An uncertainty principle on Sturm-Liouville hypergroups
By Radouan DAHERÞand Takeshi KAWAZOEÞ (Communicated by Heisuke HIRONAKA,M.J.A., Nov. 12, 2007)
Abstract: As an analogue of the classical uncertainty inequality on the Euclidean space, we shall obtain a generalization on the Sturm-Liouville hypergroupsðRþ;ðAÞÞ. Especially, we shall obtain a condition on Aunder which the discrete part of the Plancherel formula vanishes.
Key words: uncertainty principle; Sturm-Liouville; hypergroup.
1. Sturm-Liouville hypergroups. Sturm- Liouville hypergroups are a class of one-dimensional hypergroups on Rþ¼ ½0;1Þ with the convolution structure related to the second order differential operators
L¼ d2
dx2þA0ðxÞ AðxÞ
d dx;
where A satisfies the following conditions (see [1,2]):
A >0 onRþ¼ ð0;1Þ, and is inC2ðRþÞ; ð1Þ
on a neighborhood of 0;
ð2Þ
A0ðxÞ
AðxÞ ¼2þ1
x þBðxÞ;
where 12 and (a) if >0, B and B0 are integrable, (b) if ¼0, logxB and xlogxB0 are integrable, (c) if12 < <0,x2Bandx2þ1B0are integrable, (d) if¼ 12,B0is integrable,
A0
A 0 onRþ and limx!1A0ðxÞ AðxÞ ¼2;
ð3Þ
1 2
A0 A
0
þ1 4
A0 A
2
2 is integrable at1: ð4Þ
SinceA0=A¼ ðlogAÞ0, (3) implies thatAis increas- ing, and thus, Að0Þ<1. Under the conditions (1) to (3), the second order differential equation:
Luþ ð2þ2Þ ¼0, 2C, has a unique solution satisfying uð0Þ ¼1, u0ð0Þ ¼0, which we denote by . Furthermore, under (4), if =0, then there exists another solution ðxÞ, which behaves as
ffiffiffiffiffiffiffiffi p=2 ffiffiffiffiffiffi
px
Hð1Þ at 1, where Hð1Þ is the Hankel function. Similarly, we have ðxÞfor =0, and for2Rþ, there existsCðÞ 2Csuch thatðxÞ ¼ CðÞ ðxÞ þCðÞ ðxÞ.
LetC1c;eðRÞdenote the set ofC1even functions f onR. Forf2Cc;e1ðRÞthe Fourier transformff^is defined by
f^ fðÞ ¼
Z 1 0
fðxÞðxÞAðxÞdx:
Then the inverse transform is given as fðxÞ ¼X
2D
ffðÞ^ ðxÞ
þ 1 2
Z 1 0
f^
fðÞðxÞ d jCðÞj2; where D is a finite set in the interval ið0; Þ and ¼ kk2L2ðRþ;AdxÞ. We denote this decomposition as
f¼fþfP
and we call fP and f the principal part and the discrete part of f respectively. We denote by FðÞ ¼ ðFðÞ;fagÞ a function on Rþ[D defined by
FðÞ ¼ FðÞ if¼2Rþ a if¼2D.
We putFðÞ ¼ ðFðÞ;fagÞand define the product ofFðÞ ¼ ðFðÞ;fagÞandGðÞ ¼ ðGðÞ;fbgÞas
ðFGÞðÞ ¼ ðFðÞGðÞ;fabgÞ:
Letd denote the measure onRþ[Ddefined by Z
Rþ[D
FðÞd¼X
2D
a
þ 1 2
Z 1 0
FðÞjCðÞj2d:
#2007 The Japan Academy
2000 Mathematics Subject Classification. Primary 43A62, 43A32, 42A38.
Þ De´partement de Mathe´matiques et d’Informatique, Faculte´ des Siences, Universite´ Hassan II, BP. 5366, Maarif, Casablanca, Maroc.
Þ Department of Mathematics, Keio University, 5322 Endo, Fujisawa, Kanagawa 252-8520, Japan.
Nos. 9–10] Proc. Japan Acad.,83, Ser. A (2007) 167
Forf 2Cc;e1ðRÞ, we put
ff^ðÞ ¼ ðff^ðÞ;fffðÞgÞ:^
Then the Parseval formula onCc;e1ðRÞcan be stated as follows: Forf; g2C1c;eðRÞ
Z 1 0
fðxÞgðxÞAðxÞdx¼ Z
Rþ[D
ff^ðÞggðÞd:^ ð5Þ
The map f !ff^, f2C1c;eðRÞ, is extended to an isometry between L2ðAÞ ¼L2ðRþ; AðxÞdxÞ and L2ðÞ ¼L2ðRþ[D; dÞ. Actually, each function f inL2ðAÞis of the form
fðxÞ ¼X
2D
ff^ðxÞ
þ 1 2
Z 1 0
f^
fðÞðxÞjCðÞj2d
¼fþfP
and theirL2-norms are given as Z 1
0
jfðxÞj2AðxÞdx¼X
2D
jff^j2;
Z 1 0
jfPðxÞj2AðxÞ ¼ 1 2
Z 1 0
jff^PðÞj2jCðÞj2d:
Therefore, if we define ff^ðÞ ¼ ðffðÞ;^ fff^gÞ, then kfkL2ðAÞ¼ kff^kL2ðÞ holds. In particular, if f 2 Cc;e1ðRÞ, thenff^¼ff^ðÞfor all2D.
