Proc. Indian Acad. Sci. (Math. Sci.),Vol. 101,No. 3, December 1991, pp. 169-177.
9 Printed in India.
A multiplier theorem for the sublaplacian on the Heisenberg group
S THANGAVELU
TIFR Centre, Indian Institute of ScienceCampus, Bangalore 560012, India
MS received24 January 1990
Abstract. A multipliertheoremfor the sublaplacianon the Heisenberggroup is proved using
Littlewood-Paley-Steintheory of g-functions.
Keywords. Multiplier theorem; sublaplacian; Heiscnberg group; Littlewood-Paley-Stein
theory; 0-functions.
1. Introduction
Consider the Heisenberg group H, and the sublaplacian fie on H,. fie is a formally
non-negative hypoelliptic differential operator which has a unique self-adjoint
extension to L2(H,). If ~p is a function defined on R then using spectral theorem one
can define the operator ~o(fie). If ~p is a bounded function, then ~p(.fie)will be bounded
on L2(H,). In the same spirit one likes to fred sufficient conditions on r so that the
operator r
will be bounded on LP(H,,).
This probIem was studied by Mauceri [4] and the following result was proved.
If the function ~p is n + 3 times differentiable and satisfies the estimate
]~p~k)(t) j ~< C(1 +ttt) -k, k = 0, 1.... (n + 3), then ~p(fie)is bounded operator on L P ( H , )
for all l < p < ~ .
This result was proved using the theory of singular integrals on homogeneous
spaces developed by Coifman and Weiss [1]. Later Mauceri improved the above
result replacing the smoothness condition on ~p by a fractional order condition of
the order s > n + 2 (see [5]). Here we propose to give a different proof of the multiplier
theorem. We prove:
Theorem. L e t ~p be v times differentiable and satisfies Icptk)(01~<C(1 + Itl) -k f o r
k = O, l . . . . v where v = n + 2 i f n is even and v = n + 3 i f n is odd. T h e n cp(.~) is a
bounded operator on LP(H,), I < p < co.
Our proof of this theorem is based on Littlewood-Paley-Stein theory of Ok and
0~' functions. We adapt this method which was originally employed by Stein [6] to
prove the Hormander-Mihlin multiplier theorem for the Fourier transform, to the
present case. The same technique was successfully employed by Strichartz [7] and
by the author [9], [10] to prove some multiplier theorems. One good thing about
this approach is that the proof is simple and also we get a sharper result when n is even.
169
t70
S Thanoavelu
2. Preliminaries
The main reference for this section is [3]. See also [4]. The (2n + 1)-dimensional
Heisenberg group H. is the nil potent Lie group whose underlying manifold is C" x R.
The group structure is given by
(z, t)(~, s) = (z + ~, t + s + 2 Im z'~)
(1)
where t, s~R and z, ~eC". The Haar measure on H. is simply the Lebesgue measure
dzds on C" x R. For w = (z,s) the homogeneous norm Iwl is defined by Iwl 4 = Izl 4 + s 2.
We next recall the definition of the Fourier transform on H.. The infinite
dimensional representations of H. are parametrized by R\{0}. If 2 4:O, then all
the representations ~ can be realized on the same Hilbert space L2(R"). For
(z, s)eH., its(z, s) is the operator acting on Lz(R ") by the prescription
~x(z, s)tp(~) = exp(i2s)exp [i22(2~ - x)'y] tp(~ - x ) ,
(2)
where z = x + iy and ~eR".
The Fourier transform f of an L t function f on H. is then the operator valued
function
f(2)= ~
f(w)~z~(w)dw.
(3)
J H,~
Then we have the following Plancberel formula:
II/ll~ = ~2"-1 f I~,l"IIf(2) ll~sd;t,
(4)
where ll'lIHs is the Hilbert-Schmidt norm. We also have an inversion formula
f (w) =
f
da,
(5)
where tr is the canonical semifinite trace.
For each 2 ~ 0 we can select an orthonormal basis for L2(R'). Let
9 ~(x) = (21;q0 "12 ~,((21~ltx)) where ~ are the Hermite functions on R".Then { ~ } is
an orthonormal basis for L 2 {Rn). Let PN(2) denote the projection of L2(R ") onto the
eigenspace spanned by {~:10t[ = N}. Using these operators PN(2) we can write the
Fourier transform of a zonal function in a siml~le way.
