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Equivalence of the Gelfand–Shilov Spaces
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
203, 828]839 Ž1996.
0414
Equivalence of the Gelfand]Shilov Spaces Jaeyoung Chung Department of Mathematics, Kunsan National Uni¨ ersity, Kunsan 573-360, Korea
Soon-Yeong Chung Department of Mathematics, Sogang Uni¨ ersity, Seoul 121-742, Korea
and Dohan Kim Department of Mathematics, Seoul National Uni¨ ersity, Seoul 151-742, Korea Submitted by H. M. Sri¨ asta¨ a Received April 26, 1995
We prove that there is a one to one correspondence between the Gelfand]Shilov N space WMV of type W and the space SM pp of generalized type S. As an application we prove the equality WM l W V s WMV , which is a generalization of the equality Sr l S s s Srs found by I. M. Gelfand and G. E. Shilov Ž‘‘Generalized Functions, II, III,’’ Academic Press, New YorkrLondon, 1967.. Q 1996 Academic Press, Inc.
INTRODUCTION The purpose of this paper is to investigate the relations between the Gelfand]Shilov spaces of generalized type S and of type W which were introduced by I. M. Gelfand and G. E. Shilov in wGSx. They used these spaces to investigate the uniqueness of the solutions of the Cauchy problems of partial differential equations. Although the Gelfand]Shilov spaces of generalized type S are defined by means of the positive sequences M p and Np and the spaces of type W are defined by means of the weight functions M Ž x . and V Ž y ., we show that the class of the spaces of type W is exactly the same as a class of the spaces of generalized type S. 828 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
GELFAND ] SHILOV SPACES
829
In Section 1 we state definitions of the Gelfand]Shilov spaces of generalized type S and type W, some relations between sequences and their associated functions, and basic concepts of Young conjugates and their properties. In Section 2 we prove that the spaces WM , W V , and WmV of type W are equal to some spaces SM p, S Nq , and SMNqp of generalized type S, respectively, under the natural condition M ; V and vice versa. In Section 3 as an application of the above result we show the equality WM l W V s WMV
Ž W.
under the natural condition M ; V Žsee Section 1 for definition.. For this we first show the equality N
SM p l S N p s SM pp .
Ž S.
The above two equalities are generalizations of the equality Sr l S s s Srs
Ž S0.
which was suggested by Gelfand and Shilov in wGSx and proved affirmatively by Kashpirovsky in wKax in 1979. An equality of type ŽS. was also proved in Pathak wPx under the condition ŽP.. But, this result cannot be applied to prove the equality ŽW. as the defining sequence Np of VU Ž y . satisfies ŽM.1.U Žsee Lemma 2.5. and ŽM.1.U implies the reverse inequality ŽP.U of ŽP.. See Section 3 for definitions of the conditions ŽP. and ŽP.U and for an example. So we replace ŽP. by a natural condition M p Np > p! which is always satisfied if M Ž x . ; V Ž y ., and we modify the proof in wPx in order to make use of the condition M p Np > p! instead of ŽP.. Finally we N also prove the triviality of the spaces S M p l S N p and SM pp under the condition M p Np ; p! s, 0 - s - 1, which will complete the generalization of the equality ŽS0..
1. PRELIMINARIES Let M p , p s 0, 1, 2, . . . , be a sequence of positive numbers. We impose the conditions denoted ŽM.1., ŽM.2., and ŽM.2.X which denote logarithmic convexity, stability under ultradifferential operators, and stability under differential operators. We assume that M0 s 1 for simplicity and refer to wK, p. 26x for details. We also use the multi-index notations < a < s a 1 q ??? qa n , a !s a 1! ??? a n!, and a s Ž r x 1 . a 1 ??? Ž r x n . a n for a s Ž a 1 , . . . , a n . g N 0n.
830
CHUNG, CHUNG, AND KIM
DEFINITION 1.1. For each sequence M p of positive numbers we define the associated function M Ž r . on w0, `. by M Ž r . s sup log p
rp Mp
.
Ž 1.1.
DEFINITION 1.2. Let M p and Np be sequences of positive numbers satisfying ŽM.1.. Then we write M p ; Np Ž M p $ Np , respectively. if there are constants L and C Žfor any L ) 0 there is a constant C ) 0, respectively. such that M p F CL p Np , p s 0, 1, 2, . . . . M p and Np are equi¨ alent if M p ; Np and M p > Np hold. For example, M p s p! s and Np s p p s are equivalent if s ) 0, by Stirling’s formula. DEFINITION 1.3. Let M Ž x . and N Ž x . be real functions. Then we write M Ž x . ; N Ž x . Ž M Ž x . $ N Ž x ., respectively. if there are constants L and C Žfor every L ) 0 there is a constant C ) 0, respectively. such that M Ž x . F N Ž Lx . q C. M Ž x . and N Ž x . are equi¨ alent if M Ž x . ; N Ž x . and M Ž x . > N Ž x .. DEFINITION 1.4. If M Ž r . is an increasing convex function in log r and increases more rapidly than log r p for any p as r tends to infinity, we define its defining sequence by Mr s sup r)0
rp exp M Ž r .
,
p s 0, 1, 2, . . . .
DEFINITION 1.5. Let M : w0, `. ª w0, `. be a convex and increasing function with M Ž0. s 0 and lim x ª` xrM Ž x . s 0. Then we define its Young conjugate M U by M U Ž y . s sup Ž xy y M Ž x . . . x
We now refer to wGS, Vol. II, Chap. IVx for the definitions of the Gelfand]Shilov spaces of type S and to wGS, Vol. II, Appendix 1, Chap. IVx for the Gelfand]Shilov spaces of generalized type S. Let M Ž x . and V Ž y . be differentiable functions on w0, `. satisfying the condition ŽK. M Ž0. s V Ž0. s M X Ž0. s VX Ž0. s 0 and their derivatives continuous, increasing, and tending to infinity. We refer to wGS, Vol. II, Appendix 2, Chap. IVx for the Gelfand]Shilov spaces of type W.
