case dist(cp) equals the Radon extension of v. Thus it remains to prove # = v where # is the cylindrical measure on E with

/~(a) = exp (-IlZa]pP), a~E'.

But this follows from

for all atE'.

f(a) = (~bT) (a) (1) = exp( - IJ Ta II p) =f i (a)

F r o m this we get

Theorem 2. Av(E' , Lv) is a linear space.

Proof The only difficulty is to show that Ap(E', Lp) is closed under summation. But this follows directly from L e m m a 2 above.

Remark. It is easy to see that TEAp(E',Lp) and S~L(E,F), F Banach space, implies TS'~Ap(F',Lfl. Unfortunately , this is false for A~L(Lv) , i.e. there are T~Ap(E',Lfl and A~L(Lp) such that ATCAp(E',Lfl (cf. [24]). This shows the difference with the Gaussian case.

Next we want to define some norms, resp. quasi-norms on Ap(E', Lfl. To do so we put for T6Ap(E', Lp)

and

2r(T) = {~ II x FI r d#r(X) } i/r, E

2 v ( r ) = s u p t# r{ I[xll > t} l/v, t > 0

I(T) = lim t#r([Ix]l >t} ~/p.

O < r < p ,

It follows from the results of [-1] that all expressions make sense and are finite.

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Operators Generating Stable Measures on Banach Spaces 175

Easy calculations give us

l(T) <= 2v(T),

2~ ( T) <= p/(p - r) ,~v( T),

II Tall--(C~ 1 )~,(T)Ilall,

0 < r < p ,

0 < r < p , acE', and

II ra[[ < c~ 1 l ( r ) [lalL,

where c v = lim tP {]01 > t} 1/v. t ~ o O

Using Lemma 2 one easily proves

acE',

Theorem 3. 2 r is a norm (1 < r < p ) , resp. a quasi-norm ( 0 < r < 1) on Ap(E', Lp). 2, and l are quasi-norms as well. Moreover, Ap(E', Lp) is complete with respect to 2~, O<r<=p.

Remark. In general Ap(E', Lp) is not complete with respect to l. We will treat this problem later on.

As a consequence of Theorem 3 we get

Theorem 4. For 0 < r < p there exists a constant c only depending on r and p such that for all Banach spaces E and all #cRp(E) the following estimation is valid:

sup t#{lLxll >t}l/v _<c{j ILxll'd#(x)}l< t > 0 E

Proof. It follows from Theorem 3 that there is a constant c for all # = #T where T maps into some fixed space Lp. Given a sequence {#,}_~Rv(E ) one can choose a common space L v and operators T, cAv(E' , Lv) such that # ,=# r~ , n = 1, 2 . . . . . This follows by taking the /v-product of the corresponding spaces L v. From this easily follows that c can be chosen independently of the repre- sentation of #. Finally, since # is Radon we may assume E to be separable. Imbedding E into C[0, 1] we get the independence of E.

Remark. Theorem 4 generalizes a result of J. Hoffmann-Jorgensen in the case of p-stable measures having discrete spectral measure (cf. [6]). But it is known that even in R 2 there are p-stable symmetric measures without discrete spectral measure (1 < p < 2 ) . Let us recall that #cRp(E) has discrete spectral measure if

t h e r e a r e x i ' x 2 . . . . eE such that #=dis t ( ~ xiOi)

Corollary 1. I f 0 < r < p then there is a constant c > 0 only depending on r and p such that for all Banach spaces E and all #eRp(E) the following estimation is valid:

~(~ U)I/P < c {~ IIxFd#} ~/~. g

cr denotes the spectral measure of #.

Proof. Because of l(T) =- % a(~ U) 1Iv

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176

for # = #7- (cf. [-3]) the result follows from

a(O U) lip = c; a I(T) < c; 12v(T ) < c' c~- 1 2r(T )

where c' is the constant of Theorem 4.

Remark. If #=d i s t xiO~, Xa,. . . ,x, eE, this inequality implies i

known estimation (cf. [-15])

w. Linde

the well-

3. Further Properties of Ap(E', Lp)

We want to prove that each operator T~Av(E',Lv) is decomposed by a r.v. tp~L;(f2, P,E). We give two different proofs of this. But before we need several lemmas.

Lemma 3. Let v be a Radon measure on E such that

exp ( - ~ I(x, a)f dv(x)) E

is the ch.f of a Radon measure # on E. Then

IIx F dr(x) = G(a U) E

where a denotes the spectral measure of # on 0 U.

