A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
J EAN -P HILIPPE A NKER
E WA D AMEK
C HOKRI Y ACOUB
Spherical analysis on harmonic AN groups
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4
esérie, tome 23, n
o4 (1996), p. 643-679
<http://www.numdam.org/item?id=ASNSP_1996_4_23_4_643_0>
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http://www.numdam.org/
JEAN-PHILIPPE ANKER - EWA DAMEK - CHOKRI YACOUB
0. - Introduction
Harmonic AN groups and
analysis
thereon have been studiedby
several au-thors in the past 10 years
([Bo], [BTV], [CDKRI], [CDKR2], [Daml], [Dam2], [Dam3], [Dam4], [DR1], [DR2], [Ric2], [Sz2]
...).
Recallthat,
as Riemannianmanifolds,
these solvable Lie groups include allsymmetric
spaces of noncom-pact
type and rank one,namely
thehyperbolic
spacesHN (R), HN (C), HN (IHI)
and
H2 (®) (the
realhyperbolic
spaces which are somehowdegenerate,
are often
disregarded),
but that most of them are notsymmetric,
thusproviding
numerous
counterexamples
to the Lichnerowiczconjecture [DRl].
Despite
the lack of symmetry,spherical analysis
i.e. theanalysis
of radialfunctions on these spaces is
quite
similar to thehyperbolic
space case. Thiswas
already apparent
in thepioneer
works[DR2], [Ric2]
and will be made clearin this paper. First of all we shall
emphasize
thatspherical analysis
isagain
aparticular
case of the Jacobi functionanalysis,
anelementary
observation which isquite
useful and seems to have been unnoticed. The mainbody
of the paper will be devoted toestablishing
a series ofanalytical results,
which illustrate the actualsimilarity
withsymmetric
spaces: a Kunze-Steinphenomenon, sharp
and
simple
criteria for the LP - LP and the weakL
1 --~L
1 boundedness ofpositive
convolutionkernels,
the LP behavior of(functions of)
theLaplace-
Beltrami
operator,
a detailedanalysis
of the heatkernel,
the weakL
1 -L
1boundedness of both the heat maximal
operator
and the Riesztransform,
....The
(modified)
waveequation
will be studiedseparately [AMPS].
Our paper is more
precisely organized
as follows. In Section 1 we recallthe basic structure of harmonic AN groups and in Section 2 the basic
spherical
harmonic
analysis
thereon. In Section 3 we establish theanalogues
of the Herzand the
Str6mberg
criteria for the LP - LP and the weakL 1 -~ L boundedness
of
positive
radial convolutionkernels,
and get as first consequences aspherical
Kunze-Stein
phenomenon
and the weakL
1 ~L
1 boundedness of theHardy-
Littlewood maximal function. In Section 5 we obtain an
explicit
formula for the heat kernel based on the inverse Abeltransform,
deduce from itoptimal
Pervenuto alla Redazione il 2 gennaio 1995 e in forma definitiva il 11 ottobre 1996.
upper and lower
bounds,
both for the heat kernel and itsgradient,
and get asa consequence the weak
L
1 -~L
1 boundedness of the heat maximal operator.We conclude in Section 6 with some further results and open
problems.
This paper is the result of two
independent
researches. On onehand,
J.-Ph.Anker and E. Damek worked out most of the material in Sections 2 to 5
during a(n enjoyable) stay
of the first author in Wroclaw in theSpring
of 1993. On the otherhand,
C. Yacoub obtained parts of Sections3,
5 and 6 in the firstchapter
of his thesis
[Ya].
All three authors aregrateful
to N. Lohou6 forsuggesting
tolook at the weak
L
1 -~L
1 boundedness of the heat maximal operator in thissetting,
which was thestarting point
of theirstudy.
1. - Basic structure of S = AN
Harmonic A N groups form a class of solvable Lie groups,
equipped
witha left-invariant metric. More
precisely,
S = A x N is a semi-directproduct
ofR with a
Heisenberg
type Lie group N.H-type
Lie groups were introducedby Kaplan [Kal]
and studied later onby
several authors(see [BTV], [CDKRl], [CDKR2], [DR2], [Ka2], [Ka3], [KR], [Kor], [Ricl], [Rie]
...).
Recallbriefly
their structure. Let n be atwo-step nilpotent
Liealgebra, equipped
with an innerproduct ( , ).