GELFAND ] SHILOV SPACES
831
˜ of Fourier transforms wˆ Ž j . s DEFINITION 1.6. A space F yi x? j HR n w Ž x . e dx of the functions w g F is called the Fourier-dual of F. 2. EQUIVALENCE OF THE GELFAND]SHILOV SPACES In this section we prove the equivalence of the Gelfand]Shilov spaces of type W and the spaces of generalized type S. Let M p , p s 0, 1, 2, . . . , be a sequence of positive numbers. We impose the following conditions on M p ; ŽM.1.X ŽStrong Logarithmic Convexity. m p s M prM py1 is increasing and tending to infinity as p ª `. ŽM.1.U ŽDuality. p!rM p satisfies ŽM.1.X . PROPOSITION 2.1 wK, p. 49x. If M p satisfies ŽM.1., then M p is the defining sequence of the associated function of itself, that is, M p s sup pG0
rp
,
exp M Ž r .
where M Ž r . is the associated function of M p . We can easily obtain the following Lemma 2.2 from the Lemma 2.3 in wCCK, p. 370x. LEMMA 2.2. Let M : w0, `. ª w0, `. be the function satisfying the condition ŽK.. Then M Ž x . is equi¨ alent to the associated function of the defining sequence of itself, i.e., M Ž x . ( sup log p
xp Mp
,
where M p is the defining sequence of M Ž x .. LEMMA 2.3 wCCKx. Let M Ž r . be a function satisfying the condition ŽK.. Then the defining sequence M pU of the Young conjugate M U Ž r . of M Ž r . is equi¨ alent to p!rM p , where M p is the defining sequence of M Ž r .. In fact, M pU s Ž pre. prM p . Con¨ ersely, if M p satisfies ŽM.1.X and ŽM.1.U , then the associated function aŽ . M r of p!rM p and the Young conjugate M U Ž r . of the associated function M Ž r . of M p are equi¨ alent. LEMMA 2.4. Let M p be a sequence of positi¨ e numbers satisfying the conditions ŽM.1.X and ŽM.1.U . Then the associated function M Ž r . of M p is equi¨ alent to a function M0 Ž r . which satisfies the condition ŽK..
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CHUNG, CHUNG, AND KIM
Con¨ ersely, let M Ž r . be a function satisfying the condition ŽK.. Then the defining sequence M p of M Ž r . satisfies the conditions ŽM.1.X and ŽM.1.U up to equi¨ alence. Proof. We may assume that m p s M prM py1 and prm p are strictly increasing. Let w Ž t . be the line segments connecting Ž m p , prm p . and Ž m pq 1 , Ž p q 1.rm pq1 ., p s 1, 2, . . . , where Ž m p , prm p . s Ž0, 0. for p s 0 and let M0 Ž r . s H0r w Ž t . dt. Then the function M0 Ž r . satisfies the condition ŽK.. We claim that the associated function M Ž r . of M p is equivalent to M0 Ž r .. Denote by mŽ l. the number of m p F l. We obtain the following inequalities for m p F l - m pq1 prm p F w Ž l . - Ž p q 1 . rm pq1 prm pq 1 F m Ž l . rl - prm p .
Ž 2.1. Ž 2.2.
Combining Ž2.1. and Ž2.2. we have 1 2
w Ž l . F m Ž l . rl F w Ž l .
Ž 2.3.
for all l ) 0. Integrating Ž2.3. and applying the formula M Ž r . s H0r mŽ l.rl d l in wK, p. 50x and the convexity of M0 , we conclude that M0 Ž 12 l. F M Ž l. F M0 Ž l. for all l G 0. To prove the converse, let M Ž r . be a function satisfying the condition ŽK.. Then the defining sequence M p of M Ž r . satisfies ŽM.1.X . Since the Young conjugate M U Ž r . of M Ž r . also satisfies the condition ŽK. the defining sequence Ž pre. prM p of M U Ž r . also satisfies ŽM.1.X . We complete the proof by Stirling’s formula. LEMMA 2.5 wGS, p. 245x.
Let M p and Np satisfy ŽM.1. and ŽM.1.U . Then
&
SM p s S M p
and
&
S N pS N p .
N
DEFINITION 2.6. We call S M pp a space of type S0 if the sequences M p and Np satisfying the condition ŽM.1.X , ŽM.1.U , and M p Np > p!. Also, we call WMV a space of type W0 if the functions M and V satisfy the condition ŽK. and the relation M ; V. Remark 2.7. Ži. Let M p s p! r , Np s p p s, 0 - r, s - 1, r q s G 1. Then is a space of type S0 . Žii. Let M Ž x . s e x y x y 1, V Ž y . s ye y y y. Then M Ž x . and V Ž y . satisfy ŽK. and M Ž x . ; V Ž y . and, WMV is a space of type W0 . In fact, M Ž x . > V Ž y . also holds and WVM is a space of type W0 . N SM pp
GELFAND ] SHILOV SPACES
833
Žiii. Let M p and Np be the defining sequences of the above funcN tions M Ž x . and VU Ž y ., respectively. Then SM pp is a space of type S0 . Also, let M p and Np be the defining sequences of the above functions M Ž x . and V Ž y .. Then SMp!rp N p is a space of type S0 by Lemma 2.3 and Lemma 2.4. We now prove the main theorem in this section. THEOREM 2.8. There is a one to one correspondence between the spaces of N type S0 and type W0 . In other words, for any gi¨ en space SM pp of type S0 , there Np V V is WM of type W0 such that SM p s WM and ¨ ice ¨ ersa. N
N
U
Proof. For any given spaces SM pp of type S0 , we claim that SM pp s WMN where M is the associated function of M p and N U is the Young conjugate N of the associated function N of Np . Let w g SM pp. Then for every a , b g N 0n we obtain
j a bw Ž j . F CA < a < B < b < M < a < N< b <
Ž 2.4.
for some A, B ) 0. Since Np satisfies ŽM.1.U or p!rNp satisfies ŽM.1.X , it is easy to see that Np $ p!. Hence the function w Ž j . can be continued analytically into the complex domain as an entire analytic function. Applying the Taylor expansion and the inequality Ž2.4. we have
j aw Ž j q ih . F
Ý
< j a gw Ž j . <
ggN 0n
FC
Ý
ggN 0n
g!
