Uniformly Bounded Representations and Harmonic Analysis of the 2 x 2 Real Unimodular
Group
Author(s): R. A. Kunze and E. M. Stein
Source: American Journal of Mathematics, Vol. 82, No. 1 (Jan., 1960), pp. 1-62
Published by: The Johns Hopkins University Press
Stable URL: http://www.jstor.org/stable/2372876 .
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UJNIFORMLY BOUNDED
REPRESENTATIONS
AND HARMONIC
ANALYSIS OF THE 2 X 2 REAL UNIMODULAR
GROUP.*1
By R. A. KUNZE and E. M. STEIN.
TABLE OF CONTENTS
INTRODUCTION
CHAPTER1. OPERATOR
VALUEDFUNCTIONS
? 2. Lp spaces of operatorvalued functions.
? 3. Interpolationin the generalcase.
? 4. The main interpolationtheorem.
C(HAPTERII.
UNIFORMLY BOUNDED REPRESENTATIONS
? 5. Uniformlyboundedrepresentations.
? 6. Some lemmasfromFourier analysis.
? 7. Proofsof Theorem5 and Theorem6.
CHAPTER III.
THE FOURIER-LAPLACE TRANSFORM ON THE GROUP
? 8. Hausdorff-Young
theoremfor the group and certainof its
implications.
? 9. The discreteseries.
CHAPTER
IV.
APPLICATIONS
? 10. Boundednessof convolutionoperator,
?11. Characterization
of unitaryrepresentations
of G.
*
Received November 13, 1958.
s This research was supported by the United States Air Force under contract No.
AF-49 (638) -42, monitored by the Air Force Officeof Scientific Research of the Air
Research and Development Command.
1
2
R. A. KUNZE
AND E. M. STEIN.
Introduction. This paper deals with a study of the real 2 X 2 unimodular group. Our study of this particulargroup is motivatedby two
factors. First, this group has an intrinsicinterest,especiallyin view of its
connectionwith severalbranchesof Analysis. Secondly,the 2 X 2 real unimodulargroupaffordsan illuminatingexampleforthe studyof othergroups.
We constructa familyof uniformly
boundedrepresentations
of the group,
and considerits implicationwithregardto the Fourieranalysisof the group.
These representations
withthe followingproperties. They all
are constructed
act on a fixedHilbertspace S9; theyare determinedby a complexparameter
S, 0 < R (s) < 1, and dependanalyticallyon the parameters; finally,when
R(s) - 1, theserepresentations
are, up to unitaryequivalence,the continuous
principal series.
The above properties,
in particularthe analyticity,
togetherwith certain
convexityargumentsapplied to operatorvalued functionsyieldthe following:
1) The "Fourier-Laplace" transformof a functionf in L1 (G) exists
as an operator-valued
function5, whosevalues 5 (s) act on , and whichis
analyticin s, 0 < R(s) < 1.
2) When f E Lp (G), 1 ? p < 2, the Fourier-Laplacetransforma can
still be defined,and is an operatorvalued functionanalyticin the strip,
1 -/p
< R(s) < i/p.
3) A detailedanalysisof the proofsof the above revealsthe remarkable
5 of f is unifact: If fE Lp(G), 1 ? p < 2, the Fourier-Laplacetransform
1'
formlyboundedin the operatornormalong the line R (s)
In conjunctionwithan analysisof the discreteseriesof representations,
3) impliesthe followingsignificant
fact concerningharmonicanalysison the
group: Let icbe a functionin Lp(G), 1 ? p < 2. In contrastwiththe (noncompact) abelian situation,the transformation
f -> f *k,
of convolutionby 7c,is a boundedoperatoron L2(G).
We shall now discusscertainof thesefactsin greaterdetail. The representationswe considerarise as follows. Let
=[c
d]caddbc=1
be an elementof the group. We then consider,for each complexs, the
multiplierrepresentations,2
These representations may be put in the form originally obtained by Bargmanu
[11 bv means of the transformationa - tan (0/2).
(1.1)
(1. 2)
~
~
f(x)
t(x)
+ d 128-2
->Ibx
sgn(bx+ d)c)/(bx
I bx+
3
GROUP.
REAL UNIMODULAR
c)/ (bx + d))
f((ax+
d 128-2
f
((ax +
+ d)).
The two continuousprincipal series are obtained from these by setting
the functionsf to lie in L2 (- oo, oo).
s j + it and restricting
boundedrepresentations
of the uniformly
We are led to the construction
describedaboveby the followingconsiderations.In the groupwe distinguish
a particularsubgroup,namely,the subgroupof lowertriangularmatricesof
the form
9
? ],
a 0+ .
of either of the principal
It may be shown that when the representations
seriesare restrictedto this subgroupthen they are all unitarilyequivalent.
This raises a natural problem. Can one finda Hilbert space X and representationsU+( , - + it), U-( ,1 + it) unitarilyequivalentto (1. 1), (1. 2)
+ it) such that U+( ,1 + it) when restrictedto the lower tri(for s
angular subgroupare independentof t? The answeris in the affirmative;
U+( , s), 0 <R (s) <1,
boundedrepresentations
the uniformly
furthermore,
which we construct,are characterizedas the analytic continuationsof the
U+(*, 1 + it). It should be added that the representations
representations
series.
U+( ,), 0 < a < 1, are unitarilyequivalentto the complementary
In solvingthe above problemand in the actual constructionof these
it is natural to considerthe inducedaction of (1. 1), (1. 2)
representations,
here are
F of the functionsf. The considerations
on the Fouriertransforms
ratherinvolvedbut have an intrinsicinterest. For the analysisrevealsconnectionswith both the so called "iHilbert transform"and the notion of
"fractionalintegration."
(1. 1) or (1. 2) afford(at
It is clear that the multiplierrepresentations
-+ it.
least formally)an analyticcontinuationof (1. 1) or (1. 2) when s
The problem,however,is howto makethis precise,i. e., the problemof finding
act and depend
an underlyingHilbert space on whichthese representations
analyticallyon s. Ehrenpreisand Mautnerhave also dealt withthe problem
(1. 1) to values of s 7& I + it. Their result
of extendingthe representations
concerningthe analyticityof the Fourier transformof an L1 spherical
functionon G was one of the motivatingfactsin our work. In addition,it
was broughtto our attentionby Ehrenpreisthat a similarresultmighthold
betweentheirresultsand
for Lp. However,thereare significantdifferences
arisingfrom
representations
bounded
uniformly
ours. In [5] theyconstruct
act on different
these representations
(1. 1) when s 7, i + it; nevertheless,
=
4
R. A. KUNZE
AND E. -M. STEIN.
Hilbertspaces dependingon s. In [6], and [7] theyconsider,at least implicitly,representations
whichact on a fixedHilbert space; but in this case
the representations
are not uniformlyboundedwhen s 7/=1?+it.
The precedingconsiderations,
in particular3) above, lead to characterizationsof the representations
of the group. This may best be understood
in thefollowingcontext. As a resultof theworkof Bargmann[1], Godement
[8], and Harish-Chandra[14], attentionhas been focusedon a particular
class of representations,
thosewhichare "square integrable"in the following
sense; a representation
g -->U0 on a Hilbert space X is of this type if the
function
=
cp=cp(g)
(Ug4
is in L2(G) for everyX, in 59. The square integrableirreducibleunitary
representations
of the 2 X 2 real unimodulargroup are essentiallythe representationsofthe discreteseries. We are able to give a similarcharacterization
of the representations
of the continuousprincipal series. An irreducible
unitaryrepresentation
g -> U0 is equivalentto one of the latterif and onlyif
) E Lq(G)
(Ugq$,?
holdsforall E,V C 9l and all q > 2, but not for q - 2. An analogouscharacterizationholds for the representations
of the complementary
series. (See
Theorem10 and its corollary,in ? 11.)
One of the main ideas motivatingthispaper was the desireto extendthe
classical Hausdorff-Young
theoremto the group. We recall the formof the
Hausdorff-Young
theoremfor Fourier transforms,
as given by Titchmarsh.
Let f C Lp, 1 ? p ? 2, and let
F(x) =- (2r)-f
e$ixY
f(y) dy.
Then
F J<
11
(27r)1-/P11
f tIIp
wherel/p+1/q=1.
The mostconvenientmethodforprovingthis theoremis by using a convexityprinciplefor linear transformationsintroducedby M. Riesz. This
principleallows one to "interpolate" betweenvariousboundsof linear transformations. For a general discussionof this methodof proofwe referthe
readerto [3]. An extensionof the above theoremto locallycompactabelian
groups,via the Riesz convexityprinciple,is given in Weil's book [23]. An
abstractgeneralization
of thistheoremto arbitrary
locallycompactunimodular
groupshas been givenby one of us, [16]. This generaltheoremwas proved
REAL UNIMODULAR
GROUP.
5
by what amountsto an extensionof the convexityprincipleto linear transbetweenoperatorvalued functions.
formations
boundedrepreDue to the analyticstructureof the familyof uniformly
seintations
of G, it is possibleto provea versionoftheHausdorf-Youngtheorem
whichis much strongerthan its classical analogue (see Theorem7 in ? 8).
The proof of Theorem 7 necessitatesyet another extensionof the Riesz
convexityprinciple-fromthe case of a single fixedlinear transformation
to
a familyof transformations
dependinganalyticallyon a parameter.3
It seems quite likelythat many of the resultsdescribedabove hold not
only for this group but for certainothergroups as well (e. g. the comiplex
classicalgroups). We hope to returnto this matterat a later time.
We now proceedto describethe organizationof this paper.
In Chapter I, which consistsof ?? 2, 3, and 4, we consideroperator
valued functionsand we provethe basic convexity(interpolation)theorems.
?? 2 and 3 are quite generalin nature. However,in ? 4 the subject matter
is tailoredto fitthe situationwhicharisesin the 2 X 2 real unimodulargroup.
of our familyof
ChapterII concernsitselfwiththe actual construction
representations.In ? 5 thegeneralbackgroundand theoremsare stated. Their
proofs,however,requiresomeextensiveFourieranalysis. This is done in ? 6.
rn ? 7 we returnto the proofsof the statedtheorems.
Combiningthe resultsof ChaptersI and II, we study the "Fourierforthegroupin ChapterIII. This leads to our extension
Laplace " transform
of the Hausdorff-Young
theoremwhichis containedin ? 8 . In ? 9, we complete the Fourier analysisof a functionon the group by a considerationof
the discreteseries.
ChapterIV containssome applicationsof the above. In ? 10 we are
mainly concernedwith the theoremthat convolutionby a functionin LI,
1 ? p < 2, is a boundedoperatoron L2. Some implicationsof this result
are also deduced. Finally,in ? 11, we deal with characterizations
of various
of the group and with a relatednotion--the"extendability"
representations
of a representation
to Lp.
We shouldlike to observethat,exceptfor some notation,the contentsof
ChaptersI and II are independentof each other. Since ChapterI is of a
more technicalnature,the readermightwell begin with ChapterII which
deals with uniformlyboundedrepresentation
of the group.
8 In the case of numerical valued functions this extension was obtained by one of
us in [20].
6
R. A. KUNZE
CHAPTER
AND E. M. STEIN.
I. OPERATOR VALUED FUNCTIONS.
2. Lp spaces of operatorvalued functions. In this part we provetwo
purelytechnicaltheorems. In theseresultswe have ignoredvariouspossible
and have restrictedour attentionto a rathersimple situation
generalizations
whichappears to be adequate for our purpose.
We beginby introducingenoughterminology
to state the theorems.
a
will
denote
Throughoutthe paper X4
complexseparableHilbertspace.
The ring of all boundedoperatorson & will be denotedby S. If A is any
is any orthonormal
basis of 59,
non-negativeoperatorin 13and
*
then
1,
(2. 1)
tr(A)
>: (A&i
n=l
is non-negative
and independentof the choice of basis. The bounidof ain A
in B will be denotedby 11A jl, and we shall put i A
(A*A)1. The p-th
normof A is thengivenby
=
(2.2)
IA II
(tr(j A
jP))1/1,
where 1 _ p <oo. The letter M will always stand for a regular measure
space4 overa locallycompactspace witha countablebasis for open sets; the
underlyingtopologicalspace will also be denotedby 311. We shall consider
functionson M whose values are boundedoperatorson St. If F is such a
function,we say that F is measurableprovided
(2. 3)
t >(F (t) d, ),
tC
E M.
is a measurablenumericalfunctionon 11M
for each pair of vectors@,q in S1.
If F, G are measurable,our assumptionsimplythe measurabilityof F H-G.
FG, and F*, these being definedin the obviousway; thus for example F*
is givenby F*(t) = [F(t)]*.
An operatoron 69 is said to be of finiterank if it is reducedby a finite
limensionalsubspace. The set of all such operatorsis a two sided * ideal in
B and will be denotedby S.
(2. 4) By a simple functionwe shall mean a functionF on M to E
having only a finitenumberof distinctvalues, each non-zerovalue being
assumedon a set of finitemeasure.
4 For a general discussion of measure theory on locally compact spaces see Halmos
[10, Chapter 101.
REAL UNIMODULAR
THEOREM
F(t)
norms11
1. If F is a measurableoperatorvalued functionon M, the
1?
IIP,
p ? oo, are measurableas functionsof t, and therelations
F 11-
(2.5)
IIF fP
(2.6)
,
GROUP.
(f iF(t)
esssup 11F(t) 11X
IIPdm(t) )1/P,
1?p <oo,
definenormsrelativeto whichthe collectionLp(M, 3) of all measurableF
with 11
F IIP
<co is a complexBanach space (one identifiestwo functionsif
theydifferonlyon null sets). Moreover,the formula
fF=
(2. 7)
f trF(t) dm(t)
is meaningfulforF in L1(M, 3) and definesan integralsatisfying
fFjI 11F111.
(2.8)
2. If F is any measurableoperatorvalued functionon M and
vanishingoutsidecompact
1 ? p ? oo, thereexistsimplefunctionsSi, S2,
sets such thatFSn is integrable,
THEOREM
(2.9)
lir fFSnr
n
and 1iSn lq _ 1, i/p + l/q
that
lr
(2. 10)
= l/p
IF Ilp
1. Furthermore,
if p and q are indices such
+ l/qC? 1 and F, G are measurable,then
F IlP11
G lq
FG Iir? 11
11
Finally, the simple functionsvanishingoutside compact sets are dense inl
Lp(IM,3) for all p such that 1? p < oo.
Except for the minor complicationthat we are dealing with operator
valued functions,the proofsof theseresultsinvolvenothingnew and proceed
includedmost of the details for
along staildardlines. We have nevertheless
thebenefitof thereader. We mentionthatone mightobtainsimilaralthough
less explicittheoremsas consequencesof knownresultsfromdirectintegral
integration;however,it seems
theoryand the theoryof non-commutative
situation
inappropriateto complicatean essentiallysimplermeasure-theoretic
by such considerations. Furthermore,in our applicationwe require these
resultsin the ratherexplicitand concreteformgiven above.
We begin by recallingsome of the facts about the trace and the p-th
normsmentionedearlier. As a generalreferenceto this part, we referto a
paper [4] of Dixmier.
gE 5
Let t, D be fixedvectorsin &l and defineE,7 by En,,?(g)
=
8
R. A.
KUNZE
AND
M.
E.
STEIN.
An operatorE is of finiterank if and only if it is a finitesum of operators
of the formEe n and moreover
(2. 11)
tr(E4,-q)=-
Let S., 1 C p :0oo denote the collection of all bounded operators A on
A IIP< 00. A positive operator A is in B., 1 ? p < oo, if and
Sl such that 11
only if its spectrum Ak,A2,
is discrete and
00
E XnP <
00.
n=l
In this event,IfA 11
p=
(2. 12)
whichimplies,
vP)1/p,
(' 2
n=1
1A IlPis a non-decreasing
functionof p.
