Real Hardy spaces on real rank 1 semisimple Lie groups

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Japan. J. Math. Vol. 31, No. 2, 2005

Real

Hardy

spaces

on real

rank

1 semisimple

Lie groups

By

Takeshi

KAWAZOE*

(Received May 11, 2004)

(from Tokyo Journal of Mathematics)

Abstract. Let G be a real rank one connected semisimple Lie group with finite center. We introduce a real Hardy space H1 (G//K) on G as the space consisting of all K-bi-invariant functions f on G whose radial maximal functions MƒÓf are integrable on G. We shall obtain a relation between Hl (G//K) and H1(R), the real Hardy space on the real line R, via the Abel transform on G and give a characterization of H1 (G//K).

1. Introduction

In the study of the classical Hardy spaces on the unit disk and the upper half plane, real variable characterizations of their boundary values are called the real variable method. In the 1970's these boundary values were completely characterized

by various maximal functions without using the complex variable method and their

atomic characterizations were also given at the same time. This was a significant

breakthrough in harmonic analysis. Nowadays, this fruitful theory of real Hardy

spaces, which are defined by maximal functions and atoms, has been extended

to the spaces of homogeneous type: A topological space X with measure ƒÊ and

distance d is of homogeneous type if there exists a constant c>0 such that for all x •¸ X and r>0

where B(x, r) is the ball defined by {y •¸ X| d(x, y)<r} and ƒÊ(B(x, r)) the volume of the ball (cf. [2, •˜1]).

When the space X is not of homogeneous type, little work on real Hardy spaces

on X has been done. In this paper, looking at the example of a semisimple Lie

group G as a space of non-homogeneous type, we shall consider real Hardy spaces

on G. Actually, when X=G, ƒÊ(B(x, r)) has an exponential growth order as r

goes to infinity and hence G is not of homogeneous type. We shall introduce a real Hardy space on G by using a radial maximal function on G. Our goal is to give an

2000 Mathematics Subject Classification. Primary 22E30; Secondary 43A30, 43A80.

*Supported by Grant-in-Aid for Scientific Research (C), No. 13640190, Japan Society for the

Promotion of Science.

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atomic characterization of the space and to obtain a relation with the real Hardy

space Hl (R) on R.

We shall overview some results obtained in the previous papers [7], [8], [9]

and state our main theorem. Let G be a real rank one connected semisimple Lie

group with finite center and G=KAN=KCL(A+)K respectively an Iwasawa

and the Cartan decompositions of G. Let dg=dk da do=ƒ¢(a) dk da dk' denote

the corresponding decompositions of a Haar measure dg on G. In this paper we

shall treat only K-bi-invariant functions on G. Since A is identified with R as

A={ax;x•¸R}, all K-bi-invariant functions can be identified with even functions

on R. We also denote ƒ¢(ax) by ƒ¢(x) for x•†0 and extend it as an even function on

R. Then the one dimensional space R with normal distance and weighted measure

ƒ¢(x) dx is not of homogeneous type, because ƒ¢(x) •` e2ƒÏ|x| as |x| •¨ •‡, where ƒÏ is a positive constant determined by a group structure (see (2), (4), (5)). Let Lp(G//K) denote the space of all K-bi-invariant functions on G with finite Lp-norm:

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and L1loc(G//K) the space of all locally integrable, K-bi-invariant functions on G. We shall introduce a real Hardy space on G by using a radial maximal function. As in the Euclidean case, to define a radial maximal function we need to define a dilation ƒÓ, t>0, of a function ƒÓ on G. Let ƒÓ be a positive compactly supported C•‡, K-bi-invariant function on G such that

We define the dilation ƒÓt of ƒÓ as

Clearly, ƒÓt has the same L1-norm as ƒÓ: ||ƒÓt||1=||ƒÓ|| and, for 1•…p•…•‡, it gives an approximate identity in Lp (G//K) (see [3, Lemma 16]). Since this dilation has the same properties as in the Euclidean case, it is quite natural to introduce a radial

maximal function MƒÓf on G as

As shown in [7, Theorem 3.4], this maximal operator MƒÓ satisfies the so-called

maximal theorem: MƒÓ is bounded on Lp (G//K) (1<p<•‡) and satisfies the

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Real Hardy spaces on rank 1 semisimple Lie groups 283

analogously as in the real Hardy space H1 (R) on R:

and the norm

is given by

The aim of this paper is to characterize Hl (G//K).