A < a < M < a < N< g < < Bh < < g
F 2 n CA < a < M < a < exp N a Ž 2 B
Ž 2.5.
Dividing < j < < a < in both sides of the inequality Ž2.5. and taking infimum for < a < in the right hand side of Ž2.5., we have < w Ž j q ih . < F 2 n C exp yM Ž < j
U
Ž 2.6.
It follows that SM pp ; WMN where M is the associated function of M p , N a is the associated function of p!rNp , and N U is the Young conjugate of the associated function N of Np .
834
CHUNG, CHUNG, AND KIM U
By Lemma 2.4, WMN is a space of type W and also the fact M p Np > p! U implies M ; N a ( N U . Hence WMN is a space of type W0 . Conversely, let U w g WMN . Then the inequality Ž2.6. is satisfied. Hence by the Cauchy integral formula together with the inequality Ž2.6. we have < bw Ž x . < s
b!
Ž 2p i .
FC FC sC
n
j
b! R< b <
b! R< b <
b! R< b <
H< z yx
sup < z jyx j
w Ž z . d z 1 . . . d zn 1
y x1 .
b 1 q1
??? Ž zn y x n .
b n q1
exp yM Ž a < j < . q N U Ž b
exp N U Ž bX R .
sup < z jyx j
exp yM Ž a < j < .
exp N U Ž bX R . exp yM Ž a < j 0 < . ,
Ž 2.7.
where the quantity M Ž a < j <. attains its minimum at j s j 0 . In fact, we can write j 0 s x q u R where u s Ž u 1 , . . . , un ., u i s 0 or "1, i s 1, 2, . . . , n. By the convexity of M and the relation M ; N U , we have exp yM Ž a < j 0 < . s exp yM Ž a < x q u R < . F exp yM Ž a Ž < x < y < u R < . . F exp yM F exp yM
a
< x < exp M Ž a1 R .
ž / ž / 2 a 2
< x < exp N U Ž a2 R . .
Hence the inequality Ž2.7. is reduced to < bw Ž x . < F C b ! F Cb !
exp 2 N U Ž a2 q bX . R R
exp yM
exp N U 2 Ž a2 q bX . R R
exp yM
a
< x<
ž / ž / 2
a
2
< x<
.
Ž 2.8.
Multiplying < x a < in both sides of Ž2.8., taking the supremum for < x <, and taking the infimum for R in the right hand side of Ž2.8. we have < x a bw Ž x . < F C1 b ! inf R
exp N U Ž cR . R< b <
sup x
< x<
GELFAND ] SHILOV SPACES
F C1 Ž 2ra.
F C1 Ž cra.
ž
M < a < b ! sup
c
R
M< a < b !
ž
835 y1
R< b < exp N U Ž cR .
Ž < b
/
y1
/
F C1 A < a < B < b < M < a < N< b < , where A s 2ra, B sUce. N It follows that WMN ; SM pp. Now, for a given space WMV of type W0 let M p and Np be the defining sequences of M and VU . Then M p and Np satisfy ŽM.1.X and ŽM.1.U by Lemma 2.4. Furthermore, the relation M ; V N implies M p > p!rNp or M p Np > p!. Thus SM pp is a space Uof type S0 . But by Np the first part of the proof, the space SM p is equal to WMN11 where M1 is the associated function of M p and N1U is the Young conjugate of the associU ated function N1 of Np . Also WMN11 is equal to WMV since M1 and M are equivalent and N1 and VU are equivalent and so are N1U and Ž VU .U s V. N Therefore we have WMV s SM pp which completes the proof. Using the similar method as in Theorem 2.8 and Lemma 2.5 we obtain the following theorem. THEOREM 2.9. Let WM and W V be spaces of type W. Then there exist spaces SM p and S N p of generalized type S such that WM s SM p and W V s S N p . In this case, the sequences M p and Np satisfy the conditions ŽM.1.X and ŽM.1.U . Con¨ ersely, if M p and Np satisfy the conditions ŽM.1.X and ŽM.1.U , then the spaces SM p and S N p are equal to some spaces WM and W V of type W, respecti¨ ely. 3. EQUALITY FOR THE SPACES OF GENERALIZED TYPE S AND TYPE W Applying the results of the above section we prove, in this section, the equality ŽW. under the non-triviality condition M Ž x . ; V Ž y .. For this equality we first prove the equality ŽS. under the conditions which are satisfied by the defining sequences M p and Np of M Ž x . and VU Ž y ., respectively, where M Ž x . ; V Ž y .. First we state Pathak’s result on the equality ŽS.. THEOREM 3.1 wPx.
Suppose that there exists a positi¨ e constant C such that
Npq q G C
ž
pqq Np Nq , p
/
p, q s 0, 1, . . .
Ž P.
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CHUNG, CHUNG, AND KIM
and that ŽM.2. holds for Np and ŽM.1. and ŽM.2. hold for M p . Then the equality ŽS. holds. But, we cannot apply this result to prove the equality ŽW. as the defining sequence Np of VU Ž y . satisfies ŽM.1.U Žsee Lemma 2.5. and ŽM.1.U implies the following reverse inequality
Npq q F C
ž
pqq Np Nq , p
p, q s 0, 1, . . .
/
Ž P.
U
for some constant C. For example, the condition ŽP. is not satisfied by Np s p! s, 0 - s - 1, which is the defining sequence of some VU Ž y . where V Ž y . satisfies the condition ŽK.. So we replace ŽP. by a natural condition M p Np > p! which is always satisfied if M Ž x . ; V Ž y ., and we modify the proof in wPx in order to make use of the condition M p Np > p! instead of ŽP.. Let M p , p s 0, 1, 2, . . . , be a sequence of positive numbers. We impose one of the following conditions on M p : ŽM.0. ŽNontriviality. M p > p!; ŽM.0X ŽTriviality. M p ; p! s, 0 - s - 1. N
We first prove the equality SM p l S p s SMN pp for the case M p Np > p!, which is a generalization of the equality Sr l S s s Srs for the case r q s G 1. Making use of integration by parts, the Leibniz formula, and the Schwarz inequality we can obtain the following: LEMMA 3.2. If M p and Np satisfy the condition ŽM.2.X , then the supremum norm 5 ? 5 ` and the L2-norm 5 ? 5 2 are equi¨ alent for the spaces of type S. THEOREM 3.3. If M p and Np satisfy the conditions ŽM.1. and ŽM.2., and if M p Np satisfies ŽM.0., then the equality N
SM p l S N p s SM pp holds.