(2.13) If A E 3p, B E ,3q,and 1/r=1/p+1/q?
A11p 11
B 11q,where1 _ p, q C 00.
1, then j]AB llr
(2. 14) If A E B and 1 ? p coo, thereexist operatorsE1,E2,
finite'ranksuch that 11
En IIq ? 1 and
limtr(AE,)
n
wherel/p +A1/q
-
of
A 1v,
1. In case A is of finiterank,thereexistsanothersuch
- 11
A IIP.
Iq ? 1 and tr(AE)
operator E with 11E
is a Banach space under the normgiven by (2. 2), and the
(2. 15)
f3p
collectionE of operatorsof finiterankis densein Sp for 1 ? p < oo.
(2.16)
If 1
p<oo
and B E3q, where1/p+ l/q= 1, then
A--tr(AB),
A E 3p
is a boundedlinearfunctional,kB on Op, and the map B withthe conjugatespace of S3p.
B
identifies3q
Given an orthonormal
basis 4,
of 9 we formthe set 0 of all
finiterationallinear combinationsof the operatorsEij and (- 1)Ei , where
(2. 17)
Eij
=
Ets @J.
?Z is denumerable,and one easily verifiesthat the productof two members
of i
is again in ?Z.
LEMMA 1.
?) is dense in R,,, 1 < p < oo.
9
GROUP.
REAL UNIMODULAR
Suppose E is an operatorof finiterank and that P is a projectionof
E. Then for 1 ? p < oo and A, B in 9D we have
finiteranksuch that EP
E-AB
11
P-B
112
11B 112
+11 A 2 11
111
C11E-A
EP-AB
v 11
11
112.
The inequalitiesfollowfrom (2.12) and (2.13). Now because .Z is dense
in thep-th
in B2, we see that any operatorof finiterank can be approximated
normby elementsof i). An applicationof (2. 15) finishesthe proof.
As a corollarywe obtainthe fact that S , 1 _ p < so, is separable.
The collectionof all measurableoperatorvalued functionson M will be
denotedby 3(M).
LEMMA
2. If F C 3 (M) and E is an operatorof finiterank,
t -* tr(F (t) E)
is a measurablefunctionon M.
There exist vectors
,
*,
,*
E
in X such that
n
=
Eti.d
n=1
n
E EF(t), 7 and by (2. 11),
Thus F(t)E=
n=l
tr(F(t) E)
whichimpliestr(F(t)E)
LEMMA
-, r
)
E(F(t)
n=1
is measurableas a functionof t.
3. If 1? p?oo
functionon M.
n
=
and FCE3(M), t- 11F(t)I p is a measurable
Let A belongto BZ. From (2. 13), (2. 14), and Lemma 1 we see that
(2.18)
A IIp= Sup{ tr(AE) I: E C 0, 11E l1qC 1},
11
wherei/p -- 1/q 1. ReplacingA by F(t) and applyingLemma 2 we see
that 1F (t) 1p is measurable;this followsfromthe fact that the least upper
boundof a countablecollectionof measurablefunctionsis again measurable.
4. A functionF on M to p,,1 ? p < oo, is measurableif and
t
-tr
onlyif
(F (t) B) is measurableon M forall B in 3q (l/p + 1/q= 1).
LEMMA
For everypair of vectors4, -qin *, B=Ee,, is in Sq; henceFCE3(M),
providedtr(F(t)B) is measurableas a functionof t for all B in S3q.
Conversely,suppose FCE 3(M). Let BCESq. By (2.13) tr(F(t)B)
existsand is finitefor each t in M. If p 1,
=
10
R. A. KUNZE
AND E. M. STEIN.
n-*o
=
iSany orthonormalbasis of 9, and is thereforemeasurable.
where is,
4
If 1 <p <oo, thereexist, by (2. 15), operatorsE1,EF,
, of finiterank
O 11
such that B-E
0.
Thus
B
q
,
Itr(F(t)B) -tr(F(t)E,)
|
11
F(t) IIP11
B- Nn 1q > 0.
By Lemma 2, tr(F(t)E,) is measurablein t foreach n, and hencetr(F(t)B)
is also.
The resultjust establishedtogetherwith (2. 16) showsthat a measurable
functionF on ,Alto B. is weaklymeasurableas a functionon If to the
separableBanach space B.; thus F is also stronglymeasurable.5
The proofof Theorem1 now followsfromthe precedinglemmasand the
wellknowntheory5 of theLebesgueintegralextendedto functionswithvalues
in a Banach space.
II1 provingTheorem2 it is convenientto establish
LEMMA
5.
Suppose F
n
=
i=l
fiA1, where fi is a measurable numerical
function on M, fifj= 0, i=/=j, and Ai E S.
F llP (
11
(2. 19)
n
Then for 1? p <oo,
fi IPP
11
1A1 j)I'/P,
and in case 11
F 11,< ??,
fF
(2. 20)
n
E (f fi(t)dm(t) (trAi).
i=l
0 (i7=j),
Since f (t)fj(t)
f F(t) IIPP
r1
dm(t)
IIIP
F
n
Ai IIPP)
dnm
(t)
f ( j=1 Ifi(t) P11
n
~E(fr fi(t)|Pdm (t) ) 11AjJJIPP
If 1FI1< 00,then
fJF ftr(F (t) ) dm(t)-
6
tr( E f5(t)Ai) dm(t)
i=1
For a discussion of these points see Hille and Phillips [15, Chapter 3].
11
GROUP.
REAL UNIMODULAR
zf(t)trAt)dm(t)
f( j=1
n
=Y. (rff(t) dm (t) ) tr(At).
i-l
sets are
LEMMA6. The simple functionsvanishingoutside comnpact
densein Lp(M, J) for 1?5 p < oo.
Let f,g be measurablenumericalvaluedfunctionson MI,and let A, B E S.
By simpleestimatesand the precedinglemmawe have
(2.291)
A -B
A IIP+ 119
gIP 11
f - gIP 11
~1fA -gB IIP' 11
IIP,
funcwhere1 ?< p < oo. Thus if e > 0, A E B., and if f is the characteristic
tionof a set of finitemeasure,we can choosea compactset withcharacteristic
g aindan operatorB of finiterank such that
fuinction
fA-gB 11p< e.
11
(2. 22)
The conclusionof the lemma followsfromthe fact5 that finitelinear combinationsof functionsof the formfA are dense in Lp(M1,i).
LEMIA 7. If F is a simplefunctionwithcompactsupportthereexists
a simplefunctionS with compactsupportsuch that S 1l _ 1, and
F IIp,
fFS
(2. 23)
1/p+1/q==1.
where1?poo,
, A, of finiterank and mutuallydisjoint
There exist operatorsA,
f5such that
functionsf
measurablesubsetswith characteristic
n
F ===EfAi.
i=1
By (2. 14) thereexistsan operatorEi of finiteranksuch that Ei
A11 p. Let
tr(A5E)
ci
(2. 24)
11F KllplP
11Ai
jq
? 1 aild
JpP-1
and put
S
(2. 25)
Then
fF5
=X
f
n
cifiE.
j=1
c,f,(AgE )
n~~~~~~~~
==ci(ffj(t)dm(t)tr(AiEi) =
i=1
=
F II.
c 11
fi IIPP1Al IIP= 11
12
R. A. KUNZE
AND E. M. STEIN.
Also
n
qo
11sJS
jjgqq
- ECjqi
j=1
ii
q 11
Iff,llq~1E
E* lgqq
n
<_
E 11F !(p1-p'q
11
Ai jp(p-1_)q
11
f
j=1
n
z
f, IIPP
Ai IIP
11
11
-11 F1 P_P
lipp
1.
j=1
Finally,sinceF has compactsupport,so does S.
Proof of Theorem 2. SupposeF, G are measurable. To establish(2. 10)
we use (2. 13) whichimplies
11FG
lr?_f (1F(t) IIP' G(t)
fF(
(t
11
F)IPP
IIq)r dm(t).
dm (t) ) "P (f 11G (t) 11
qqdmn(t)) I-q
providedp 74oo and q 74oo. The other two cases arse treated by similar
arguments.
As the case p oo is somewhatexceptionaland requiresseparatetreatment,we shall prove (2. 9) onlyfor p such that 1 ? p < oo.8 Suppose first
of all that 11
F,1 P< oo. By Lemma 6, thereexistsimplefunctionsF1, F,
with compact supportssuch that 11
F -F
IIP-> 0. Choose S,, for Fn in
accordancewithLemma 7. By (2. 10) FS, is integrableand
I fFSnf-
FnSn I
11
F-Fn
IIp?.
If 11
F liP=- , let F. be the productof F and the characteristic
functionof
a set of finitemeasure containedin {t: F(t) lpc n} and chosen so that
Fn 1p- oo. Then iiF,, J1p
Thus we can choose a simple function Sn
<oo.
with compact support contained in the support of F, such that, 1iS,,lq ? 1
and
II, < 1/n.
fFnSn F11F
Then FS,===
FnSn
and
f FSn
1F lIP.
3. Interpolationin the general case. In this sectionwe provea rather
general interpolationtheoremfor operatorvalued functions. Let 4 be a
complexvalued functionwhose domain contains a strip, X< Rz ?< /. We
shall say that 1 is admissible on the strip if (D is analytic in c < Rz </,
continuousin a ?< Rz _/, and satisfiesthe growthcondition
6 We
do not need the exceptional case p - oo in our application.
REAL UNIMODULAR
(3.1)
13
GROUP.
Sup log Ij> (x + iy) I
CekIvI,
whereC and /iare constantsdependingon d1; we requirealso that ,usatisfies
the additionalcondition
(3.2)
If
<7r/
12
dii,
are admissible on a given strip and if v1, v2 are complex numbers
it is easilyverifiedthatthe combinations
v11 +
v2d12, )142
are also admissible.
A complexvalued functionon a measurespace will be called a simple
functionif it can be expressedas a finitelinear combinationof characteristic
functionsof measurablesets of finitemeasure.
Now let M1, M2 be measurespaces, and let D be a strip, -? Rz?<,l8.
Suppose B,, z E D, is a complexvalued bilinearformdefinedfor all simple
functionsfl, f2 on M1, M2. We shall say that the collection{B,} is an
admissiblefamilyof bilinearformson D if
(3. 3)
4)(z)
Bz (flyf2)
is admissible
onD foreachpairofsimplefunctions,
fl,f2 on M1,M2.
We now introducesome notationand terminologywhich will remain
fixedthroughout
this part.
The stripa < Rz ? ,Bwill be denotedby D and we shall put
(3. 4)
y-==(1
-
) (X+ Tfl
0<
T<1.
We supposepo,piL, qo, q1 are givenindicessuchthat1 ?
or q1# o. The indicesp, q are then determinedby
(3. 5)
1/p=- (1-r)
(3. 6)
1/q
pi,
qi C oo, and qo
so
l/po + 71/pi,
(1-r) 1/qo + 71/q1.
The conjugateindicesof q0,q1,q will be denotedbyqo', q1',q'. Finally,Ao,Al
will denotenon-negativefunctionssuch that
(3. 8)
logAi(y)
A
Ae6ll,
8 < 7r/
a).
Withminorchanges,the proofof Theorem1 [20] yields the following
convexityprinciple.
LEMMA 8.
and suppose
(3. 9)
Let {Bz} be an admissible family of bilinear forms on D,
IBa+iy(fl,f2)I Ao(y)11
11
f,IIPo
f2
liqo
R. A. KUNZE
14
AND E. M. STEIN.
f2 IjqL'
11
Bp+i8(f1if2)I ?A1(y) 1If. IIPO
(3. 10)
for all simple functionsf,,f2 on MA1,
M2. Then for simple functionsf. f,
we also have
(3.11)
|jY(fl
f2)
IAT-
jj q'
11
fll P 11f2
The constantA, is given explicitly,in termsof the Poisson kernelfor the
strip,by
logA,
(3.12)
- logAo [(-)y]
w(1
r
+ f logAl[(
, y) dy
a)y]w(r, y)dy,
where
w(1, y) -
sin7r/
(cosh7ry+ cos7r).
By a boundedsubsetof a regularmeasurespace we mean any measurable
subsetof a compactset. Now let N be an arbitrarymeasurespace. Suppose
fromsimplefunctionsf on N to measurTZ,z C D, is a lineartransformation
able operatorvaluedfunctionsT, (f) = F, on AM. We shall say that { T,} is an
admissiblefamilyon D if (F, (t)$,y) is locallyintegrableon M1and
(3.13)
?)(Z)
=--j (Fz(t) , n)dm(t)
is admissibleon D for each choiceof vectors$, 7 in 94, simplefunctionf on
N, and boundedsubsetK of M.
THEOREM 3. Let N be a measurespace, and suppose {Tj}, z C D, is an
admissiblefamilyof linear transformation
fromsimplefunctionsf on N to
measurableoperatorvalued functionsTz(f) F-F on M. Suppose further
that the followingtwo conditionsare satisfiedfor each simple functionf.
(3. 14)
(3.15)
T+iy(f) l oC Ao(y) 11 IIP.
11
T (+iY()
11
IIf i C A1(y)11
f IY1.
Then it is also true that
(3-16)
q AT11
11Tny(f)11
f
fIP
In provingthe theoremit is convenientto establishthe followinglemma.
LEMMA 9. If {Tz}j, z ED, is an admissiblefamilyand S is a simple
operatorvalued functionon M whichvanishesoutsidea compactset in M.
Then trP,(t)S(t) is integrableand
15
REAL UNIMODULAR GROUP.
(3. 17)
1(z)
=
f trFz(t)S(t)
dm(t)
is admissibleon D for each simplef on N.
functionof a
Suppose firstthat S == kE , wherek is the characteristic
boundedset. Then
t-rFz(t) S(t) -= kc(t)(Fz(t)t,t)
and the resultfollowsby assumption. The generalcase followsby linearity.
Proof of the theorem. Our assumptionsimply q #oo.7
that
Thus to show
Ty(f)11q
11
C A7 11fIIP
in view of Theorem2, to showthat
it suffices,
I ftrlFy(t)S(t) dm(t) C? A, 11
S 11k
f IIP
11
(3. 18)
for each simple functionS vanishingoutsidea compactset.
The idea of the proofis to reduce this problemto one concerningan
admissiblefamily of bilinear forms. We shall then apply Lemma 8 to
completethe argument.
Supposethenthat S is a simplefunctionwithcompactsupport. We can
n
expressS as E k1Ej,wherekI,1c2,
j=1
functionsof
, cknare the characteristic
mutuallydisjoint boundedsubsetsK1 and each EI is an operatorof finite
rank. Now let El U=
Ei I be the canonical polar decompositionof Ei,
and let
(3.19)
j
A,P>O
X,ij> ?,
be the spectraldecompositionof I Ei
The pairs of indices i, j will then
rangeovera finitesetwhichwe shall call M2. To each complexvaluedfunction
g = {gij) definedon M2 we associate an operatorvalued functionG on M
whichis given by
(3. 20)
n
G (t)- _k.7i(t) Y: g{jUxP+j.
f
j=1
Then G is a simple functionwith compactsupport,and by an elementary
computationwe get
(3.21)
7
G*(t)G(t)
itv
7
ls(t)IgHI'2Pi,.
The case q-=oo couldbe dealt withby a moreinvolvedargument.
R. A. KUNZE AND E. M. STEIN.
16
Now for 1 Cp
11G IPP==ftr[(G*(t)G(t))I/2]dm(t)
<oo,
which implies
kijjpP)"P.