Denoteby 3
thecenter of n and
by
v theorthogonal complement of 3
in n(so
that[0, 0]
C3).
Let
JZ :
u - to be the linear map definedby
Then n is
of Heisenberg
type if thefollowing equivalent
conditions are satisfied:(1.2)
ad X is anisometry
from(Ker
adX) -L (C to)
onto J, for every unit vector Z~t);Recall
[Kal]
thepossible
dimensions k =dim 3
and m = dim b :where K >
0 and it > 0 arearbitrary integers.
Inparticular m
isalways
even.
(As
waspointed
outby
A.Koranyi,
there isactually
asimple
and clearexplanation
for this fact: anyJZ corresponding
to a unit vector Z E 3 inducesa
complex
structure onb.)
The
corresponding (connected and) simply
connected Lie groups N are calledof Heisenberg
type. We shallidentify
them with their Liealgebra
n viathe
exponential
map. Thusmultiplication
in readsConsider
([Bo], [BTV], [CDKR1], [CDKR2], [Daml], [Dam2], [DRl], [DR2], [Sz2]
...)
the semi-directproduct
S = N xRj
definedby
S is a solvable
(connected and) simply
connected Lie group, with Liealgebra
~. = to
fl3 3
fl3 R and Lie bracketS is
equipped
with the left-invariant Riemannian metric inducedby
on .~. The associated left-invariant
(Riemann-Haar)
measure on S isgiven by
Here
Q = 2
+ k is thehomogeneous
dimension of N.Most Riemannian
symmetric
spacesG/K
ofnoncompact type
and rank one fit into this framework.According
to the Iwasawadecomposition
G =NAK, they
can be realized indeed as S = NA = AN, with A -- R. N is abelian for realhyperbolic
spaces =H N (R) (which
are therefore oftendisregarded)
and of
Heisenberg
type in the other casesG/K
=HN (IHI), H2(O).
Notice that these classical
examples
formonly
a very small subclass of harmonic AN groups, as can be seenby looking
at the dimensions:It should be
pointed
out that different notations and normalizations are used in thesetting
ofsymmetric
spaces on one hand and in thesetting
of A N groupson the other hand. In this paper we shall stick to the latter conventions.
Hyperbolic
spaces have a well known realization as the unit ball in IFn . Ourgeneral groups S
can be realizedsimilarly
as the unit ballin 5,
via aCayley
typetransform:
where
and
conversely
Letting a ~
0 i.e.1,
this transform extendsnaturally
to a stereo-graphic
typeprojection
a C between the boundaries a S - N andf (o, 0, 1 ) } .
Notice that the Jacobian of 8C is related to the Poisson kernelon N
([Dam3], [Dam4], [DR2]). Precisely,
In the ball model
~(e),
thegeodesics passing through
theorigin
are thediameters,
thegeodesic
distance to theorigin
isgiven by
and the Riemannian volume writes
where d~ denotes the surface measure on the unit
sphere
in sandn = dim S = m + k + 1. In
particular:
( 1.17)
The volumedensity
in normal coordinates at theorigin,
andby
translationat any
point,
is apurely
radialfunction,
which means that S is a harmonicmanifold ([DRl], [Szl]).
(1.18)
The ball volume has theasymptotic
behaviorLike all harmonic
manifolds,
S is an Einsteinmanifold.
Alengthy
compu- tationyields
the actual constant:The sectional curvature, as far as it is
concerned,
isnonpositive,
with mini-mum = -1
[BTV] (see
also[Bo], [Dam2]).
Notice that it mayvanish, contrarily
to the
hyperbolic
space case.Eventually
let us mention aninteresting inequality
between the
A-component a(x)
of x and thegeodesic
distancer(x)
from x tothe
origin,
which follows from(1.12)
and(1.15),
and which is theanalogue
of a classical relation between the Iwasawa and the Cartan
projections
in thesymmetric
space case.2. -
Spherical
harmonicanalysis
on SSpherical
harmonicanalysis
on S wasdeveloped
in[DR2]
and[Ric2],
inclose
analogy
with the classical(rank one) symmetric
space case. We recall it in thissection, place
it in thegeneral
framework of Jacobianalysis [Koo]
and introduce the Abeltransform.
Harmonic
analysis
onsymmetric
spacesG/K
relies on thecommutativity
of convolution on bi - K -invariant
objects
on G. A similarphenomenon
occurson
general
S, if onereplaces
bi-K -invarianceby radiality.
