2y~aj(xj~ -yjxj)]. xj + iyj, [z,t]. [w,s] = [z+w,t+s+ < z...

E S T I M A T E S F O R S P E C T R A L P R O J E C T I O N O P E R A T O R S

O F T H E S U B - L A P L A C I A N O N T H E H E I S E N B E R G G R O U P

By

DER-CHEN CHANG AND JINGZHI TIE

Abstract. In this paper, we use Laguerre calculus to find the LP spectrum (),, #) of the pair (Z:, tT). r/ere c = - �89 ~]=1 (Zj~j + ZjZj) and T = O/Ot with

{Z 1 . . . . , Zn, Z I , , . . . , Zn, T} a basis for the left-invariant vector fields on the Heisenberg group. We find kernels for the spectral projection operators on the ray )~ > 0 in the Heisenberg brush and show that they are Calder6n-Zygmund-Mikhlin operators. Estimates for these operators in LP(Hn), I-IP(Hn), and ~k'~(Hn) spaces can therefore be deduced.

1. I n t r o d u c t i o n

The Heisenberg group is a non-commuta t ive nilpotent Lie group, with underlying

manifold R 2n+1 and the group law

(1.1) n

Ix, Y, t]. [x', y ' , s ] = [x + y , x ' + y ' , t + s - 2 y ~ a j ( x j ~ -yjxj)]. j = l

Here aj, j = 1,2 . . . . ,n, are all positive numbers . The group law (1.1) should be

looked upon as the non-Abel ian analogue o f Euclidean translation on R 2n+1. I f

we identify R 2n+l with C n • R by zj = xj + iyj, the Heisenberg group law can be

written in complex coordinates as

(1,2) [z,t] . [w,s] = [ z + w , t + s + < z , w >],

where z = ( z l , z 2 , . . . ,zn) ~ C", w = ( w l , w 2 , . . . ,Wn) E C n, and the symplect ic

fo rm < .,. > is defined b y

(1.3) t /

< z , w > = 2 I m E a j z j C v j. j = l

It is wel l -known (see Beals, Gaveau, Greiner, Vauthier [4]) that

0 0 0 0 0 (1.4) Xj = u.~j'a-5"- + 2ajy j~ , Y j = Oyj 2ajxj-~ and T =

0~ '

315 JOURNAL D'ANALYSE MATHEMATIQUE, ~1. 71 (1997)

316 D.-C. CHANG AND J. TIE

f o r j = 1, . . . ,n, form a real basis for the Lie algebra On of left-invariant vector fields on t t , . Note that we have the commutation relations

(1.5) [Yj, Xj] = YjXj - XjYj = 4ajT,

and all other commutators vanish. Next, we turn to the complex vector fields

1 0 . 0 1 0 . 0 Zj = ~(Xj + iYj) = 0~j mjzj-~ and Zj = 5(Xj - iYj) = ~ + taj~j-~=~,~

fo r j = 1 ,2 , . . . , n. Here

a 1 ( 0 _ i 0 ) 0 l ( a i 0 ) and o j-2 "

The commutation relations (1.5) then become

[Zj, Zk] = 2iaj6#T

with all other commutators among the Zj, Zk and T vanishing. The Heisenberg sub-Laplacian is the differential operator

(1.6) I•L• 1 n 2 (X) Y}) + ic~T. Ca = - (ZjffSj + ZjZj) + iaT = --~ E +

j = l j = l

We set s = E0 .

In the case of aj = 1 for all j 's , the operator s was first introduced by Folland and Stein [10] in the study Of 0b complex on a non-degenerate CR manifold. They found the fundamental solution of E~. Beals and Greiner [5] solved the case in which, ay's may be different.

There are many works devoted to the theory of harmonic analysis on the isotopic Heisenberg group (i.e. all aj's are equal). In [19], R. S. Strichartz proposed a different defirfition of harmonic analysis on the isotropic Hn, namely, that it is the joint spectral theory of the two operators - s and iT, where

n

j----1

where Xj, Yj and T are defined as in (1.4) hut with aj = 1/4 for a l l j = 1 ,2 , . . . , n. Since - s and zT are essentially self-adjoint strongly commuting operators, there

is a well defined joint spectrum. We first summarize some of Strichartz's results.

SPECTRAL PROJECTION OPERATORS 317

His results are for the the case aj = 1/4 for all j = 1 ,2 , . . . , n. In this case the spectrum is the closed subset of the plane, which he called the Heisenberg fan, consisting of the union of rays

(1.7) Rk,~= (A,#): # = n T 2 k ' A > O f o r e = + l , k = O , 1,2,...

and the limit ray

(1.8) Roo={(A,U): U = 0 , A_>0}.

Here A refers to the spectrum o f - s and # to the spectrum of iT.

Among the functions o f - s and t"r, he singled out for special study the operator i s -1. This operator has a discrete spectrum consisting of the values e(n + 2k)

for e = + 1, k = 0, 1,2, .... The spectral projection operator associated to the point e(n + 2k) is essentially a convolutionf ~ f �9 Pk,, where

, ..k2n-l(n+k)! ([zl2-4iet) k { k Izl2+4iet~ (1.9) Pk, ,(z , t )=(-- l) 1 + n+klzl2-_4--~etj-- .

It is easy to see that Pk,, is homogeneous of degree -2n - 2 and satisfies the mean value zero property. By the theory of singular integral operators, we know that the spectral projection operator is bounded from LP(Hn) into itself for 1 < p < c~.

In this paper, we use a different approach to study some harmonic analysis

problems on the group Hn. More precisely, we will look at those problems from the point of view of the Laguerre calculus. This method was first studied by Greiner [12] on I l l and generalized to I-I,, by Beals, Gaveau, Greiner and Vauthier [4]. They computed the kernels for the generalized Cauchy-Szeg6 projection in their paper. In fact, this method provides us with a powerful tool to study left-

invariant differential operator whose Laguerre tensors are of form. For example, we can use the Laguerre calculus to obtain the fundamental solutiorl for powers o f the Heisenberg sub-Laplacian s and Schrfdinger propagator e -isz. We give a

detailed discussion of these two operators in a forthcoming paper [7]. The main contribution of this paper is to give a complete solution for the fol-

lowing question relating to the joint spectrum of the pair ( - s tT): what is its 1_P spectrum? The Lp spectrum of - s is defined as the set of complex numbers A for

which AI + s is not invertible on Lp. The Lp spectrum of the pair ( - s z'r) may be defined as the complement of the set o f (A, #) ~ C 2 for which there exist/f l

bounded operators A and B with A(AI + s +B(# I - iT) = I. As mentioned before, we can use the Laguerre calculus to answer this question. Instead o f the operators

--s and tq', we consider the slightly different operators s and iT to obtain more

compact formulas. Note that s = - �88163 Since the Laguerre tensors o f AI - s and

318 D.-C. C H A N G A N D J. T I E

#I - iT are simply diagonal, their joint spectrum can be eas~y determined, and it is the Heisenberg brush introduced in Agranovsky, Berenstein and Chang [1]. To find the corresponding eigenfunction, we only need to take the inverse Fourier transformation of a distribution related to the Laguerre functions. Once we get the explicit kernels for the projection operators, it is straightforward to investigate them as Calder6n--Zygmund-Mikhlin operators. Therefore, this spectrum is equal to the L 2 spectrum as long as I < p < oo. This answers the question proposed by Strichartz in [ 19]. Moreover, we can easily deduce the regularity properties for the projection operators in other function spaces, e.g. Hardy as well as Morrey spaces. As a byproduct of our calculation, we also find the projection analogous to (1.9) directly from the Laguerre series expansion.