G IIP ( X I gij I11
11
(3.22)
i,j
is independentof p being,in fact,equal to the measureof KE,
Since 11
kIIpPP
G IIP. We
we can introducea measure in M2 relativeto which1 9
g IIP 11
observethat this relationis also valid for p - 0o. Because the maps f-F,
g -* G are linear it followsthat the equation
Bz(f, g)
(3. 23)
ftr(Fz(t) G(t) ) dm(t)
=
definesa bilinearformfor each z in D. In this formulaf is an arbitrary
simplefunctionon M=-N, and g is any complexvalued, obviouslysimple,
functionon M2. By Lemma 9, in particularby (3.17), {B4 is an admissible
familyon D. Now
Fa+iyG1i1 C
Ba+iy(f,g) : I 11
Fa+1Yl o 11G lIqo',
11
and using (3.14) we get
I Ba+iy(f,g) _Ao(y) 1If|IPO
11
g 1Iqo'.
(3. 24)
By similarestimateswe obtain
g 1a
I B:+V(f,g) ? A1(y) || f ||P111
(3. 25)
Thus by Lemma 8, I B7(f, g) I ? AT11
g IIq'. As this holds for all g,
f1IP11
we may take g
Then G-=S
{Aij}.
f trFy (t) S (t) dm(t),
By(f,g)
(3. 26)
and
whichimplies (3. 18).
4. The main interpolationtheorem. In orderto proveour resultsfor
the 2 X 2 real unimodulargroup,we use, in additionto factsaboutthe group
certain convexityarguments. The basic and most
and its representations,
importantfact along theselines is establishedin this section;withthe intent
of clarifyingthe situation,we have presentedit in a slightlymore general
formthan our applicationrequires.
An operatorvaluedfunction5 definedon an open strip,aO < Rs </3o, in
the complexs-planeis said to be analyticif (J (s) d,-) is analyticfor all 6,s
in N9. We shall say that a is of admissiblegrowthin the stripif
(4.1)
sup log1
THEOREM
(+
i)
:-SCell
,u<.ii/(go
- a).
4. Let N be a measurespace and T be a linear map fronm
REAL UNIMODULAR
17
GROUP.
oniN to analyticoperatorvaluedfunctionssuch that5 = Tf
simplefunictions
is of admissiblegrowthon the stripz, < Rs < ,P3 for each simplefunctiont
f.
Suppose that for c, < a <,B <,38 we have
(4.2)
sup
-oo <t<oo
(4. 3)
(}I
Ja (I +
11a(A +
oo
it) 1o (1 +
it) 1122It j24(
t l)ec
-t I)2b
AO1 f 11,
dt)
Al?A1If 112
forall simplef, wherea, b, c are real and a ? 0. Then we may contclude
(
(4.4)
00
00
|| a (y+it) gq(1 + t )qddt)1/
*?< AT||f IIP,
where 1 <p<2,
1, y=-+T(/31/p+1/q
and theparameterr is determinedby i/p = 1
a), d =c+r(a+br/2.
c),
Remarks. Beforewe provethe theorem,we noticethat the result (4. 4)
is intermediate-inthe sense of Riesz-Thorinconvexity-betweenthe hypotheses (4. 2) and (4. 3). It shouldbe notedthat the singularityat t 0 of
the measure tI12a( + I t |)2b dt does not persistin the conclusion;only the
influenceof I t |2a for t near infinityremains.
The proofgivenbelowcouldbe generalizedin severaldirections. We may
begin with a generalpair of indices (po, q0), (pi, q1) instead of (1,oo) and
(2, 2). We might also considermore general measuresthan those of the
form tI (12a( + I t I) 2b dtgivenabove. Weshallnotconsider
thesegeneralizations here.
It should be pointed out that the proof given below would be much
simplerif a, b, and c were zero. In that case the left-handsides of (4. 2)
and (4.3) would be translationinvariantin t. Since the basic methodof
the proofconsistsof translationalong verticalliiies of the strip,we are forced
to overcomethe lack of translationinvariance by somewhatcomplicated
devices.
At severalpointsin the proofit will be convenientto referto the easily
verifiedresultgivenbelow:
LEMMA10. If v is real and 8 > 0, thereexistsa constantA > 0 such
that
(4. 5)
for-
(+
o<y,t
I y + tl) < A (1 + I y l)v(1 + I t )
<o.
Proof of the theorem. We shall obtain the result as a consequenceof
Theorem 3. To do this we set M1 (-oo, oo) and put
2
18
R. A. KUNZE
dm
(4.5)
=
AND E. M. STEIN.
(1 + I t 1)2(a+b-c) dt,
wheredt is Lebesgue measure. Given a simple functionf on N we form
5= Tf and set
Fz(t) =5(z
(4.6)
+ it) (1 + I t | )c-a(z
+
it)a
for c-?Rz ?< /. Since a> 0 we may choose a single valued branchof the
factor (z -,l + it) a which is analytic in ac< Rz < / and continuouson
is analytic in z for each t and is jointly
-<?Rz?_,8. Thus (Fz(t)>-q)
continuousin z, t forall vectors4, - in 54. Furthermore,
the transformation
T0 definedby Tx(f)
F, is linearand maps simplefunctionson N to measurable operatorvalued functionson M.
We shall now estimate Fz (t) 1.o By (4. 6) and the condition (4. 1)
that 5 is of admissiblegrowthin oc? Rz ?< /, we find,
=
(x+i(y+
+ I t)c-a I - ,+i (y + t)Ia
)II.(1
t)
IIFz(t)II1.=11
ll
<A
(x+i(y+
t))1 0(1 + t )c(1 + IYI)a
(4.7)
Hence,
log 11Fz (t) 11.o
CeAIY+t I +
log (1 + I t l c + log (1 + I y | a
+
log A.
This estimatetogetherwiththe aboveimpliesthe condition(3. 13) that {T.}
be an admissiblefamily. Now, for z a + iy we find,using (4. 2), (4. 7),
that
=
+ i(y + t))110(1+ I t c)(1 + I Y I)a
A IIflI1(1+ I y+ t )-C(1 + It C)(1 + y
5( (a
< A 11
||Fa+iy(t) lIoo
?
)a
Thus by Lemma 10, we obtain (4. 8).
A (1
11Ta+iy(f) 11.o0
(4.8)
f K1
+ Iy la-cI
11
Next we shall estimate11T+iy (f) 112.
jjFp+y112'
00
5 (/3+ i(y +
11
We have,
Iy + t 12a(1+ I t I)2c-2adm.
t)) 1122
Now makinguse of (4. 3), we obtain
jj
.+y 1122
?(A1
If112)2- x0sup
<t<x0
[(1 +
? (A1 f/112)2 sup [(1 +
-co<t<9o
y + t I)-2b(1+
I t 1)2c-2a(1+
y + t I)--2b(1+ I t 1)2b].
I t 1)2(a+b-c)]
19
REAL UNIMODULAR GROUP'.
Thus by Lemma 10,
(4.9)
11l2< A(1 +
||F+iy
I yj)II71 If 112.
Having (4.8) and (4.9) we can apply Theorem3 and concludethat
1ITy(f) IIqC AT1If II
(4.10)
Now
Tey(f)Iq
11
f 115 (y+it)l1qq1( + I t I),c-qaIy00
CAT11f IJll.
Hence
IL
_00
+ I t I)qc-qa(j +
5?|(y + it)q(1
Since Tq
Thus
3+it Igaddm
=
t I),qa(l +
It 1)2(a+b-c) dt ? A 1ff 1q
2,
=qc+2(a+b-c).
qd=gc+qT(a+b-c)
oo1
qq( + 1t I))qd dtC< A 11f JjVq,
(-y+ it) 11
whichprovesthe theorem.
5. Uniformlybounded representations. We now considerthe group
G of 2 X 2 real unimodularmatrices,and we firstrecall some of the known
of G.
factsconcerningthe representations
We representan elementg C G, by
g=[c da
]
ad -bc
1,
and denoteby g (x) the fractionallinear transformation
Then
(5.1)
g(x)
(ax + c)/(bx + d),
-oo < x < oo.
(9192) (x) =92(g1x)
(bx + d)-2, bx+ d=/=0.
and dg(x)/dx
We now introducetwo "multipliers" 4+ and 4v. These are definedby
(5.2)
(5.3)
p+(g,x,s) =
-(g, x, s)
bx+d
I28-2
sgn (bx + d)p+ (g, x, s),
wheres is an arbitrarycomplexnumber.
20
R. A. KUNZE
AND E. M. STEIN.
Next we considerthe "multiplierrepresentations"
g -v
(g, s)
givenforfunctionsf on the real axis by
v+(g, s) :
(5. 4)
f(x) -- q+(g,x,s)f(g(x) )
From these, one may obtain the irreducible unitary representationsof G.
They fall into threeclasses.8
a)
The two continuous principal series
oo00<t <oo
g -> v+~(g,1/2+it),
wherethe Hilbert space is the space L2 of square integrablefunctionson
K< x<po, with the usual measure.
b)
The complementaryseries
g ->v+ (g,v) ,
O <
of<
12
The Hilbertspace,in this case, is definedby the innerproduct
(5. 5)
wherea,
c)
au
(f,h)u=
00
f
00
00
f(x)h(y)
00
I
x -y2
dxdy,
P(2a)cos(o-r)/7r.
The two discrete series,
0,1,2,
g --> +~(g, k)
k=-O,
We shall not need the exact formof theserepresentations.
The Plancherelformulafor G was derivedby Harish-Chandra[13]. It
involvesrepresentations
of type a) and c) and not of typeb). To state it
we make the usual definition
U(f)
f(g)U(g)dg
for uniformlybounded representations
g -- U (g) and f in L1 (G). Using
thisnotationthePlancherelformulaassertsthat,wheneverf C L1 (G) nL 2(G),
f
11
1122
1/2
.i:i v+(f, 1/2+
+ 1/2
00o
t tanhrtdt
it) 1122
V-(f, 1/2 + it)
2'
tcoth7rtdt
8Except for notation, these representations are those of Bargmann [1]; the differenceof notation is discussed more fully in the proof of Theorem 10 in ? 11.
21
REAL UNIAMODULAR GROUP.
+
+
oo
E 11
D+(f, k) 1122(k
k=O
E
k=O
+
1/2)
D-(f,k)l22(k+ 1).
11
Here 11112means the usual Hilbert-Schmidt norm for operators as used
in ?2 above.
One of our main results is contained in the following theorem.
THEOREM 5.
tations
There exists a separable Hilbert space X9 and represeng-
U+(g,s)
of G on t with the followingproperties:
1) 9-- U+(g,s) is a continutous
representation
of G on &9 for each
complexs in the strip0 < R(s) < 1.
g --
2) g-- U+(g, 1/2 + it) is unitarilyequivalent to the representation
v+ (g, 1/2 + it) of the continuousprincipalseries definedabove.
3) g-- U+(g. u). 0 <a<
1/2 is unitarily equivalent to the representation g -> v+(g, ) of the complementary series.
4)
If e and - are two vectorsin 91, thenthe functions
s
(U?(g,s),),
g fixed,
are analyticin 0 < Rs < 1.
5)
sup 11U+(g,s)
? Au(I + I t
)2,
g
s-=
+ it, 0 <Ko- 1.<
Furthernmore,
the constantA(y is bounded on any
intervalof theform0 < a ?r ? /3K 1.
It is known that for each t, the representations v-(
1/2 + it) and
v+( 1/2 - it) are unitarily equivalent. Hence the same fact holds for the
representations U+ (, 1/2 + it) and U+(, 1/2 -it).
As the next theorem shows, these equivalences are to some extent already
inherent in the " analytic structure" of the representations g -* U? (g, s);
the theorem also describes some adidtional, and rather interesting, relations
among the representations U? ( *,s).
THEOREM 6.
The following symmetries exist:
9 As in ? 2, the ordinary bound of an operator A is denoted by 112K11X.
22
R. A. KUNZE
AND E. M. STEIN.
1) The representations
U+( , s) and U+( *,1s- ) are contragredient.
Similarly,U-( , s) and U(*, 1-s)
are contragredient.
2) U+( , s)
contragredient.
U+(, 1- s).
Hence U+( ,s) and U+(.,s)
are also
3) There existsa fixednon-scalarunitaryoperatorS such that for all
s in 0 <Rs< 1,
SU-(,
s)S-1l
U-(
, 1-s).
Thus U( ,s) is unitarilyequivalentto the contragredient
of U-( ,s).
Remarks. (i) It should be observed that the known result [1] concerning
the reducibility of the representation U- (, 1/2) is implied by 3).
(ii)
The representations U+ ( , s) for s / 1/2 + it and s 7/4 are unitarily equivalent to representations introduced by Mautner and Ehrenpreis
[5]. These they show are not equivalent to unitary ones. They also assert
that the representations are uniformlybounded. However, the more definite
statement contained in 5) of Theorem 5 is crucial for our purposes.
The proof of Theorem 5 is lengthy and requires some vigorous
(iii)
classical Fourier analysis. This is contained in ? 6, which is, for the most
part, somewhat technical. At first reading the reader may prefer to pass
on to ? 7.
6. Some lemmas from Fourier analysis. We shall begin by introducing
a class of Hilbert spaces, which will be seen 10 to be related to the Lp spaces
via the Fourier transform. These spaces San
are indexed by a parameter
O< f < 1, and are given by the norm
00
The spaces &9(r, 9,ru corresponding to any pair of indices Orl,U2, such that
< fT1,
U2 < 1, are naturally related by a family of unitaries which we shall
now exhibit. Let s, = ol + it, and S2 = (2 + it2 where -c < tl, t, <oo. Now
o
let
V (s1, S2)
(6. 2)
be the mappingwiththe domain6V0, givenby
F (x)-
F(x)
Ix
812,
FE
U1.
10Although many of the results of this section are probably known,
they do not
seem to be accessible in the literature in the manner in which we need them.
REAL UNIMODULAR
23
GROUP.
Then
00
W (sl,
S2)F
I
11qf22
12 1 X
F(X)
2(T121-2)
1X
12LT2-1
dx
-
11F
11012.
This facttogetherwith (6. 2) showsthat W(S2, sl) is the inverseof W(sl, S2).
In what follows,we shall be mainly concernedwith the pair of spaces
9u and &
For the sake of convenience we shall set TV,=
W (s, 1 -
s).
The mapping W( is of particularinterestbecause it implementsa duality
between 9u and 9(1u. In orderto make this,statementprecise,we shall
introducesome additional notation. Throughoutthis section and the one
that follows,it will frequentlybe convenientto put
(6. 3)
00
F(x) G(x) dx.
(F, G)
This notationwill be used with the understanding
that F, G are measurable
complexvaluedfunctionsdefinedon -so < x < oo such thatFG is integrable.
The innerproductin 9( will be denotedby (, ) and we shall sometimes
set 1-o a=o'.
LEMMA
(6.4)
11.
If F E94a and GE
<, O
<i,1 then
(F7,G) = (F,WuG)an
(WuF,G)0u=
FIfu 1G
II0'.
(F, G) I _ 11
(6. 5)
Furthermore,
if A is a boundedoperatorontSa, the operator
(6. 6)
A'-
WuA*Wu-l,
whereA* is the (Hilbert space) adjoint of A is characterizedas the unique
boundedoperatoron 59u' such that
(6. 7)
G)
(A(F),
(F A'( G))
forall F in 9u and all G in 59ga.
To prove (6. 4) we firstobserve that 2oa'-
(WuF, G)ua=
00
F (x) 2X
x 1G2(1
1
1-
X)
21.
x 11-2c dx
(F, G)
JF (x) G(x)
(F, Wa G)a.
X
11-2o
1
X
12a-1
dx
Thus
24
R. A. KUNZE
AND E. M. STEIN.
