FOURIER TRANSFORMS OF LIPSCHITZ FUNCTIONS ON THE HYPERBOLIC PLANE H =

VOL. 21 NO. 2 (1998) 397-402

FOURIER TRANSFORMS OF LIPSCHITZ FUNCTIONS ON THE HYPERBOLIC PLANE H =

M.S. YOUNIS Departmentof Mathematics

YarmoukUniversity Irbid,JORDAN

(ReceivedApril 24,

1996)

ABSTRACT. The purposeof the present work is to study the order of magnitude of theFourier transforms

f(A)

^for

_larse ^A

^of

complex-valued

^functions

,f (z) sating

certain

Lipschitz

conditions in the non-Euclideanhyperbolic plane

H .

KEY

WORDS AND PHRASES: Fouriertransforms, Lipschitz functions, absoluteconvergence of Fouriertransforms.

1991AMS^SUB/ECTCLASSIFICATION CODES: Primary 42B10.

1. INTRODUCTION

Therelation between smoothness conditions imposed on functions

f(z)

^andthe behavior of its Fouriertransforms

f

^nearinfinity is well knowninthe literature.

In fact, the Fourier transforms of Lipschitz functions defined on various domains have been extensivelystudiedover the lastdecades. Thepurposeofthe present researchis totracethe behaviorof the Fouriertransformsofcomplex-valuedfunctionssatisfyingLipschimconditions inthehyperbolic plane

H 9-.

This will pavethe ground forhandling the transforms ofLipschitz functions defined on other domainssuchas

SL(2.R)

^and

SL(2.C)

inparticular.

2. DEFINITIONS AND NOTATIONS

Our main reference on the Fourier analysis on

H

² is the book by Helgason

([2],

p. 29 infra);

reference[7]willbe consulted (especially Chapter 10)aswell. Inthe beginningwewouldlike tomake clear an idea which will befrequentlyencoumered in thesequel. Let

D

betheopendisc

Izl ^<

^1inthe

plane

R 2.

Then ahorocyclein

D

is a circletangentialtothe boundary

B

ODat acertainpointb. This horocycleisdenotedby

.

^{Ifzisa point on}

^,

the distance

d(0, z)

from the origin0of

D

tozisdenoted by

(z, b).

Let

d(0, z)

^{r. Then}

r

d(0, z)

log

-iz----

1

Thisgives

z[

^tanhr.

^We

^indicate here that tanhr is ofthe order ofr nearr 0and that it approaches¹asrgoestoinfinity. Thesetwolimits will occurfrequentlyin duecourse. Inthis section weintroduce the basic definitionsnecessary fortheproofof themaintheorems. Other definitionswillbe givenwhenthey are needed. Thus westart with

DEFINITION 2.1. Let

f(z)

^{be a} complex-valued function defined on the unit disc

D

when endowedwiththe Pdemannianstructure. Then

f(z)

is said tobelongtothe Lipschitz class Lip(c,

2)

if

II.f(z + h) .f(z)ll O(Ihl) (2)

398 M.S.YOUNIS

ash 0, 0

<

^a

<

1. Where

I[-[[2

^{isthe usual}

L

norm. The generalization ofthisdefinitionfor higher differences

Ahlf(z

^{of orderk in}step h takestheform

ll/khl](z)ll2 ^{O(lh[ ()}

⁰

^<

^a

^_<

^k

⁺

^1.

⁽³⁾

DEFINITIOM2.2([2],p.3). If

.f(z)

is acomplex-valuedfunctionon

D,

its Fouriertransformis definedby

, ^b) _=/ ()-’/l)I"b)d ()

for all

C,

b

B

forwhichthis integralexists. The Parseval’s identityinthis case iswrittenas

/o ^{If(z)l 2dz}

¹

JoB ^{]fl 2A}

^{tanh dAdb. (5)}

3. MAIN

THEOREMS

Ourmainresultmaybe viewed as the non-Euclideananalogueofthefollowing theorem.

THEOREM3.1([5],Theorem85). Let

f(z)

belongto

L2(R).

Then the conditions

Ill(=

^/

h) f(=)l12 o(h ) ()

as h^--,O, 0

<

^c,

<

1and

+ ]}12dz O(X-2(’) (7)

as

X

ooareequivalent. Themaintheoremofthis section isstated as

THEOREM3.2. Let

f(z)

beacomplex-valuedfunctionontheunit discD. Thenthe conditions

IIf(z + h) f(z)ll2 O(]hla) (8)

aslh

^--,0,O<a

^<

^land

as ovareequivalent.

oo ^ilddb O("x-2-’) (9)

PROOF.