The paper is organized as follows. In Section 2, we recall the definitions and some basic properties of the twisted and principal value (PV) convolution operators on the Heisenberg group. In Section 3, we list the basic facts about the Laguerre calculus which we shall need later. In Section 4, we give a complete solution for Strichartz's question, i.e. we find the spectrum of the pair (s t'F) for spaces LP(Hn), LP'~(Hn), 1 < p < o0, and 0 < v < 1 as well as HI (H ~) by finding the

eigenfunctions corresponding to (A, #) E a(s t'I') and discuss pointwise spectral projection as well as projection operator Pk,, on the ray A > 0 in the Heisenberg brush. In Section 5, we discuss estimates for the operator Pk,, in Hardy and Morrey spaces. In the last section of this paper, we obtain the Abel summability of the decomposition

(1.10) f = lim ~ ~ rkPk, * f r--+ 1 -

k=O ~=4-1

f o r f E LP(Hn), 1 < p < oo a n d f E H 1 (Hn). We obtain Pk,, from the generalized Cauchy--Szeg6 kernels, and Pk,, differs from (1.9) only by a factor 4 if we set aj = 1. In the Appendix, we summarize our results for the isotropic Heisenberg

group.

2. T h e twis ted and P.V. c o n v o l u t i o n o p e r a t o r s

Twis ted Convolu t ion . In order to develop Laguerre calculus on I-I~, we have to recall the basic properties of the Fourier transform in the Heisenberg group.

F o r f E L 1 (Hn), denote by

(2.1) .f:~(z) = f ( z , .)^ (~) = f s f ( Z , t)e-i~tdt.


The functionfx is well-defined for almost all z E C n and belongs to L l (Cn).

We denote by)?(rh, . . . , rln, A) the full Fourier transform of the func t ionf as a function defined on R 2n x R:

f ( r / l , . . . , 0n, A) =/R2n+' f ( z , t)e -iRe(z'~) e-i;~tdzdt.

Here dz = I-ISn__ 1 dzj A cl~j is the Euclidean measure on C n. I f f E L1 (I-In) n L2(Itn), then the Plancherel formula shows that

F o o

IlfllL2(Hn) = IlY IIL2(c.)d , o o

and then, as usual the Fourier transform can be extended to a Hilbert space isomorphism from L2(I-ln) onto itself.

Def ini t ion 2.1 For A E R* = R \ {0}, we define the twisted convolution of

two functionsf and g by

(2.2)

thus

(f*,x g)(z) = f c f ( z - w)g(w)e-i;~<z'w>dw.

Notice that in view of the antisymmetry of < .,. >, we have that

< Z - - W , W > = - - < W , Z > ;

g *xf = f *-;~ g,

and the twisted products are not commutative in general.

The twisted convolution occurs when we analyze the convolution of functions on the Heisenberg group in terms of the Fourier transform in the t-variable. To see

this, l e t f and g be two test functions on I-In. Suppose f �9 g is the convolution of these two functions, that is

(2.3) ( f * g)(x) = fn. f(y)g(y-1, x)dy.

Then

(2.4) ( f *'~g);~ =.fx *x g;~.


Indeed, with y = [w, s] and x = [z, t], then by the multiplication law of Hn,

( f ~ g ) ~ = fH e - i A t f ( w , s ) g ( z - - w , t - s + < z , w > ) d w d s d t nXR

= -**./".xRf(W , s)g(z -- W, t -- s)e-iA(t-S)e-i:~SeiA<z'w>dwdsdt

= ~ ,~ ~,~.

This proves (2.4).

P.V. Convolution Operators We now introduce principal value convolution operators on H, . These operators are the analogies of Calder6n-Zygmund principal value convolution operators on R ". As we know, the underlying manifold of Hn is R2n+l; but the role of the additive structure in R 2n+l is supplanted by the Heisenberg group multiplication law (1.1). Moreover, the group law forces us to use non-isotropic dilations on Hn, i.e.

x ~ 6 o x = 6 o [z , t ] = [ t z , 62t]

for all 6 > 0. These dilations are automorphisms of the group Hn:

6o ( x . y ) = ( 6 o x ) . (6oy);

but the standard isotropic dilations of S 2n+1 a r e not automorphisms of Hn. A func t ion f defined on Hn is said to be H-homogeneous of degree m on Hn if

f (6 o [Z, t]) = f ( t z , 62t) = 6mf(z , t)

for all 6 > 0. Next we introduce the norm function p given by

(2.5) p ( x ) = (llzll 4 + / 2 ) 1/4 , where Ilzll = : ~ ajlzjl 2. j=l

Obviously, we have p(x -1) = p ( -x ) = p(x) and p(6 o x) = 6p(x). In addition, the function p satisfies the triangle inequality

p ( x . y ) < C l { p ( x ) + p ( y ) }

for some universal constant C1. The distance function, d(x, y) of points x, y E Hn, is defined to be

d(x, y) = p(y-1, x).


It is clear that d(x, y) satisfies the symmetric property: d(x, y) = d(y, x). Suppose that K E C~176 \ {0}) is of H-homogeneous of degree 7. Then K is

locally integrable near the origin if~f > - 2 n - 2. See Folland and Stein [10] for the proof of this statement. This paper is mainly concerned with a special kind convolution operators on Hn, spectral projection operators, induced by kernels which are H-homogeneous of the critical degree - 2 n - 2. But if the kernel K only satisfies the homogeneity condition Ig(x)l < Cp(x) -z'-2, then the P.V. limit of the integral f . K would fail to exist whenever f ~ 0. The key point here is that the very definition of the P.V. convolution depends on a cancellation property in the integral.

Def in i t ion 2.2 Let K E C~176 \ {0}) be H-homogeneous of degree - 2 n - 2, K is said to have mean value zero if

(2.6) / K(x)da(x) = O, ao (x)---1

where da(x) is the induced measure on the Heisenberg unit sphere p(x) = 1.

Using Theorem 3 and Corollary 5.24 of Chapter XII in Stein [18], we can get the basic estimate concerning P.V. convolution operators on Hn:

Theorem 2.1 Let K E C~176 \ {0}), H-homogeneous o f degree -2n - 2 with mean value zero. Then K induces a principal value (P.V.) convolution operator, given by

f (2.7) K ( f ) ( x ) = ( f , r l K)(x) = lira t / ( y ) K ( y - 1. x)dy,

~--~0 Jd d(x,y)>~

for f E C~~ Moreover, the operator K given by (2.7) can be extended to a bounded operator from the lP-Sobolev space LPk(Hn) into itself, for 1 < p < oo and k ~ Z +.

The P.V. convolution operators can be composed, and the composition of two P.V. convolution operators yields another P.V. convolution operator. The best known example o f the left-invariant P.V. convolution operators on Hn is the Cauchy--Szeg6 operator S+ ( f ) = f * H S+ with the kernel

2n-ln[ l-I j= 1 aj (2.8) Sa:(z, t) = ~rn+ 1 n n+l"

a, lz,: .]

The fact that the Cauchy-Szeg6 operator is a projection operator implies that

S~ ,H S~ =S~.


This is in sharp contrast with the Euclidean convolution in which ca se f .E f = f does not admit a nontrivial solution. The Cauchy-Szeg6 kernel gives the first example of the nontrivial solution o f f * r i f = f . This is another interesting feature of the Heisenberg convolution which distinguishes it from the Euclidean convolution.

In fact the Cauchy--Szeg6 operator turns out be the simplest one of a large number of the basic operators which are induced by Laguerre functions.

3. Laguerre calculus

Laguerre calculus is the symbolic tensor calculus induced by the Laguerre functions on the Heisenberg group Hn. It was first introduced on HI by Greiner [12] and extended to I-In and I-In x R a by Beals, Gaveau, Greiner and Vauthier [4]. The Laguerre functions have been used in the study of the twisted convolution, or equivalently, the Heisenberg convolution for several decades. For instance, Geller [11] found a formula that expresses the group Fourier transform of radial functions on the isotropic Heisenberg group (i.e. functionsf(z, t) that depend only on Izl 2 and t) in terms of Laguerre transform, and Peetre [17] derived the relation between the Weyl transform and Laguerre functions. The connection between Laguerre functions and Fourier analysis on the isotropic I-In has been exploited in the study of various translation-invariant operators on I-In by de Michele and Mauceri [8], Jerison [13] and Nachman [16]. The Laguerre functions also played an important role in the Fock-Bargmann and Schr6dinger representations of the Heisenberg group (see Folland [9] for details). But it was in Greiner [12] that, for the first time, Laguerre functions were connected with the left-invariant convolution operators on Hi, and were used to invert some basic differential operators on Hi, namely the Lewy operator and the Heisenberg Laplacian.