For functions onS,
this notion has the obviousmeaning.
Fordistributions,
invariant differentialoperators,
...radiality
is definedby
means of anaveraging
operator overspheres,
which writes
in the ball model and
generalizes K -averages
on rank onesymmetric
spaces Thefollowing
fundamentalproperties
were established in[DR2]:
(2.2)
Convolution preservesradiality.
(2.3)
Convolution is commutative on radialobjects.
Inparticular,
the radialintegrable
functions on S form a commutative Banachalgebra L1(S)a
underconvolution.
(2.4)
Thealgebra D(5’)~
of invariant differentialoperators
on S which are radial(i.e.
which commute with theaveraging operator)
is apolynomial algebra
witha
single
generator, theLaplace-Beltrami
operator ,C.From there one can
develop ([DR2], [Ric2])
atheory
ofspherical
func-tions, spherical
Fouriertransform,
...quite
similar to the classical(rank one) symmetric
space case, for which we refer for instance to[An5], [Fa], [Hel].
Let us sum up:
(2.5) Spherical functions ({J
on S are characterizedby
the set of conditionsis a radial
eigenfunction
of ,C(and
thusautomatically analytic),
t
= 1 .(2.6) They
are all obtainedby
a Harish-Chandra typeintegral formula:
Moreover £ wx
= -(
4 +k2) and wx
4== ±it -
(2.7)
ForRe(i ,)
= - ImÀ >0,
we have thefollowing asymptotic
behavior:where c
(2.8)
Thespherical
Fouriertransform
is definedby
for radial functions
f
=f (x )
on S, which we shall oftenidentify
with functionsf
=f (r)
of thegeodesic
distance to theorigin
r =d (x, 0)
E[ 0,
.-(2.9) Paley-Wiener
theorem:?~ is an
isomorphism
from the space of smooth radial functions with compactsupport
on S onto the space of even entire functions ofexponential
type.(2.10)
Plancherelformula:
We have the
following
inversion formulawhere co =
2k-2 ~ - 2 -1 1 r(~).
Moreover 1í extends to anisometry
from thespace a of radial
L2
functions on S ontoL2«0, +(0),
Notice that the constant
given
in the inversion formula[Ric2 ; (4.1)]
isslightly
incorrect. The error is due to adiscrepancy (already
alluded to inSection
1)
between standard normalizations in the harmonic AN groupsetting
onone hand and in the
symmetric
spacesetting
on the other hand. See also(5.61).
As for
hyperbolic
spaces, all thisanalysis
fitsactually
into thegeneral
framework of Jacobi function
analysis [Koo].
and this is thepoint
of view wewant to present now. The radial
part (in geodesic polar coordinates)
of theLaplace-Beltrami operator
,C on S writesBy substituting t = L,
4 rad £ becomes the Jacobi operator[Koo; (2.9)]
with indices a =
and f3 = k2l.
Notice that we are in the ideal case a >f3 > - 2
for Jacobianalysis.
Acomparison
of thepreceding
formulaswith
[Koo; § § 2.1-2.2] yields
thefollowing
identifications:(2.13) Spherical
functions on S are Jacobifunctions:
(2.14)
Thespherical
Fourier transform coincides with the Jacobitransform:
As a consequence, all above-mentioned results follow from the
general theory developed
in[Koo].
And much moreactually.
For instance there is an
analytic
Abeltransform,
which can beinterpreted again geometrically
as an orbitalintegral,
as in thesymmetric
space case, and which leads to an effective inversion formula for thespherical
Fourier transform.Let us
consider,
for radial functionsf
on S, theintegral
transformIt follows from the
integral expression (2.6)
that(2.16)
i.e.where Y denotes the classical Fourier transform on the real line. Hence
(2.17)
Abel tranformsA f
are even functionsand
inverting effectively
thespherical
Fourier transform 1t or the Abel trans- form .,4 amounts to the sameproblem.
Let us
compute (2.15) explicitly.
Write n =(X, Z)
as usual.(1.12)
and(1.15) yield
theexpression
for the
geodesic
distance r’ between x =(X, Z, a)
and theorigin.