Now, by Schwartz'sinequality,
lu'11G
I(F G) C 11WuF]?
t',
and (6. 5) followsfromthe fact that Wa is an isometry. Suppose A is a
boundedoperatoron 9(u, and that F E San G E 9 . By (6. 4), the fact that
preservesinnerproducts,and a secondapplicationof (6. 4) we findthat
WoJ
(A (F), G)
=
(A (F), Wa G) a
-(F, A *Wu,G)a
(WuF, WuA*Wa/G)a
(F, WuA*Wa G).
Thus (6. 7) is satisfiedby the operatorA' WuA*Wu-l. That A' is the
unique operatorwith this propertyfollowsfromthe easily establishedfact
that,GE 59(, and (F, G) =0 forall F E &u impliesG(x) =0 a.e. It should
A.
also be observedthat (A')'
In additionto the tu spaceswe shall considerthe Lp spaces,1 < p < o,
of functionsf definedon -o < x < so and normedby
=
11
fIIP -
00
"P.
I f(x) IPdx)
Since the parametera ranges between0 and 1 and 1 ? p < co thereshould
be no confusionbetweenthe norms 11 lI, and 11IIP
For a functionf definedon- Ko x <oo, the Fourier transformF is
definedby
(6. 6)
F (x)
(27)-f
eiav f(y)dy
and the inverse Fourier transform is given by
(6. 7)
f(x)
(2i)-feixyF(y)dy.
Here and throughoutthis sectionwe shall adhere to the followingconvention. Pairs of functionswhichare relatedto each otherby either (6. 6)
lowercaseand capital letterssuch
or (6. 7) will be denotedby corresponding
we take for grantedsuch standardfacts as
as f,F or g, G. Furthermore,
the Planchereltheorem,and the sense in which these transformsexist for
functionsin Lp, 1 < p ? 2, as well as the equivalenceof (6. 6) with (6. 7)
on f or F. (See e. g. [21] ). To be morespecific,
undersuitablerestrictions
a theorem
we shall make use of two well knownresults on Lp transforms,
of Titchmarsh(the so-called Hausdorff-Young
theorem),and the Parseval
formulafor L.,Lq. These resultsmay be stated as follows:
REAL UNIMODULAR
25
GROUP.
LEMMA 12. If f,G E Lp, 1 ? p ? 2, and theirFourier transforms
F, g
are given by (6. 6), (6. 7), then
a)
F 11q?_
A 11
f IIP,
11
b)
lglq?A
IIG 1p,
wherel/p + l/q == 1, and
c)
(f, g)
(F, G).
Now the relationbetweenthe L. spaces and the SL, spaces mentioned
earlieris containedin the followinglemma.
LEMMA
form (6.6).
13. Let f E Lp, 1 < p 2, and let F denoteits Fourier transI-1l/p.
Let a
Then O <o1/2, F E9tU. and
iF IIa?AuIfil,
(6.8)
I(;
1l<p_2.
The class of FE St9uwhichare Fourier transforms
of f E Lp is dense in
0< a
< p?2,
1/2.
Analogously, let FEW u
1/2 ? (r <1,
and let
=
1 -1/p.
2 ? p < co, the inversetransform(6. 7) existsin Lp norm,and
Then
1/2 --- < 1.
u
Au11
F lII
11
f IIP-::
Considerfirstthe case 1 <p ? 2. By a theoremof Hardy and Littlewood (see [11], p. 375),
(f
F(x) |P x
Now.,
Ap 1jf IIP,
Ip-2 dx)1/P ?
1 < p2.
00
F
I F(x)
ulf2
I ' IF(x)
n 00
(
00
F (x) 1Iqdx)/a
I IX
12,y-l
s
dx
~~~~~~~~00
00
| F (x) I I X I f-1P dx) ll/
by Holder's inequality. Furthermore,(2u 1) p = p -2.
Thus usilig the
inequalitiesof Titchmarsh(Lemma 12) and Hardy and Littlewoodwe obtain
1JF j2?A1 A
11
fJ2,
1
<p
2.
This proves (6. 8).
If F is the characteristic
functionof a finiteinterval,then f given by
(6. 7) is in Lp forall p > 1. Hence finitelinearcombinations
of characteristic
functionsof finiteintervalsare contained among the Fourier transforms
(6. 6) of f E Lp. Thereforethe image of Lp, 1 < p 2, under the Fourier
26
R. A. KUNZE
AND E. M. STEIN.
transform
is dense in the corresponding
space Nu,
1 1/p. This concludesthe considerationof the case 1 < p ? 2.
The second part of the lemma,which deals with the case 1/2 <r < 1,
2 ? p < so, followsfromthe firstpart by duality. We shall brieflyindicate
1
1 - 1lp', where1/p+ 1/p'
the argument. Put &' 1- o. Then '
and 1 < p'?`2. By Lemma 11, Nut and No, are dual and it is well known
thatLp and Lp are dual. The secondpart of the lemma then followsupon
identifying(6. 7) with the adjoint of (6. 6) consideredas a mappingfrom
Lp,to Stu (properlyspeaking,thiscan be doneonlyon a densesubsetof N,).
LEMMA
14. Let
00
K(F) (x)
Then
{J K(F)
-e
I x/y11 I x-y1-kF(y)dy.
,00
^00
00
(X)
12
dx Aa jI
00
F(x)
12 dx
if -1/2 < c < 1/2.
This lemmais known. The prooffollowseasily fromTheorem319 of [12].
There,a moregeneraltheoremon integraloperatorswhosekernelsare homogeneousof degree-1 is given.
The maill discussionof this sectionis containedin the lemma below.
We shall deal with operatorsacting on &1(J. It will be convenient,however,
to specifytheactionoftheseoperatorsbyexhibitingtheiractionon the Fourier
of the functionsin question.
transforms
Thus we considerthe multiplicationoperators
(6. 9)
Mt +:
(6. 10)
mSt-: f(x)
f (X)
I
X 121t f (X)
sgn(x)
IX
x
2it
f (X)
Now if F is the Fourier transformof fC L1 n L2, we shall denotethe
Fouriertransforms
of mt+(f), mt-(f) by M1It+(F),Mt-(F).
It will also be convenient
to introducethefollowingclass 2Z of functions:
F C i) if F is C? and vanishes in a neighborhoodof zero and outside a
compact set. Clearly 2Z is dense in each N9(u, 0 < a < 1. Furthermore,
2) is containedin the image of L1 n L2 underthe Fouriertransform,(6. 6).
LEMMA
15. If F C O, thenMt' (F) C 59L7
foreachqf such that0 <
and the transformations
h
i
n
tF
Mto(F)l
F
C9
have unique bounzded
extensionsto all of Nu,.
0
< 1.
REAL UNIMODULAR
27
GROUP.
The extensions,
whichwill also be denotedby t+,Alt-are unitaryon &I
and, in general,the bound,11
MIt 11aof Mt' consideredas an operatoron
satisfies
(6. 11)
It IIa-A(1 + t ),
M1
0 <u < 1.
Since the restrictions
of mt+, mt-to Lp, 1 ? p < co, map Lp isometrically
ontoLp, the Planchereltheoremimpliesthat Mt' extendto unitaryoperators
on j.
To treat the case a 7?&1/2, one would like to expressMt+ and Mt- as
integraloperatorsof convolutiontype. However,the kernelsin questionare
not locally integrable,and we must thereforeproceedratherindirectly."
We introducethe transformation
mre: f (X)
O < e<
_> X I-e+2itf (X) 1/2,
whichmaps L1 n L2 into L1 n L2.
PuttingF forthe Fouriertransform(6. 6) of fC L1 n L2, we denotethe
Fourier transformof mte(f) by Mte(F). Now let FE 5Z'. Since F is the
Fouriertransformof an f C L lnL2, we can formMtI(F), and, as is easily
verified,by the Planchereltheorem,
(6.12)
11MtE(F) -Mt+ (F) 112
0,
as
E ->
.
Next, we claim that
(6.13)
MlIte(F) = ae,t
F(y) I x-y
le-1-2it
dy
forF in Z),where
ae,t = Jr(1 -E
+ 2it)cos[Ir/2(1 -E + 2it)].
This followsfromthe fact [2, p. 43] that the Fouriertransformof
e-bixaI X J-e+2it
b> 0
is
( 6.14)
(2r)-r(1e+2t
+
iX),e-1-2 it +
(_bix)e-1-2it].
This convergesto
(6. 15)
(27r)-la. t I X
12it
"The following observations may help clarify the situation. When t = 0, M,+
reduces to the identity transform. This may be regarded as convolution by the Dirac
kernel. When t = 0, Mt- reduces to the so-called " Hilbert transform," which apart
from a constant factor may be viewed as a convolution by the function 1/x. In this
case our result was proved by Hardy and Littlewood [11], whose argument we extend
to the general case.
28
R. A.
AND
KUNZE
E.
M.
STEIN.
as b-> 0+,and is boundedbyA I x j'-1,withA independentof b. Now (6. 13)
theoremsand the Planchereltheorem.
followsby standardconvergence
Togetherwith FC Z, considerG(x)
Ix 16-IF(x). Since FC CO and
vanishesin a neighborhoodof zero and outside of a compactset, the same
may be said of G(x). Thus we may apply formula (6. 10) to G as well.
Call
-
(x)
A6(x) =Mte(G)
(6. 16)
x I f-1MItE(F)
(x).
-
Then by (6. 13),
+00
+
a,x
Ae(x)=ae,t
(6.17)
]
2E-1-2it
F(y)lxaF(y)
[|Y
y
dy.
It is easy to verfy,(by the Lebesgue dominatedconvergencetheorem)
that
(6.18)
If we use (6.12)
with G in place of F, (6. 18) anid (6.16),
x
xIX'-MAIt+
(F)
(6.19)
(FcC
asE --0,
lAE(x)-AO(x)12-*0,
112
M1It+
(G) 112+
we obtain
|| AO (X) 112.
As has already been noted
|| Mt+ (G) 112 =
G 112
||
while
11
and
G 112
11 I X |ff-'F 112
i
(F) 11
11I 1A-IMt+
11F jjgn
Mt+(F)1
Substitutingthe above in (6. 19) leads to
(6. 20)
|| MIt (F) ||
_ || F
II u+
|| AO(X) 112-
to estimate11AO(X)
therefore
It remains
+ 00
Ao(x) =ao,t
+
=ao.t
y I-F(y)
-
oo
1
-
X 1/1 y
Ix
I
112.
1-IF(y)I
I X-y
x If-"F(x)
x
1-1-2it
y I-'-2It dy
j y I0yF(y)dy.
, and with j y IjAF (y) in place
We now applyLemma 14, witha-f=u
of F(y). We thenhave
11AO(X) 112C Ag I aoIt|
Recalling(6. 17) we have
112=A(
| ao,t F (x) llu
However
ao,t=- (1/)r(1
+ 2it)cos
17r(1
+ 2it).
REAL UNIIODPULAR
29
GROLUP.
Hence by well-knownestimatesin the theoryof the r function,see [22],
p. 151, it followsthat
Iaot 6A(l
+
t).
Combiningthis withthe abovewe obtain:
|| Ao(X)
+
112 AA(l
t F)
11Fo.
Togetherwith (6. 20), this implies
_- Ag(1 + I t 1 1F
11Mft+(F) 11ag
11aor
This was our desiredresultfor Iit+.
The prooffor Mt- is verysimilar. The onlychangethat occursis that
we use the fact that the Fouriertransformof sgn(x) I x 1-,+2it is
I
(27r) b,tsgn(x)
where
bet
(i/r) r(1 -E+
xle-1-2it
2it) sin 27r[1 -,+
2it]
This concludesthe proofof the lemma.
LEMMA
16. The estimatesfor Mt+ and Mt- may be strengthenedas
follows. Let e> 0, then
_ A,Je(L +
Mt+ 11,g
I
t I)(l+)k-l
O<
f< 1
withAor,,independentof t.
Proof. Let us consider[t+, and assumethat 1 _ < 1; the othercases
are treatedanalogously. We have alreadynoted that MIt+is unitaryon 6Vi.
Thus we have
+
+00
(6. 21)
I Mt+(F)
12
dx)
+00
+
I|Ff12 dx).
By the lemmawe have just proved,we have,if a?
(6. 22)
(4
Mt+(F) 12 1 X 120To-1
dx)^
+
f)
_Aao(l + I t I)l(
< 1,
i()2 1 1 x 12ao-l x
Notice that the above inequalitiesare of th-esame nature,exceptfor the
weightfunctionswhich determinethe measuresin question.
Now it is possibleto "interpolate" betweenthese two inequalities,and
obtainintermediateones fromthem. Of coursewe have alreadyused many
30
R. A. KUNZE
AND E. M. STEIN.
varianltsof this type of argumentin ? 3 and ? 4 above. The particular
theoremwe need is containedin [20], (Theorem 2). To apply it we argue
as follows:
Choose ao, so that ur<
2u- 1=
,o < 1.
(1-0)
We may then write
*0 + 0(2o -1)
0(2uo-1),
with 0 < 0 < 1. Notice that in the above, u =4 when 0 =0,
when0 1. The resultof applyingTheorem2 of [20] is
H-owever,
+ co
ao
0 =- (2,o'-1) /(20r0-1).
Thus we chooseco-close enoughto 1 so that.0? (2a - 1) (1 +,E).
resultbecomes
oo
(f M|t+(F)
I
and a
Hence the
dx)
12 j x 12,r-I
?Aci (1
@ + I't I) (oIr1) (I(
I F(x)
12
j
X 12a-I
dx).
Our lemmais therefore
proved.
We observethat the above proofyieldsthe inequality
Ruwemarkc.
Aeue ?eAco6m
A simple argumentthen allows us to deduce the followingfact: The
constantA,, whichappears in (6. 11) maybe takento be uniformly
bounded
in everyclosedsubintervalof a lyingin 0 < o < 1.
This observation
will be of use later.
CHIAPTER
II.
UJNIFORMLY
BOUNDED REPRESENTATIONS.
7. Proofs of Theorem5 and Theorem6. Beforepresentingthe details
of the argument,we shall brieflydiscussthe main steps involvedin the cong --- U- (g, s).
structionof the representations
Our representations
v. constructedon the space
are
fromrepresentationsg -whic (g, s) on
,12 s m be+it. The operatorsm(g, s) and
s) are relatedby
Thi(g,
12
For the definitionof the Hilbert space 8W
a see ( 6. 1 ) .
REAL
(7.
1)
U
(g,s)
UNIMODULAR
=
W(s,j
)V
31
GROUP.
(g,s)W(1,s),
where W(s, 1) is the unitarytransformation
(6. 2) of 9a onto 4
The
1
+
are
Vobtained
the
representations
it)
g
(g,
by simply transferring
representations
g v-(g, + it) of the continuousprincipalseries fromL2
to S, by means of the Fourier transform. We also obtain the operators
V' (g, s), 0 < Rs < 1, via the Fouriertransform
in a similar,but technically
more involved,fashionfromthe representations
g -* v' (g, s). To definethe
"
operatorsVI (g, s) for <R (s) < 1 it is convenientto extendthe notation
r'= 1-cr to complexs with 0 < Rs < 1 by settings' 1 s; the transformations -- s' is then simplyreflectionabout the line a =.
Now the
1
to an s with < Rs < 1 is definedto be the
representation
corresponding
of the representation
to s'. Thus we put13
contragredient
corresponding
(7.2)
V+(g,s)=[TJ(g-1,s')]',
<Rs<1.
It followsthat
I
[V,(g-,~ s) ]'
(7. 3)
-V(g, sI),
O < Rs < 1.
It will be shown in the course of the proof that the apparentlyarbitrary
definition(7. 2) is the natural one to make.
As a firststep in the proofwe shall establishthe followinglemma.