L a g u e r r e Func t ions The generalized Laguerre polynomials L~'~)(x) are defined by their usual generating function formula:

= , { x w ) e x p '

(3.1) e=-I

f o r a = O , 1 ,2 , . . . , x_>O, andIwl < 1.

From the Laguerre polynomials, we can define the Laguerre functions:

I r (k+ 1) [r(~p+- 1)J x P l 2 L ~ ) ( x ) e - x l 2 , w h e r e x >_ O a n d p , k = O, 1 , 2 , . . . .

It is well known that {s : k = O, 1,2,.. .} forms a complete orthonormal basis of the space L2([O, oc)) for eachp = O, 1,2, .... (See Szeg6 [20] for a proof.)

S P E C T R A L P R O J E C T I O N O P E R A T O R S 323

Def in i t ion 3.1 The exponential Laguerre functions ~-~P~ )(~, r) on H~ are defined by

(3.2) W~)(f, r) 0 0')` ~ ' = k t ~ ) , fork, + p = 0 , 1 , 2 , . . . ,

where ~ ) ( ~ ) is defined as follows, for ~ = I~1 d~

(3.3) " ~ ) ( ~ ) = 2(-iy~(-1)l'e~k)(l~lz)dPa and Wk(-P)(~) = ~ ) ( ~ ) .

Then an elementary calculation yields

Propos i t i on 3.1 Le t z = [zle ia andk, p = 0, 1,2,. . . . Then

~-~ ~( -P) rz ra ~-~ )(21rllzl2)e -ip~ W~)(z, r) = e~)(2lrllzl2) e'p~ "-k , , , = (-1)Pe~ (3.4)

L a g u e r r e C a l c u l u s on H1 The most important property of the W~)(z, r) is the following theorem by Greiner [ 12]:

T h e o r e m 3.1 Letp, k, q, m = 1,2,. . . . Then

( 3 . 5 ) ~'~(p-k) ~ ( q - m ) = r ~ ( p - m ) ~ ( p - k ) ~'~(q-m) = ~m) ~ ( q - k ) �9 "pAk- 1 *[r[ - "qAm- 1 " "pAre- 1 ~ " ",oAk-- I *- - [r[ , "qAm-- 1 " " "qAk-- 1

where a A b = min(a, b) and 8(k q) denotes the Kronecker delta function, i.e. 8(~q) = 1 i f k = q and vanishes otherwise.

Thus the twisted convolution of two functions of W~)(z, r) is another function of the same type. This surprising result justifies the use of Laguerre function expansion on the Heisenberg group in analogy with Mildalin's use of the spherical harmonics on R n.

Let W ~ ) (z, t), +p , k = 0, 1,2, . . . , be the inverse Fourier transform OfWk 0') (z, r) with respect to r , i.e.

s 1 e i t r~ ) ( z , r )d r " W~)(z, t ) = ~

These are the kernels of the generalized Cauchy--Szeg6 operators on Hi . In particular,

Wo (~ (z, t) = S+ + S- , 1 1

where S+ = 7r-- ~ �9 (iz[ 2 q: it)2

denotes the Cauchy-Szeg6 kernels in HI and is given by setting n -- 1 and al = 1 in (2.8).


The following result implies that the generalized Cauchy-Szeg6 kernels indeed induce the principal value convolution operators.

T h e o r e m 3.2 The generalized Cauchy-Szeg6 kernels W~)(z, t) are in C~176 \ {0}) and have zero mean value.

L a g u e r r e Ca lcu lus on Hn We now extend the Laguerre calculus on H1 to H~. First, we define the n-dimensional version of the exponential Laguerre functions on C n by the n-fold product

W(k p) (~) = ~ a W~J) ( ~j ~ i l ~" kj k ~ ) ' j=1

where k = (kl ,k2,. . . ,k,,) E I~ and p = (Pl,p2,.. . ,Pn) ~ Z"; and the , ,~ ~,,/ are given by (3.3). Then we define the exponential Laguerre functions on H,, as follows:

n ) j = l

As in the one-dimensional case, we also have

(3.6) n

Wk(P) (Z, 7 - ) ]-Ta.~(PA(v/'~zj,7- ) = l l ~] kj j = l

where ~j~oj) (x/-~zj, r) ' s are given by (3.4).

We will compose two functions of the type (3.6) via twisted convolution on Cn:

I lr ] --(~)

=l LJ=l n

j~_la2/R .~2ia, r~m(z, Cc,)~'~(P,)[ /-~.(~.. Wj),T)~Ly)(v/~Wj,.r)dwj n

j = l

Consequently, we have the n-dimensional version of Theorem (3.1):

T h e o r e m 3.3 Let~., pj ,mj andqj = 1 , 2 , 3 , . . . f o r j = 1,2, . . . ,n. Then

~ p ( p - k ) ~"i((q-m) = ~(k q) ~ ( p - m ) ~ '~(p-k) * ~ '~(q-m) /~(p) ~"~(q-k) Ak--I ~' "rl vVqAm--1 ", VpAm_ 1 a n d vVpAk-I -[~'1 YVqAm--I = vm "" rqAk--I


where k = (kl,k2,.. . ,kn), p = (Pl,P2,...,pn), m = (ml,m2,.. . ,mn), q = (ql, q2,.. . , qn). Here p A q = (min(pl, ql) , . . �9 min(pn, q~)) and

j-----I

is the n-fold Kronecker delta function.

Let K induce a left-invariant (not necessarily P.V.) convolution operator K on I ' In ,

K(~)(x) =/__ K(y)~(y -~ x ) d y . d l t ' l n

Now J~(z, ~-) has a Laguerre series expansion:

o o n

(3.7) K ( z , T ) = Z K(kP)(l")I-[aJ~A(v/'~zJ ''c)" IPl,lkl=l j = l

The coefficients K(P)(~ -) in (3.7) are determined by the orthonormality of {s :k = 0, 1,. . .} on L2([0, oo)) forp = 0, 1,.. . , and are given by

(3.8) n n

_ = xo,, ,p, , . . - ,p.) , , -~ i - i a j i s g ~ ( p j - ~ . ) y , , -~ ~ : ~ P ) ( " ) 1 - [ a j i s ~ ' J ~. ) ] " - '~ ' ,,,,k, ..... ~. , J j = l j = l

= fn k ( z l zn ) f i e~-k:A(lzjl2)e_i~,:_kj)OidZl .dz,. v c a V �9 �9 " ' v , ~ ; ~ . . ," j = l

Defini t ion 3.2 Let K induce a left-invariant convoluation operator K on Hn.

For r E R* we define the positive Laguerre tensor .M+(K) by

A4+(K) = l,~k~ ..... k~ ~-J) for r > 0

where pj, kj E Z + for j = 1, . . . , n. The negative Laguerre tensor .M_ (K) can be defined as

A4_ (K) = { e-(pl,...,pn):..~ t \"k, ..... g, ~.j] for r < 0.

We say that U t = :Tf(P','",P.)~ /r /(Pt, . . .~v.)V

~"kl,...,k~ / = ~ " k l ..... k~ )

is the "transpose" of the (n, n)-tensor U :r~',."~n)~ if ~-- k ~ . J k , , . . . , k n 1

(rf(P,,...,P.)~ ..tk~ ..... ~) " k , ..... k. j = ( u . ~ , , . . . ~ , . ) .