Thus(2.15)
becomes
1 y
Elementary integral
calculus(introduce polar
coordinatesX = 2013
in b,Z
= ç 2 2013 in 3
and substitutesuccessively cosh 1-
=cosh § + , cosh2 S2
=cosh2 2 + ç
i.e. cosh s2 =cosh sl
+2 )
leads to the formulaexpressing
the Abel transform as acomposition
of twoWeyl
typefractional integral transforms [Koo; (5.61 )],
definedby
for T
> 0,
In order to invert the Abel
transform,
we use the fact(for
fixed t >
0)
aone-parameter
group of transformations let say of withThus
Explicitly,
when k is even and
when k is odd
(recall
that m isalways even).
The
mapping properties
in thesetting
of smooth functions withcompact support
can be summarizedby
thefollowing
commutativediagram
where each arrow is a
(topological) isomorphism.
Thispicture
can be extendedto
larger
function spaces.Inspired by
thesymmetric
space case, let us introduce for instance thefollowing
radial Schwartz type space on S:which is
nothing
else but(cosh . )i S(R)even (with
the usualidentification), denoting
the classical Schwartz space on R. Then we haveagain
acommutative
diagram
where each arrow is a
topological isomorphism [Koo; § 6].
Let us make tworemarks:
(2.28)
There isactually
a whole range of radial LP Schwartz type spacesSP(S)q
(0
p2),
which are definedby substituting e for
in(2.26).
TheirAbel transform is and their Fourier
transform,
forp
2,
is the space of evenholomorphic
functions h =h(h)
inside thestrip
I Im h [ (t - ~) Q,
whichsatisfy
for any
nonnegative integers
M, N[Koo;
Theorem6.1 ] .
(2.30)
These Schwartz spaces can be defined moreintrinsically by replacing
the radial derivatives in
(2.26) by
powers,CM
of theLaplace-Beltrami operator,
as in thesymmetric
space case(see
for instance[An2; § 4,
Remark(i)]).
For the sake of
completeness,
let usgive
theexplicit expression
of thePlancherel measure:
depending
whetherNotice that all
possible
cases are covered. See the list(1.4).
Eventually
let usrelate,
as in thesymmetric
space case, thespherical
functions on S with a natural series of
representations
of Sliving
on itsboundary, namely
theanalytic
continuation of therepresentations unitarily
induced from the characters a -+
(X
ER)
of A = These inducedrepresentations
are realized onL2(N) by
where x =
(n, a)
with n EN
and a E A. Via thestereographic projection, they
can also be realized on
L2(aB(s)) by
Both realizations admit obvious
analytic
continuation in À E C. We want toshow that
where
Pi
is the Poisson kernel on N(see ( 1.13)
or( 1.14))
and c = =n
2
20137T.. Contrary
to thesymmetric
space case, this cannot be obtained hereby
an
elementary
substitution. Consider instead the normalized kernelon S:
A direct
computation
as in[Dam3] yields £ Px
= -( Q2
4 +A)7.
HenceFrom now on, let us assume 0.
Inspired by
the determination of the Martinboundary
of S in[DR2; § 7],
we shallapproximate ’PX (x) by
ratiosx
>involving
theGreen function G,
ofLx
= L+ 2
4 +À 2. Gx
wasdetermined in
[DR2;
Theorem7.8]
for À E i[ o, -f-oo) ,
but this restriction is not essential. Noticethat,
as in thehyperbolic
space case([An5], [Fa], [MW]), G~,
= is amultiple
of the fundamental solution atinfinity [Koo; (2.15)]
of the Jacobi
type equation (rad £x)
(D = 0.Precisely,
Now,
let y =(X,
Z,a )
E S tend to(Xo, Zo, 0)
in N xR+. Then,
as it was shown in theproof
of[DR2;
Theorem7.11 ],
I
is bounded above(and below)
if x remains in acompact
subset of Sandd(y, e)
--~ +00.We are now
ready
for theproof
of(2.34). Take y = (o, 0, a )
E S with0.
By using (2.39)
and dominated convergence,(2.36)
becomesSince
Gx
isradial,
where
(by)~
a is the normalized surface measure on thegeodesic sphere aB(e, r)
of radius r =d (y, e)
in S.Thus,
Using again (2.39)
and dominated convergence, we obtainwhere na
=1).
This lastexpression
iseasily
seen to beequivalent
to
(2.34).