2. Uncertainty inequality. We retain the notations in the previous sections. We put for x2Rþ,
aðxÞ ¼ Z x
0
AðtÞdtand vðxÞ ¼ aðxÞ ð6Þ AðxÞ
and for2C,
wðÞ ¼ ð2þ2Þ1=2:
Theorem 2.1. For allf 2L1ðAÞ \L2ðAÞ, kfvk2L2ðAÞ
Z
Rþ[D
jff^ðÞj2wðÞ2d1
4kfk4L2ðAÞ; ð7Þ
where the equality holds if and only iffis of the form
fðxÞ ¼ce
Z x 0
vðtÞdt
for somec; 2Cand < <0.
Proof. Without loss of generality we may suppose that f 2Cc;e1ðRÞ. Since ðLfÞ^ðÞ ¼ f^
fðÞð2þ2Þ ¼ffðÞwðÞ^ 2 and wðÞ is positive onRþ[D, the Parseval formula (5) yields that
Z
Rþ[D
jff^ðÞj2wðÞ2d¼ Z 1
0
ðLfÞðxÞfðxÞAðxÞdx
¼ Z 1
0
jf0ðxÞj2AðxÞdx:
Hence it follows that Z 1
0
jfðxÞj2vðxÞ2AðxÞdx Z
Rþ[D
jff^ðÞj2wðÞ2d
¼ Z 1
0
jfðxÞj2vðxÞ2AðxÞdx Z 1
0
jf0ðxÞj2AðxÞdx
Z 1 0
<ðfðxÞf0ðxÞÞvðxÞAðxÞdx
2
¼1 4
Z 1 0
ðjfðxÞj2Þ0aðxÞdx
2
¼1 4
Z 1 0
jfðxÞj2AðxÞdx
2
:
Here we used the fact thata0¼A(see (6)). Clearly, the equality holds if and only if fv¼cf0 for some c2C, that is, f0=f¼c1v. This means that logðfÞ ¼c1
Z x 0
vðtÞdtþC and thus, the desired
result follows.
Remark 2.2. WhenðRþ;ðAÞÞis the Bessel- Kingman hypergroup, the equality holds for ex2,
< <0. However, when it is the Jacobi hypergroup, each function satisfying the equality has an expo- nential decayex.
Since w2ðÞ ¼2þ2, (7) can be rewritten as follows:
Corollary 2.3. Let f be the same as in Theorem 2.1.
kfvk2L2ðAÞ
Z
Rþ[D
jff^ðÞj22d ð8Þ
1
4kfk2L2ðAÞ
Z 1 0
jfðxÞj2ð142vðxÞ2ÞAðxÞdx:
3. Vanishing condition of the discrete part. We shall prove that under the assumption:
0vðxÞ 1 2; ð9Þ
it follows thatD¼ ;. We suppose thatD6¼ ;and we takef ¼,2D. Then, sinceff^ðÞ ¼1if¼ and 0 otherwise, it follows from (8) that
kfvk2L2ðAÞ2 1
4kfk2L2ðAÞ
Z 1 0
jfðxÞj2ð142vðxÞ2ÞAðxÞdx:
168 R. DAHERand T. KAWAZOE [Vol. 83(A),
Here we recall that2<0, becauseDið0; Þand 142vðxÞ20 by (9). This is contradiction.
Therefore, we obtain the following Theorem 3.1. If0v 1
2, then D¼ ;. For example, ifAsatisfies the inequality:
aðxÞA0ðxÞ ¼ Z x
0
AðxÞdxA0ðxÞ A2ðxÞ;
ð10Þ
thenAsatisfies (9). Actually, (10) implies
v0ðxÞ ¼A2ðxÞ aðxÞA0ðxÞ A2ðxÞ 0:
Hence v is increasing on Rþ and vðxÞ ¼aðxÞ=
AðxÞ AðxÞ=A0ðxÞ because A=A0>0 by (3). Then it follows from (3) thatAsatisfies (9).
Corollary 3.2. If A satisfies the inequality (10), thenD¼ ;.
Remark 3.3. It is well-known that D¼ ; for Che´bli-Trime´che hypergroups where A0=A is decreasing and (4) is not required (cf. [1]). This fact easily follows from our argument. Since A=A0 is increasing and 0A=A01=2 by (3), we see that aA=2 by integration and thus, (9) holds.
HneceD¼ ;by Theorem 3.1.
4. Uncertainty principle. We suppose thatD¼ ;. Then (8) is of the form:
kfvk2L2ðAÞ
1 2
Z 1 0
jff^ðÞj22 d jCðÞj2 1
4kfk2L2ðAÞ
Z 1 0
jfðxÞj2ð142vðxÞ2ÞAðxÞdx:
Sincevis increasing,vð0Þ ¼0, and142vðxÞ20 by (9), it follows that f and ff^ both cannot be concentrated around the origin.
In general, if D6¼ ;, then we must pay atten- tion to the discrete part off to consider uncertainty principles. We refer to [3] for the Jacobi hyper- groups.
References
[ 1 ] W. R. Bloom and H. Heyer,Harmonic analysis of probability measures on hypergroups, de Gruyt- er, Berlin, 1995.
[ 2 ] O. Bracco, Fonction maximale associe´e a` des ope´rateurs de Sturm-Liouville singuliers, The`se, Univ. Louis Pasteur, Strasbourg, 1999.
[ 3 ] T. Kawazoe, Uncertainty principles for the Jacobi transforms, Tokyo J. Math. (to appear).
Nos. 9–10] Uncertainty principle on hypergroups 169