Defini t ion 3.3 Let K induce a leff-invariant convolution operator K on Hn. We define its Laguerre tensor, .M (K), by

A.4(K) = A4+(K) ~ A.4_ (K).

Before we state the main result for Laguerre calculus, we recall the notion of tensor contraction. Let

U = \ k, ..... k. } and V = \ k, ..... k. )

denote two infinite (n, n) tensors. Their product, U- V, is defined to be

fw?, U . V = W = \ kl ..... 1,. } ' where oo

kl,...,~ : Z m l , . . . , m n = l

U • m l l , . . . , p n ) V ( m l . . . . . m n ) , . . . , r a n k l , . . . , kn "

The tensor W is the contraction of the tensor U and V. In the case of n = 1, the tensor contraction is simply the product of two infinite matrices.

Theorem 3.3 yields the following result.

T h e o r e m 3.4 (The Laguerre calculus on Hn) Let F and G induce the convo-

lution operators on Hn. A4(F) and A4(G) denote the Laguerre tensors o f F and

G respectively. Then

M ( F *n G) = = M + ( F ) . M + ( G ) , M _ ( F ) . . M _ ( C ) ,

where the product on the right hand side denotes the tensor contraction.

A simple consequence of Theorem 3.4 is

C o r o l l a r y 3.1 The identity operator In on C~(I-In) is induced by the identity

Laguerre tensor:

A4• = (8~ ') x(P.)~ �9 " " V k n 1"

In turn this corollary suggests the following result.

T h e o r e m 3.5 Let f E LP(Hn). Then

lim y]~ r k f , r l W~ ~ = f in L p norm. r - - - * 1 -

k=O

We shall need this result to decompose the LP(Hn) in the sense o f Abel summability.


Left- invariant differential operators A left-invariant differential operator 79 on Hn is a polynomial 79(X, Y, T) with constant coefficients, or in complex coordinates, a polynomial in vector fields T and Zy, Zj. We can have the following representation for 79 as a convolution operator on H,:

(3.9) 79 : 7911-I = Z 79W~:.'.'.',~ )*I'I' wherelH = ~ Wk(~,'.'.'.',~2*H Ikl:0 Ikl:0

is the identity operator on C~(H,). In particular, T and Zj, Zj, j = 1,2, . . . , n can be represented as convolution operators and written in the Laguerre tensor forms. This is our next proposition.

Proposit ion 3.2 (1) .M(T) is the i7- multiple o f the identity Laguerre tensor:

.M(T) = i~-(8~ ') x(P,)~ �9 " " ~ ' k n 1"

(2) Zj, j = 1 , 2 , . . . , n, has the fol lowing Laguerre tensor representation:

M(z:) =

where

A4:~.,O~,,...#,) ~ [ p J / 2 ~j) 6~j+l) 6 0~") and .M+(Zj ) = .A4_(Zj)t.

(3) Z,j, j = 1 , 2 , . . . , n, has the fol lowing Laguerre tensor representation:

= -.M(z:)'.

Theorem 3.6 Let 7 9 = 79(Z,Z,T) = 79(Z1,... ,Zn,Z1, . . . ,Zn,T) denote a

left-invariant differential operator on Hn, i.e. 79 is a polynomial in the vector fields

T and Zj, Z,j, j = 1 , 2 , . . . , n. Then

(3.10) = 79(.M(z), M(2),

where we set .M(Z) = (.M(Z1),..., .A4(Zn)) and .A4(Z) = ( A4(Z,1), . . . , .M(Z,n) ).

The Heisenberg sub-Laplae ian As an application of the Laguerre calculus, Beals et al. [4] obtained the fundamental solution of the Heisenberg sub-Laplacian:

j=l


via its Laguerre tensor. They started with

( 3 . 1 1 ) = = -

[k[=0 j=l

Then a simple calculation yields

1 - -~ - - - --(o) (3.12) - ~(ZjZj + ZjZj)Yr (v~Zj, r) = (2k + 1)[rlajW(k~ 7-).

Thus, (3.11) and (3.12) imply, as a convolution operator,

(3.13) • = (2kj + 1)17-laj - ,~7- aj . Ikl=0 \ j = '

Consequently the Laguerre tensor of the convolution operator induced by s is

(3.14) .M(Ea) = [7-[ [ ~ ( 2 k y + 1)aj -asgn(r )]5~ ' ) . . . ~ ) , \ j = ,

which is invertible as long as + a ~ ~]=,(2kj + 1)aj where k = (k , ,k2, . . . ,kn) E (Z +)n. According to Theorem 3.3, the inverse Laguerre tensor of (3.14) is

.x~(s = 17-1-' (2kj+ 1)a j -asgn( r ) 6~ ' ) . . . 6~ ") .

I f we write it in the Laguerre series expansion:

=

[kl=0 j= l j : l

we can sum up the right hand side of (3.15) to obtain the fundamental solution of L~. In fact, this approach can be used to find the fundamental solutions of the differential operators whose Laguerre tensors are o f the diagonal form; see [7] for the power of s and Schr6dinger propagator e -iszo, and [21] for the solution of the 0-Neumann problem in the non-isotropic Siegel domain. In this work, we shall consider the application of Laguerre calculus to the harmonic analysis on the Heisenberg group.


4. T h e LP s p e c t r u m o f (12, zT)

We define the joint spectrum of the pair (12, iT) as the complement of the set of (A, #) E C 2 for which there exist LP bounded operators A and B with A(AI - 12) + B(#I - iT) = I. This implies that the spectrum should be the set of (A, #) E C 2 for which neither (AI - 12) nor (#I - ~T) is invertible.

From the Laguerre calculus, we know that a convolution operator is invertible if and only if its Laguerre tensor is invertible. So we reduce the invertibility of an operator to that of its Laguerre tensor, and it is natural to first compute the Laguerre tensors of (AI - 12) and (#I - iT).

T h e o r e m 4.1 (i) The Laguerre tensor of the operator (hi - 12) is

A4(AI-12)= [A-I~-I aj(2kj+ 1)l~k(Pll)'' '(~k(~ ") .

( i i ) The Laguerre tensor of the operator ( # I - i T ) is

P r o o f (i) The proof is just a simple application of the results in the previous section. Take the Fourier transform with respect to t for AI - 12 and write it in the form of the Laguerre expansion:

tl

(AI-~_ Ikl=0 j=l

From the previous section, we have the identity

_ l (~j~j + ~j~,j)~(ko)(x/~zj, r) = (2k + 1)ajMW~~ ).

This implies that the Laguerre expansion of (AI - 12) is

n n

( AI'-~- s = ~ ['~ - N y~(2kj + 1)aj] I I aJW(k~ , 7") ,.~. Ikl=0 j=l j=l

This yields that the Laguerre tensor of (),I - 12) is

" ) �9

j= l


This proves (i). To prove (ii), we can simply take the Fourier transform with respect to t. []

The Laguerre calculus and the above theorem imply

T h e o r e m 4.2

n

(Al - s is invertible ~ A - [~-1Z(2kj. + 1)aj ~ 0 j = l

f o r all k E (Z +) ' , 7- E R.

(# - iT) is invertible r I~ + r 7~ 0 f o r all r E R .

Hence the spectrum of s alone is the set of nonnegative numbers {A E R : A > 0}, the spectrum of iT is the set of real numbers R, and the joint spectrum of (~, l'I') is the union of

n

(4.1) {(A,u) e C 2 : A = l # [ ~ - ' ( 2 k j . + l ) a j and u E R } . j = l

over the set k E (Z +)n. We can also write the joint spectrum in the form

{ } (4.2) a(Z~,iT)= U ( A ' # ) E C 2 : A > O ' e = + l ' # = ] ~ , j n _ l a j ( 2 k j + l ) " k~(z+).

The set (4.2) was called the Heisenberg brush and first introduced by Agranovsky, Berenstein and Chang [1].