Now that we have identified with a matrix coefficient of it becomes clear that
(2.44)
is apositive
definite function on S when h E R.See
(6.14)
for further information.3. - LP - L p and weak
L 1 --+ L 1 boundedness
ofpositive
radial convolution kernelsThe
problem
consists infinding
minimal conditions on anonnegative
lo-cally integrable
radial function K =K (r) ensuring
LP - LPboundedness, respectively
weakL
1 -~L
1 boundedness of thecorresponding right
convolu-tion
operator T f
=f *
K, i.e.respectively
We shall first establish the
analogue
of the Herz criterion forsymmetric
spaces
([Her; Theorem 7], [LRy; Anhang 3]),
which takes care of(3.1).
Aswe shall see later
(Proposition 4.10.i),
we canalways
reduceby duality
toI p2.
(3.3)
THEOREM.(3.1 )
holdsif and only if
Moreover this
quantity
coincides with the LP operator normof T.
Notice that this
integral
condition reducessimply
to K EL 1 (S)a
for p =1,
as should be
expected,
and that it can be madequite explicit
in all cases,namely
by introducing polar
coordinates in S andusing
the basicbehaviort
t The symbol between two positive expressions means for us that
Cl x second expression first expression C2 x second expression
for some positive constants Cl and C2.
The
proof of
Theorem 3.3 is different from thesymmetric
space case andactually simpler.
Consider theright regular representation
of S on On one
hand, given f
E andf ’ E
with norms s1,
we have
hence
On the other
hand,
since S isamenable,
there existfn
E andfn
ELP’ (S)
with norms 1 such that
locally uniformly
in X E S.Hence,
In order to
conclude,
notice thatTheorem
(3.3) implies
thefollowing
Kunze-Stein typephenomenon
forright
convolution with radial functions. This is an
amazing
result at the firstglance,
since
(any
versionof)
the Kunze-Steinphenomenon
is known to be false forgeneral
functions on S(see
for instance[Col], [CF], [Li], [Lo; pp. 414-415],
...).
for every F E Lp (S)
andfELq(S)Q.
Let us turn our attention to the weak type estimate
(3.2)
forT f
=f
* K .By taking
forf
anapproximate
unit in L1 (S),
such an estimateimplies
that Kbelongs
to theweak L 1 space
Since the volumedensity
behaves atinfinity
like(a positive
constanttimes) e Qr ,
atypical
radial function which isL 1, ’
atinfinity
isThe
following
theorem asserts that thistype
ofdecay
atinfinity
isactually
sufficient to
imply
weakL
1 -L
1 boundedness. Both its statement and itsproof
areadapted
from theanalogous
resultfpr symmetric
spaces[Str].
(3.14)
THEOREM. Let K be alocally integrable radialfunction
on S withthe following
behavior at
infinity
-Then the associated
right-convolution
operatorT f
=f
* K on S isof
weak typePROOF. We can assume that
f
> 0 andK (r)
=e -Qr . By definition,
r
Write y =
r rz,er)
= arnx,z, where ar =(0,0,er)
and nx,z =Write y - =
where ar - (o, 0, e r )
and(X, Z, 1).
On onehand,
the left-invariant Riemann-Haar measure on S in thisdecomposition
isjust d y
= dr dX dY. On the otherhand, (2.18) implies
thefollowing
control of thegeodesic
distance r’ between thepoint
y and theorigin:
Thus
where
is the
right-convolution operator
on S associated with the Poisson kernelPI
on N. Since T’ is
trivially
bounded on we are left with the weakL 1 ~ L
1 boundedness of theoperator
which is
just
theright-convolution
on S with the function 1 on A n R.Writing
this time x = nas
(with n
E N and s ER),
we haveHence
-f -00
where e d r = t . This
brings
us to the desired conclusion:where
eQso
201300
dre-Qr !(nar)=t.
Thisbrings
us to the desired conclusion:(3.22)
COROLLARY. TheHardy-Littlewood
maximal functionis
of weak
typeL - L 1 on
S.Since M is
trivially
bounded on it is also bounded on all inter- mediate spacesLP(S) (1
p Forgeneral
Riemanniansymmetric
spaces of
noncompact type,
this LP result was first obtained in[CS].
The weakL 1 --~ L1
1boundedness,
as far as it isconcerned,
was established in[Str].
PROOF OF COROLLARY
(3.22).
Asusual,
we considerseparately
the localHardy-Littlewood
maximal functionand the
large
scaleHardy-Littlewood
maximal functionOn one
hand, Mo
can be handled as in the Euclidean case. On the otherhand, .Moo
is dominatedby
theright-convolution operator T f
=f
* K associated to the radial kernelwhich is of weak
type L 1 -~ L
Iby
Theorem(3.14).