Proof Without loss of generality we may assume v({0})=0. Defining ~ by 9(B) = 2 - ~ v ( B ) + 2 -a v ( -B) we get a symmetric measure ~ on E with

From

with

we get

Defining

fi(a) = exp ( - j'l(x, a ) f d~(x)). E

oo

- I s f = A ; 1 S (cos (s t ) - 1)t- l-P dt, s~R, 0

oo

Ap= S (1-cost) t-a-P dt>O 0

O E

2 ( B ) - g(sx~B)s- l-Vdx O

=p-1 ~ ~(sl/PB)ds=(Rp)-i ~ v(sign(s)lsll/PB)ds 0 - - o o

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Operators Generating Stable Measures on Banach Spaces 177

it follows

fi(a) = exp (A~- 1 j'(cos ((x, a ) ) - 1) d2(x)). E

Since 2 is symmetric and because of 2({0})=0, A~-12 is the uniquely de- termined Levy measure of # (cf. [25]). It is known that the Levy measure is finite on E\U(O) for each open neighborhood U(0) of zero. Consequently,

,~{llxll > l}

is finite and independent of the special choice of v. On the other hand,

~{ Hx II > 1} = p - 1 ~ ~{ ILx II > s l/p} as 0

=p-1 ~ v{llxl] > s l / ' } d s = p - l j HxllPdv(x). 0 E

If a denotes the spectral measure of # on 0 U then

fi(a) = exp ( - j' I{x, a)l p da(x)) OU

which implies

j IlxllPdv(x) = j IlxllPda(x)=a(oe), E OU

Corollary 2. Let O be an E-valued r.v. such that the operator T defined by

Ta = (t)( . ), a), acE', belongs to Ap(E', Lp).

Then ~pELp(Q, P, E) and 11 ~II p = c)- 1 I(T). Consequently,

]10Up_-< e2r(T)

where c depends only on r and p.

Lemma 4. Let tp and T be defined as in Corollary 2. Then

P {O(o)eN(T) ~ = 1

where

and

N (T) = {acE', Ta=0}

N ( T ) ~ ( x , a ) = O for aeN(T)} .

Proof. Let v = dist (0) be the distribution of 0. Then v is a Radon measure on E. Particularly,

v( (~ F~)= inf v(F~)

for each decreasing family F~ of closed sets. N o w ,

N(T) ~ (~ {xeE; (x, a l ) = ... = (x , a , ) =0} {at ..... an} ~-- N ( T )

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178 W. Linde

which implies

v(N(T) ~ = inf v {(x, a l ) . . . . . (x, a , ) =0}. { a l . . . . . an} c_ N(T)

Because of

for a e N ( r )

we get

proving the lemma.

v { x e E ; <x, a ) =0} = p { r a = O} = 1

v(N(v) ~ = 1

Lemma 5. Suppose E has the metric approximation property (cf. [13]) and E is separable. Then the finite rank operators in Av(E',Lv) are dense in Ap(E',Lp) with respect to 2 r.

Proof Let S, eL(E) be a sequence of finite rank operators with IIS.II __<1 and lim S , x = x for all xeE. Then we get

n ~ o o

lim 2~(TS' n - T ) = 0 n ~ 09

because of

2r (TS' , - T) = {j" II S, x - x II r d#T (x)} 1/r E

using Lebesgue's convergence theorem.

Remark. Lemma 5 remains true without any restriction to E (personal com- munication of J. Rosifiski).

Now we are able to prove the main result of this section.

Theorem 5. Assume Te A v ( E' , Lv). Then there exists a r.v. g, e Lv ( f2, P, E) such that

T a = ( r aeE'.

Moreover,

II ~' II p = c~ 1 l(T) < c2r (T).

Proof Using Lemma 4 we can and do assume E to be separable. Furthermore, E can be isometrically embedded into a separable space having the metric approximation property, for instance C[0, 1]. Again using Lemma 4 it suffices to prove Theorem 5 for separable Banach spaces having the metric approxima- tion property. If TeAp(E',Lp) is a finite rank operator then it is ~c(E',E)- continuous. Consequently, there exists a r.v. ~keLv(f2, P,, E) such that

Ta = ( ~ ( . ) , a), aeE'.

By Lemma 3 (Corollary 2) we get

IIr <c2r(Z).

Now the general case follows from Lemma 5 approximating an arbitrary operator TsAv(E', Lv) by finite rank operators.