For a compactly supported C•‡, K-bi-invariant function f on G, we shall

define the Abel transform Fsf, s •¸ R, of f as

This integral over N is explicitly given by a composition of generalized Weyl type

fractional integrals (see [11, Corollary 3.3] and (14)). Especially, we put

and denote by W- the inverse operator of W+, which is given by a composition of Weyl type fractional derivatives (see (18), (19)). Let f denote the spherical Fourier

transform of f on G and F•` the Euclidean Fourier transform of F on R. Both f

and (Fsf)•` are regarded as functions on the dual space F of the Lie algebra of A,

which is identified with R. Then they are extended to holomorphic functions on

Fc, the complexification of F, of exponential type and

(cf. [11, (3.17)]). Let C(ƒÉ) denote Harish-Chandra's C-function (see (9)) and MCp

the Euclidean Fourier multiplier corresponding to CƒÏ(ƒÉ)=C(ƒÉ+iƒÏ):

We here define

and also W- (MCƒÏ (H1 (R))) by replacing the condition that W+(f) •¸ Hl (R) in the above definition with M-1CƒÏ o W+(f) •¸ H1 (R).

THEOREM

A. Let notations

be as above.

This is one of the main results in [8]. However, the proof was a little bit

complicated, because to obtain the first inclusion we used the Harish-Chandra

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expansion of the zonal spherical function and also the Gangolli expansion (see (8), (10)). Thereby, to sum up the estimates of each expanded terms we required a sharp estimate and a deep theory of H1 Fourier multipliers on R. In this paper we shall give a simple proof based on fractional calculus on G. Actually, Theorem

A follows from the next Theorem B: We define a maximal operator MRƒÓ and a

fractional operator WRƒÁ on R by Definition 3.2 and (30) respectively, and n e N and the set D by (37) and (38) respectively.

THEOREM B. There exist 0<c1<c2 such that for all f •¸ H1(G//K)

where F=W+(f).

We next introduce atomic Hardy spaces on G. In the Euclidean space the

atomic Hardy space H1,•‡,0(R) coincides with H1(R) (cf. [6, Theorem 3.30], [15, •˜2 in Chap. 3]). However, it may not be true in our setting, because the Lebesgue measure dx is replaced by the weighted measure ƒ¢(x) dx. We denote the interval

[x0-r, x0+r] by R(x0, r) and set the volume by

We say that a K-bi-invariant function a on G is a (1, •‡, 0)-atom on G provided that there exist x0•†0 and r>0 such that

(i)

(ii)

(iii)

Here a is identified with a function on R+. Then H1•‡

,0 (G//K) is defined by

and the norm is given by

where the infimum is taken over all such representations f=ƒ°iƒÉiai. Moreover,

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above definition of (1, •‡, 0)-atom a on G, respectively, with

(ii)•¸

and

(iii)+

Clearly, for ƒÃ•†0,

We define a truncated maximal operator MlocƒÓ on G as

In [9] we essentially proved that MlocƒÓ is bounded from H1,+•‡,0(G//K) to L1(G//K). As for MƒÓ, we shall prove that MƒÓ is bounded from H1,+•‡,0(G//K) •¿ W- (H1(R)) to L1(G//K). This means that H~ o (G//K) fl W_ (H1(R)) C H1(G//K). Finally,

as a refinement,

we have the following

main theorem.

THEOREM

C. Let notations

be as above.

Then

Furthermore,

H 0 (G//K) n W_

(H1(R)), e > 0, is dense in W- (H1(R)) and

especially,

H1(G//K) is dense

in W- (H1 (R)).