Next we will find the eigenfunction corresponding to (A, #) E a(s iT), i.e. we want to find the function ~(kX)(z, t) such that

eA (4.3) (A - s t) = 0, (# - fr)~bxk,~(z, t) = 0 where # = 2.,j=r"~tnl + 1)aj

with e = + 1 and A > 0. From Theorem 3.9, we have

(4.4) n [ n I n (A - ~,) H ~(k~ r) = A - [ r [ 2 ( 2 k j + 1)aj l rX~~ r), j = l j = l j = l

(4.5) n n

j = l j = l

S P E C T R A L P R O J E C T I O N O P E R A T O R S 331

Hence, if we set u = - r and A = Irl ~"~jn=l (2kj + 1)aj,

n

(4.6) (A-~) l - I~(?) (v~yZj , v ) = O and j = l

n (u- i~) -(o) I I w ; : ( ~ j z j , , ) = o. j = l

This yields that

(4.7) -(a) 5 ( r r (Z, 7") -~- --[--

B ~(o)

EJ "=l(2kj + 1)aj j=l

satisfies the condition (A - Z~)~(kx,2(z, r) = 0 and (/~ + r)~(k~2(z, r) = 0. Therefore the eigenfunction corresponding to (A, #) �9 a(s tT) is just the inverse

Fourier transform of (4.7):

F "~(X)rz t ) = 1 eitr~p(k~,~(z,.r)dr 'e'k,r k ~ ~ oo

n

I f ~ . ( eA ) 1-[~o)(v~_~. ,r)dv = 2--'~ e'tr~ 7-+ Ejn__l(2kj + 1)aj x oo j = l

1 { ieA, t h ~ , ( o , ( eA ) Da. f 11 ""# v/~ZJ' n = ~ exp Ejn__, (2kj + , j ) j= , Ej=,(2kj + 1)aj

Apply the definition of W(k? ) ( v~Zj, r):

W ( ? ) ( V ~ , r) = 2He-aArllzA2Li~ 7f

to the last formula; one first obtains

n

' ) 1 1 " % , n = Y]j=I (2kj + 1)aj j = l

{ )" (4 .8) 2), e x p / - -~-II~ 2- I~-[r(~ 2AajlzJ[-- 2 ~EkL,(2~ + 1)aj [ E]=,(2~ + 1)aj f~A "~*' ~E]=,(2# + 1)a/'

then substitutes (4.8) into the last formula for 4~(k~(z, t), and this yields

(x)r7 t ) =

(4.9)

(2a-)-n-lA- [~n__ 1 (2kj + 1)aj] n

{ " A " z 2~i I-IL(O)[ 2Aajlzj[ n II I I ] 1 1 k. n ~ - - , exp Ej=](2kj+ l)aj(tet+ I j=l ' kE~='(2kj + l)aj ]


with Ilzll 2 = ~ n 1 asl-~.l 2.

Pointwise spectral projection We now consider the pointwise spectral projection induced by the eigenfimction q$(i,~,~ �9

(~) t '~(:~) ~}k,e(f)(z, ) = f * H V'k, e (4.10)

:" L_ ~b(~ ~([w' S]-I" [Z, t ] ) f ( w , s ) a w a ~ , dlt n

f o r f E i f ( H . ) .

The variables t and z in r are separable, and we can write 4b(k~ = ~1 (t)~b2 (z) with

(4.11) ~bl ( t )=e -izt and ?1

"g,2(z) = 1 _lul,,e_i,,,i. i l, i i ~ ]--[L~O)(21ulmlz:12), (27i')n+1 j= l

where Ae

u = E ~ I m(2~. + 1)"

To simplify the problem, we consider suchf(z, t) E/] ' (Hn) t ha t f = f i (t)j~(z) with .fi(t) ~ / P ( R ) andJ~(z) ~ / f ( C " ) . Then we obtain

(~) ~k,~(f)(z, t)= f:(s)e-'.<'-')es Scf i (w) ,2(z _ w)e-iU<z'W>dw

=fl (-#)e-i~'t[(f2 *. lP2l(z)],

wheref l is the Fourier transform of),] and [( j~ *u Ip2)(z)] is exactly the twisted convolution of)~ and ~P2. : k , ~ , f ) ( z , t ) l = I A( -z ) l l [ ( . f i *. r which is

independent of the variable t, yields that (~) ~k,~(f)(z, t) cannot be in LP(H,) for any

finitep.

Projection operators on the ray A > 0 We compute the spectral projection operator associated with the ray Rk,, of the Heisenberg brush. This is just the operator

io:i f= f Tk,~(f)(z,t) = *H "~(x)dx ~b ([W,S] -1 �9 [z,t])f(w,s)dwdsdA. Wk'~--" ---- dO JI-In

Formally, this is the convolution on the Heisenberg group with the kemel Tk,~ (z, t) =

f0 ~ .~(~)d~ Wk,~ . . . . We will show that Tk,~ is homogeneous of degree - 2 n - 2 with respect to the Heisenberg dilations, and has mean value zero.


First, apply the generating function formula of the Laguerre polynomials (3.1) and use the usual multi-variable notation r k = ~ ' . . . ~ ; we have

e~ rk f0e~ (4.12) ~ ~ k r k=O E~=~(2 ~+ 1)a:

fO00 O0 n (kOj =(2a-)-(n+') A.e-a(i,t+llzII 2) ~ r k IX L )(2Aa:[zjl2ldA k=0 j=l

=(2rr) -(n+1) f0 ~176 Ane -a(m+llzll2) ~Lk(2Aajlzjl 2) dA j=l

=(2rr)-(n+l) fo~Ane-a(i't+llzll=' h exp{-l~-rj -~--~j) dA

=(27r) -(n+l) (1 -- rj) -1 Anexp -A(iet+ ajlzj[ 2) dA j=l

n!

(2re) n+' [rljn__,(1- ~-)] [iet + ~in_, a;Iz:l=J This in turn yields that Tk,~/~5~=1 (2kj + 1)aj is the coefficient of r k of the Taylor series of the last formula. After going through some lengthy but elementary calculation, we obtain

fO ~ Tk,~ = "~(~)'~

= 2kj+ 1 L,=I

1 ~., (n + Ill)!(-1)Ill I-I~l(2ajlzjl2)lJ (27r)n+l ~!=0 [JlZl] 2 + iet] n+lil+l

~ 7 1 (2kj + a)aj k f i ( 2ajlzj[2 ~ ~-'~(n + III)! t~

[2~'(llzll 2 + i~t)] n+~ ~=o Ix\.= ll~t2 "

It is clear by inspection that Tk# is in C~176 n \ {0}) and is homogeneous of degree - 2 n - 2 with respect to the Heisenberg dilations.

Next, to show that Tk# induces a P.V. operator on the Heisenberg group, we only need to prove that Tk# has zero mean value:

(4.13) f~ Tk:(X)d~(x) o. (x)=l


From the calculation of (4.12), we can see that in order to prove (4.13), it suffices to show that

__[~" 1 + 1 - rjrJajlzjl2 t]-n-1 (4.14) fja<p(x)<b I~=l + is dzdt = 0,

where 0 < a < b < oo. (4.14) has been proved in [4] by introducing spherical coordinates.

In fact, the operator induced by Tk,, is closely related to that induced by the W(k ~ We shall find the exact relation between generalized Cauchy-Szeg6 kernel

them. Let A

w = y.~=l(2kj + X)aj'

then we write ~b (~) k,~ as follows:

l aj, e - i t ewN )(z, 1 (~) W(k ~ -ew). ~bk, e (Z, t) = 20r l-I;

This implies that

irk,~ (z, t) = f0 ~ }"~= 1 (2ky + 1)a/ e_it,o,~(kol(z,_ew)dw.