4. - LP radial
multipliers
In the
previous section,
we have consideredright
convolution operatorson
LP(S)
associated to radial functions which areessentially positive.
Thissection is devoted to an
actually
widerclass, consisting
of all functions of theLaplace-Beltrami
operator,C,
which are at least bounded onL 2 ( S) .
Let us first make clear
that,
as in thesymmetric
space case, thespherical
Plancherel formula
(2.10) provides
anexplicit
realization of thespectral
decom-position
of C onL2(S)
and hence of all(spectrally defined)
functions of S.For every h E
L°°(R)even
consider theright-convolution
operatorlet say from
C°° (,S)
toC°° (S) .
Theright-hand
side of(4.1 )
should be understood in thefollowing
way:Notice that
Th
isentirely
determinedby
its action on radialfunctions,
whichcorresponds simply
topointwise multiplication
on the Fourier transform side:(4.3)
LEMMATh
is a bounded operator onL2 (S),
with operator normequal
tollhlll- -
PROOF. To be
safe,
we may assume that hbelongs
to thePaley-Wiener
space
PW(C)even .
Thegeneral
case followsby
a routinecompletion argument.
We have
Since is
positive
definite when h E R(2.44),
Hence
By (2.10)
theright-hand
side of(4.6)
isequal
toThus
Th
is bounded onL2 (S),
with operator norm On the otherhand,
it is clear from(2.10)
thatTh
is bounded onL2(S)a
with operator norm= IIhllLoo .
It now follows
easily
from(2.6), (2.10)
and Lemma(4.3)
that thespectral decomposition
of ,C onL2 (S)
isgiven by
for
O:s Àl À2 :S +00.
Moreover all(spectrally defined)
functions of £ canbe written
for some h E
The classical LP
multiplier problem
consists infinding properties
of hreflecting
the boundedness ofTh
onLP(S).
As forsymmetric
spaces[CS],
wehave the
following
necessary conditions.(4.10)
PROPOSITION.(i)
Assume thatTh
is bounded onLP (S), for
some 1 p ThenTh
is bounded on the dual spaceLP’ (S)
and hence onL2(S).
(ii)
Assume thatTh
is bounded onLP(S), for
some 1 p 2. Then h = extends to an even boundedholomorphic function
in thestrip I (-L - 1)
p 2Q
(continuous
up to theboundary
in the case p =1).
The
proof
is similar to thesymmetric
space case[CS].
Let us recall it.Since h =
h, Th
coincide with the dualoperator (Th)t,
whichimplies
immedi-ately (i).
For(ii)
we use the fact that, q 1 1
(4.11) belongs
to %when [ (- - -) Q p 2 (respectively
in the casep = 1).
.(
P YI
2 p
)
This follows from the
integral
formula(2.6),
whichyields by convexity
(4.12) when I Imk
and from the
asymptotic
behavior(2.7).
Our next step is the crucial formulawhich is
easily
established for À E R:B
holds indeed for every
f
E Theanalytic
continuation of h followsnow from
(4.13),
moreprecisely
fromwith suitable choice of test functions
f
E The uniform boundedness of h is obtainedby taking
theLp’-norm
in(4.13):
and
dividing by 11 LP’ :A
0. This concludes theproof
ofProposition (4.10).
Reciprocally
one canshow,
as in[An3], [ST], [Ta],
... , thatTh
is boundedon
L P (S) provided
h = extendsholomorphically
to thestrip I Im X I
1 - 4) Q
and satisfies theresufficiently
many uniformsymbol
type conditions.p
Here is a
sharp
Hörmander typemultiplier theorem,
which wasrecently
obtainedin an
actually
wider context[AS].
(4.17)
THEOREM. Let 1 p 2. Assume that h =h(.X)
is an even boundedholomorphic function
in thestrip I
1m ÀI (i - 4) Q
whoseboundary
valueh (,)
= limh(À
+i tt) satisfies 1 1)
Qp I
(i)
rioh belongs
to the Besov spaceBq,1 (II~) ,
(ii)
r~h (t . ) belongs
toBq,1 I (R), uniformly
in t »0,
where i7o =
respectively
17 = is an even(smooth) bump function
= 0, respectively
À = :f: 1 and a = n(1 - _21)
>!!