Let f(z, s ) = f(Izl, s) be a zonal function and f(z, 2) be the Fourier transform in the
s-variable.
2) = f exp (i2s)f(z, s) ds.
(6)
Define RN(2,f) by the formula
R~(2,f) = C~ (N + N!
n - 1)! I= f(r, 2)L~-Z(2l,~lr2)exp(_
Jo
iAlr2)r2. - x dr, (7)
Multiplier theorem
171
where L~v-1 are the Laguerre polynomials of type (n - 1). Then one has
f(2)= ~, RN(2,f)PN(2 ).
(8)
N=0
And the Plancherel formula takes the form
2"-~ f ~
(N+n-1)!
Ilfll2 = ~
jN~__o [RN(~.,f)I2
-~[
121"d2.
(9)
On H. consider the following left invariant vector fields.
Z~ = ~zj + igj -~, • = d~
izj ~ .
(10)
The sublaplacian L~' is then defined by
1 ~ (Zj2j+LZi) '
(11)
Each repres~nfation n~ determines a Lie algebra representation dlt~. It can be shown
that d~(.~) is a closable operator. Its closure is denoted by H(2) and it has the
following spectral decomposition:
H(2) = ~ (2N + n)IAIPN(2).
(12)
N=O
For any reasonable function ~ on R, using spectral theorem, one can define the
operator q~(Le). It can be shown that q~(~) is a convolution operator with kernel k
i.e. ~o(~)f= k,f. The Fourier transform of k is given by
/~(2) = ~ ~o((2N + n)I2I)PN(2).
(14)
N=O
All these things will be made use of in the following sections.
3. Littlewood-Paley-Stein theory on H,
In [2] Folland has shown that the sublaplacian .Z generates a contraction semigroup
T t which satisfies all the conditions required to develop a Littlewood-Paley-Stein
theory (see [6]). As in Stein [6] we define, for each positive integer k, the following
functions
(gk(f, W))2 ----fO t2k- 11~ Tt f (w)[2 dt
(g~(f' w))2 =
ftl.fo t-'(1+ t-2lvl*)-'l~ Ttf(v-lw)12dtdv"
For these functions we will prove the following theorem.
(15)
(16)
172
S Than#avelu
Theorem 3.1. (i) For k >/1, IIgk(f)[12 = 2-k Ilfl12.
(ii) For 1 < p < ~ , Cx IIf IIp ~< tl gk(f) lip ~< C2 IIf lip.
(iii) I f k > (n + t)/2 and p > 2, then IIg~ (f)]1 p <~C IIf ItpProof. The inequality II#~(f)lip ~ C2 Itflip follows from the general theory. The reverse
inequality can be easily deduced once we have (i). When k > (n + 1)/2, the function
(1 + Iv14)-k is integrable and hence one can prove (iii) using (i). This is routine and
well known. So, it remains to prove (i).
We prove (i) when k = 1. The case k( > 1 is similar. From the definition it follows that
Ilgl(f) ll~ =
fo~
tlO, Ttf(w)12dwdt.
(17)
n
In view of the Plancherel formula (4) the integral becomes
(18)
Since Tt.f = e x p ( - t.CP)f, we see that
(0, T'f)A(2)= -- H ( 2 ) e x p ( - tH(2))f().)
(19)
and hence its squared Hilbert-Schmidt norm is given by the expression
~((21~1 + n)l).l)2 exp( - 2t(21~1 + n)121)(~a,f(A)*f(~.)~).
(20)
If we use this in (18) and integrate with respect to tdt, we will get
n--I
)l.,(s> ii,
And this proves that II01(f)112 = 2-11LfOL2-
4. The multiplier theorem
Let us set M f = ~(LP)f. To prove the multiplier theorem what we need is the following
pointwise inequality.
Ok+1(My) <~Cg~ (f)
(21)
for some integer k > (n + 1)/2. For then the multiplier theorem for p > 2 will follow
immediately from Theorem 3.1. For p < 2 one can use duality to conclude that M is
bounded on LP(H,).
So, we proceed to prove the inequality (21). Let us set ut = T'f, Uz = Tt(Mf). Then
it is easy to see that
u, §
= Ca,, u,)(w)
(22)
where the Fourier transform of Gt is given by
G,(2) = ~ e x p ( - (2N + n)121t)q~((2N + n)l~.l)Ps(;~).
