A generalization of Miyachi's theorem

doi: 10.2969/jmsj/06120551

A generalization of Miyachi’s theorem

By Radouan DAHER, Takeshi KAWAZOEand Hatem MEJJAOLI (Received June 9, 2008)

Abstract. The classical Hardy theorem on R, which asserts f and the Fourier transform offcannot both be very small, was generalized by Miyachi in terms of L¹þL¹ and log^þ-functions. In this paper we generalize Miyachi’s theorem for R^d and then for other generalized Fourier transforms such as the Che´bli-Trime`che and the Dunkl transforms.

1. Introduction.

Classical Hardy’s theorem [7] asserts the following: let a; b >0 and f a measurable function on R satisfying jfðxÞj Ce^ax2 and jfðyÞj ^ Ce^b2. Then f0 if ab >1=4, f is a constant multiple of e^ax2 if ab¼1=4, and there are infinitely manyf ifab <1=4. Considerable attention has been devoted to finding generalizations to new contexts for Hardy’s theorem. Especially, M. Cowling and J. Price [5] obtained anL^p version of the theorem. As further generalizations, A.

Bonami, B. Demange, P. Jaming [2] extend it in a Beurling form, and Miyachi [9]

obtains anL¹þL¹ version: Letab¼1=4and f2L¹ðRÞsatisfy e^ax2fðxÞ 2L¹ðRÞ þL¹ðRÞ

and

Z þ1 1

log^þjf^ðÞe^b2j

C d <1

for someC >0. Thenf is a constant multiple ofe^ax2, where L¹ðRÞ þL¹ðRÞis the set of functions of the formf¼f1þf2,f12L¹ðRÞ,f22L¹ðRÞ, andlog^þx¼ logx ifx >1 andlog^þx¼0 ifx1. In this paper we shall generalize Miyachi’s 2000Mathematics Subject Classification. Primary 43A85; Secondary 43A62, 43A32, 44A12, 42A38.

The second auther was supported by Grant-in-Aid for Scientific Research (C) (No. 20540188), Japan Society for the Promotion of Science.

Key Words and Phrases. Hardy’s theorem, Miyachi’s theorem, Radon transform, Dunkl transform, Che´bli-Trime`che transform.

J. Math. Soc. Japan

Vol. 61, No. 2 (2009) pp. 551–558

theorem forR^dand furthermore, we shall obtain an analogue in a general measure spaceðX; d!Þ equipped with Fourier, Radon, and dual Radon transforms. As a special case of this general setting, we can deduce Miyachi’s theorem for the Che´bli-Trime`che, the Dunkl transforms, and the Jacobi-Dunkl transform.

Especially, we can obtain alternative proofs in [6] for the Jacobi transform and in [3] for the Dunkl transform.

2. Miyachi’s theorem onR^d.

For l¼1;2; , we denote by Fl the Fourier transform on R^l. Miyachi’s theorem is generalized onR^d as follows.

THEOREM1. Let a_i; b_i>0 and a_ib_i¼1=4 for 1id. We put A¼diagða1; a2; ; adÞ and B¼diagðb1; b2; ; bdÞ. If a measurable function on R^d satisfies

e^hAx;xifðxÞ 2L¹ðR^dÞ þL¹ðR^dÞ ð1Þ

and

Z

R^d

log^þjFdfðÞe^hB;ij

C d <1 ð2Þ

for someC >0, then f is a constant multiple ofe^hAx;xi.

PROOF. For ðx₁; x⁰Þ ¼ ðx₁; x₂; ; x_dÞ and ⁰¼ ð₂; ; _dÞ, we shall con- sider

Gðx1; ⁰Þ ¼Fd1ðfðx1;ÞÞð⁰Þ ¼ Z

R^d1

fðx1; x⁰Þe^ih0;x0idx⁰;

wheredx⁰¼dx₂ dx_d. Sincef 2L¹ðR^dÞby (1),Gðx₁; ⁰Þis well-defined and is in L¹ðRÞ L¹ðR^d1Þ. Moreover, since f¼e^{ða1x21þa2x22þþadx2dÞ}ðu1þu2Þ, where u₁2L¹ðR^dÞandu₂2L¹ðR^dÞby (1), it follows that

jGðx₁; ⁰Þj e^a1x21X²

k¼1

Z

R^d1

e^{ða2x22þþadx2dÞ}ju_kðx₁; x⁰Þjdx⁰:

and thus, as a function of x1, e^a1x21Gðx1; ⁰Þ 2L¹ðRÞ þL¹ðRÞ. We note that F1ðGð; ⁰ÞÞð1Þ ¼FdfðÞand substitute it in (2). Then Fubini’s theorem implies that there exists a subset E of R^d1 with positive measure such that for