GELFAND ] SHILOV SPACES
837
Proof. Using integration by parts, the Leibniz formula, and the Schwarz inequality as in wKax we obtain 5 x a bw Ž x . 5 22 s F
HR
x 2 a bw Ž x . bw Ž x . dx
n
Ý
kF2 a kF b
F C2 Ý k
b k
2a k! 5 2 bykw Ž x . 5 2 5 x 2 aykw Ž x . 5 2 k
b k
2a k! A2Ž < a
ž /ž /
ž /ž /
F C 2A2Ž < a
F C 2 Ž 2 AH .
2 Ž < a
b k
y1 2a k! Ž M < k < N< k < . k
ž /ž /
M < 2a < N< 2b < .
Ž 3.1.
N This implies that w Ž x . belongs to SM pp in view of Lemma 3.2. The reverse inclusion is obvious, which completes the proof. N
Remark 3.4. Let SM pp be a space of type S0 . Then M p and Np satisfy ŽM.1. and ŽM.1.U . Since ŽM.1.U implies ŽP.U and ŽP.U implies ŽM.2., the equality ŽS. holds by Theorem 3.3. THEOREM 3.5.
Let WMV be a space of type W0 . Then the equality WM l W V s WMV
holds. Proof. For given spaces WM , W V , and WMV of type W0 there exist SM p, N S , and SM pp of type S0 such that Np
WM s SM p ,
W V s S Np ,
and
N
WMV s SM pp
by Theorem 2.8 and Theorem 2.9. N N N Since SM pp is a space of type S0 , the equality SM p l S p s SM pp holds by Remark 3.4. Consequently, we have the equality WM l W V s WMV . N
We now prove the triviality of the spaces SM pp and S M p l S N p under the condition M p Np ; p! s, 0 - s - 1, which generalizes the equality fSr l S s s Srsor the other case r q s - 1, which will complete the generalization of the quality ŽS0..
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CHUNG, CHUNG, AND KIM
THEOREM 3.6. If M p and Np satisfy the condition ŽM.2.X and M p Np N satisfies ŽM.0.X , then both the spaces SM pp and SM p l S N p are tri¨ ial. N
Proof. If w Ž x . belongs to S M pp or SM p l S N p , then by the condition ŽM.0.X , w Ž x . is continued analytically into the complex plane as an entire analytic function. For w Ž x . g SMN pp we have 5 x a aw Ž x .5 ` F CA < a < < a
ž
Ý
5 x aykq1 aykq1w Ž x . 5 2
kF a kF Ž1, . . . , 1 .
q
Ý
5 x ayk aykq1w Ž x . 5 2
kF a kF Ž1, . . . , 1 .
/
for some constants C and A. Replacing a , b by a y k q 1 in Ž3.1., we have 5 x aykq1 aykq1w Ž x . 5 22 F C 2 Ž 2 A . 4 < a <
Ý
j!M < 2 aq2yj < N< 2 aq2yj <
Ý
< j < !1ys Ž < j < ! <2 a q 2 y j < ! .
jF a q1
F C 2 Ž 2 A.
4
s
jF a q1
F C12 A21 < a < < a < !1q s . Similarly, we have 5 x ayk aykq1w Ž x .5 22 F C22 A22 < a < < a
0 - s - 1,
then w Ž z . degenerates to a constant function. In fact, by the Taylor expansion we obtain
bw Ž 0 . s
Ý aq b Ž j . Ž yj . a
a
ra !.
Ž 3.2.
GELFAND ] SHILOV SPACES
839
By letting < j < ª `, Ž3.2. implies that bw Ž0. s 0 for b / 0, hence w is N constant. Therefore the spaces SM pp and SM p l S N p are trivial. ACKNOWLEDGMENTS This research is partially supported by Ministry of Education and GARC. Also, we thank the referees for helpful suggestions, especially for the proof of Theorem 3.6.
REFERENCES wBMTx
R. W. Braun, R. Meise, and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Resultate Math. 17 Ž1990., 206]237. wCCK1x J. Chung, S.-Y. Chung, and D. Kim, Une caracterisation de l’espace de Schwartz, C. ´ R. Acad. Sci. Paris Ser. ´ I Math. 316 Ž1993., 23]25. wCCK2x J. Chung, S.-Y. Chung, and D. Kim, Characterizations of the Gelfand]Shilov spaces via Fourier transforms, Proc. Amer. Math. Soc., in press. wCKx S.-Y. Chung and D. Kim, Representation of quasianalytic ultradistributions, Ark. Mat. 31 Ž1993., 51]60. wCKKx S.-Y. Chung, D. Kim, and S. K. Kim, Equivalence of the spaces of ultradifferentiable functions and its application to the Whitney extension theorem, Rend. Mat. Appl. Ž7. 12 Ž1992., 365]380. wGSx I. M. Gelfand and G. E. Shilov, ‘‘Generalized Functions, II, III,’’ Academic Press, New YorkrLondon, 1967. wHx L. Hormander, ‘‘The Analysis of Linear Partial Differential Operator, I,’’ Springer¨ Verlag, BerlinrNew York, 1983. wKax A. I. Kashpirovsky, Equality of the spaces Sab and Sa l S b , Functional Anal. Appl. 14 Ž1978., 60. wKx H. Komatsu, Ultradistributions, I, J. Fac. Sci. Uni¨ . Tokyo Sect. IA 20 Ž1973., 25]105. wPx R. S. Pathak, Tempered ultradistributions as the boundary values of analytic functions, Trans. Amer. Math. Soc. 286 Ž1984., 537]556. wTx F. Treves, ‘‘Linear Partial Differential Equations with Constant Coefficients,’’ Gordon & Breach, New York, 1966.