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Operators Generating Stable Measures on Banach Spaces 179

Remark. Theorem 5 becomes false in the case p = 2. More precisely, it is true in this case iff E has cotype 2 (cf. [15]).

Second Proof of Theorem 5. Here we need a result of [8] which is a con- sequence of a deep theorem due to Rudin (cf. [23]).

Theorem ([8]). Let p be a real number with p~=2k, k= 1, 2 . . . . . I f T~L(E',Lp) satisfies

IITallP=j'l(x,a)lPdv(x), aeE' E

where v is a Radon measure on E with ~llxllPdv(x)<oo, then there is a E

@~Lp(f2, P~ E) such that Ta = (~b(. ), a). Moreover, II0ilp-- {~ Ilxll p dr(x)} lip E

Using Remark 2 of Theorem 1 together with this theorem we get the conclusion of Theorem 5.

Remark. We included the first proof since it is elementary and uses only some well-known properties of Levy measures. Besides it provides us with a formula for the calculation of the Levy measure for a given TSAp(E', Lp) (Lemma 3). Next we want to state two corollaries of Theorem 5.

Corollary 3. I f 1 < p < 2 then each operator T in Ap(E',Lp) is p-nuclear in the sense of [18].

Proof This follows from Theorem 5 and Theorem 1 of [17].

Corollary 4. Let q be a real number with 0 < q < p and let Tbe an operator from E' into Lp such that

exp( - I I Zall~p)

is the ch f of a (q-stable) Radon measure on E. Then

TeAp(E', Lp).

Proof Because of

II Zall~ = c J I[ �9 Tall~

it follows dPT~Aq(E', Lq). Then Theorem 5 and Lemma 2 imply T~Ap(E', Lp).

Remark. The converse of Corollary 4 is also true (cf. [9]).

4. Banach Space of Stable Type p

We saw that each operator T~Ap(E', Lp) can be written as

Ta=(~b( . ) ,a) , acE',

with OeL p(f2, P,E). One may ask now for which spaces this defines a sur- jection from Ap(E', Lp) onto Lp(f2, P, E). But this question is already answered (cf. [2] and [16]).

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180 W. Linde

Theorem 6. A Banach space E is of stable type p iff for each tp~Lv(~2, P, E) the operator T o defined by

T o a = ( ~ ( . ) , a ) belongs to Av(E', Lv).

Proof Using the results of this paper one can give a rather easy proof. If E is of stable type p then

2~(To)<=cHOlIp

for step functions ~ where c is the stable type p constant with respect to the r- th moment. Then the result follows by the completeness of Ap(E',Lp) with respect to 2r approximating an arbitrary OffLp(f2, P,E) by step functions. The converse is an easy consequence of the closed graph theorem.

Now we want to characterize Banach spaces E for which l defines a complete quasi-norm on Ap(E', Lv).

Theorem 7. The following are equivalent: (i) E is of stable type p.

(ii) [(ii)'] For one leach I infinite dimensional space Lp Ap(E', Lv) is complete with respect to I.

(iii) There is a constant c > 0 such that for all #~Rp(E) the estimation

sup t#{rlxll > t} l/p <cl im t#{llxtl > t} ~/p t > O t ~ o o

holds.

Proof. Using the same ideas as in the proof of Theorem 4 it follows the equivalence of (ii)' and (iii).

Let us now assume E satisfies (ii) for one infinite dimensional space Lp. By the closed graph theorem there is a constant c > 0 such that

2r(T) <= cl(T)

for all TeAv(E', Lv). Given x 1 .. . . , xneE we construct an operator T~Ap(E', Le) with

Because of

]A T = dist x i 0 i . \ i = 1

l (T)=cv [[xilf ) lip i

we get (i). Suppose now (i) is satisfied and let/z be in Rp(E). Then we choose T and

as in the proof of Theorem 1, i.e. T a = ( . , a ) a ( c 3 U ) lip and ~9(x)=x.a(c3U)l! v where a is the spectral measure of/~ = #r.

(i) and Theorem 6 imply

2~ (T) =< e IF ~ ]1 p = co(c3 U)1/p = c c; ~ l(T)

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Operators Generating Stable Measures on Banach Spaces 181

where c is the constant appearing in the definition of stable type p. Con- sequently,

sup t# { I1 x 11 > t} 1/p = 2;(T) < c' 2r(T ) < c" I(T) t > 0

=c" lim t~{llxl] >t} 1/" t ~ o o

with c" independent of/~. This proves Theorem 7.