2"It I-[jn=l aj

Hence, we obtain

(4.15) Tk, l(z,t)+ Tk,_l(z,t)= ~Jn--l(2~'+ l)ajw(k~ n, aj

We summarize the above calculation in the following theorem:

Theorem 4.3 The spectral projection operator on the ray 0 < A < oo has kernel

Tk,e(z,t)= Y~n=l(2ky+l)aj k n ( 2ajlzsl = ,lj [2~r(llzll 2 + iet)]n+l y~(n -t-Ill)! l ' I - S=l 111112 + iet]

Furthermore, Tk,~ (z, t) is H-homogeneous of degree -2n - 2 with mean value zero.

Now we may apply Theorem 2.1 to obtain estimates for the spectral projection operator in Lr-Sobolev spaces L~(Hn), for 1 < p < oo and k E Z + .

Corollary 4.1 The spectral projection operator on the ray 0 < )~ < oo,

Tk, J f ) = f * n rk, (Z, t),

is bounded on L~(Hn) for 1 < p < oo and k E Z + .


5. Estimates for the operator Tk,E in Hardy and Morrey spaces

In this section, we discuss estimates for the projection operator Pk,~ in Hardy and Morrey spaces. First of all, let us recall some basic properties of Hardy and Morrey spaces.

Hardy Spaces The estimates for the Cauchy--Szeg6 operator on L p spaces, in fact, can be extended to Hardy spaces/-P(Hn) for 0 < p < 1. Let us recall some basic definitions for the Hardy spaces.

Definition 5.1 A Heisenberg p-atom (0 < p < 1) is a compactly supported function a(x) such that

(1) (size condition) there is a Heisenberg ball B whose closure contains supp(a)

such that la(x)l _< I/~1 - l / p , (2) (moment condition)

fH, a(x)p(x)dx = 0

for all monomials p(x) where p(x) = p(z, t) = ,"~'-~2... ~ , t~,+, and "1 "2

Here Is] is the integral part ofs.

Using the idea of atoms, we now give the definition of liP space on I-In.

HP(Hn) = E S'(H~) : f = ~ ,hkak, where ak arep-atoms, k=l

and we define

IAkl p < oo , k=l

Theorem 5.1 Suppose the kernel K satisfies the following condition:

fp IK(y -1. x) - K(x)ldx < c l . E -~ (x)<~p(y)

V/3 > 2,

[[fl[n,(ri,) = inf At,

where the infimum is taken over all possible atomic decompositions o f f . The "norm" IlY]i r is comparable to the lp norm of the sequence {,~k}.

The following theorem can be found in Chang [6] which gives us a sufficient condition for a P.V. convolution operator bounded on/-/P(Hn).


for some 0 > O. We define the mapping K by

K(f ) (x) = f*H K(x).

Assume that I l K ( f ) l l v m . ) _< e2 �9 llgqlL2m~) holds. Then K extends to a bounded operator on I-lP(Hn), for po < p < oo, where po is an index depending on 0 such

that 0 < Po < 1. The operator norm o f T only depends on el and e2.

Let K be a distr/bution on the Heisenberg group smooth away from the origin and satisfies the size conditions

IK(x)l <_ A

I,(x,p(x))l'

oz?' . . . o z ~ ~ < IB(x,p(x))l'

~K(x) [ Ap(x)-:: [---gg-- < IB(x,p(x))["

I f we assume further that K has the mean value zero property, then we know from

Theorem 2.1 that K ( f ) ( x ) = f *H K(x) is bounded from LP(Hn) to itself. Fur- thermore, the above size conditions provides K satisfying the assumption o f 2.1. Therefore, we may conclude that the operator K can be extended as a bounded operator from/-P'(Hn) into itself. In particular, the Cauchy--Szeg5 operator S• ( f ) (z , t) maps HV(Hn) into/-P'(Hn) for 0 < p < 1.

M o r r e y Spaces We denote Money spaces on H,~ as follows:

LP,~(H,) =

{: ( )" ) 1 [ f ( y ) ~ ' d y < ~ , ~ L f o c ( H n ) : Ilfllz~,~ = xe~,r>0sup iB(x~r)] ~ (*,,)

for 1 _< p < oo and 0 < v < 1. It is easy to see that/J' ,~ =/-P(H,,) and

/ : ' l (Hn) = L~176 But when v > 1,/-P'~(Hn) = {0} is the trivial space. We may define the Morrey-Sobolev spaces ~'~'(Hn) in a similar way:

5~k'V(Hn) = {f C LPS'(Hn), P m ( Z , Z ) f E LP:'(Hn)).

Here Pm are any polynomials in Z and Z of degree m < k with the norm

[[f[[~'~(~) = E H~m(Z'Z)f]Itv'~(H.)" m<k


The A 1 (dx) weights are defined as non-negative Borel measurable functions w satisfying .M(w)(x) < c . w(x), for almost every x ~ H~. Here M(w)(x) is the Hardy-Littlewood maximal function of w defined as follows:

1/. m ( f ) ( x ) = ,>0 It3(x,r)l (x,,) sup ~ If(y) ldy.

We denote by ALw the smallest constant c for which the above inequality holds. Consider the following subclass of A1 (dx) indexed by a > 1:

Al(dx, a) = {w EAl(dX): Ilwlloo 1 and Al,w < a},

where Ilwll~o = inf{t > 0 : I{x E Hn : w(x) > t}} = 0}. In the paper [2], Arai and Mizuhara have proved the following theorem;

T h e o r e m 5.2 Suppose f and K ( f ) are two non-negative Borel measurable

functions on Hn. Suppose that for every a > 1, there exists a positive constant c( cx) such that the following inequality holds:

fHn K(f)(x)w(x)dx <_ c ( a ) / l l f ( x ) w ( x ) d x ,

where w E A1 (dx, a). Then there exists a universal constant C t such that

llK(f)l[v,-fn.) < C'. II~I~,~<i~),

for l < p < oo and O < v < I.

In fact, under the assumption that the operator K is bounded on/-P(Hn) for 1 < p < ~ , we can generalize the above t h ~ r e m to 29~k'v(Hn) directly, i.e. we don' t have to use the weighted norm inequality (see [3]). More precisely, we have the following theorem.

T h e o r e m 5.3 Suppose 1 < p < oo and 0 < v < 1. Let K be a P.V. convolution

operator induced by a kernel K E C ~ (Hn \ {0} ), H-homogeneous o f degree - 2 n - 2

with mean value zero. Then there exists a poative constant Cp,v,k such that for all

f E ~'V(Hn),

(s. i) II K(:3 ll ,'ca.) -<

for all k E Z +.

As a corollary of the above theorem, we ~ that tim Cauchy---Szeg~ operator

maps ~'"(Hn) into itself for I < p < oo, 0 < u < I and k E Z + .


In order to prove the above Theorem, we need the following two lemmas.

L e m m a 5.1 Let x E Hn and 6 > O. Then there exists a constant/3 > 0 depending only on C1 such that

(5.2)

3-1 t " Js If(u)ldu < M(fxB(x,6))(Y) p(x, y)2n+2 (x,6)

< 3 [ l / (u) ldu , - p(x,y) 2n+2 JB(x,~)

f o r every y E Hn with p(x, y) > 2C16. Here Xe is the characteristic function o f the measurable set E.

P r o o f The lemma is a consequence of the definition of the Hardy-Littlewood maximal function and the doubling property of Heisenberg balls, i.e. B(x, 2r) < CI �9 22n+2B(X, r). []

As a corollary, we may assume f = XB(x,~); then we obtain the following inequality:

IB(x,6)l (5.3) M(XB(x'6))(Y) ~ p(x, yi 2n+2"

L e m m a 5 . 2 Let 1 <_ p < oo and 0 < u < 1 and let f E LP'"(Hn). Suppose "y is a positive number satisfying g < 7; then for any balls B(x, 6),

(5.4) fH. [ f ( x ) f {M(xB(x'a)(x)}Tdx <- CT. Ilfl[~,~" 6.(2.+2),

where C is a positive constant depending only on u, 7, and n.