17. The multipliersp+ given by (5. 2), (5. 3) satisfy
LEMMA
a)
p+(g,x,s) = 4(g,x,s),
b)
+(9192,
)c
X,S) -=0?(g1
X,S) +(g2,glx
(9, g-lx,s ) dg-I(x) ldx
s),
?> (9-l x, 1 -s)
The firstrelation,a) is immediate,b) is essentiallya consequenceof
the chain rule for derivativesapplied to (5. 1), and c) followsby simple
computationsfromb) upon setting92 =- g and g1 g-1.
As the followinglemma shows, it is natural to restricts so that
=
0<Rx<
1.
LEMMA
18. Suppose s = cr+ it, where0 ? a ? 1. Then foreach g C G,
the operatorsv (g,s) are isometricon Lp, wherep= (1
)-1.
Silnce the case p = oo is easily verified,we shall suppose 1 < p < o0.
Making the transformation
x -* g (x) we findthat
13If A is an operator on
a, A' is the operator on Sa' given by (6.6)
and (6.7).
32
R. A.
3
x00
I f(x)
_00
Now Iv?(g,s)f(x)P
KUNZE
AND
00
IPdx
I bx+ d
E.
M.
STEIN.
I bx+ d 1-2 1 f(g(x) ) IPdx.
00
f(g(x)) P, and since (2-
1(2a-2)p
2)p=
it followsthat 1vV (g, s) f IIP= 11
f IP.
It is interestingto observethat whenp
p' is given by p'
=
2,
(
o) -1 its conjugateindex
Thus the operators ve(g, s) and v (g, s') give
(1 -o')-1.
riseto a pair of isometricrepresentations
of G on Lp, Lp wherep) (l
)-1
and s
+ it. Moveover,as thefollowinglemmashows,theserepresentations
are contragredient.
=
LEMMA 19. Let s =a + it and p
any gE G, fE Lp, and hC Lp,
(v (g, s) f, h)
(7. 4)
=
(l
<
r,0
< 1.
Thzen for
(f, v+(g-1,s') h).
To provethis we make the tranisformation
x - g-1(x) and findthat
^00
(v+ (g, s) f,h)
(g, x, s) f (g (x) )h(x)
dx
oo
=
(g, g-lx, s)f(x)h(g-lx)
(dg-1(x)/dx)dx.
Thus, by c) of Lemma 17,
00
f (x) ++(g-1, x, 1
(v+ (g, s) f,h)
s) h (g-lx) dx,
and now part a) of the same lemmashowsthat
(v+ (g, s) f, h)
=
(f, v (g-1,s') h).
We iiow considerthe representation
spaces HIa of the complementary
series. These spaces are describedin the followinglemma.14
LEMMA 20. Let 0 <r
<
2
and p =(1
-)
.
Then the innlerproduct
(5. 5) is well definedfor f in Lp, and the completionHa of Lp with respect
to the norm f j2 =- (f, f) is unitarilyequivalentto S9( via a mapping
which coincides with the Fourier transformn
on Lp.
To provethis, suppose firstthat fC L1 n L2 and that F is its Fourier
transform.By (6.14), whichis valid for 0 <,E < 1, and the dominatedconvergencetheoremwe obtain
700
.00
(7.5)
l
_00
14
F(x)
12 1 X 12uf-1
dx
ac,
f*
00
f (x)
I
X 1-26
dx.
Lemmas 20, 21, and 22 are essentially restatements of known facts.
REAL UNIMODULAR
33
GROUP.
By Lemma 13 theleftside of (7. 5) is finiteforf C L., and by simpleapproximationarguments,it followsthat the rightside of (7. 5) exists and equals
the left side for all f in L.. This showsthat the formula(5. 5) definesan
innerproducton Lp. Now observethattheFouriertransform
ofLp, 1 < p ? 2,
includesthe characteristics
functionsof finiteintervalsand theirlinear combinations. This observationtogetherwith (7. 5) establishesthe final statementof the lemmaand concludesthe proof.
As a consequenceof Lemma 18 and Lemma 20, we obtainthe fact that
the representations
g-- v+(g,s) are definedonla dense linear subset of Ha,
0 <~ <K 2. Moreover,as the followinglemma shows,the operatorsv+(g, u)
extenduniquelyto unitaryoperatorson Ha.
Let 0 <,
LEMMA 21.
< 1 and p
(7. 6)
(1 -)-1.
11v+(g,a)f 11a
Then for f in Lp,
1 f 11a
In provingthis,we use the fact that
g (x)-
g (y)
(x -y) (bx + d) -1(by+ d)-l,
=
which follows by straightforward
computation. Then making the transformationsx-* g(x) and y-*g(y) we see that
00
f Ila42
11
00
jJ
*00
-
00
f f(x)f(y)
so0
00
00
-~
I x
(g(x))f(g(y))
y 1-2adxdy
+ d 12a-21by+ d 12a-2dxdy
1-2 I bx
X -y
00
=j v+(g,o )f Il12.
Next we shall showthat thereexistsa uniformbound independentof g
for the operatorsv+(g, s) in Ha ; s g + it, 0 <Ka< . In doing this, we
considerthe lowertriangularsubgroupof G consistingof elementsg of the
form
=
9
[a
],
a 7&?0
We make essentialuse of the fact that there are only two distinctdouble
cosets of G modulo this subgroup. To be explicit,we introducethe group
element
i
and provethe followingresult.
3
[
1
0]
34
R. A. KUNZE AND E. M. STEIN.
LEMMA 22. If g C G and is not lowertriangular,thereexist lower triangulargroupelementsg1 and g2 such that
gg1jg2
9
(7. 8)
We provethis by exhibitingsuch a decomposition.If g C G and is not
lowertriangularwe may write
b]
[a
b#0
Then as is easily verified
9
r1
0n
Ldb-1 1L
1 b-i
O0 a
o
-1
on
bj
In viewof thisresultand the factthatv? (g1g2,s)
v? (g1,s) v? (g2, s) for
all g1,g2 in G, it is natural to considerthe operators,v+(g, s), firstfor g in
the lowertriangularsubgroupand then for g j.
=
LEMMA
1)
+ it, where0 < r-< I and -oo < t < oo. Then
23. Let s5=
if g is lowertriangular,v+(g,s) has a unique unitaryextensionto
all of Ha, and 15
2)
_- A,(1
11v+ (j, s) 11,g
+ I t ).
[
In proving
1), we supposethatg =
(7. 9 )
v+(g, s) :
-->
f (x)
a
], a #0.
12s-2f
(a
-2X
+
a-1c)
and v-(g, s) =- sgn(a) v+(g, s). Furthermore,
v+(g,s)
relationstogetherwith Lemma 21 establishpart 1).
operatorsv+(j,s) we findthat
(7.10)
(7.11)
v+(j,s):
v- (j, s):
f(X)
f (X)
_
I
-_Sgn(x)|I
128-2f(
x
Thenbydefinition
-
a |2itV+ (g, ; these
Turning now to the
JI)
12s-2f(_
JIX)
Now with the aid of the operatorsmt+and mt-given by (6. 9) and (6. 10)
we can writev+(j, s) =mt+v+ (j, a) and v-(j, s) =mt-v+ (j, a). Since v+(j, a)
has a unitaryextension,it followsthat the bounds of the operatorsv+(j, s)
are exactlythe boundsof mt+,mt-,considere-d
as acting in H,. Now using
the factthatH, is unitarilyequivalentto S9( and the definitions
of Mt+,Mtwe obtain2) as a consequenceof Lemma 15.
16 The
symbol 11-il1designates the bound of the operator on S.
REAL UNIMODULAR
35
GROUP.
Finally,using Lemma 22 and Lemma 23, we findthat
(7.12)
sup I v+(g,s)
g
-_A,(1 + I t )
< 1.
0 <l
Because of (7. 12) we may,and shall fromnow on, assumethat the operators
v+(g,s) are everywhere
definedon H,.
Since Ha and 5V, are unitarilyequivalentwe may transferthe representations9 -- v+(g,s) to S9( and obtainequivalentrepresentations
g -*> V+(g, s).
The operatorsV+(g, s) are obtainedas follows: Let 5,, 0 <a <I, be the
unitarytransformation
fromHa to S9( that coincideswiththe Fouriertransform (6. 6) on Lp, p =
(1 -)'.
In addition let
be the Fourier trans-
52
formrestrictedto L2; we note that 52 is unitarybetweenL2 and X9i. We
now defineV+(g,s) for s=-F+it by
(7. 13)
0 <o_
V+(g,s)=5av+(g,s)5gr',
.
From (7. 12) and the definitions(7. 13), (7. 2) we obtainthe bounds
(7. 14)
Sup1 V+(g,
9
S)
A,
A(
+ t
0
<
<
1.
This resulttogetherwith (7. 1) implies
(7.15)
Sup 11<U<(g,1s).1 A( 1+
9
t
0<
< 1
Moreover,as the remarkat the end of ? 6 states,we may assume A, is
boundedon any closed subintervalof (0, 1). Hence we have proved 5) of
Theorem5, and conclusions2) and 3) followfrom (7.13).
To show that (7.2) is a natural definitionwe consideronce again the
class of functions0 introducedin ? 6. Recall that FC 0 if F is Cw and
vanishesin a neighborhood
of zero and outsidea compactset.
LEMMA
24. Suppose F,HC 0 and that f,h are their Fourier transforms. Then forall s in the strip0 < Rs < 1,
(7. 16)
(v+(g, s) f,h)
V (g, s)F, H).
=(
To provethiswe supposefirstof all that0 < Rs ? 1. Thenf,v+(g,s)f C Lp,
(1 - )', and 1 < p 2. Our result, (7. 6), now followsfrom the
p
definition
of Th(g, s) F and the Parseval formulafor Lp,Lp', whichis stated
in Lemma 12. In case 1 <Rs < 1, VT(g,s)
[T(g1,s')]'.
V
Thus
(V-(g, s)F, H)
(F, V7(g-1,s')fH.)
By the resultjust established,
(F, T7(g-1,s')H)
=
(f,v(g-1, s')h).
36
R. A.
KUNZE
AND
E.
M.
STEIN.
Now applyingLemma 19, we see that
(f,v (g-1,s')h)
(v (g, s)f, h).
Thus (7. 16) also holdsfor 2 < Rs < 1, and henceforall s in 0 < Rs < 1.
To prove that the representations
g -* U' (g, s), definedby (7. 1), are
continuous,it suffices
to provethat the representations
g -* VT(g, s) are; and
forthis,it is sufficient
by (7. 2) to considerthe case 0 < Rs 1. Now if f
is continuousand has compactsupport,it may be shownthat for bounded
functionsh,
(g, x,s)f (g (x) ) h(x) dx
f (x) h (x) dx
as g -* e, e beingthe identityin G. Because the representations
g -* v' (g, s),
o <RS ? 2, are uniformlyboundedon H, this is sufficientto insure their
continuity. Hence the equivalent representations g9-->VT (g, s) are also con-
tinuous.
It remainsto proveconclusion4) whichrefersto the analyticityof the
operatorsUe(g, s). For this purpose we prove a result which has some
interestin its owInright.
LEMMA 25. If g is a lowertriangularmatrix in G the operatorsU (g, s)
are independentof s, 0 < Rs < 1.
Let
and chooseFE SV,. It then followsfrom (7. 9) and well knownproperties
of the Fouriertransform
that
V (g,s): F (x)
(7. 17)
e'a I a1
We also obtain the relation V- (g, s)
F E .94,we have by definition
that
Ue (g, s) F
28
F (a2X).
sgn(a) V+(g, s).
=W (s, 1)V
(g, s) W(1
Startingnow with
s) F.
Hence by (7.17),
V (g, s) W(1, s)
F(x)
eixacI a
F(x)
ex'
e
128 1a2X
and applyingW(s, 1) we get
(7. 18)
U+(g, s)
a F (a2x).
I-8F(a2X)
REAL
UNIMODULAR
37
GROUP.
Similarly,we obtain the relationU (g, s)
sgn(a) U+(g, s). Thus we have
provedthe lemma.
This resultshowsthat the inner products(U- (g, s) ,7) are constantas
functionsof s, and henceanalytic,forany fixedlowertriangularg C G. Now
if g is notlowertriangular,it has a decomposition
g = g1jg2of thetype (7. 8).
Since
U+(g,
s) =
S) U+(j,
U+(gl,
S) U(g2,s
whereU (gi,s), i -1, 2, are independentof s and have boundedinverses,it
is sufficient
to showthat (U+ (j, s) $, -) is analyticin s foreach pair of vectors
Recall the uniformbound, (7.15a) for the representations
in
N9.
e,
-*
g U+(g, s). Since the constantAa whichappearsis boundedas a function
of o- over any closed subintervalof (0, 1), it is sufficient
to prove that
(U (j],s) e,-) is analyticfora dense collectionof vectorsin 51. Choosethis
collectionto be theset 9) of functionswhichare C and vanishin a neighborhood of zero, and outside a compact set. Pick g = F and B = H in .
Let F, (x)
| x I'--3F(x) and put
x1 J-it"H(x).
H1s(x)
It is theineasilv verifiedthat
V
T (j, s) F, HIs)
(U`+(j, s),e7-)
Denote the Fouriertransformns
of F8, Hs by f8,h,. Then as F8, He belongto
i), Lemma 24 applies,and we see that
(U+(j, ,s) e, ) == (v+(j.
s) fs,h.)
Now using (7. 10), (7. 11) we obtain
00
(7.19)
(7. 20)
(U+(j, s)4r
(U-U,s)e-)=
) - 1n
f
x
12s-2 fs(_
Jx)h8(x)dx.
co
sgn (x)
oo
I X
12s-9-8
llx)h (x) dx
Since
f. (x) = (2Tr)
f eieyI y I- WF(y)
dy,
00o
and in view of the variousrestriction
on F, we may concludethat f8(x) has
the followingproperties:it is jointlycontinuousas a functionof x and s;
it is analyticin s for each fixedx; and if s is restrictedto any compact
subset of the strip, 0 < Rs < 1,
negativepowerof I x 1. Since
fB(x)
decreases as I x |
oo as fast as any
38
R. A. KUNZE AND E. M. STEIN.
e00
hs(x)
=
(27rk-f
Ie-
yIs-l-H (y)dy
it has the same properties. It is now a very straightforward
matterthat
(7. 19), (7. 20) can be obtainedas uniformlimitsof functionsanalyticin s.
Hence the inner products (U+ (j, s) $,v ) are analyticin s. This concludes
the proofof Theorem5.
Conclusion 5) of Theorem 5 may be strengthenedas
follows. Givenany E> 0, then
COROLLARY.
sup 11U(g, s) 1 _ A,a,e(1 + I t j) io-(14+e)
9
O<o-<1.
for s =+it,
In provingthe theoremwe made use of the estimategiven by (6. 11).
If, however,we had used the estimategiven by Lemma 16, we would have
obtainedthe above.
We shall now Prove 1) of Thetorem6, whichassertsthat the representations U+( ,s) and U+( ,s') are contragredient.
In orderto do this,we firstcombine(7. 3) and (6. 6) to obtain
(7.21)
V-(g,S)
WV-(g-1, s)*Wa-1.
=-
It thenfollowsby definitionthat
U+ (g, s')
=
W(S'
12) WaV+
(g-1,
S) *Wa-lw
(1,
sI).
Using the definitionsof W(s', 1),2 W4,togetherwith the fact that s'- ~~~~~~~~~~~~~
+cr Irts
1we findthat
W(S
1
) Wa
WV (S5
1)
Substitutinginto the above we obtain
whichimplies,
(7.22)
U+ (g, St) Z_w (S, 2 ) V+9?
)*
1
)
U+(g, s') U=U?(1, s).
Hence we have provedpart 1).
The second statementof Theorem6 is easily seen to follow fromthe
fact that the representations
g -* U+(g, u) are unitaryfor 0 <u < 1.