By

definitionof

f ^{(A, b)}

^{itiseasilyseen}that the transform of

(f(z + h) f (z))

isgiven

(e

(-iA+l)(-h’b)

1) f(A, b) (0)

by

observe that

h,b) (h,b) d(O,h)

rand hence r

--

^{0 with}

^[hi

^where

^Ihl

^{tanhr. So, the}

factore^{h,b) e tendsto oneasrgoestozeroandthereforecanbesuppressedwithoutharm. Also, since E

R

+ inthe right side of the Parseval’s identity([7],p. 376), (10)could be simplified sothatthe transform of

If(z

^/

h) .f(z)l

^isgivenas

12

^sin

^]’1.

^Thus

2sin-- ^f

^{tanh dAdb}

^O(tanhr)

^2a

^O(r2a).

Observethat

tanh(-)

^{tendstobeaboundedconstantforlarge A. Thisyields}

^(as

^{in theproof of}

Theorem3.1)

Z-Z, ^{.l,aeae o( -o) o(-o),}

equivalently

Fromthisoneobtains

ITl2dAdb ^{+ +... +}

(11)

(12)

Itisclear that the integral db Iwhenthe normalizedHaarmeasureisin action. The lastquantityis bounded as

, --

^{oo if1-}

^{a/-/ <}

^{0, so that}

^{</3 <_}

^{2, givingthe} condition a

>

⁰

for/9

1.

Thisshows that theFouriertransform

f(,, b)

of

f(z)

convergesabsolutely forany^c,greater than zero.

This reflects the strength of the conclusion in Theorem 3.2 in contrast to those ofsimilartheorems provedfor Lipschitz functionsin

R, R

and

T

forexample. Thisis mainlydue tothepresenceof the weight

A

inthe inversionformula aswell as in one sideofParseval’s identity. Hadit notbeenforthis extrafactorinthe present situation, the conclusion of Theorem3.2would have been exactlythe same as that of Theorem3.1 asfar asthe order of magnitude and the absoluteconvergenceof

f

^{areconcerned.}

Weshallencounterthe same situation whendealingwiththe spherical

Furier

transforms ofLipschitz functionsin

H

²whichisthe subjectmatterof thenextsection.

4. SPHERICALTRANSFORbISIN

H

²

In this section we prove an analogue of Theorem 3.2 for the spherical Fourier transforms of Lipschitz functions. These are related to spherical functions which (by theirvery

nature)

^{are radial} eigenfunctions of the Laplacian^ontheunit disc

(see [2],

pp.38,39fordetails). Theyaredefinedby

x(z) f el-’x+l><z’b>db. (13)

If

f(z)

is aradialcomplex-valuedfunctionin

D,

thenitsspherical-Fourier transform^isgiven by

JD (14)

with this inhand,we statethe following

THEOREM4.1. Let

f(z)

beradial inD. Thenthe conclusion of Theorem 3.2 holdswiththe integral

as

hl "

^{0and theproofiscomplete.}

REMARK

3.3. Weindicateherethatifthe k-th difference

Ahlf(z

^{isemployedinDefinition2.2}

of the Lipschitz condition, thenthiswould resultintheappearanceof the factor

[rAI

^2kinplace of

[rAI

in theestimatesgivenby(10), (11),butthis will notaffect thefinal conclusionof Theorem 3.2. This is due tothewayin which rand aretiedupin theirvariation.

REMARK

3.4. Applying HOlder’s inequalityto

(12) for/ff <

^2weget asAtendstoinfinity,thisprovesthefirst part of the theorem.

Ontheotherhand,given

(12),

then by following the reverse argumentasintheproofof Theorem 3.1 onecanarriveeasilyatthe estimate

IIf(z + h)-/(z)ll O(r ) O(tanhr 9-0)

O(Ihl )

400 M.S. YOUNIS

beingreplacedwith

PROOF. Theproofwill carried briefly;wemainlypointoutthenecessarymodifications which will suitthe presentsituation. Oneessentialaspect here is thepresenceof theHarish-Chandra c-function bothintheinversionformulaaswellasintheParseval’s identity. Thus one has

c(a)=r ^{r (a+), c}

whereasforA6

R

t

TheParseval’s idemity readsin thiscase

If(z)ldz I]’1 IC(A)

^{dA. (15)}

Thetransform of

f (z + h)

is

fo.f(z ^{+ h)_x(z)dz} fo ^{f(w) [f} e(-iX+l){u’-h’b>db]

^dw

e(-iA+l)(h.b)

T()"

Tng

imo

aoum e or

ofe^(h,b) e rtendstorodtt

IC()l -= _o()

forl$e

A,

eprfof

eorem

3.2 could be appfi(moword forword)to

@eld

^the

r

^{reslt of}

^e

theory.