> 0. ThenTh
is ap 2 q
bounded operator on
L P (S).
MoreoverBq,1
can bereplaced by B2,1
whenp n+3°
As in the
symmetric
space case([LRy], [Ta]),
onegets
as a consequence thefollowing
LPspectral
result.(4.18)
COROLLARY. For 1 p +00, the LP spectrumof
L consistsof
theparabolic region
5. - The heat kernel
Consider the heat
equation
associated to the
Laplace-Beltrami operator
L on S. Its solution isgiven by
the heat
semi-group
with
corresponding
heat kernelht (x, y).
Like all kernels of functions ofL, ht
is a radial
right-convolution
kernel on S:Its
analysis
fits into thesetting (2.27):
By
the inversion formula(2.10),
A better
expression
is obtainedby applying
first the inverse Euclidean Fourier transformT-1
and then the inverse Abel transformhence
by (2.24.1 )
when k is even andby (2.24.2)
when k is odd. Such formulas werepreviously
obtained in[LRy]
for all
hyperbolic
spaces.Most of this section will be devoted to
establishing
thefollowing sharp
heat kernel estimate.
(5.9)
THEOREM. We havefor t
Let us make some
bibliographical
comments about Theorem(5.9).
Forgeneral hyperbolic
spacesHN (1F) (and
somehigher
ranksymmetric spaces),
the upper bound was
given
in[An 1 ] . Simultaneously
andindependently,
theupper and lower estimates were established in
[DM] (see
also[Dav; § 5.7])
for real
hyperbolic
spacesHN (1R).
Theproof
can beactually adapted,
withsome care, to the
general hyperbolic
space case([GM; § 3], [Mu]).
Here we shall extend it further to harmonic A N groups.Recently
such upper and lower estimates were establishedby probabilistic
methods for a much wider class of radialLaplacians [LRo].
We think however that ourproof
is not without interestand in any case
simpler
in oursetting.
We shall first estimate
expressions
of the formwhich constitute
essentially
theright-hand
side of the even case(5.7)
and appearalso inside the
integral
in the odd case(5.8).
In order toanalyze (5.10),
we shallborrow and
improve
somearguments
from[CGGM; pp. 647-648].
Consider themodified
Jacobi operatorStraightforward computations yield
thefollowing shift
or transmutation formulas.(5.12)
LEMMA. For any a,fJ
E (C,Set
where p and q
arenonnegative integers with p
-+- q > 0.(5.14)
COROLLARY.fp q (r ) () .f or some constant c
> 0.PROOF.
r
and 2 ’ q -2 ()
are even smooth functions on 1Rsatisfying
the same differentialequation:
For
fp,q
this followsby
induction from Lemma(5.12)
and from the trivialidentity d3 r2 =
0.Equation (5.15)
has aregular singularity
at theorigin
with..
(p+q-i,q-1)
....indices 0 and
120132201320.
is known to be(stricly) positive.
For instance
and the
hypergeometric function
ispositive
ifa 2:
0, b 2: 0,
c > 0 and0 :S
z 1. Hence -c~~ ~’~ ~ (~) .
It remains to show that c > 0. We shall assume p > 0 and p -~ q > 1
(the degenerate
case p =0, q
> 0 is handledlike p
>0, q
= 0 and the case= is
trivial).
Thenfp,q (r)
isequal
to’ sinh 1
plus
a linear combination ofproducts
with
By expanding
one sees that
as r -~ and
consequently
for some constant c’ > 0. In
particular fp,q (r)
> 0 for r » 0 and therefore the constant c ispositive.
(5.21) Corollary.
(5.22)
PROPOSITION. Let p, q benonnegative integers
with p + q > 0. Thenfort
> 0 and r > 0.PROOF. We have
where a 1
= f p, q ,
ap+q = 4-p-q and moregenerally aj
is a linearcombination,
withnonnegative coefficients,
ofproducts
... withp + ...
+ p~ -
p , q + ...° + qj ~ q
°Therefore,
Hence
which finishes the
proof
ofProposition (5.22).
This settles the even case
(5.7).
The estimate(5.9)
in the odd case(5.8)
is obtained
by combining (5.22)
with thefollowing Proposition,
which handles theintegral part.
(5.26)
PROPOSITION. Assume p > 0and q
>1.