In fact, suppose that g -* U (g, s) are any representations
of G on S
such that
U (g, s')
U(g1
U
s)*
39
GROUP.
REAL UNIMODULAR
and forwhichthe innerproducts(U (g, s) e, -) are analyticin s. Then
1 -s)
s) =- U(,
U(,
if and onlyif the representations
U(
, a)
are unitary for eachla, 0 < cr< 1.
To prove this, we observe that the condition U ( , s) = U ( ,1 - s) is,
for0 < a < 1.
U, 1 -)
byanalyticity,
equivalentto theconditionU(, a () U
Hence the
a'
so that U(g',ar)*
On the other hand, 1 U(g, 1 a).
g
E G.
U (g-1,a)
above is equivalent to the condition U (g, a)
U( ,s) do not satisfy
It is interestingto note that the representations
2). This is a reflectionof the knownfact that theyare not unitarywhen
to showthat the representations
af7/ 1. In orderto provethis,it is sufficient
0 < a < 2, are not unitary. Withoutgoing into detail,we remark
v( (a),
that this is a consequenceof the relation
(7. 23)
|| v- (j, C)
f ||a
ao
J
||
f || a"
00 00
J (sgn(x)sgn(y) -1)
O00
00
which is valid for all f in Lp, p=
(1
f(x) f(y)
x
-y
2"dxdy,
a)1.
We supposenow that S is a bounded operatorwith a boundedinverse
such that
SU(
U(,
s)S-1
1-s).
Replacings by 1 - s and makingsimplecalculationswe findthat
S2U- (> s)
=UQ
(, s)
S2.
We shall assumetheknownfactthattheunitaryrepresentations
+ it),
U- (,
t # 0, are irreducible. It then followsthat S2 is a scalar multiple of the
identity. For lowertriangulargroup elementsg of the form
9
g= a-1
we knowthat
and
U- (g, s)
ca
0]
?a]
F (x)
SU (g, s) =U
a
,
a =/-Z
0,
>eiac aF (a2x)
(g, s) S.
Setting a = 1 and then settingc =- 0 we find that S is the operationof
K (a2x).
multiplicityby a function,say K, with the propertythat K (x)
Since S2 is a scalar multipleof the identity,we obtainthe additionalrelation
(const.)sgn(x).
(K(x))2=
const.,which implies K(x) =const. or K(x)
=
40
R.
A.
KUNZE
AND
E.
M.
STEIN.
As thefirstalternativeholds if and onlyif U-(, u) is unitaryfor0 <
we concludethat K (x)
(const.)sgn(x).
We shall now defineS by
rf<
1
S: F(x) -sgn(x)F(x),
(7.24)
and prove that SU-(g,s)S-1= U-(g,1 -s) for all g in G. This may be
showndirectlyfor all g; however,such a proofdoes not exhibitthe crux of
the matter,which,as it turnsout, is the relation
)-1
SU-(j,
=-- U-(j, 1 -O).
We thereforeproceed along different
lines and firstof all recall that the
operatorsU (g, s) are independentof s forlowertriangulargroupelementsg.
For such g, the above relationbecomes
SU- (g, s)
=
U(g, s) S.
To verifythis,suppose
9
0]
[al
a 40.
Then
SU-(g, s): F(x)
and
-*
sgn(x) eixac aF(a2x)
U- (g, s) S: F (x) -> eixac a sgn (a2x) F (a2x).
In view of the decomposition(7. 8) it is thereforeseen to be sufficient
to
provethe relation
SU-(j,s)
=
U(j, 1 -s)S;
a, 0< u <
Moreover,by analyticity,
it is sufficient
to provethis for s
Now
S U= (j, a) S-
U (j,a')
S-1 W(a',)
W (1,"')
if and onlyif
W('2
,')
S U- Gj)
U-(j,
a')
Thus using the fact that S commuteswith WI(V,c')
sufficient
to prove
(7. 25)
V-(j, ')
SW(, U') V-(j, a)W(',
r)S-1,
0 <a
W(a',1
we see that it is
<
2'
In proving(7. 25) we use the followingconsiderations. The operation
SW(o, a') is multiplicationby sgn (x) I x 12*i1. Going over to the Fourier
this correspondsto convolutionby b /(27r)isgn(x) I X2f, where
transform,
REAL UNIMODULAR
b=
41
GROUP.
(2ou) sinvu.
i/7rr
(This fact may be establishedin the same way as (6. 13) was; for further
discussionsee the proofof Lemma 15, ? 6.)
Recalling the definitionof V-(j, u) in termsof the Fourier transform,
it then sufficesto prove the following: the operation of convolutionby
ba/(27r)lsgn(x)j x 1-2j, followedby the operationf(x) -> sgn(x)I x J-2f(- I/x)
is equal to the operationf(x) -> sgn(x) I x 126-2f( 1/x) followedby convolution with ba/(27r)'sgn(x) Ix L2f. This leads to the verification
00
fsgn (x) sgn (
1/x
sgn(y)sgn(y -x) Iy
=f
f(y)dy
y) I X 1-2 1 -/x
_ y
1-2a
12-2
_00
y
Ix-
1/y)dy.
2f(
That this holds may be checkedby the obviouschangeof variables.
The argumentabove needs to be made precise. We thereforeargue as
follows.
In proving(7. 25) it clearlysuffices
to showthat
(WV-(j, ')SW ( F, ')F,H)
=-- (S W (r, u') VW(j,o)
Let f be the Fouriertransformof F.
for F, H C 0.
f1(x)
=
(27r)
f
ei$y
_00
(F), H)
Put
sgn(y)j y j21F(y)
dy.
Then by what has been said before,
4
f (x) = ba/(21r)
Ix
sgn (x-y)
y 1-26f(y)dy.
We defineh, and h, similarly;thus it followsthat
hi(x) = bu/(27r)
Sggn
(x
y)
I x-y
1-2a
h (y) dy.
Since
(SW(a, or')V-(j, a) (F), H)
=
(W(j, cr)(F), SW(o,
')H),
in view of Lemma 24 to showthat
it su-ffices
(7. 26)
Now,
(v(j,
a)f, hi)
-00
(v-(j,')fl,
=4
-ba/(27r)i
h)
=
(v-(j,)f, hi).
00
sgn(y)I
y
I2T-2
f(-
1/y)hi(y)dy
h(x)f(- 1/y)sgn(y)sgn(y - x) y
2-
Xy
1-2odydx.
42
R. A.
KUNZE
AND
E.
M.
STEIN.
On the otherhand,
(v(,'n,h)
00
,*00
J
boJ/(27r)f
00
h(x)f(y)sgn(x)sgn(-1/x-y)
00
j x-y
I x 1t2o
1-2?dydx.
If we makethechangeofvariablesy >-I/y in the firstdouble-integral,
thenit is easilyverifiedthatthisfirstdouble-integral
equals the seconddouble
integral. This proves (7. 26) and concludesthe proofof Theorem6.
CHAPTER III.
THE
FOURIER-LAPLACE TRANSFORM ON THE GROUP.
8. Hausdorff-Young
theoremforthe group and certainof its implications. Let f C L, (G), and let us definethe Fourier transform
of f on G as
follows:
( 8.1)G
+?iJ-(s)= U+(f,s)
-(
U+(g,s)f(g)dg,
O< R(s) < 1, f CLI(G).
U+( , s) is the analyticfamilyof representations
which act on 91, and
whichwerestudiedin ?? 5, 6 and 7. Because foreach fixeds, 0 < R(s) < 1,
UT( , s) is a uniformlybounded representation,
the integral appearing in
(8. 1) is well defined. Moreover, if e, - C
, then
(51+(S)t)
s)e,q) f(g)dg.
jw_0=(U+(g
An applicationof Fubini's theorem,and the analyticity
of U+( , s) showsthat
js(F (s)e,-q)ds-O
,
for any closed curve C in 0 < R(s) < 1.
Thus the Fouriertransform
7+(s) is not onlywell-defined
whenfC L,(G),
and 0 < R (s) < 1, but is also an analyticoperator-valued
functionof s in
that strip.
The resultsof this sectionshow,in a very preciseway, that one may
obtain similar results for the Fourier transformof functionsin L,(G),
1 ? p < 2. These facts are contained in the followingtheoremtogether
withits corollaries.16
THEOREM
7. Let 1 < p < 2, and q be its conjugateindexI/p + l/q = 1.
16The norms 11- jq, 1 < q ?< ?, are those introduced in ? 2. We recall that
the " Hilbert-Schmidt" norm while 11O
* 1i denotes the operator bound.
11 112is
REAL UNIMODULAR
43
GROUP.
There existsa measured/q,t,,(t)
so that
00
(fIIq(
(8. 2)
oo
a(C+dt)q,d,(t)
it)
)1/Q
1 f 11,
+ it, and 1/q <,c < I/p. For the measured/q,t,
f simple,s
(t) we have
the followingestimate: Givenany 8 > 0, then:
=
djqu,(t) ? Aq,,, (1 + I t
COROLLARY
so that
)1-Qi-i-Odt.
1. For eachfixedp, 1 < p < 2, thereexistsa u0, l/q < (o
(f
+ 00
11 a|(cr
<
1,
+ it) 1jqQdt)l/9 < Aq,7 11f 11
wheneveru0 <a
< 1 - a0, and f is simple.
COROLLARY
2. For each p, 1 ? p < 2,
sup 11
5 (I + it) IojC Ap 11flIP,f simple.
COROLLARY
3. For each p, 1? p < 2, l/q < R (s) < l/p, s
+ it,
11af (s) jj.,? Ap,o,t 11f j1p, if simple.
COROLLARY 4. The Fourier transform,
initiallydefinedfor f C L, nLp,
has a unique boundedextensionto all of Lp(G), 1 p < 2, withthefollowing
property:U+(f, *) is for each f C Lp(G) analyticin s, for l/q < R(s) < I/p.
Moreover,the extensionsatisfies(8. 2) as well as the conditionsof Corollaries
1 through3.
Remarks. A strictanalogue of the classical Hausdorff-Young
theorem
would have been a resultlike (8. 2), but onlyfor r= -1. The above results
show,however,
thatthesameconclusionholdsfora properstripwhichcontains
the line uc
in its interior. This, togetherwiththe analyticityof 5, has
far-reachingconsequences. Once (8. 2) has been proved, the results of
" type
Corollaries2, 3, and 4 followby ratherstandard" Phragmen-Lindel6f
arguments.
It is possibleto obtainsomewhatstrongerversionsof Corollaries2 and 3
by replacingthe 11IIo operatornormby the norm 11IIq Since these latter
resultsdo not seemto have any immediateapplications,we have not bothered
to give theirproofs.
A completeFourier analysis of an arbitraryfunction (in the class
L2( G) ) necessitatestogetherwith the continuousprincipal series also the
discreteprincipalseries. The discussionof the discreteprincipal series is
much simpler,and is taken up in the next section.
=
44
R. A. KUNZE
AND E. M. STEIN.
Proof of Theorem 7. Let us considerthe case U+, that of J-being
entirelysimilar. On accountof the corollaryto Theorem5 (see ? 7) we may
writedownthe followinginequality:
(8. 3)
(1 4+
sup
t
+4-it) IIoC AAie 11f 1ll,
)-ii-la(+e)
0 <U <1, and c>
0.
This inequalityfollowsfromthe above quotedcorollaryand the observation that
it)11_sup 11U+(g,s) II'X1f 1ll
~~~~~~~~
5l+ (a +
We knowthat U+( ,+ it) is unitarilyequivalentto the representation
v+,
+ it) of the continuousprincipal series. This series, however,is
containedin the Plancherelformula (see ? 5"). Hence we may write down
the followinginequality:
(8. 4)
(j
00
1 + (1 +
I t I)-1dt)?-<A 1f 112.
it) 1122t2(1 +
Here we have used the semi-trivialobservationthat,
t2(1 +
I
t I) -1
Attanhrt,
-00
<
t < 00.
We shall apply Theorem3 to inequalities (8.3) and (8.4) above. We
argueas follows. Assumethatcris givenand l/q < f < i/p. We assumefirst
thatUr< 1. Let a be a fixedreal numberwith 0 < a < a <
but otherwise
arbitrary.Rewrite(8. 3) with acinsteadof u. It becomes
27
(8.5)
sup
-x0< t< x0
Our given p, I < p
Now if l/q
(8. 6)
<
<
( -T)
l/p=
u
<
+(a+it)
I + It
withc
1-,
or
||-
Aa,e
|| f II
(a -)(I+
2, determines a parameter Tr 0
and
+ T/2 =-T/2,
<T
<
1, with
/q =T/2.
therealwaysexistsan a, 0 < a < u, so that
(-)
I
+PT,
(+
3=)r
The above relation determinesac uniquely,which ac we now fix. In
applying Theorem 3 to (8. 4) and (8. 5) we make the followingfurther
identifications:
(8.7)
r
c=
a
b
Now the resultof Theorem3 is
(
I
1
-
+(E)
45
REAL TJNIMODULAR GROUP.
(8. 8)
(
1+(
+ it) 11
qq(I + I td|)
dt) llq C
AE,T
11f Ilp
whenever f is simple.
A straightforwardcalculation leads to
(8.9)
dq=1-
u-1
I
q(
( >O)
+E),
.
Now given any 8 > 0, we can choose an E> 0, small enough so that
(8.10)
dq
I
I-Iu-
q -
Substituting this value of dq in (8. 8) proves (8. 2), wheneverf is simple.
The consideration of the case 1 < o-< i/p is carried out in the same manner
once one defines
a5+ (1
a1+(s)
The consideration of 5-(s)
of Theorem 5.
s).
is analogous to 5+(s).
This concludes the proof
Proof of Corollary 1. Consider the quantity 1-q
. This
2-a
is the exponent that occurs in the measure d1tq,
Recall that 8 was
0(t).
arbitrary,except 8 > 0. Notice that if q is fixed we can make the quantity
non-negative by choosing 8 small enough and a sufficientlyclose to 1. However cr is also restricted by l/q <Ka < I/p. Thus it is clear that we can
realize the conditions of the corollary if we take
UO
1- - lq).
max (llq,
Hence for this choice of uo, the corollary is proved.
The proofs of the other corollaries necessitate the following lemma which
is along very classical lines.
LEMMA 26. Let ? (s) be a (numerical-valued) function analytic in an
open region which contains the strip
a ?R
Suppose that for some c>
and furthermore,for some q, q>
O(It),
as ItIoo,
1,
. 00
so<
I((a+it)
Let a<
a<
0
sup I4(o+it)I
L
,
(s)?
y < 8.
Is dt C- 1I,
I
(,3 + it) 1Ivdt _- I.
46
R. A. KUNZE
AND E. M. STEIN.
Conclusion:
sup
? + it)
(e
-x <t<xo
_A.
A dependson , /3,Y, and q, but does not otherwisedependon c or c.
Proof. Let p be the index conjugate to q, l/p + i/p 1. Choose 4
to be a continuousfunctionon (- oo, oo) which vanishes outside a finite
interval,and satisfies
=
(8.11l)
z
00
I + (t) |Pdt_ 1,
but let 4) be arbitraryotherwise.
Define ,(ai + it) by
b1a+ it)
J
_00
a+ it +
a _f
it,) 0(ti) dti,
Then it is easy to verifythat ID (s) is analyticin an open regionwhich
containsa ?< R (s) < f8; that
sup I 1(oI + it)I = O(I t c), as I tI
a-'a',B
;
and in view of the assumptionson c1 and (8. 11) that
sup I ii (a + it) I _ 1, and
-xo<t<xo
sup I (I (/ + it) I1.