203, 828]839 Ž1996.
0414
Equivalence of the Gelfand]Shilov Spaces Jaeyoung Chung Department of Mathematics, Kunsan National Uni¨ ersity, Kunsan 573-360, Korea
Soon-Yeong Chung Department of Mathematics, Sogang Uni¨ ersity, Seoul 121-742, Korea
and Dohan Kim Department of Mathematics, Seoul National Uni¨ ersity, Seoul 151-742, Korea Submitted by H. M. Sri¨ asta¨ a Received April 26, 1995
We prove that there is a one to one correspondence between the Gelfand]Shilov N space WMV of type W and the space SM pp of generalized type S. As an application we prove the equality WM l W V s WMV , which is a generalization of the equality Sr l S s s Srs found by I. M. Gelfand and G. E. Shilov Ž‘‘Generalized Functions, II, III,’’ Academic Press, New YorkrLondon, 1967.. Q 1996 Academic Press, Inc.
INTRODUCTION The purpose of this paper is to investigate the relations between the Gelfand]Shilov spaces of generalized type S and of type W which were introduced by I. M. Gelfand and G. E. Shilov in wGSx. They used these spaces to investigate the uniqueness of the solutions of the Cauchy problems of partial differential equations. Although the Gelfand]Shilov spaces of generalized type S are defined by means of the positive sequences M p and Np and the spaces of type W are defined by means of the weight functions M Ž x . and V Ž y ., we show that the class of the spaces of type W is exactly the same as a class of the spaces of generalized type S. 828 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
GELFAND ] SHILOV SPACES
829
In Section 1 we state definitions of the Gelfand]Shilov spaces of generalized type S and type W, some relations between sequences and their associated functions, and basic concepts of Young conjugates and their properties. In Section 2 we prove that the spaces WM , W V , and WmV of type W are equal to some spaces SM p, S Nq , and SMNqp of generalized type S, respectively, under the natural condition M ; V and vice versa. In Section 3 as an application of the above result we show the equality WM l W V s WMV
Ž W.
under the natural condition M ; V Žsee Section 1 for definition.. For this we first show the equality N
SM p l S N p s SM pp .
Ž S.
The above two equalities are generalizations of the equality Sr l S s s Srs
Ž S0.
which was suggested by Gelfand and Shilov in wGSx and proved affirmatively by Kashpirovsky in wKax in 1979. An equality of type ŽS. was also proved in Pathak wPx under the condition ŽP.. But, this result cannot be applied to prove the equality ŽW. as the defining sequence Np of VU Ž y . satisfies ŽM.1.U Žsee Lemma 2.5. and ŽM.1.U implies the reverse inequality ŽP.U of ŽP.. See Section 3 for definitions of the conditions ŽP. and ŽP.U and for an example. So we replace ŽP. by a natural condition M p Np > p! which is always satisfied if M Ž x . ; V Ž y ., and we modify the proof in wPx in order to make use of the condition M p Np > p! instead of ŽP.. Finally we N also prove the triviality of the spaces S M p l S N p and SM pp under the condition M p Np ; p! s, 0 - s - 1, which will complete the generalization of the equality ŽS0..
1. PRELIMINARIES Let M p , p s 0, 1, 2, . . . , be a sequence of positive numbers. We impose the conditions denoted ŽM.1., ŽM.2., and ŽM.2.X which denote logarithmic convexity, stability under ultradifferential operators, and stability under differential operators. We assume that M0 s 1 for simplicity and refer to wK, p. 26x for details. We also use the multi-index notations < a < s a 1 q ??? qa n , a !s a 1! ??? a n!, and a s Ž r x 1 . a 1 ??? Ž r x n . a n for a s Ž a 1 , . . . , a n . g N 0n.
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CHUNG, CHUNG, AND KIM
DEFINITION 1.1. For each sequence M p of positive numbers we define the associated function M Ž r . on w0, `. by M Ž r . s sup log p
rp Mp
.
Ž 1.1.
DEFINITION 1.2. Let M p and Np be sequences of positive numbers satisfying ŽM.1.. Then we write M p ; Np Ž M p $ Np , respectively. if there are constants L and C Žfor any L ) 0 there is a constant C ) 0, respectively. such that M p F CL p Np , p s 0, 1, 2, . . . . M p and Np are equi¨ alent if M p ; Np and M p > Np hold. For example, M p s p! s and Np s p p s are equivalent if s ) 0, by Stirling’s formula. DEFINITION 1.3. Let M Ž x . and N Ž x . be real functions. Then we write M Ž x . ; N Ž x . Ž M Ž x . $ N Ž x ., respectively. if there are constants L and C Žfor every L ) 0 there is a constant C ) 0, respectively. such that M Ž x . F N Ž Lx . q C. M Ž x . and N Ž x . are equi¨ alent if M Ž x . ; N Ž x . and M Ž x . > N Ž x .. DEFINITION 1.4. If M Ž r . is an increasing convex function in log r and increases more rapidly than log r p for any p as r tends to infinity, we define its defining sequence by Mr s sup r)0
rp exp M Ž r .
,
p s 0, 1, 2, . . . .
DEFINITION 1.5. Let M : w0, `. ª w0, `. be a convex and increasing function with M Ž0. s 0 and lim x ª` xrM Ž x . s 0. Then we define its Young conjugate M U by M U Ž y . s sup Ž xy y M Ž x . . . x
We now refer to wGS, Vol. II, Chap. IVx for the definitions of the Gelfand]Shilov spaces of type S and to wGS, Vol. II, Appendix 1, Chap. IVx for the Gelfand]Shilov spaces of generalized type S. Let M Ž x . and V Ž y . be differentiable functions on w0, `. satisfying the condition ŽK. M Ž0. s V Ž0. s M X Ž0. s VX Ž0. s 0 and their derivatives continuous, increasing, and tending to infinity. We refer to wGS, Vol. II, Appendix 2, Chap. IVx for the Gelfand]Shilov spaces of type W.