P r o o f Now we may apply inequality (5.3) to prove this lemma. Let f~0 = B(X, 2Cl 6), and

(5.5) Ok = {y E I-In : 2kC16 < p(x, y) < 2 k+l C16}, k E N.

Denote B = B(x, 6). Then, by Corollary 4.1 and Lemma 5.1, we have

/rl If(Y)~'{M(xB)(Y)}TdY

L oo = [f(y)lP{M(x6)(y)}~dy + Y~ / [f(Y)~{M(xB)(Y)}~dY

k=-I JN~


<C~ [ [f(y)lPay + E f If(Y)~'p(x,y)(2n+2)naY J~o k=l J~2k

~ r L II fllL,v 6v(2n+2) + (72 If(Y) IPdy ~-I (2kg)(2~+2)'Y *

f'# II ,Clip ,r co l =...711JllD,,.v + C2 E 2k(2n+2)7 Ilfl l~ '~(2k6)~(2"+2)

k=i o o

II r at . C2 II ~v(2n+2) . E 2k(v-7)(2n+2)

k=l

=C-r-Ilfll~,,,,6 ''(2"+2),

since u - 7 < 0. []

Now we are in a position to prove Theorem 5.3. We just need to prove the case for k = 0. The case for k E N follows easily from the identity

P(Z, 7.)K = K75(Z, Z).

Here ~' and 75 are two polynomials of degree k in Z and Z. The polynomials P and 75 are not the same in general (see [10]). The reason for this is the operators

f ~ Z i ( f ) o r f ~ Z j ( f ) a n d f ~ K ( f ) do not commute. However, given any PV convolution operator K (induced by a kernel K), there exists PV convolution operators Mk and Nk for k = 1 , . . . , n (induced by kernels Mk and Ark respectively) such that

n

Zj(K(f)) = Zj(f, K) = E {Mk(Zk(f)) + N~(Zk(f)) } k=l n

= E {Z~(f) �9 Mk + ZkCf) * Ark}. k=l

Fix a ball B = B(x, 6). Let f~k, k = 0, 1 ,2, . . . , be as defined in (5.5). Denote fi =fxao andJ~ = f - f t . Then for almost every x E B, we have

K ( f ) ( x ) = K(f i ) (x) + / I ~ K(x . y-1)j~(y)dy.

SinceJi ~ LP(Hn), by the assumption of the theorem, we know that

]IK(A)iI~ <~ CpllAl~ S Cp,~,llfllz~,~" ~,<2,,+2),


where the constant Cp depends on the kernel K and p only. It remains to look at the second term. From (5.2), we have for v < 7 < 1,

n . [K(x .y - l ) j~ (y ) ]dy <Cn/n.\~0 [J~(Y)I d~ - p(x, y)2n+2 Y

C3 fH.\ao M(xB(y))IJ2(y)ldy

= ~ [ / H , \ a 0 }f2(Y)IM(xB)'r/P(Y)M(xB)I-'r/P(Y)IdY

Now by (5.4), we obtain

1 [j~(y)~'M(xB)'Y(y)dy _< C~,~,. [[f][~,~. b ("-I)(2"+2)/p

Moreover, since (p - 7)/(P - 1) > 1, then we have

1 I n M(XB)(P-'r)/C~ < ~ [ ][1Hzp'v "62n+2 -< Cp,~,. .\f/o

I t follows that

n . IK(x-y- l )y~(y) ldy < Cp,v " llflirp,~ " 6 <"-1)<2"+2)/p

Consequently, we get

.~ ]K(f)(x)~ ~ < ]IK(J~ )]15 at- ~'p,~," IIJ~l/.r," " ~(v-1)(2n+2)~ dx

<_ C p , v " [[J]l/p,~" 6v(2n+2).

This completes the proof of the Theorem. Now we may apply results in this section to our operator Tk,~. We have the

following corollaries:

C o r o l l a r y 5.1 The spectral projection operator on the ray 0 < A < oo,

rk, (f) =f*a rk, (Z, t),


is bounded on ~'V(Hn) into ~'"(Hn) f o r 1 < p < oo, 0 < v < 1 and k E Z +. Moreover, the operator Tk,, originally defined on C~(Hn) can be extended to a bounded operator from HP(Hn) into itself for 0 < p < 1.

Corollary 5.2 L e t f E// '~(Hn). Then

lim ~ rk f *H , ~ ( 0 ) = f , r---* l -

k=O

in L p'v norm.

Proof According to Theorem 5.3, the operator norm of the operatorf .n wCk ~ is bounded by (n + Ikl)l/Iklt and C(1 + [k[) 2nll/p-I/z[+e (see Strichartz [19]). Now the result follows immediately. []

6. The spectral projection operator

Strichartz found the spectral projection operators (1.9) associated to the point e(n + 2k) and the operator iT-1s via the representation of the isotropic Heisenberg group. The main result in Strichartz [19] is the summability in Lp of the decomposition

o o

(6.1) f = lim Z ~ -~?nPm,e* f f ~ l < p < o o . r---, 1 -

m=0 e=4-1

See Section 3 of [ 19] for details. In this section, we will take a different approach. Basically, we obtain the

same projection decomposition without any reference to the eigenfunction of the special operator iT- 1Z;. We will derive these projection operators directly from the Laguerre calculus. In fact, the decomposition is nothing but Theorem 3.8 and the projection operator Pro(z, t) is simply the sum of the generalized Cauchy--Szeg6 kernels W(k ~ over Ikl = m. The Abel summability is an easy consequence of the Laguerre calculus.

We begin with any functionf(z, t) and proceed formally. We will impose the conditions which f must satisfy later. We first take the Fourier transform with respect to t and let

// .f(z, r) = e-i~'ge(z, t)dt. o o

We require that f(z, 7-) can be written as the Laguerre series expansion:

?!

(6.2) f(z, r ) = ~-'] F(kP)(r) n aj~p~-k~)(v~zJ ' r), k,p j = 1


and the left hand side will converge tof(z, r) in the Abel sense ofsummability. We now do the twisted convolution of the above equation with the exponential Laguerre function ]-/)~1 ,,:~,(q/-mj), __ r) from both sides, then we apply Theorem (3.3) t~j I~ VqjAmy_ 1 {, ~/ajzj,

and obtain n

I-[ a'~(q~-m:) jT(z,r) ,~- ii ~ qj^m,-'(V~zY 'r)

j=l

n n

= E F(P)(r ) y'f a.~.~,-~) ~'~'~(qj-m,) II ~ p/^~-1*r HaJW~^mj-l(V~zJ 'r) k,p 1=1 1=1

n

---- E F(kp)(T) ]-]"., ,~(q./){~(py-m/) I I "/'% ''p:Amj - I ( V ~ Z j ' T ) k,p j=l

n

= ~ F ~ P ) ( ~ ) - ~ , - m ~ ) l l aj~%^m~-~(v~Zj, ~). p j=l

We now set qj = mj for a l l j = 1,2,... ,n in the above calculation, and this will yield

(6.3) n I1

f(z, r )*r H ajW(m~)-' (V'~zj' r) = E F(mP)(r) H aJW:̂ mj-'(V~"J'--(rJ-m/) a.z r). j=l p j=l

Applying equation (6.3) to (6.2) leads to

(6.4)

n

y(z, ~)= E E r(~)(~) M ajWp~), (v~zJ, ~) k p j=l

n

: Fj(z,,, . II k j=l

with the summation over k ~ N n. If we change to summation over (Z+) ~, then (6.4) becomes

I1

(6.5) f(z, ~-)= ~ ' f(z, r)*~ ]'I ajW(?)(v~ZJ, ~)" kE(Z+) n j=l

One can rearrange the summation in (6.5) and obtain

n

kE(Z+)n j=l

m=O [k[=mj=l


We compute the sum over Ikl = m and obtain

oo

(6.6) .f(z, 7-) = Z . f ( z , r) ,~- Pm(z, 7.), m---O

where n

em(z,T) def Z H aj~(kO)(~v/'~Zj'T) Ikl=mj=l

= - - aje.-ajlrllzA2 ~ HLi~ 2) Ikl=mj=l

=(~-)n~=I'Iilaj)e-l~'i'ilzll2L(mn-U(2lTl.llzll2 ).