⁰¼ ð2; ; _dÞ 2E, Z þ1

1

log^þjF1ðGð; ⁰ÞÞð1Þe^b121j

Ce^{ðb222þþbd2dÞ} d1<1:

Therefore, Miyachi’s theorem onRyields thatGðx₁; ⁰Þ ¼Cð⁰Þe^a1x21for ⁰2E.

Hence FdfðÞ ¼F1ðGð; ⁰ÞÞð1Þ ¼Cð⁰Þe^b121 for ⁰2E. Since FdfðÞ has a holomorphic extension onCⁿ(see (1)), we can prolong the precedent relation as FdfðÞ ¼Cð⁰Þe^b121 for all⁰2R^d1. HenceGðx1; ⁰Þ ¼Cð⁰Þe^a1x21 for all⁰ 2 R^d1. Here we put hðxÞ ¼e^a1x21fðxÞ. Then, as a function of x⁰¼ ðx2; ; x_dÞ, it belongs to L¹ðR^d1Þ (see (1)) and Fd1ðhðx1;ÞÞð⁰Þ ¼e^a1x21Fd1ðfðx1;ÞÞð⁰Þ ¼ e^a1x21Gðx1; ⁰Þ ¼Cð⁰Þ for ⁰2R^d1. Since Cð⁰Þ is independent of x1, it follows that h is also independent of x₁. We obtain that f is of the form fðx1; x2; ; xdÞ ¼e^a1x21hðx2; ; xdÞ. We can obtain the desired result by

repeating this argument.

3. Generalization of Miyachi’s theorem.

Let ðX; d!; Þ be a topological space with a positive measure d! and a distance function :X!R_þ. In what follows we assume the existence of an integral operatorR that satisfies the following properties: R is of the form

RfðyÞ ¼ Z

X

fðxÞdyðxÞ; f2C_cðXÞ; ð3Þ

(A1) yis a positive measure with the support infx2Xj kyk ðxÞg, (A2) R is bounded fromL¹ðX; d!Þto L¹ðR^dÞ,

(A3) Rgives an isomorphism betweenSðXÞandSðR^dÞ, whereSðXÞis a suitable Schwartz class onX and SðR^dÞis the Schwartz class onR^d. We here define the dual operatorR:S⁰ðR^dÞ !S⁰ðXÞofR by

Z

R^d

RfðyÞgðyÞdy¼ Z

X

fðxÞRgðxÞd!ðxÞ: ð4Þ

We furthermore assume that (A4) R is of the form

RfðxÞ ¼ Z

R^d

fðyÞdxðyÞ; f2CðR^dÞ; ð5Þ

where _x is a positive measure on R^d with support in fy2 R^dj kyk ðxÞg.

Then we see thatRð1ÞðxÞ C0for allx2Xbecause of (4) and (A2). We denote the Euclidean Fourier transform onR^d byFd and put

FX¼FdR:

We call FX a generalized Fourier transform on X, which is an isomorphism betweenSðXÞand SðR^dÞ(see (A3)). We define Gaussian type functions haðxÞ andh_aðxÞ,x2X; a2R_þ, as follows:

haðxÞ ¼ ðRÞ¹ðe^akk2ÞðxÞ and h_aðxÞ ¼Rðe^akk2ÞðxÞ:

Since e^akk22SðR^dÞ, h_aðxÞ is well-defined (see (A3)), positive on X and FXhaðyÞ ¼Ce^kyk2=4a. Since h_aðxÞ e^aðxÞ2Rð1ÞðxÞ C0e^aðxÞ2, h_aðxÞ is well- defined and positive onX. Especially,ðh_aÞ¹ðxÞis positive onX.