N=0
(23)
173
Multiplier theorem
Differentiating (22) k times with respect to t and once with respect to s and putting
t = s we obtain
~k+ l T2t(Mf) = Ft*dt Tt.f,
(24)
where the Fourier transform of Ft is given by
fit(2) = ( - 1)k ~. e x p ( - (2N + n)lAlt)(2N + n)klRlk~P((2N + n)lAI)PN(2).
N=O
(25)
Therefore, we have
I~,~
+ 1
T2t (Mf)(w)l ~<
flF,(v)ll~, T ' f ( v - t w) ldo.
Applying Cauchy-Schwartz inequality
Iokt + 1 TZ,(Mf)(w)lz <<At'B,(w),
(26)
where we have written
A, =
Bt(w)
flFt(v)l=(1+t-21vl4)kdv
~(1 + t-2lvl4)-klO, T'(v-lw)12dv.
(27)
Now to complete the proof we need the estimate of the following Lemma.
Lemma. Under the hypothesis of the theorem the estimate A t <~C t -n-2k- 1 is valid
when k is the smallest inteoer 9reater than (n + 1)/2.
Assuming the lemma for a moment it is easy to establish inequality (21). Indeed,
from (26) we have
]O~+1 T2,(Mf)(w)[2 <~C t - " - 2k- 1Bt(w).
Integrating this against t z~+ t we get
Ok + 1(Mf, w) <<.Cg*(f, w).
This completes the proof of the multiplier theorem modulo the above lemma.
5. Proof of the Lemma
To prove the Lemma let us write
[Ff(w)[Z(1 + t-21Wl4)kdw
(28)
J = 1"
IFt(w)I2(1 + t-2lwl4)kdw.
Ji ~l>v4
(29)
I= ~
Ji
174
S Thangavelu
Estimating the integral I is easy. We note that since [wl ~<x / t
I <<.C flF,(w) 12dw
and hence in view of Plancherel formula
l<<.C 121"
d
\N=o
(2N+n)2kl212kexp[--2121(2N+n)t](N+n--1)!
N!
)
d2
<~Ct-,-2k- 1(~(2 N + n)-2) ~< Ct-,-2k-1.
This proves the estimate for the integral I.
Next consider J. Let us write w = (z, s). We observe that
s<.ct-2'ff(s2+
=Ct-2kffl(is-,zl2)kFt(z,s)12dzds.
,zl4)klF,(z,s)12dzds
(30)
If we can show that the integral in (30) is bounded by t - ' - 1 then we are done. If we
write the Fourier transform of G = (is- r2)kFt(z, s) in the form
G(2) = ~ RN(2,(is--IZI2)kF:)PN(~.)
N=0
then we need to show that
f N=o
~ tRs(2,(is_r2)kF,)12(N +N!
n-- 1)!lXi,d;t ~< Ct_,_ 1
(31)
where we have set Izl 2 = r 2.
Let us write
~(N, 2) = ( - 1)k(2N + n)~12lkexp[ - (2N + n)lXit]tp((2N + n)12l)
so that Rs(2,Ft)" ~k(N,2). We define ek(N,2) to be Rs(2,(is-r2)kFt). Then the
following estimate is valid.
Lemma 5.1. Under the hypothesis of the theorem there is an e > 0 such that
I~,k(N,2)t ~<C e x p [ - e(2N + n)]2lt.
(32)
If we use (32) in (29) then the estimate J ~<t - ' - 2 ~ - t is immediate. So we proceed
to prove Lemma 5.1.
Recall the definition of Rs(2,f) for a zonal function f.
Rs(2,f) = C,(N +N!n- 1)! f : 7(r, 2) L~- 1(2[2[r2)exp (--t21r2)r 2"- 1dr,
(33)
where .f(r, 2) is the Euclidean Fourier transform of f in the s variable. We will prove
Multiplier theorem
175
(32) when 2 > 0. The case 2 < 0 is completely similar.
Since (isf)'(r,).) = (d/d2)f(r, 2) we obtain
RN(A, isf)= ff_~RN(2,f} - C"(N + N'n - 1)! f :
](r, 2)
x d { L~- 1(2itr 2) exp ( - 2r 2) } r 2~- i dr.
Now
~d( L .N- 1( 2 2 r2) e x p ( - A r 2)
2d . 1
2
2
= 2r drr LN- ( 2 2 r ) e x p ( - - 2r ) -- r 2 L~v- 1 (22rZ)exp(_ itrZ).