After taking the inverse Fourier transform with respect to r, we have

oo

f ( z , t) = ~-~ f * Pm(z, t) m=O

1 ~ c ~ with Pm(z, t) = -~ J-Io ~ eit~pm(z, r)dr.

We now compute the functions Pm(z, t). First,

(6.7) n 2n_ 1 L~o

Pm(z,t) = ( H aJ)~-'A-~ [7.lneitr-lrl'[Izll2L(mn-1)(2M " ][z[12)d7." j--1 oo

Introduce the new variable s = 21rl ~ff~l ajlzjl ~ = 211"1 "Ilzll 2, and let

X,(z)=~ l+ll-~ll~ ) with~=+l;

we can write (6.7) in the form

Pm(z,t) = E Pm,~(z,t), e==t=l

with

Pm,e(z,t)= l-I~=laj fo~176 4(~rllzl12),+l We now compute the integral in Pm,, by applying the formula

(6.8) f o O0 ~ , e-WS~r<k):s~ds=a ~mk l (It+k)] ( w - 1) n m! w re+k+1 "


We have

fo ~sne-A,sL~n-l)(s)ds- fO ~

0 sn_le_A.sL(n_l)(s)ds OA,

oA~O L l) ' ' (A~m+~)m] [ (m + n -

m! ( m + n - 1)! [ ( A , - 1) m

mf [(m + n) Am+n+ 1 (Ae - 1) m-I ]

m -A-~mT- ff j.

Note that 1 - Ae = A_~. Hence we have

5 [ ] (6.9) sne-A~SL(mn-1)(s)ds m(m+n)! Am-~ m A~ = ( - 1 ) ~.. Am+.+le 1 + ~.m+n ~__~ "

Finally, we obtain

(6.10) Pm,~(z,t) = ( -1) m II~in=l aj (m + n)! Am_~ [ m A~ ]

(TrHzll2) n+a m! A re+n+1 1 + ~ + n ~_~ "

Applying the definition of A~ to (6.10), we obtain

1,m,~-r ,2"-2(m§ m [ m Ilzll2+i t] em~, (Z, t) = ( - - ) ( H aj) ~n--~T~ml~ ~st~+--h-- ~ " [1 + - - -- .

j= l m + n Ilzl[ 2 ietJ

I f we set aj = 1 in the above equation, we have

2n+l(m+n)[(lzl2-iet) m [ m Izl2+iet] (6.11) Pm,~(z,t) = ( -1) m ~ ( ] - ~ ' ~ ' ~ ' ~ - ' f i - ~ �9 1 + m +---n Izl =

where [z[ 2 = ~ n 1 Izj[ 2. This is exactly (1.9) except for a factor 4. We now consider the convergence of(6.1) f o r f E /2 (Hn) , 1 < p < oo. First,

the property of Laguerre calculus (Theorem 3.5) yields that

(6.12) lim y ~ rlkrvP (~ , H f = f Ikl=0

uniformly f o r f E C~~

where Wk (~ is the inverse Fourier transform of l-[~n__l aj~(~~ x/-~zy) with respect

to r. Since Wk (~ E C~176 \ {0}) has mean value zero and is H-homogeneous o f degree - 2 n - 2, Theorem 2.1 implies that

(6.13) W(k ~ * n f E LP(Hn), for a l l f E LP(Hn).


(6.12) and (6.13) imply that

(6.14) lim Z rlklW(k0)*Hf=f r--*l-

Ikl=0 for all f E LP(Hn) in LP-norm,

since C~ ~ is dense in Lp. We can also prove the covergence of(6.1) in Hardy and

Morrey spaces by the same method. In particular,

(6.15) lim E r[klWk(0)*rIf=f r--* 1- [kl=0

for all f E HI(Hn),

in H l-norm.

Since we obtained Pm by summing up Wk (~ over [k[ = m and Pm = Pm,+l +Pro,-1, (6.14) yields the Abel convergence of (6.1). It is easy to see that Pm,~ induces a projection operator from the definition of Pro. Indeed, by the twisted convolution law of Laguerre functions,

~:~mt*'r~Pm2(Z'T)= Z E aj~2?)(M/~Zj,T) *'r aj~I~O)(v/~Z),T) Ik [=ml Ill=m2 =1

n X X +-,o) = ( , / a ; + , , - )

Iki=ml Ill-m2 j---- 1 n

Ikl=ml Ara2 j = 1

where ml A m2 = min{ml, m2}. This yields that Pm *~- P m = Pm and Pro,. induces a projection operator on/_/'(H.), H 1(H.) and ~ 'V(H.) .

Appendix: The isotropic Heisenberg group

We can get the isotropie Heisenberg group by setting all aj = a for j = 1 ,2 , . . . , n. a = 1 in Stein [18] and a = 1/4 in Strichartz [19], Folland [9], and Miiller and

Rieci [ 14], [ 15]. We will set a = 1 and compare the formulas of the eigenfunetions

we get to those of Strichartz [19].

In the ease ofaj = 1 for a l l j = 1,2, . . . ,n,

~A ~A where A > 0, e = •

U = Ek=l(Zkj + 1)aj " Ej=I(2 + 1)

Let Ikl =kl + k2 +.. . + kn =m; then we obtain # =eA/(2m + n) and the Heisenberg

brush

cr(L, iT) = (A,#) E C2 : A > 0 , e = • g = Ejn__laj(2kj + 1 ) kE(Z+)n "


becomes the Heisenberg fan

cr(s U ( A ' # ) E C 2 : A > O ' e = + l ' # = 2 m + n " m E Z +

The eigenfunction corresponding to # = e A / ( 2 m + n) is

I.~LCO)( 2AIzjl (6.16) ,~(~)rz t~ = An A(IzI2 + iet) n 2 '~k,ek ' I (21r)n+l(2m+n)neXp (2re+n) J j l J ~ ~(2-'mm'~)J"

EA The geometric multiplicity of the eigenvalue (A, 2m~- n) is

E 1 = ( n + m - 1 ) n - 1 "

kl +k2 +'" +kn =m

Since # depends on m only, we can sum up ~b (x) k,, over I k] = m by applying the following property of the Laguerre polynomials:

/I n

(6.17) E 1-[L(k ~ ( Alzjl2 ( 2~1zl2" 3 \2(2m + n)) = L(mn-1)k(~m ~ n ) ) ' where Izl 2 = ~-'~ t~12. [k i=mj=l j = l

This yields

An { A(lzl2+/---et) "[L(,-I)(. 2AIzl 2 (6.18) ~b~)(z,t)= (27r)n+1(2m+n)nexp (2m+n) J m \ ( 2 m + n ) ) "

The difference between (6.18) and Strichartz's result (see (1.6) of [19]) is just a constant factor.

ACKNOWLEDGEMENT

A significant part of this paper was written while we visited the Hong Kong University of Science and Technology. We would like to thank the Mathematics Department at HKUST, especially Dr. Bing Yi Jing and Dr. Jing Song Huang, for their warm hospitality.

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Math., to appear (1997).

Der-Chen Chang DEPARTMENT OF MATHEMATICS

UNIVERSITY OF MARYLAND COLLEGE PARK, MD 20742, USA

Jingzhi lie DEPARTMENT OF MATHEMATICS

UNIVERSITY OF TORONTO TORONTO, CANADA, M5S 3G3

(Received September 16, 1996 and in revised form December 8, 1996)

2y~aj(xj~ -yjxj)]. xj + iyj, [z,t]. [w,s] = [z+w,t+s+ < z...

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