-xo<t<xo
We are now in a positionto apply the classical Phragmen-Lindel6f
principle
to D1,7 The conclusionis that I
is bounded by 1 in the entire strip
a < R (s) ?fA. In particular,
I <DI(0f) I
-
a -<f-A
Going back to the definitionof cIv, we obtain
00
Consideringthe arbitrariness
of 4 (exceptfor condition(8. 11)) the converse
of lld1der'sinequalityshows:
^00
(8. 12)
1D
|
(,T + it) Iq
dt_ ,
if a
f_.
For functionswhichare analyticin a stripand satisfya uniformestimate
like (8.12) thereis a knownvariantof Cauchy'sintegralformula. It is
17
See e.g. Titchmarsh [22; p. 181].
REAL UNIMODULAR
47
GROUP.
+it)
(,ID
(a,+
(8. 13)
it,) / (a+it,
-y
-
t) )dt,
^00
2..,'(
-y-it
it,
+ it, ) / p+
(
) ) dti,
a<
Ky < P.
In Paley and Wiener ([18] pp. 3-5), (8.13) is demonstratedunder
to q 2 in (8.12). However,the proof in
the assumptioncorresponding
the generalcase, q 1, is no different.
If one applies Hldider'sinequalityto each of the integralsin (8. 13) one
obtains:
>
(8. 14)
sup
where
Aopeq 21 [ (
<
[Aa
+
J(y+it)
-oo<t<x
yq,
00
dtl ( _(
a)
Aaj~~~~~~~~~~~~~qso
+
2
t2 ) p/2 ) Il/p
+
dt/(- (_
(J
(1/p
+
1/q
)2
+
t2)p/2)
/p]
)
A simplecalculationshows,
(8. 15)
Aaf,>yq-_C[ (_/
,)-11q+
-llq
/a
and with c some absoluteconstant. This concludesthe proofof the lemma.
Proof of Corollary2. Let us assumefor simplicitythat 11f IIP 1.
Consideringthe indexo, definedin Corollary1, chooseuo < a1 < 1, and
keep u1 fixedthroughout
the rest of this argument. Now by the choiceof cr1
(and the normalizationimposedon f) we have
^00
*1a-
U,f
(8. 16)
-
+
it)
+
11qq
dt_<-Aq,
it) Jqq dt? Aq,
withA independentof f, for some appropriateA.
Choosee and -qto be twovectorsin 4, subjectto the restriction14
v 1, but otherwisearbitrary. Now
I
I <11a|(s) 11)'11 a+(S)JJq.
I (9 (s)t)
Hence if we let
1,
R. A. KUNZE
48
then
(8.17)
AN)D E. M. STEIN.
f
00
00
fI4
(rl+it)lqdt<1
+it)fqdt?1.
14D(1-al
However @F(s) is clearlyanalytic in an open region containing 1c?R (s)
< 1 - oh-it is analyticin 0 < R (s) < 1. Moreover,it satisfiesthe growth
conditionspecifiedin Lemma 11, withc 1. We also noticea, < 1 < 1 -a,.
We thenconclude:
sup JC(j+it)j
-Xo<t<o
A 11,q.
Going back to our definitionthis leads to
sup
2(5?(+it)4,q)1?<3q.
-ox<t<ox
Notice that Al,q and hence Bq is independentof 4,,q. Taking the sup over
?1,
all 4,, 11411
?1 we obtain
-X
sup
<t< oO
II+(?+it)I?<
Bq.
If we now dropthe normalization11
f IIP- 1, we obtainthe conclusionof
Corollary2. This concludesthe proof.
Proof of Corollary3. The proofis similarto that of Corollary2 but is
somewhatmorecomplicated.
Let f be a simple function. We use inequality (8. 2) which we have
alreadyprovedforsuch f. We fixsome8 > 0, and assumemomentarily
that
f IIP=1. Let us call
11
wherewe choose1/q <
<1K
Then (8. 2) becomes
I
r0
ll5+(of1
+ it) J]q(1 + ItI)A
Goo
dt ? (Aq,ci1JX)q.
Choose4,fE
C , with11411?1,11 ? 1, and let
(s)
=
Then
* (a + it) (
1([a|(r+
( F+(s), 4,).
it) llo_1
(
+
it) 1q-
The above then becomes
I
OD
IO
*((J1 +
it) I2
(l + I t )| dt _<(Aq, al,X) q.
REAL UNIMODULAR
49
GROUP.
Since the formula (8. 2) is symmetricin
and 1-
oi
01,
one also obtains
00
o
We let
I*(l
o1 + it) Iq(1 +I
I 1) Xdt<(Aq,
01
x)2q.
(D(s)=- cl (2 + s) x/,g(s) .
If we choosecl as appropriateconstant(dependingon q, ul, and A) thenthe
aboveinequalitiesbecome
I f + it),7dt --<I,
00
and fHGo,(1 -,
+
it) Iq dt? 1.
Moreoverit is an easy matterto verifythat '1 (s) satisfiesthe growthconditionspecifiedin Lemma 11. We may thus conclude,(see (8. 15)),
|I(?+it)
o-O
?C2[I
-1l/q]
< or <
u1
+
1 +a7
[|
1 l/q]
1 -'r,
Goingback to the definitions
of 1Dand I the above becomes
12 + s
X/q I5r(S)e5,l)
I<
s-a=
o
+
9J rl|l/a+|
C3[I|v
it,
a,
<
af <
Il-l-
1
1r|11a]5
0-al.
Notice that the right-hand side is independent of @, and 7.
If we
rememberthat $ and v are arbitraryexcept 11J
11 1I,1111_ 1, and we take
the sup of the left-handside, droppingthe restriction11
f II 1, we then
obtain
2 + s IX/q115?(s) <C3[1
-i -l/q] f
(8. 18)
-l1/q +
I 1
C-1
I 1/
IIPI
wheres-o=u+ it,
<i
of< 1- ar,
C3
l/q <o
C3 (q,
al0
<
12,
A).
Notice that this formulaactually holds for everys in the open strip
l/q < R(s) < i/p. In fact,for such an s, we need onlychoosea o-, so that
ul < u < 1 -
, and 1/q < o- < 1.
If we now fix our A and s, it is clear that (8. 18) implies Corollary3.
Proof of Corollary4. It is clear from Corollary3, that whenever
l/q < R(s) < l/p, 5- (s) has a unique boundedextensionto all of
L.(G).
Inequality (8. 18) shows that the bounds are uniformwhenevers is
restrictedto a compactsubsetof l/q < R(s) < i/p. But we know that f;
is analyticin the strip 0<1R(s) <1, when fcL1(G)n Lp (G).
4
50
R. A. KUNZE
AND E. M. STEIN.
Hence a simple limitingargumentalso shows that 91 is analytic in
1/q <R(s) < l/p, for each fixedfELp(G).
Other limiting arguments (which we will not give) show that the
extension9Jt to all of Lp also satisfiesthe inequalities(8. 2) and thosecontained in Corollaries1, 2, and 3.
This concludesour discussionof Corollary4.
9. The discreteseries. We now intendto investigatethe formof the
Hausdorff-Young
theoremfor our group, so far as it involvesthe discrete
series.
As contrastedwith the case of the continuousseries consideredabove,
we do not concernourselveswith an analyticstructurein the discreteseries.
This lack is mitigatedby the fact that in the Plancherelformulafor the
group,elementsof the discreteseriesoccurwithweightsboundedaway from
zero.
We beginby provingthe followingtheorem.
THEOREM
(9.1)
8. Let 1 p<2,
(7(k+
k=O
and 1/q+1/p=
1. Then
D+ 7,)11q + (k +1) D- (f,7)
ll
1,q)l
wheneverfE L1 (G) n Lp(G).
f II
Proof. We considerthe measurespace M, definedas follows:The points
of M are the pairs (k, ?), wherek runs overthe non-negative
integers,and
the secondcomponentis either+ or - as indicated.
On M we definethe measuredm as follows:The point (k, +) is assigned
the measurek +I; the point (k,-) is assignedthe measurelk+ 1.
We let 9& denote a separable infinite-dimensional
Hilbert space. In
accordancewiththe discussionof ? 2 we considerfunctionsfromM to bounded
operatorson S. In view of the discretenessof M, all such functionsare
automaticallymeasurable.
We now definea mappingfromsimplefunctionson G to operatorvalued
functionson M. The mapping,whichwe denoteby T, is given by
T: f- F={F
(c, + )},
and with
D?)(f,k) ifD?(g,
k)f (g)dg.
As explicitlygiven,the representations
D+ ( ., 7c) act on different
Hilbert
REAL UNIMODULAR
GROUP.
51
spaces. However,since all separableinfinitedimensionalHilbert spaces are
unitarilyequivalent,we may assumethat we deal with appropriateunitarily
all of whichact on our given S4.
equivalentrepresentations,
Using the definitionsof ? 2, (9. 1) becomes
1IT(f) II
(9. 2)
f 1lq.
11
This is whatwe mustprove
Observethat by definition,
T(f)
11
co
1122=
kFO
(k
+ i) IID+(f, k) 1122 + (k + 1) 1D-(f, k) 1122.
Hence, in view of the Plancherelformulafor G, (see ? 5), we have
(9.3)
T(f) 112Cli
11
f 112.
Notice also that
=
sup 11D (f, k) ||x
11T(f) JIM
k,+
while
11D (f,
since D( 7,)
k) lloo_
|| f l1,
is unitary. We therefore
have,
1IT(f) 11X,11
f Ii.
(9. 4)
We now use the general interpolationtheoremof ? 3. In the present
case the operatorT is independentof z, and so a fortiorisatisfiesthe conditionsof analyticityand admissiblegrowth.
We let (p0,qo)
(2, 2), and (p,, q,)
(1,oo). Then it is easily
verifiedthat l/p + 1/q 1, and that we may chooseany p, 1 _ p _ 2, by an
appropriatechoice of ,0 <-- T < 1.
It is apparentthat in the presentcase A, (y)
1 because of (9. 3), and
also A1(y) =1 because of (9.4).
The resultof Theorem3 is
=
=
11
T(f) IIq?_AT1If IIP
Since Ao(y) =A1(y) =1, it followsthat logA,==0, and henceA,=1.
Thus we have demonstrated(9. 2), and therefore(9. 1), wheneverf is a
simplefunction.
The extensionof the inequalityto all L1 (G) n Lp (G) followsby standard
limitingarguments. This concludesthe proofof the theorem.
The followingcorollaryis basic forour applicationsof the abovetheorem.
52
R. A. KUNZE AND E. M. STEIN.
COROLLARY. The mapping f-* D+(f,k7) has a unique extensiont
to all
of Lv(G), and this extensionsatisfiesthe following:
(9. 5)
sup 1jD-(f, 7c)110? 21-1/P1I
f
whenever1 _ p ? 2.
Proof. We considerfirstthe case when f C L1 n L,.
obtain
(k + 1lD+ (fnk7)
1qqCk
Hence,
k=O
(7c+1) 11D+ (f|7)
Using (9. 1) we
llq qC-- 11f IlPq.
D1 (f,7k)llJqC 1/(lk+ -1) 11
f ?IP
f jlj_
C 2 11
A similarargumentfor D- (f,7k) showsthat
sup 11D (f,k) llqC 2/q 11f IP- 21-1/P
f 11p.
11
kC,I
Since theoperatornormsused aboveare non-increasing
withq, (see (2. 2)),
we conclude (9. 5), wheneverfC L1 n Lp.
In view of the inequality just proved it follows that the mapping
f-*D+ (f,k) has a unique extensionto Lp which again satisfies(9. 5).
CHAPTER IV.
APPLICATIONS.
10. Boundedness of convolutionoperator. We are now in a position
to obtain an importantapplicationof the analysis of the previoussections.
We shall findit convenientto adopt a slight change in our notation.
In this sectionlettersx,y,z, - - will denoteelementsof the group G, and
f,g,h, * functionson the group.
We recall the operationof convolutionof two functionsf and g, defined
as follows
.
(f * g) (x)
dy
f
f(xy-1)g
(y) dy,
Haar measure.
Now if f C L2, and g E Lp, 1 ? p ? 2, then by Young's inequality (see
[23]), f*g is well definedand is in Lr, where1/r=J+1/p -1.
TIIEOREM
h E L2, and
(10.1)
9.
Let fE L2, and gLEL
h
11
112 C
1? p < 2; if h
f11211LIP,
A 11
f*g, then
REAL UNIMODULAR
53
GROUP.
whereAp does not depend on f or g. Ilence the operationof convolution
bya functiong E Lp, 1 ? p < 2, is a boundedoperatoron L2.
Remarks. Inequality (10. 1) fails when p = 2. This is not surprising
for manyreasons; we indicateone such reason. Inequality (10. 1) is essentially a statementof the fact that the Fouriertransformof a functiong in
LP, 1 p < 2, is uniformlybounded. But a functionin L2 may be given
by appropriatelyassigningits Fourier transform,and this may be done so
that the Fourier transformis not uniformlybounded.
The statementwhich correspondsto (10. 1) when G is, for example,a
non-compactabelian groupis false,as long as p # 1. This is so even in the
case when G is the additive group of the real-line. We postponefurther
discussionsof these mattersto the next section.
to prove inequality(10. 1) for a dense class of
Proof. It is sufficient
assume
that f and g are in L1 (in additionto the fact
functions,and so we
also in L2 and Lp).
that f and g are respectively
f * g, and x
Noticethat if h
then
U$
Uh = U
(10. 2)
Here U,
=
is any (say unitary) representation,
UV.
f(x)Ux,dx, with similar definitionsfor U,, and
Uh.
Moreoverby (2. 13) and (10. 2) we obtain
U7I
11
(10. 3)
112?-11 U, 11211Ug 110
We apply (10.3) successivelyto the cases when U U-(,2 + it), (the
continuousprincipalseries), and U D+(, ck), (the discreteseries).
For the continuousprincipalserieswe apply Corollary2 of Theorem7,
(withg in place of f) ; forthe discreteserieswe similarlyapplythe corollary
of Theorem8. The result for the continuousseries is
(10.4)
U(h,
11
g IIP,
4+ it) 112? Ap 11U+(f, 21+ it) 11211
1?S p < 2, withA. independentof t.
The resultforthe discreteseriesis
(10. 5)
1ID-(h, 7C)112
? 21-1/P1i D+(f, 7k)112 1j9
I,
1? p ? 2.
h I12and 11f 112via thePlancherelformula,(see ? 5).
Finally,we calculate11
54
R. A. KUNZE
AND E. M. STEIN.
It is to be notedthatin computingthe requirednorms,it makesno difference
whetherwe use the representations
v@( , j + it), or the unitarilyequivalent
representationsU-( , 1 + it). Using (10.4) and (10.5) we then easily
obtain
1Ih11
2C
AP11
2 119 IIPi
f 11
1C~P < 2.
This proves (10. 1), and hencethe theorem.
From theabovetheorem,and withthe use of variousdevices,it is possible
to prove otherinequalitieslike (10. 1). All of these have in commonthe
remarkablepropertythat they hold for the group we are consideringand
also forcompactgroups,butfail in the simplestnon-compact
abelianinstances.
We shall limitourselvesto the proofof onlyone moresuch result.
and
COROLLARY. Let fE L2, andgE L2. If h = f * g, then hE Lq, 2 < q<
IIh 1q1 Aq 11
f 11211g
(10. 6)
oo,
112,
WhereAq does not dependon f or g.
Remark. By the results of the next section it will be seen that this
corollaryand the theoremfromwhichit is derivedare essentiallyequivalent
results.
Proof. Let le L1 n Lp, where i/p + 1/q= 1, but let k be arbitrary
otherwise. Then,
f
h(x)kl(x)dx
IC
k(x) ff(xy-1l)g(y)dydx
g(y) fk(x)
where I =*
*k, with f* (x)
f (xy-1)dxdy= f
g(y)l(y)dy,
f(x-1).