GELFAND ] SHILOV SPACES
831
˜ of Fourier transforms wˆ Ž j . s DEFINITION 1.6. A space F yi x? j HR n w Ž x . e dx of the functions w g F is called the Fourier-dual of F. 2. EQUIVALENCE OF THE GELFAND]SHILOV SPACES In this section we prove the equivalence of the Gelfand]Shilov spaces of type W and the spaces of generalized type S. Let M p , p s 0, 1, 2, . . . , be a sequence of positive numbers. We impose the following conditions on M p ; ŽM.1.X ŽStrong Logarithmic Convexity. m p s M prM py1 is increasing and tending to infinity as p ª `. ŽM.1.U ŽDuality. p!rM p satisfies ŽM.1.X . PROPOSITION 2.1 wK, p. 49x. If M p satisfies ŽM.1., then M p is the defining sequence of the associated function of itself, that is, M p s sup pG0
rp
,
exp M Ž r .
where M Ž r . is the associated function of M p . We can easily obtain the following Lemma 2.2 from the Lemma 2.3 in wCCK, p. 370x. LEMMA 2.2. Let M : w0, `. ª w0, `. be the function satisfying the condition ŽK.. Then M Ž x . is equi¨ alent to the associated function of the defining sequence of itself, i.e., M Ž x . ( sup log p
xp Mp
,
where M p is the defining sequence of M Ž x .. LEMMA 2.3 wCCKx. Let M Ž r . be a function satisfying the condition ŽK.. Then the defining sequence M pU of the Young conjugate M U Ž r . of M Ž r . is equi¨ alent to p!rM p , where M p is the defining sequence of M Ž r .. In fact, M pU s Ž pre. prM p . Con¨ ersely, if M p satisfies ŽM.1.X and ŽM.1.U , then the associated function aŽ . M r of p!rM p and the Young conjugate M U Ž r . of the associated function M Ž r . of M p are equi¨ alent. LEMMA 2.4. Let M p be a sequence of positi¨ e numbers satisfying the conditions ŽM.1.X and ŽM.1.U . Then the associated function M Ž r . of M p is equi¨ alent to a function M0 Ž r . which satisfies the condition ŽK..
832
CHUNG, CHUNG, AND KIM
Con¨ ersely, let M Ž r . be a function satisfying the condition ŽK.. Then the defining sequence M p of M Ž r . satisfies the conditions ŽM.1.X and ŽM.1.U up to equi¨ alence. Proof. We may assume that m p s M prM py1 and prm p are strictly increasing. Let w Ž t . be the line segments connecting Ž m p , prm p . and Ž m pq 1 , Ž p q 1.rm pq1 ., p s 1, 2, . . . , where Ž m p , prm p . s Ž0, 0. for p s 0 and let M0 Ž r . s H0r w Ž t . dt. Then the function M0 Ž r . satisfies the condition ŽK.. We claim that the associated function M Ž r . of M p is equivalent to M0 Ž r .. Denote by mŽ l. the number of m p F l. We obtain the following inequalities for m p F l - m pq1 prm p F w Ž l . - Ž p q 1 . rm pq1 prm pq 1 F m Ž l . rl - prm p .
Ž 2.1. Ž 2.2.
Combining Ž2.1. and Ž2.2. we have 1 2
w Ž l . F m Ž l . rl F w Ž l .
Ž 2.3.
for all l ) 0. Integrating Ž2.3. and applying the formula M Ž r . s H0r mŽ l.rl d l in wK, p. 50x and the convexity of M0 , we conclude that M0 Ž 12 l. F M Ž l. F M0 Ž l. for all l G 0. To prove the converse, let M Ž r . be a function satisfying the condition ŽK.. Then the defining sequence M p of M Ž r . satisfies ŽM.1.X . Since the Young conjugate M U Ž r . of M Ž r . also satisfies the condition ŽK. the defining sequence Ž pre. prM p of M U Ž r . also satisfies ŽM.1.X . We complete the proof by Stirling’s formula. LEMMA 2.5 wGS, p. 245x.
Let M p and Np satisfy ŽM.1. and ŽM.1.U . Then
&
SM p s S M p
and
&
S N pS N p .
N
DEFINITION 2.6. We call S M pp a space of type S0 if the sequences M p and Np satisfying the condition ŽM.1.X , ŽM.1.U , and M p Np > p!. Also, we call WMV a space of type W0 if the functions M and V satisfy the condition ŽK. and the relation M ; V. Remark 2.7. Ži. Let M p s p! r , Np s p p s, 0 - r, s - 1, r q s G 1. Then is a space of type S0 . Žii. Let M Ž x . s e x y x y 1, V Ž y . s ye y y y. Then M Ž x . and V Ž y . satisfy ŽK. and M Ž x . ; V Ž y . and, WMV is a space of type W0 . In fact, M Ž x . > V Ž y . also holds and WVM is a space of type W0 . N SM pp
GELFAND ] SHILOV SPACES
833
Žiii. Let M p and Np be the defining sequences of the above funcN tions M Ž x . and VU Ž y ., respectively. Then SM pp is a space of type S0 . Also, let M p and Np be the defining sequences of the above functions M Ž x . and V Ž y .. Then SMp!rp N p is a space of type S0 by Lemma 2.3 and Lemma 2.4. We now prove the main theorem in this section. THEOREM 2.8. There is a one to one correspondence between the spaces of N type S0 and type W0 . In other words, for any gi¨ en space SM pp of type S0 , there Np V V is WM of type W0 such that SM p s WM and ¨ ice ¨ ersa. N
N
U
Proof. For any given spaces SM pp of type S0 , we claim that SM pp s WMN where M is the associated function of M p and N U is the Young conjugate N of the associated function N of Np . Let w g SM pp. Then for every a , b g N 0n we obtain
j a bw Ž j . F CA < a < B < b < M < a < N< b <
Ž 2.4.
for some A, B ) 0. Since Np satisfies ŽM.1.U or p!rNp satisfies ŽM.1.X , it is easy to see that Np $ p!. Hence the function w Ž j . can be continued analytically into the complex domain as an entire analytic function. Applying the Taylor expansion and the inequality Ž2.4. we have
j aw Ž j q ih . F
Ý
< j a gw Ž j . <
ggN 0n
FC
Ý
ggN 0n
g!