This paper is organized as follows. We recall some basic notations in •˜2. Then we shall define a radial maximal function MƒÓf on G in •˜3 and obtain a key formula which reduces MƒÓf to a Euclidean maximal function MRƒÓ (W+(f)) on R (see Propo sition 3.3). Theorem B follows from this formula and fractional calculus on G and R in •˜4 (see Theorem 4.6). As an easy consequence of Theorem B, we can obtain a simple proof of Theorem A (see Remark 4.7), and moreover, a norm-equivalence

between ||f||H1(G) and ||MlocƒÓf||1+||W+(f)||H1(R) (see Theorem 4.9). In •˜5 we in

troduce atomic Hardy spaces on G. Then the inclusion H~ o (G//K) f1 W_ (H'(R)) C H1(G//K) follows from the estimates obtained in •˜4 and [9] (see Proposition

5.9). Next we shall consider a (1, •‡, +)-atomic decomposition of functions f in

H1(G//K) (see Proposition 6.2). Then, combining with Theorem A, we can de

duce our main Theorem C (see Theorem 6.3 and Theorem 6.4). In •˜7 we shall

consider (H1, L1)-boundedness of other operators on G; singular integrals, mod

ified heat and the Poisson maximal operators (see Definition 7.2) and the Riesz

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comment on the dual space of W- (H1 (R)) in •˜8. In •˜8.A, we shall obtain a sharp

estimate of ƒ¡m(ƒÉ), meromorphic functions appearing in the Gangolli expansion of

a spherical function (see (10)), and in •˜8.B, we shall obtain a norm-equivalence

among some Euclidean maximal operators. In •˜8.C, we shall introduce a dual

space of W- (H1(R)) as a pull-back of BMO(R) via the complex Fourier-Jacobi

transform on G (see (91) and Definition 8.7). In this paper we introduce many real Hardy spaces on G as subspaces of L1(G//K). We refer the reader to the figure in •˜ 8.D for the understanding of main relationship among them.

2. Notations

Let G be a real rank one connected semisimple Lie group with finite center

and G= KAN an Iwasawa decomposition of G. Let a be the Lie algebra of A and

F=a* the dual space of a. Let ƒÁ be the positive simple root of (G, A) determined

by N and H the unique element in a satisfying ƒÁ(H)=1. Let m1 and m2 denote

the multiplicities of ƒÁ and 2ƒÁ respectively. We put

(2)

We parameterize each element in A, a, and F as ax=exp(xH), xH, and xƒÁ(x •¸ R)

respectively. In what follows we often identify these spaces A, a, F with R and their

complexifications with C. We put A+={ax; x •¸ R+}. Then, according to the

Cartan decomposition G=KCL(A+)K of G, every K-bi-invariant functions f on

G are determined by their restrictions to CL(A) and hence, they are identified with even functions on R. We denote them by the same letter, that is, if g •¸ KaƒÐ(g) K with ƒÐ(g) •¸ R+, then

(3)

Let dg denote the Haar measure on G. We denote by L1loc(G//K) and Lp(G//K), re

spectively, the spaces of locally integrable and p-th power integrable K-bi-invariant

functions on G. Let C•‡c (G//K) denote the space of compactly supported C•‡,

K-bi-invariant functions on G. Then for f •¸ L1(G//K) the Cartan decomposition of G yields that

where

(7)

We note that the order of ƒ¢(x) is given by

(s)

where the symbol "•`" means that the ratio of the left side and the right side is

bounded uniformly below and above by positive constants. We extend ƒ¢(x) as an

even function on R. By using this weight function ƒ¢(x) the Lp-norm ||f||p of f on G can be rewritten as in (1). We denote the Euclidean Lp-norm on R by |•E| ||LP(R).