Weretookat

s pot tt

the

deee

^of

sim

^{bnthe 1}

^o r

isnot

t

mdsg

ew

ofthe ft

e te

^overthe

d ^B

^of

e t

didoesnotplayy

sifit

^role

pnen

to the ord of

mde

^{d inthe ablute nvergen of bo}

e

FoYer

dthesphtrsfo of

ons

in

H 2. Ts

imeg cod

y suppress out

dgetothe coseof theproof. Thus asf as

e

problof

e

orderof

mde

^d

e

solute nvg e

nm e or

of

f(A, b)

d

f(A)

^is

ost

the e

4.2. Wewouldfike to

t

^out

^{t e prous}

ysis appfies

ost exay

tothe sphec sfo on theLobaschowsspacesociat th

e

complex goup

SL(2.C)

(s [7], pp.

40002). r

^tridenthe11 be

ve

^brief.

In

ts

cethesphtrsfoisdetobe

I ^{() I(t) itei’dt}

so that thetransformof

f (t + h)

is

f ^{(x) sinh(x} h)eU’(-h)

^dx.

Ash 0

sinh(x h) O(sinhx)

and the transform of

f(t + ^{h) f(t)}

^isequalto

^(e

^-’vh

1)]’Iv

^as

usualand thisleadstothefinal estimate

L ^{ITl,2d, o(, -,)}

TRANSFORMSOF LIPSCHITZ FUNCTIONS 401 orequivalently

as

,

_goestoinfinity. The otherpart of the proofisvery clear bynow. Moreovertherangeof for which

f [f[d,

^isboundedisbetter than that foundinthe case the spherical transforms for

functions

on

H2.

Inthiscase HOlder’s inequalityfor/

_<

2leadsto

which inturngives

2 <_<2 for the boundedness of the last quantityasv oo.

.

CONCLUDING REMARKS

Itwould beconvenient toend with afew comments-ofarather heuristicnature-whichmightcast somefightonthetreatmentof the problemonotherdomains. Ourfirsttargetin thisrespectis thegroup

SL(2.R)

denoted here byGfor brevity. Thus fora functionf(g)gEGasmall translation suitablefor the Lipschitzconditions inthiscaseisgiven by

f(gh),h G, Ih

^{O. Thus}

f

^{belongstoLip(a,}

²⁾

^on

^G

^if

Ilf(gh) f(g)ll2 O(Ihl)

0

<

^a

<

1,

]hi

^--,0. With this inhand,one canapplyParseval’s identity

(see [4],

p. 346,

[6],

Vol. 2, p.53)inordertoobtain the required estimates for theFourier(spherical) transforms

f

correspondingto theprincipal and discrete seriesrepresentationsofG. The occurrenceof thetwointegralsandtheone summation onthe rightsideof the identity causesnoproblembecauseonthe onehand cothAtendsto onefor large

A,

thus thetwointegralsaretreated in the same manner. Onthe other hand the three parts aremajorizedby thesamequantity

O(Ihl);

thisenables ustodealwitheach part onits own. Thiswould be easier; besidesit will notaffect the final conclusions.

Secondly, we hintthat the previous analysis is amenable to treatment ofthe problem on other classicalgroupsandtheir alliedsymmetric andhomogeneousspaces(see [1],ChapterXfor examples of thosegroups). Itgoeswithoutsayingthat therewould besome modificationsinthemaincourseof the proof when handlingconcretesituations suchasSL(2,C) for example. Thirdly,onecould explore the validity of the presentlineofthoughtsonsemi-simpleLiegroupsingeneral. Acluetoinvestigatingthis problemlies in anestimate forthe Harish-Chandrac-functionnear infinityinthat case(see

[2],

p.450and [3], p.

183).

These points willbetakenupin aforthcomingpaper.

REFERENCES

[1] HELGASON, S., Differential ^Geometry,

^Lie

^Groups,

and Symmetric

Spaces,

Academic

Press,

New York, 1978.

[2] HELGASON, S.,

Groups

and GeometricAnalysis,AcademicPress,New York,1984.

[3] GANGOLLI,R. andVARADARAJAN, V.S.,HarmonicAnalysts

of

^SphericalFunctions on Real Reducttve

Groups,

SpringerVerlag New York,1988.

[4]

SUG1URA,

M.,

UnitaryRepresentations andHarmonicAnalysis, Wiley,NewYork, 1975.

[5] TITCHMARSH,E.C.,Theory

of

^{FourierIntegrals,}Oxford UniversityPress,1948.

[6]

WARNER,

G., HarmonicAnalysisonSemi-Simple^Lie

Groups,

Vol. 2, Springer Verlag, New York,1972.

[7] WAWRZYNCZYK,

A., Group

Representations and Special Functions, D. Reidel Publishing Company,

Boston,

^1984.