A < a < M < a < N< g < < Bh < < g
F 2 n CA < a < M < a < exp N a Ž 2 B
Ž 2.5.
Dividing < j < < a < in both sides of the inequality Ž2.5. and taking infimum for < a < in the right hand side of Ž2.5., we have < w Ž j q ih . < F 2 n C exp yM Ž < j
U
Ž 2.6.
It follows that SM pp ; WMN where M is the associated function of M p , N a is the associated function of p!rNp , and N U is the Young conjugate of the associated function N of Np .
834
CHUNG, CHUNG, AND KIM U
By Lemma 2.4, WMN is a space of type W and also the fact M p Np > p! U implies M ; N a ( N U . Hence WMN is a space of type W0 . Conversely, let U w g WMN . Then the inequality Ž2.6. is satisfied. Hence by the Cauchy integral formula together with the inequality Ž2.6. we have < bw Ž x . < s
b!
Ž 2p i .
FC FC sC
n
j
b! R< b <
b! R< b <
b! R< b <
H< z yx
sup < z jyx j
w Ž z . d z 1 . . . d zn 1
y x1 .
b 1 q1
??? Ž zn y x n .
b n q1
exp yM Ž a < j < . q N U Ž b
exp N U Ž bX R .
sup < z jyx j
exp yM Ž a < j < .
exp N U Ž bX R . exp yM Ž a < j 0 < . ,
Ž 2.7.
where the quantity M Ž a < j <. attains its minimum at j s j 0 . In fact, we can write j 0 s x q u R where u s Ž u 1 , . . . , un ., u i s 0 or "1, i s 1, 2, . . . , n. By the convexity of M and the relation M ; N U , we have exp yM Ž a < j 0 < . s exp yM Ž a < x q u R < . F exp yM Ž a Ž < x < y < u R < . . F exp yM F exp yM
a
< x < exp M Ž a1 R .
ž / ž / 2 a 2
< x < exp N U Ž a2 R . .
Hence the inequality Ž2.7. is reduced to < bw Ž x . < F C b ! F Cb !
exp 2 N U Ž a2 q bX . R R
exp yM
exp N U 2 Ž a2 q bX . R R
exp yM
a
< x<
ž / ž / 2
a
2
< x<
.
Ž 2.8.
Multiplying < x a < in both sides of Ž2.8., taking the supremum for < x <, and taking the infimum for R in the right hand side of Ž2.8. we have < x a bw Ž x . < F C1 b ! inf R
exp N U Ž cR . R< b <
sup x
< x<
GELFAND ] SHILOV SPACES
F C1 Ž 2ra.
F C1 Ž cra.
ž
M < a < b ! sup
c
R
M< a < b !
ž
835 y1
R< b < exp N U Ž cR .
Ž < b
/
y1
/
F C1 A < a < B < b < M < a < N< b < , where A s 2ra, B sUce. N It follows that WMN ; SM pp. Now, for a given space WMV of type W0 let M p and Np be the defining sequences of M and VU . Then M p and Np satisfy ŽM.1.X and ŽM.1.U by Lemma 2.4. Furthermore, the relation M ; V N implies M p > p!rNp or M p Np > p!. Thus SM pp is a space Uof type S0 . But by Np the first part of the proof, the space SM p is equal to WMN11 where M1 is the associated function of M p and N1U is the Young conjugate of the associU ated function N1 of Np . Also WMN11 is equal to WMV since M1 and M are equivalent and N1 and VU are equivalent and so are N1U and Ž VU .U s V. N Therefore we have WMV s SM pp which completes the proof. Using the similar method as in Theorem 2.8 and Lemma 2.5 we obtain the following theorem. THEOREM 2.9. Let WM and W V be spaces of type W. Then there exist spaces SM p and S N p of generalized type S such that WM s SM p and W V s S N p . In this case, the sequences M p and Np satisfy the conditions ŽM.1.X and ŽM.1.U . Con¨ ersely, if M p and Np satisfy the conditions ŽM.1.X and ŽM.1.U , then the spaces SM p and S N p are equal to some spaces WM and W V of type W, respecti¨ ely. 3. EQUALITY FOR THE SPACES OF GENERALIZED TYPE S AND TYPE W Applying the results of the above section we prove, in this section, the equality ŽW. under the non-triviality condition M Ž x . ; V Ž y .. For this equality we first prove the equality ŽS. under the conditions which are satisfied by the defining sequences M p and Np of M Ž x . and VU Ž y ., respectively, where M Ž x . ; V Ž y .. First we state Pathak’s result on the equality ŽS.. THEOREM 3.1 wPx.
Suppose that there exists a positi¨ e constant C such that
Npq q G C
ž
pqq Np Nq , p
/
p, q s 0, 1, . . .
Ž P.
836
CHUNG, CHUNG, AND KIM
and that ŽM.2. holds for Np and ŽM.1. and ŽM.2. hold for M p . Then the equality ŽS. holds. But, we cannot apply this result to prove the equality ŽW. as the defining sequence Np of VU Ž y . satisfies ŽM.1.U Žsee Lemma 2.5. and ŽM.1.U implies the following reverse inequality
Npq q F C
ž
pqq Np Nq , p
p, q s 0, 1, . . .
/
Ž P.
U
for some constant C. For example, the condition ŽP. is not satisfied by Np s p! s, 0 - s - 1, which is the defining sequence of some VU Ž y . where V Ž y . satisfies the condition ŽK.. So we replace ŽP. by a natural condition M p Np > p! which is always satisfied if M Ž x . ; V Ž y ., and we modify the proof in wPx in order to make use of the condition M p Np > p! instead of ŽP.. Let M p , p s 0, 1, 2, . . . , be a sequence of positive numbers. We impose one of the following conditions on M p : ŽM.0. ŽNontriviality. M p > p!; ŽM.0X ŽTriviality. M p ; p! s, 0 - s - 1. N
We first prove the equality SM p l S p s SMN pp for the case M p Np > p!, which is a generalization of the equality Sr l S s s Srs for the case r q s G 1. Making use of integration by parts, the Leibniz formula, and the Schwarz inequality we can obtain the following: LEMMA 3.2. If M p and Np satisfy the condition ŽM.2.X , then the supremum norm 5 ? 5 ` and the L2-norm 5 ? 5 2 are equi¨ alent for the spaces of type S. THEOREM 3.3. If M p and Np satisfy the conditions ŽM.1. and ŽM.2., and if M p Np satisfies ŽM.0., then the equality N
SM p l S N p s SM pp holds.