ThenPROOF. After
performing
thechange
of variables s = r + u, the left-hand side of(5.26)
becomesUsing
theelementary
estimatesthe
integral
in(5.27)
iseasily
seen to beequivalent
toCall
Il
= the lastintegral.
It remains to show thatTo estimate
11
fromabove,
we use theelementary inequalities
and get
Call
/2
= the lastintegral.
For t > 1 + r, we haveand,
1 + r,(5.36), (5.37)
and(5.38) give
the announced upper estimate(5.32)
of11.
Forthe lower
estimate,
we useand get
Call
/3
=13 (t, r)
the lastintegral. rnin(1 , ~ },
we havewhere T =
2
+2 Performing
thechange
of variables u’ =--,It 2-
andusing
we obtain
For
min{l, lt- 1,
we havePerforming
thechange
of variables andusing
this timewe obtain
(5.42), (5.45)
and(5.48) give
the announced lower estimate(5.32)
ofIl.
And this concludes theproof
of Theorem(5.9).
Since grad ht (r) _ - ar ht (r)
= sinh r(- Si h r a,- ~ ht (r) ,
one obtains sim-ilarly
thefollowing gradient
estimate.(5.49)
COROLLARY. For all t > 0 and r >0,
The heat kernel upper estimate in Theorem
(5.9) implies
thefollowing
refined result.
(5.50)
THEOREM. The heat maximal operatoris
of weak
typeL
1 --+L 1.
Since T is
trivially
bounded onL’(S),
it is also bounded on all intermedi-ate spaces
LP(S) ( 1
p+oo).
The weakL~ 1 -~ L~
1 boundedness of the heat maximaloperator
was established in[LZ]
forhyperbolic
spaces and somehigher
rank Riemannian
symmetric
spaces of noncompact type, in[An4; Corollary 3.2]
for
general
Riemanniansymmetric
spaces of noncompacttype,
in[CGGM]
for groups with
polynomial growth
and for adistinguished Laplacian
on someIwasawa AN groups, in
[AJ]
forgeneral
Iwasawa AN groups, ....PROOF OF THEOREM
(5.50).
Asusual,
we estimateseparately
the small timemaximal
operator
and the
large
time maximal operatorConsider first
To.
Thepurely
local maximal operatorwhere x
denotes the characteristic function of the unit ball inS,
can be handledas in the Euclidean case. For
instance,
it is dominatedby
the localHardy-
Littlewood maximal function
Mo (3.23).
Theremaining part
ofTo
is controledby
anintegrable
convolutionkernel,
thanks to the Gaussian factor e - 4tr2
in(5.9):
Too
is controledsimilarly by
a radial convolutionkernel,
which satisfies thedecay
condition of Theorem(3.14):
The last estimate follows from
(5.9) by analyzing
the function Iwhich has a maximum at
t (r ) - Q
+(as r
-~+00).
HenceToo
is ofweak
type L
1 --~L 1.
And this finishes theproof
of Theorem(5.50).
°
Eventually
let us check the actual values of the various constants in the inversion formulasand
by using
the classical heat kernel behavior:For
f
=ht
and x = e, these formulas readrespectively
For small time
asymptotics,
thefactor e 4 2
t can bedisregarded.
Consider first
(5.57). Using
either theexplicit expression (2.7)
of thec-function and
Stirling’s
formula or theexplicit computation (2.31)
of thePlancherel measure, we see that
After
performing
thechange
of variables h’ andletting t
-0,
we obtainSince the last
integral
isequal to ~r(~), (5.56) implies
which was the value
given
in(2.10).
Look next at
(5.58). By (5.23)
theexpression
under consideration has anfinite
expansion
innegative
powersof t,
withleading
term(5.56) implies
which was the
value
found in(2.24.1 ).
For the last case
(5.59),
wekeep
inside theintegral
theleading
termin t,
which was
just
identified:After
performing
thechange
of variables s’ =2~
andletting t
2013~0,
weobtain
(5.56) implies
which was the value found in
(2.24.2).
6. - Further results and open
problems
More
generally,
allanalytical
results established forsymmetric
spaces in[An4]
and[CGM]
remain valid in oursetting.
Let us be morespecific.
(6.1 )
The Riesztransform
is definedby
for smooth functions
f
EC c 0()(S)
and ingeneral
as asingular integral operator
i
where K is the
(distributional)
kernel of(-£)-2
andLy (z)
= yz denotes left translation in S. Its Lip - Lpmapping properties
are stated below.Everything
relies upon the