THEOREM2. Let a >0 and ab¼1=4. If a measurable function f on X satisfiesf ¼f₁þf₂ such that

h_aðxÞf1ðxÞ 2L¹ðX; d!Þandh¹_a ðxÞf2ðxÞ 2L¹ðX; d!Þ ð6Þ

and

Z

R^d

log^þjFXfðÞe^bkk2j

C d <1

for someC >0, then f is a constant multiple ofha.

PROOF. We show thatRðfÞsatisfies

e^akyk2RfðyÞ 2L¹ðR^dÞ þL¹ðR^dÞ:

Actually, by (6) fðxÞ ¼f1ðxÞ þf2ðxÞ ¼ ðh_aÞ¹ðxÞv1ðxÞ þhaðxÞv2ðxÞ, wherev12 L¹ðX; d!Þand v₂2L¹ðX; d!Þ. Then

ke^akyk2Rf1k_L1ðR^dÞ ¼ Z

R^d

e^akyk2Rððh_aÞ¹v1ÞðyÞdy

Z

R^d

e^akyk2Rððh_aÞ¹jv1jÞðyÞdy

¼ Z

X

Rðe^akk2ÞðxÞðh_aÞ¹ðxÞjv1ðxÞjd!ðxÞ(see(4))

¼ Z

X

jv1ðxÞjd!ðxÞ

and

e^akyk2jRf₂ðyÞj kv₂k₁e^akyk2Rh_aðyÞ ¼ kv₂k₁:

Hence e^akyk2RfðyÞ belongs to L¹ðR^dÞ þL¹ðR^dÞ. Since FXf¼FdðRfÞ, it follows from Miyachi’s theorem on R^d (see Theorem 1) that RfðyÞ ¼Ce^akyk2.

Hence we obtain thatf¼Cha.

REMARK3. If there exists a functionðxÞonXsuch that haðxÞh_aðxÞ ðxÞ;

then the conditionð6Þin Theorem 2 can be replaced by

h¹_a ðxÞfðxÞ 2L¹ðX; d!Þ þL¹ðX; d!Þ: ð7Þ

4. Examples.

4.1. Che´bli-Trime`che hypergroups.

We refer to [1], [10] and [12] for general notations and basic facts on Che´bli- Trime`che hypergroups. Let >1=2 and AðxÞ ¼x<sup>2þ1</sup>BðxÞ be a Che´bli- Trime`che function on R_þ. Then ðX; d!; Þ ¼ ðRþ; AðxÞdx;j jÞ and the Che´bli- Trime`che transform FX gives an isometry between L²ðX; d!Þ and L²ðR; dÞ wheredðÞ ¼ jCðÞj²dand supported on even functions onR. We recall that FX¼FR where R is the Weyl type integral transform

RfðyÞ ¼ Z 1

y

fðxÞKðx; yÞAðxÞdx; y0

andRis the Riemann-Liouville type integral transform

RfðxÞ ¼ Z x

0

fðyÞKðx; yÞdy; x0:

The multiplicative functions on the hypergroup coincide with spherical functions ðxÞ,2C. LetSeðRÞdenote the space of even rapidly decreasing functions on RandSðXÞ ¼0ðxÞSeðRÞ, where0 is the spherical function with¼0. Then R satisfies the assumptions (A1) - (A4) in Section 3 (see [12]), where dyðxÞ ¼Kðx; yÞ½y;1ÞðxÞAðxÞdx, y0, and dxðyÞ ¼Kðx; yÞ½0;x ðyÞdy, x0.

Hence, by introducing the Gaussian type functionsh_a andh_a, Theorem 2 holds for the Che´bli-Trime`che transformFX. We shall obtainðxÞin Remark 3. We recall from [12] that

h_aðxÞ C 1 ffiffiffiffiffiffiffiffiffiffiffi pBðxÞe^ax2:

On the other hand, sincee^ax2¼e^{aðx =2aÞ2þ x 2=4a},

h_aðxÞ Ce^{aðx =2aÞ2}Rðe ^yÞðxÞ Ce^{aðx =2aÞ2}i ðxÞ Ce^{ax2 x}:

Hence

haðxÞh_aðxÞ C 1 ffiffiffiffiffiffiffiffiffiffiffi pBðxÞe ^x

and

C 1

ffiffiffiffiffiffiffiffiffiffiffi BðxÞ

p e ^xd!ðxÞ ¼Cx^2þ1 ffiffiffiffiffiffiffiffiffiffiffi pBðxÞ

e ^xdx

In particular, if X is the Bessel-Kingman hypergroup, then AðxÞ ¼x^2þ1 and d!¼Cx^2þ1dx. If X is the Jacobi hypergroup, then AðxÞ ¼cðsinhxÞ^2þ1 ðcoshxÞ^2þ1, 1=2, and d!¼CðtanhxÞ^þ1=2x^þ1=2dx. Hence the con- ditionð7Þin Remark 3 is nothing but the one used in [6, Theorem 3.1].

4.2. Dunkl analysis.

We refer to [8] and [11] for general notations and basic facts in Dunkl transform. LetðX; d!; Þ ¼ ðR^d; !_kðxÞdx;k kÞ wherekis a multiplicity function on the root system. Then the Dunkl transform FX gives an isometry between L²ðX; d!ÞandL²ðR^d; C_kd!Þ. We recall thatFX¼FdR. HereR and its dual operatorRare given as

RfðyÞ ¼ Z

R^d

fðxÞdyðxÞ;

where_yis a positive measure onR^dwith the support infx2R^dj kxk kykgand RfðxÞ ¼

Z

R^d

fðyÞdxðyÞ;

where x is a positive measure of probability on R^d with the support in fy2 R^dj kyk kxkg(see [11]). LetSðR^dÞdenote the Schwartz space onR^d. ThenR satisfies the assumptions (A1) - (A4) in Section 3 (see [11]). Moreover, we note thath_aðxÞ ¼e^akxk2 and

h_aðxÞ e^akxk2Rð1ÞðxÞ ¼e^akxk2:

Henceh_aðxÞh_aðxÞ 1and ðxÞ ¼1inð7Þ. Hence we can obtain the correspond- ing Theorem 2 for the Dunkl transform.

4.3. Jacobi-Dunkl analysis.

We refer to [4] for general notations and basic facts in Jacobi-Dunkl transform. Let 1=2 and 6¼ 1=2. We put ¼þþ1>0 and A;ðxÞ ¼2² ðsinhjxjÞ^2þ1ðcoshxÞ^2þ1. Then ðX; d!; Þ ¼ ðR; A;ðxÞdx;j jÞ and the Jacobi-Dunkl transform FX gives an isometry between L²ðX; d!Þ and L²ðR; dÞ. Here dis of the form

dðÞ ¼ jj

8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^{2 2}

p cð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^{2 2}

p Þ²1_{Rn ; ½}ðÞd;

where 1_{Rn ; ½} is the characteristic function of Rn ; ½ and cðÞ is a meromorphic function on C. We recall that FX¼F R where R the Jacobi-Dunkl dual intertwining operator defined by

RfðyÞ ¼ Z

jxjjyj

Kðx; yÞfðxÞA;ðxÞdx

andRis the Jacobi-Dunkl intertwining operator defined by

RfðxÞ ¼ Z jxj

jxj

Kðx; yÞfðyÞdy if x2Rnf0g, fð0Þ if x¼0.

8>

<

>:

LetSðRÞdenote the Schwartz space onRandSðXÞ ¼ ðcoshxÞ² SðRÞ. Then R satisfies the assumptions (A1) - (A4) in Section 3, where dyðxÞ ¼ Kðx; yÞ½jyj;1ÞðxÞA;ðxÞdx and dxðyÞ ¼Kðx; yÞ½0;jxj ðxÞA;ðxÞdx. Hence, by introducing the Gaussian type functions ha and h_a, Theorem 2 holds for the Jacobi-Dunkl transformFX. As in Jacobi hypergroup case, the condition (7) in Remark 3 is given byd!¼CðtanhjxjÞ^þ12jxj^þ12dx.

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Radouan DAHER Department of Mathematics Faculty of Sciences and Informatics University Hassan II

B.P. 5366 Maarif, Casablanca Morocco

Takeshi KAWAZOE Department of Mathematics Keio University at Fujisawa Endo, Fujisawa

Kanagawa 252-8520 Japan

Hatem MEJJAOLI Department of Mathematics Faculty of Sciences of Tunis Campus, 1060 Tunis Tunisia