GELFAND ] SHILOV SPACES
837
Proof. Using integration by parts, the Leibniz formula, and the Schwarz inequality as in wKax we obtain 5 x a bw Ž x . 5 22 s F
HR
x 2 a bw Ž x . bw Ž x . dx
n
Ý
kF2 a kF b
F C2 Ý k
b k
2a k! 5 2 bykw Ž x . 5 2 5 x 2 aykw Ž x . 5 2 k
b k
2a k! A2Ž < a
ž /ž /
ž /ž /
F C 2A2Ž < a
F C 2 Ž 2 AH .
2 Ž < a
b k
y1 2a k! Ž M < k < N< k < . k
ž /ž /
M < 2a < N< 2b < .
Ž 3.1.
N This implies that w Ž x . belongs to SM pp in view of Lemma 3.2. The reverse inclusion is obvious, which completes the proof. N
Remark 3.4. Let SM pp be a space of type S0 . Then M p and Np satisfy ŽM.1. and ŽM.1.U . Since ŽM.1.U implies ŽP.U and ŽP.U implies ŽM.2., the equality ŽS. holds by Theorem 3.3. THEOREM 3.5.
Let WMV be a space of type W0 . Then the equality WM l W V s WMV
holds. Proof. For given spaces WM , W V , and WMV of type W0 there exist SM p, N S , and SM pp of type S0 such that Np
WM s SM p ,
W V s S Np ,
and
N
WMV s SM pp
by Theorem 2.8 and Theorem 2.9. N N N Since SM pp is a space of type S0 , the equality SM p l S p s SM pp holds by Remark 3.4. Consequently, we have the equality WM l W V s WMV . N
We now prove the triviality of the spaces SM pp and S M p l S N p under the condition M p Np ; p! s, 0 - s - 1, which generalizes the equality fSr l S s s Srsor the other case r q s - 1, which will complete the generalization of the quality ŽS0..
838
CHUNG, CHUNG, AND KIM
THEOREM 3.6. If M p and Np satisfy the condition ŽM.2.X and M p Np N satisfies ŽM.0.X , then both the spaces SM pp and SM p l S N p are tri¨ ial. N
Proof. If w Ž x . belongs to S M pp or SM p l S N p , then by the condition ŽM.0.X , w Ž x . is continued analytically into the complex plane as an entire analytic function. For w Ž x . g SMN pp we have 5 x a aw Ž x .5 ` F CA < a < < a
ž
Ý
5 x aykq1 aykq1w Ž x . 5 2
kF a kF Ž1, . . . , 1 .
q
Ý
5 x ayk aykq1w Ž x . 5 2
kF a kF Ž1, . . . , 1 .
/
for some constants C and A. Replacing a , b by a y k q 1 in Ž3.1., we have 5 x aykq1 aykq1w Ž x . 5 22 F C 2 Ž 2 A . 4 < a <
Ý
j!M < 2 aq2yj < N< 2 aq2yj <
Ý
< j < !1ys Ž < j < ! <2 a q 2 y j < ! .
jF a q1
F C 2 Ž 2 A.
4
s
jF a q1
F C12 A21 < a < < a < !1q s . Similarly, we have 5 x ayk aykq1w Ž x .5 22 F C22 A22 < a < < a
0 - s - 1,
then w Ž z . degenerates to a constant function. In fact, by the Taylor expansion we obtain
bw Ž 0 . s
Ý aq b Ž j . Ž yj . a
a
ra !.
Ž 3.2.
GELFAND ] SHILOV SPACES
839
By letting < j < ª `, Ž3.2. implies that bw Ž0. s 0 for b / 0, hence w is N constant. Therefore the spaces SM pp and SM p l S N p are trivial. ACKNOWLEDGMENTS This research is partially supported by Ministry of Education and GARC. Also, we thank the referees for helpful suggestions, especially for the proof of Theorem 3.6.
REFERENCES wBMTx
R. W. Braun, R. Meise, and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Resultate Math. 17 Ž1990., 206]237. wCCK1x J. Chung, S.-Y. Chung, and D. Kim, Une caracterisation de l’espace de Schwartz, C. ´ R. Acad. Sci. Paris Ser. ´ I Math. 316 Ž1993., 23]25. wCCK2x J. Chung, S.-Y. Chung, and D. Kim, Characterizations of the Gelfand]Shilov spaces via Fourier transforms, Proc. Amer. Math. Soc., in press. wCKx S.-Y. Chung and D. Kim, Representation of quasianalytic ultradistributions, Ark. Mat. 31 Ž1993., 51]60. wCKKx S.-Y. Chung, D. Kim, and S. K. Kim, Equivalence of the spaces of ultradifferentiable functions and its application to the Whitney extension theorem, Rend. Mat. Appl. Ž7. 12 Ž1992., 365]380. wGSx I. M. Gelfand and G. E. Shilov, ‘‘Generalized Functions, II, III,’’ Academic Press, New YorkrLondon, 1967. wHx L. Hormander, ‘‘The Analysis of Linear Partial Differential Operator, I,’’ Springer¨ Verlag, BerlinrNew York, 1983. wKax A. I. Kashpirovsky, Equality of the spaces Sab and Sa l S b , Functional Anal. Appl. 14 Ž1978., 60. wKx H. Komatsu, Ultradistributions, I, J. Fac. Sci. Uni¨ . Tokyo Sect. IA 20 Ž1973., 25]105. wPx R. S. Pathak, Tempered ultradistributions as the boundary values of analytic functions, Trans. Amer. Math. Soc. 286 Ž1984., 537]556. wTx F. Treves, ‘‘Linear Partial Differential Equations with Constant Coefficients,’’ Gordon & Breach, New York, 1966.