Hence,
IJh(x)kc(x)dx
g-
X
(y)lI(y)dy
I
1 11211
g112 11
However,by our theorem
1l1 112CAp
11A2
11
f 1k
IIIP,
since 1 < p < 2. Thus we have
I rh(x)kc(x)dxI?AD
II 112
11
f112 a 11
11kIp.
REAL UNIMODULAR
55
GROUP.
k II
Now take the sup of the left-handside over7k,such that 11
resultis
hlIq? Ap11f 1121g11j2,
11
1. The
and the corollaryis proved.
11. Characterizationof unitaryrepresentationsof G. Let g -> U. be
(not necessarilyirreducible)on a Hilbert space 54.
a unitaryrepresentation
of the
We now introducetwo notionswhichare basic for our characterization
of G.
representations
Definition. +(g) is an entryfunction,if
<<>(g)=--(Uv4,r),
(1.1)
~,q) S.
Definition. g -> Ug is extendableto Lp(G), if for some fixedp, p
1,
fIIP,everyfE L1(G) nLpL(G),
C A 11
UfIIX
11
(11. 2)
withA independentof f.
to notethe followingfacts. Theorem9, whichdealt with
It is interesting
the boundednessof the operationof convolution,can be restatedby saying
is extendableto Lp(G) foreveryp, 1 ? p < 2.
thatthe regularrepresentation
We may furthernote that the corollaryto Theorem9 statesthat everyentry
is in Lq (G), for everyq > 2.
functionof the regularrepersentation
As a preliminarymatter,we obtain the followingrelationbetweenthe
notionsdefinedabove.
27.18 The representation
g-*>Ug is extendableto Lp(G), if and
onlyif for everypair X, E 9, the entryfunction4, definedin (11.1), lies
1.
in Lq (G), where1/p+ 1/q
LEMMA
Assumefirstthat Ug is extendableto Lp. Let fE L,l nLp.
fa(g)f(g)
dg
J (Ug., r1)f(g)dy
U(
fUof(g)dg<q)=-(Uf,
q).
Thus by (11.2)
18
Ths l (g)emma
hol fr a
l c
a1171ct
c g
18This lemmaholdsforany locallycompactgroup.
p p11 11
11
Then
56
R. A. KUNZE
AND E. M. STEIN.
We now limitourselvesto thosef's forwhich 11
f II 1, and we take the
sup of the left-hand
A 1111
side. We thenobtain1pqI ?1qC-Az
and thus
1l71
11,
4 C Lq. This provesthe implicationin one direction. To provethe converse
we shall use the closed-graphtheoremseveraltimes. We argue as follows.
For fixedq, consider
themapping
*
(Ug$ q) =p(g)
as a mappingfromX to Lq (G). By the assumptionsof the lemma,it is
clear that this mappingis everywhere
definedon X ; obviouslyit is linear.
We nextnoticethat it is closed. For suposethat 4,n
-: , and
(9)
O)n
<q) -> 0o(9) in Lq norm.
(Ug$ni
However,pn(g) --'p(g)
(Ug$ q), foreverygE G. Thus p(g) ===4(g) a.e.,
and .j (g) -* (g) in Lq norm. This showsthat the mappingis closed.
Hence,
(11.3)
qf:-::
AX I
Similarly,
(11.4)
j 11
11
BCq1f1
Now let f be any functionin Lp(G). We proposeto defineUf. We
shall do this by defining(Uf$,-), foreverypair E, N.
In fact set
(U,)
X) =
(g)f(g)dg,
wherep(g) - (Ugj, q). Since c C Lq, f C L., and l/p + l/q = 1, the integral
is well-defined,,
by HSlder's inequality. Holder's inequality, (11. 3), and
(11.4) furthershow:
(11.5)
and
(11.6)
f IIp,
I(U,,) ICAX1111
11
X11
fII.P
(Ue,,))IB_Bl 11
11
Now (11. 6) showsthat the vectorU,. is well-defined
for every C1i
E
Moreover,(11. 5) and a simpleargument,provethat UJis a closed operator.
Hence, usingthe closedgraphtheorem,we obtainthat U,,foreach f e LP(G),
is a boundedoperatoron 69 (to itself).
Finally, considerthe mapping
f- Uf,
whichis a mappingfromL, (G) to 6
-Banach space of boundedoperators
REAL
UNIMODULAR
GROUP.
57
on 9 withusual norm. We have just seen that this mappingis everywhere
defined. It is clear fromthe definitionthat this mapping is linear. We
shall next see that it is closed. In fact,assumef,,-> f in Lp norm,and that
UO in the operatornorm. Then
Ufn-*
( UtntN
foreveryt and E N.
Hence
) -->(-Uo*-
1)
By (11. 5) it followsthat
v)-
( Uwf4nt7)
( Ugt
(Uf$,B)
(Uo4, ).
=
Thus Uf U,. Thereforethe mappingf-* Uf is closed. A finalapplication
of the closed graph theoremgives
IIUfJ C-A JfIIp.
This showsthat g -> U. is extendableto L., and the lemema is completely
proved.
We noticethat the lemma provesthat Theorem9 and its corollaryare
equivalentpropositions.It is to be observedthat the identityrepresentation
(on the one-dimensional
space) is not extendableto Lp if p 7 1. Thus there
are very simple representations
which are not extendableto Lp if p # 1.
We make one furtherremarkbeforewe proceed. Every entryfunctionis
automaticallyin L>>(G). Hence a simple argumentshows that if it is in
Lq,(G), it is also in Lq (G), whereq > qo. Thereforethe lemmaleads to the
factthat if a representation
is extendableto Lp,0(G),and po > 1, thenit also
is extendableto L. (G), for 1 ? p < po.
We are now in a positionto give our characterization
of the irreducible
unitaryrepresentations
of the 2 X 2 real unimodulargroup G.
THEOREM
10. Let g -e U, be an irreducibleunitaryrepresentation
of G.
Assume U is not the identityrepresentation.Theen
(a) U is unitarilyequivalentto an elementof the discreteseriesif and
onlyif U is extendableto L2 (G).
(b) U is unitarilyequivalentto an elementof the continuousprincipal
seriesif and onlyif U is extendableto everyL. (G), 1 ? p < 2, but is not
extendableto L2(G).
(c) U is unitarilyequivalentto the element of the complementary
series correspondingto the parametera-, 0 <a- < j, if and only if U is
5
8
R.
A.
KUNZE
for 1
extendable to Lp(G)
AND
E.
p <17(1
M.
STEIN.
but is not extendable to
-r),
(G;).
Ll/pi_a)
COROLLARY. Let g ---U be an irreducible
unitaryrepresentation
different
fromthe identityrepresentation.Then U is unitarilyequivalentto (1) an
elementof the discreteseries, (2) an elementof the continuousprincipal
series, or, (3)
0 <,v
(2')
the element of the complementaryseries corresponding to o,
every entry function is in L2(G),
< 1, if and only if respectively (1')
q > 2, but not every entry function
every entry function is in Lq(G),
is in L2(G),
every entry function is in Lq(G),
or, (3')
q>1/a,
but not
ev7eryentry fuuction is in L/la(G).
Before we pass to the proof of these facts we should like to clarify the
difference of notation that we have adopted for the representations of G and
that which is used in Bargmann's
paper.
The parameter a, 0 < a- < 1, which
series corres-
we have used to identify the elements of the complementary
ponds to Bargmann's
parameter 1-
metrization of the discrete series.
a.
There is also a difference in para-
We have called elements of the discrete
series those which appear as discrete summands
the Plancherel
formula of the group.
This
(with non-zero measure)
exhausts
Bargmann's
series, except for the representations which he labels D+1/2, and D-1/2.
in
discrete
In our
notation these two elements occur as follows.
The representation g -* U (g, i)
of the continuous principal series is not irreducible.
It splits into the direct
sum of D+1/2 and D-1/2.
Thus in our notation we count D+1/2, and D-1/2 as
elements of the continuous
principal
series.
It is with these definitions in
mind that the above theorem and corollary are stated.
Now to the proof.
It is known that every irreducible unitary represen-
tation of the group is, except for the trivial representation, up to the unitary
equivalence,
either an element of the discrete series, the continuous principal
series, or the complementary series.
By the corollary of Theorem 8, it follows that elements of the discrete
series are extendable to L2(G).
By Corollary 2 of Theorem 7 it follows that
elements of the continuous principal series are extendable to Lp ( G), 1 ? p < 2.
If we consider the representation g -* U (g, i) we see that it is also extendable
to Lp, for 1 < p < 2. However, this representation splits into two irreducible
representations (which we have counted among the continuous principal
series).
A simple argument
shows that each of these pieces is then also
extendable to Lp(G),
1 < p < 2.
Finally, Corollary 3 of Theorem 7 implies that the element corresponding
to a, 0 < a < j, is extendable to Lp, 1 < p < 1/(1-a).
We must now show
REAL
UNIMODULAR
59
GROUP.
thatelementsof the continuousprincipalseriesare not extendableto L2 (G),
series correspondingto a is not
and the element of the complementary
(G). We considerfirstthe continuousprincipalseries.
extendableto L1/(,-of)
to exhibitan entryfunctionwhich is not in
By Lemma 27 it is sufficient
L2 (U). We considerthe parametrization
of the group given by Bargmann
with 0 < y <oo, 0?,uc 27r,and 0 ? v< 2r. In this case Haar measure
becomes(2,r)2 dydptdv, (see Bargmann (10. 14)). We considerthe "prin" corresponding
to thisrepresentation.In Bargmann's
cipal sphericalfunction
which
has
the
asymptoticexpansion,as y-o o,
notationthis is WOO
(y),
We also have
Woo(y) ,-2y-1,1(PSoo
(it, o) ytt ) .
I,8)"(it,
0) 12= (coth7rt)/47rt,
or (tanh-rt)/47rt,
dependingon whetherwe are dealing with U+(g, + it)
These
or U-(g, 1+it),
(see Bargmann (11.4), (11.7a1), and (11.7b)).
Except for this case we
asymptoticrelationsare valid exceptfor U-(g,j).
can easilysee that the elementWoo(y) is not in L2 (0, oo; dy), becauseof the
factory-1. Thus the corresponding
principalsphericalfunctionsare not in
L2(G).
In considering
therepresentation
U (g, I) we recallthatit splitsintoD+j
and D-T (in Bargmann'snotation). It is also demonstrated
by Bargmann
to theserepresentations
are asympthatthe sphericalfunctionscorresponding
Thus
in
to
times
these
are
also
not
totic constant
L2(G). The compley-.
in
mentaryseriesis dealt withsimilarly. Taking into accountour difference
notation, we have /l3o(1 - a, 0) y4, as asymptotic expression (as y -* oo) for
the principalsphericalfunctioncorespondingto o-, (see Bargmann (11. 5)).
Now clearlythis functionis not in L,Iy(0, oo; dy). Hence the principal
is not
sphericalfunctionis not in L/1a(G), and thus the representation
extendableto L1/(,-u)(G). If we recall Lemma 27, we see that Theorem10
and its corollaryare completelydemonstrated.
We now pass to the considerationof not necessarilyirreducibleunitary
of our group on a
representations.Let g -> Ug be such a representation
separableHilbertspace 59.
Using the von Neuman reductiontheory[17], and followingSegal [19],
we may decomposethe representation
as follows.
f
The Hilbert space 5tlmay be writtenas a directintegral
xdo(A)
the representation
of Hilbertspaces .94X. With respectto this decomposition,
g -* U0 may be decomposedinto {U\,}, where g-9 UP is irreducibleand
unitary,for a. e. A.
60
R. A.
KUNZE
AND
E.
M.
STEIN.
We do not wish to go into the backgroundof these facts, or into the
sensein whichthis reductionis unique. Aside fromthe simplemanipulative
factswhichwe shall use, we shall also use the followingfact: Let A be an
operatoron X which can be decomposedwith respectto the above decompositionof X4 into the direct integral of the V9X's. We write A - {AX}.
A 110 esssup 11
AX10
Then 11
x
Our theoremis the following:It may be viewed as an extensionand
clarification
of Theorem9 and its corollary.
11. Let g-9 U. be a unitary representationof G on 69.
Considerits reductioninto a directintegralof irreducibleunitaryrepresenconditionthat (except for a set
tationsg -* Uxg. A necessaryand sufficient
of measurezero) everyUXgbe unitarilyequivalentto elementsof the discrete
g - Us,be extendable
or continuousprincipalseriesis that the representation
the conditionis equivalent
to Lp(G) for everyp, 1 ? p < 2. Alternatively,
g -> Ug be in
withrequiringthat everyentryfunction.of the representation
everyLq(G), 2 < q.
THEOREM
Proof. Assumefirstthat,disregardinga set of measurezero,everyUP
is equivalentto eitherelementsof the discreteseries or of the continuous
principalseries. Let f C L1 (G) n Lp(G). Then
U-
Now 11Uf
esssup11Uxf
{ U",}.
2 of Theorem7, and
Becauseof Corollary
x
the corollaryto Theorem8, we obtain
esssup 11Uxfll_A
x
p < 2;
1 fII,
we have disregardedthe set of measurezero which does not correspondto
eitherthe discreteor continuousprincipalseries. Hence,
11
Uf11CAP 11f IIP,5 1<p < 25
and g-* Ug is extendableto everyLp(G), 1 ? p < 2. To provethe converse,
we argue as follows. Let {fn} be a denumerablecollectionof functionson G
Now
whichlie in everyL.p(G), and are dense in everyLp(G), 1?p<oo.
have
we
to
be
can
extended
Since
g-*
Ug
L.,(G),
Ufn -Uxf.
Thus,
I1IUfnJ1o:IX_APIIfnIp5
essUx
sup II
flo
1 C-p < 2.
fnIIP,
Ag 11
1' P < 2
REAL UNIMODULAR GROUP.
(1
to the above
Let En be the exceptionalset of measurezero corresponding
esssup.
U En; then E is still of measurezero. Now
Let E
C Ap || fnIIP,
sup 11Ulf. IIOO
1 C p < 2.
Owingto the densenessof the collection{f4),we obtain
P,
f 11
11FIf IIo'< Ap 11
1C5;P <
2~,A E fE L, n LpI
ThereforeUxgcan be extendedto Lp, 1 ? p < 2, foreveryAf E. By Theorem
10, and the fact that E is a set of measurezero we obtainthat almostevery
belongsto eitherthe discreteor continouusprincipalseries. Using Lemma
UXg
27 we obtainthe alternatecondition.
This concludesthe proofof Theorem11.
Let us now considerthe additivegroupof the line. We shall showthat
the analogues of Theorem9 fails for everyp & 1, and that the analogue of
the corollaryof Theorem9 fails if q#0oo.
In fact let f(x) = (log(2+- I xI)) -, - oo<x<oo.
Then by the use
of Theorem124 of Titchmarsh[21] it may be shownf is the Fouriertransformof a positivefunctionwhichis in L1 (- oo, oo). A simpleapplication
of the Planchereltheoremthen showsthat f is the convolutionof two functions in L2(- oo, oo). However,clearly,f Lq(- oo, so), if q oo. Thus
the analogue of the corollaryof Theorem9 fails. Because of Lemma 27
appliedto theregularrepresentation
on L2 (-oo, oo), we see thatthe analogue
9
of Theorem fails if p 7A 1.
Let us considerthe problemof whetherTheorem9 would hold for our
group in the case p- 2. This, clearly,is equivalentto requiringthat the
regularrepresentation
of the groupis extendableto L2(G). By an argument
like that in the proofof Theorem11, it would then followthat the regular
representation
can be writtenas a directsum of representations
equivalentto
representations
of the discreteseries. This, of course,is not true.
MASSACIUSETTS
INSTITUTE OF TECHNOLOGY.
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R. A. KUNZE
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