Let ƒ¶ denote the Laplace-Beltrami operator on G and ƒÓƒÉ (ƒÉ •¸ F) the normal ized zonal spherical function on G, that is, the K-bi-invariant eigenfunction of ƒ¶ satisfying

By restricting ƒÓƒÉ to A, ƒÓƒÉ(x), x •¸ R, is a solution of

(6)

which satisfies ƒÓƒÉ (0)=1 and cps (0)=0. Hence, if ƒ¿ _??_ -N, it is explicitly given by the Jacobi function of the first kind with order (ƒ¿,ƒÀ):

Clearly, ƒÓƒÉ (x) is even with respect to ƒÉ, x and it is uniformly bounded on x •¸ R if ƒÉ is in the tube domain

(cf. [3, Lemma 11]). For ƒÉ -iN, let „U(ƒÉ, x) denote another solution, which is

given by the Jacobi function of the second kind with order (ƒ¿, ƒÀ):

(7)

Then for ƒÉ _??_ Z, ƒÓƒÉ(x) has the so-called Harish-Chandra expansion:

(8)

where

(8)

We denote the Gangolli expansion of „U(ƒÉ, x), x>0, as

(l0)

For some basic properties of ƒÓƒÉ (x), „U(ƒÉ, x), and the recursive definition of ƒ¡m (ƒÉ) we refer to [3, •˜2, •˜3] and [17, 9.1.4, 9.1.5].

For f •¸ L1(G//K) the spherical Fourier transform f(ƒÉ), ƒÉ •¸ F, of f is defined by

(11)

Then f (ƒÉ) is even and continuously extended on F(ƒÏ), which is holomorphic in the interior and

For f •¸ C•‡c(G//K) the Paley-Wiener theorem (cf. [3, Theorem 4]) implies that J(A)

is holomorphic

on Fc of exponential

type.

Furthermore,

it satisfies

the inversion

formula

and the Plancherel

formula

Therefore, the spherical Fourier transform f _??_ f of C•‡c (G//K) is uniquely ex

tended to an isometry between L2 (G//K)=L2 (R+, ƒ¢(x) dx) and L2 (R+,

| C(ƒÉ)|-2 dƒÉ) (cf. [3, Proposition 3], [17, Theorem 9.2.2.13]).

For f •¸ C•‡c (G//K) we define the Abel transform Fsf, s •¸ R, of f as

(12)

Then the Euclidean Fourier transform (Fsf)•`(ƒÉ) is holomorphic on Fc of exponential type, because Fsf (x) •¸ C•‡c (R), and coincides with the spherical Fourier transform:

(13)

(cf. [11, •˜3]). Especially, F0f is even on R. The integral over N in (12) can be

(9)

WƒÐƒÊ: For ƒÐ>0 and ƒÊ •¸ C, we define W ƒÐƒÊ (f) (g), y>0, as

(14)

where n=0 if RƒÊ>0 and -n<ƒÊ•…-n+1, n=0,1, 2,•c, if R ƒÊ<0 (see [11,

(3.11)]). Koornwinder [11, Corollary 3.3] obtains that for x>0,

(15)

where A(x, s) is given by

(16)

(see [11, (2.18), (2.19), (3.5)]). We note that for 0•…ƒÁ•…ƒ¿ and 0<x•…s,

(17)

In the following, for simplicity, we denote W+(f)(x)=F1F(|x|), that is,

(18)

and for a function

F on R+,

(19)

Then W- o W+(f)=f and W+ o W-(F)=F. For f •¸ L1(G//K), W+(f) belongs

to L1 (R), because (12) and the integral formula for the Iwasawa decomposition of G yield that

(20)

(cf. [11, (3.5), (2.20)]). Hence W+(f)•`(ƒÉ), ƒÉ •¸ F, is well-defined and

(21)

For f, g •¸ L1(G//K), since f * g •¸ L1(G//K) and (f * g)•È(ƒÉ)=f(ƒÉ)g(ƒÉ) (cf. [3, Theorem 5]), it follows that

(10)

where the symbol "*" in the left and right hand sides denotes the convolution on

G and on R respectively. We denote them by the same symbol. We say that a

function F on R is W+-smooth if W-(F) is well-defined and continuous. Then,

for W+-smooth functions F, G on R with compact support such that e-ƒÏxF and

e-ƒÏx G are even, it follows that

Let (Xi,||•E||i), z=1, 2, be normed spaces and T: X1 •¨ X2 a sublinear

operator. We use the symbol ||x1||1 _??_ ||T(x1)||2 if there exist constants 0<c1•…c2 such that c1||x1||1•…||T(x1)||2<c2||x1||1 for all x1 •¸ Xl. We denote by [•E] the

Gauss symbol and by f=O(g) the Landau symbol. We use letters c, C, c1, c2,•c. to denote many different constants.