Using the recursion formula (see [8])
d
n
1
r-~r Ls- (r) = NL~-~(r) - (N + n -- 1) L~-_~(r)
(34)
a simple calculation shows that
N
RN(2, isf) = ~-~RN(2, f) -- --~(RN(A,f) -- R~_ 1(it, f)) + RN(2, r2f) 9
Thus we have obtained the formula
O~(N, 4) = aO
Oit
N
2 (0(N, it) - 0(N - 1, it)).
(35)
Since 0 ( N , 2 ) = 0((2N + n)2) we can write (35) in the form
O,(N, it)
lnO0
N/O@
= 220--N + 2 - ~
\
- A0),
(36)
where A0(N, ~.) = 0(N, it) - 0(N - 1,2). Define the operators S, D and T by
~O
so = ~,
oO
nO = ~ -
aO,
rO = NDO.
So, we have
~b,(N,2)=),-'(2S+
T ) O ( N , ;t).
(37)
From this formula we can conclude that
0k(N, it) = it-~
~
i+j+mfk
auraS' T~S'O(N, it).
(38)
Now we observe that Sm0(N, it)= 0(m)((2N + n)it)(2it) m and by hypothesis of the
theorem S ~ in essence brings a factor (2N + n) -m. We will show that T j also does
176
S Thangavelu
the same thing. Then each term in the sum (38) will behave like 2-k(2N + n)-kd/(N, 2).
Recalling the definition of ~b(N, 2) we see that
I~kk(N,~-)t ~< Cexp [ - e(2N + n)2t]
as desired.
For the operators T j the following formula is valid.
Lemma 5.2.
TJ~I = E
Cp,mNPD'(Am~)
where the sum is extended over all p, q, m satisfying the relation j + p <<.2q + m <<.2j.
Proof. We prove, this lemma by induction. We first observe that from the definition
of T, the lemma is trivially valid for j = 1. N o w assume the lemma true for some j
and consider T j§ 1~b
(39)
TJ + I ~I = Z CpqmN D( NP D~(Amd/) )
where j + p ~< 2q + m ~< 2j. We need a formula for D(NPDd/).
We claim that
p-1
p-2
i=0
i=O
D(NPD~) = NPD2~ -I- ~.~ aiNtD(A~b)+ ~, biNiD~k.
(40)
Assuming the claim for a m o m e n t we have
p-1
TJ+t~b= E Cpq,.NP+'Dq+'(A"d/) + E Cpqm ~ aiNi+ l Dq(AM+l~b)
p,q,m
p,q,m
i=O
p--2
+ E Cp,,~ E b,N'+'D'(Am~k) 9
p,q,m
i=0
F r o m this formula it is clear that TJ+10/is of the desired form.
To prove the claim we first observe that
(41)
A(q~k)(N) = Aq~(N)~k(N) + q~(N -- 1)A~k(N).
In view of this formula
A(NPD~b) = A(NP)D~k + (N - 1)PD(A~k).
(42)
We also have
p-2
A(N p) = N p -- (N - 1)p = pN p-1 -
~, b,N i
(43)
i=0
p-1
(N - 1)PD(A~b) = NPO(A~b) -
~ a,N'D(A~)
(44)
i=0
t~
p
1
(45)
Multiplier theorem
177
F r o m (42)-(45) it follows that
p-1
D(NPDO)=NPD2~+
p-2
~ a, N i D ( A O ) + ~ b~NiDO.
i=0
(46)
i=0
This proves the claim.
Finally we will show that the action of T j has the desired properties. We have
T j ~ = ~ CpqmN " D q(A '~~),
(47)
where p + j <~2q + m <<.2j. N o w using Taylor's formula with integral form of remainder
we can write
D~k(N) = ~ t~b"(N - 1 + t,A)dt,
(48)
where the primes stand for the derivatives with respect to N. F r o m (48) it is clear
that the action of D is to bring d o w n the factor N - z. An iteration will show that D ~
will bring d o w n the factor of N -2q when applied to ~b. Since Am~ brings d o w n N -m
the formula (47) shows that T ~ acting on ~b brings d o w n the factor
C m,,, N t, N - 2~ -,,,.
Since p + j ~< 2q + m, essentially T j brings d o w n a factor of N -J as required.
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