5. p-stable Measures on Lq-spaces

Our aim is to describe Ap(Lr for l < q < ~ , p~=q. Let us mention that S. Kwapien recently gave a description of operators in Ap(Lp,, L,), 1 <p <2.

We start with a very useful lemma.

Lemma 6. Suppose 1 < p, q < 2. Then T6 Ap (Lq,, Lp) iff T' ~ Aq (Lp,, Lq).

Proof Of course we only have to prove one implication. We assume TeAp(Lr Lp) and choose a number r with

1 < r < min (p, q).

~bp and ~bq denote the injections of Lemma 1 from Lp into L r and from L~ into Lr, respectively. From Lemma 2 it follows that the operator ~pT~'q is decom- posed. Since this operator maps L r, into L r its dual operator ~qT'~'p is decomposed as well. Using Lemma 4 and the injectivity of ~p it follows that ~qT' iS decomposed, i.e. T'EAq(Lp,, Lq) (Lemma 2).

Theorem 8. I f max(1,p)<q< oo then Tbelongs to Ap(Lq,,L,) iff

Tg=(g,~( . ) ) with OELp(La).

I f 1 < q < p < 2 , then r~Ap(Lq,, Lp) iff

Tg= s g(to)O(co)dP(co) with OsL~(Lp). f2

Proof Since Lq is of stable type p in the first case the result follows by Theorem 6. Using Lemma 6 the second statement is an easy consequence of the first one.

Remark. Let us mention that T~Ap(Lr is equivalent to the fact that T' is q-absolutely summing (cf. 1-19] for the definition) in the first case and that Tis q-absolutely summing in the second one.

Finally, we want to describe operators in Ap(lr Lp) (cf. also 1-21]).

Corollary 5. The operator T belongs to Ap(lq,, Lp) iff

IZell q ffgp, max(l , p) <q < oG i

F, ItZeillq< o% l < q < p < 2 . i = 1

Here e~ denotes the i-th unit vector of lq,.

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182 W. Linde

6. Estimations Between Different p-stable Symmetric Measures

We want to show that a well-known proper ty of Gaussian measures is false for p-stable symmetr ic measures, p < 2 . More precisely, we treat the following problem: Given two p-stable symmetr ic measures on R" with

Ml (x , a)l _-< 1} __< v{l(x, a)l < 1}

for all aeR". Does this imply # ( C ) < v ( C ) for all absolutely convex closed subsets C__cR"? The answer is negative even if C is the intersection of two strips in R 2.

Theorem 9. I f 1 < p < 2 then there are two p-stable symmetric measures # and v on R 2 and a number t o > 0 having the following properties:

(1) #{](x, a)[ < 1} < v{](x, a)[ < 1} for all a~R 2 and (2) 12(C,)>v(Ct) , t>=to,

where C t = {x =(41, 42); max(]r 1 l, ]r -< t}.

Proof It is easy to see (estimating the cotype 2 constant) that R z provided with the norm

I]xll o~ = max(l~ ~1,1~21), x =(41, ~2),

cannot isometrically embedded into a space Lv, l < p < 2 . Using Theorem 7.3. of [-12] then there are elements xa, . . . , x , and ya . . . . ,y , in R 2 such that

(+) ~ Kx , a)l p< ~ I(yi,a)[ v, aER 2, i = l i = I

and

Now we define # and v by

and

Then ( + ) implies

Assuming

i = 1 i = l

/~ = dist yiOi \ i = 1 ]

v = dist i=~1 x~ 0 i .

#{ l (x , a ) l _-< 1} __< v{l(x, a)l < 1},

we get

a~R 2.

it(Ct~)<v(Ct~ ) with lira tk=oe k ~ ao

t~v{llxllo~>tk} <_t~#{llxl[~ >tk}, k = l , 2 , . . . .

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Operators Generating Stable Measures on Banach Spaces 183

Pass ing to the l imi t this impl i e s (cf. [3])

c; ilxi< s c; Hyill: i = l i ~ l

c o n t r a d i c t i n g the cho ice o f X l , . . . , x n a n d Yl . . . . , Y,.

Remark. As p o i n t e d o u t by the referee it w o u l d be ve ry in t e r e s t i ng to f ind

c o n c r e t e e l e m e n t s x l , . . . , x , a n d Yl, . . . ,Y, in R 2. M o r e o v e r , w h a t is the min i - m u m n for wh ich such p o i n t s exis t?

Acknowledgment. The author would like to express his gratitude to the referee for his helpful comments.

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Received April 16, 1980; in slightly revised form January 12, 1982