3. Maximal

functions

and

a reduction

formula

We shall introduce a radial maximal function Map f on G and give a reduction formula, which relates MƒÓ f with a maximal function of W+(f) on R. We suppose that ƒÓ is a positive C•‡, K-bi-invariant function on G, such that after identifying it with an even function on R, it is supported on [-1,1],

(23)

and furthermore, there exists M •¸ N such that

(24)

For t>0

we define a dilation

ct of

as

As mentioned in •˜1 this dilation keeps the L1-norm of ƒÓ and gives an approximate

identity in Lp (G//K), 1•…p•…•‡. We here introduce the radial maximal function

MƒÓf on G as follows.

DEFINITION 3.1. For f •¸ L1loc(G//K),

As shown in [7, Theorem 3.4 and Theorem 3.5], Mc satisfies the maximal

theorem and ||f||p•…||MƒÓf||p, if the both sides exist, for 1•…p•…•‡. Next, by

(11)

DEFINITION 3.2. For F •¸ L1loc (R),

Since W+(f*ƒÓt)=W+(f)*W+(ƒÓt)(see (22)) and W+ is an integral operator

with a positive kernel (see (15), (16)), it follows that

This fact and (20) yield a relation between MƒÓ and MRƒÓ:

PROPOSITION 3.3. For f •¸ L1loc (G//K),

In particular,

if the both sides exist.

In what follows we shall prove that the maximal operator MRƒÓ characterizes

H1(R), that is, F •¸ H1(R) if and only if MRƒÓ (F) •¸ L1 (R) (see Theorem 3.7

below). We first obtain some properties of W+(ƒÓ)•`(ƒÉ)=ƒÓt(ƒÉ+iƒÏ) (see (21)),

which guarantee that W+(ƒÓt) behaves like a dilation on R. Let M be the same in (24) and N •¸ N.

LEMMA 3.4. There exists a constant c such that for all t>0, ƒÉ •¸ R, 0•…

n•…N and 0•…k•…M,

PROOF. For simplicity, we denote ƒÕ=ƒ¢ƒÓ and ƒÕ[t]=ƒÓt, that is

(12)

We recall that, by identifying a K-bi-invariant function f on G with an even function on R, the Laplace-Beltrami operator ƒ¶ acts on f as ƒ¶(f)=ƒ¢-1(ƒ¢f')'=D•Ef'+f''

, where

(see (4), (6)). Then it is easy to see that for each k,

where Qku(x) is a polynomial with degree•…h of (d/dx)l-1D(x)•Exl(1•…l<k). We note that

Since ƒÓ is supported on [-1, 1], we may suppose that |x|•…t. Thereby, we can

deduce that |Qku(x)|•…c(1+t)k. Then the argument used in the proof of [8,

Lemma 6.5] yields the desired estimate. •

LEMMA 3.5. There exists a constant c such that for all t>1, ƒÉ •¸ R, and 0•…n•…N,

PROOF. Substituting the Harish-Chandra and Gangolli expansions of ƒÓƒÉ (see

(8) and (10) respectively) with (11), we can expand ƒÓt(ƒÉ) as

where

(13)

In order to obtain the desired estimate of ƒÓt(ƒÉ+iƒÏ), we first estimate Im(ƒÌ), ƒÌ=•}(ƒÉ+iƒÏ), in (25) and combine it with estimates for ƒ¡m(ƒÌ) and C(ƒÌ). Then we sum up these estimates of ƒ¡m(ƒÌ)C(ƒÌ)Im (ƒÌ) with respect to m.

As for Im(ƒÉ+iƒÏ), since (ƒÓƒ¢)(x)=O(x2M+2ƒ¿+1) (see(4), (24)), integration

by parts yields that for 0•…l•…2M,

and it is dominated

by

(26)

Let 0<ƒÂ<1 and set ƒÀƒÂ(p)=(l-p)+2ƒ¿+1+ƒÂ(0•…p•…1). We take a constant C

such that Jx(P)e °S -x < C for all x E R+ and 0•…p•…l. Then, since 2(m+ƒÏ)•†1 for all m, (d/dx)2((q5z)(x)x) n•Ee-2(m+ƒÏ)tx in (26) is estimated for 0<x<1 as

Therefore, the integral over [0, 1] is finite and (26) is dominated by tn-(2ƒ¿+1+ƒÂ)

(m+ƒÏ)-(2ƒ¿+l+ƒÂ). Hence, we have

As for Im(-(ƒÉ+iƒÏ)), we can repeat the exactly same process after replacing

e-2(m+ƒÏ)tx by e-2mtx However, when m=0, we have no exponential term in (26)

and thus, we cannot take ƒÀ(p) for I0(-(ƒÉ+iƒÏ)) as before. In this case we note that I0(-(ƒÉ+iƒÏ)) is nothing but the Euclidean Fourier transform (ƒÓƒ¢)•`(tƒÉ)of (ƒÓƒ¢)(x) and thus, it is rapidly decreasing with respect to |tƒÉ|. Hence, we have for

(14)

arbitrary n'•†0,

(27)

On the other hand, forƒÌ=•}(ƒÉ+iƒÏ), ƒ¡m(ƒÌ) and C(ƒÌ) satisfy the following estimates (see Proposition 8.2 in •˜8 and [3, Theorem 2]); For each n •¸ N

(28)

(29)

Therefore, letting n'=2M+ƒ¿+1/2 and substituting (27), (28) and (29) into (25)

we see that for t>1

This completes

the proof of the lemma.

0 LEMMA 3.6.

Let notations

be as above.

(1)

(A+ip)1

asItAl0,

(2) Ict(A+ip)I ? 1/2 if 0 < ~taI <2.

PROOF. (1) We note that ct(A+ip) = f0' ~(x)cpa+zp(tx}0(x)

dx and ~oZp

-1,

Hence ct (i p) =1 by (23) and it follows

from [3, Lemma 11, Lemma 14] that

We here put l(t, ƒÉ)=qt(A + ip) = W+(ct)"(a) and refer to •˜8. B. Then

(15)

N •¸ N (see Definition 8.4) and, furthermore, Lemma 3.6 (2) implies that l(t, ƒÉ)

satisfies the assumption (90) in Theorem 8.6. Since Ml=MRƒÓ (see Definitions 3.2

and 8.5), Theorem 8.6 yields the following.

THEOREM 3.7. Let ƒÓ be as above and suppose that M•†2. Then F E Hl (R)

if and only if MRƒÓ F •¸ L1(R):

4. Real

Hardy

spaces

on G

Let ƒÓ be the same as in the previous section (see (23), (24)) and MƒÓ, MRƒÓ the corresponding radial maximal operators on G and R respectively (see Definitions 3.1

and 3.2). In this section we shall define two real Hardy spaces H1ƒÓ(G//K) and

W-(H1ƒÓ(R)) on G and give a relation between them. Especially, we shall give a

simple proof of Theorem A based on fractional calculus.

We introduce the real Hardy space H1ƒÓ(G//K) on G as follows.

DEFINITION

4.1. We define

and ||f||H1ƒÓ(G)=||MƒÓf||1.

Since |f|1•…|MƒÓf|1, it follows that

Let Ms, s•†0, denote the Euclidean Fourier multiplier corresponding to (ƒÉ+iƒÏ)s:

We introduce a pull-back of H1(R) to G via W+ (see (18)) and Ms:

DEFINITION 4.2. For s•†0, we define

and give the norm by |Ms| o W+(f)||H1(R). We denote W-(M0(H1(R))) by

W-

(H1(R)).

An easy consequence

of Proposition

3.3 and Theorem

3.7 is the following.

COROLLARY 4.3. Let M•†2. There exists a positive constant c such that

(16)

We shall conversely control the H1ƒÓ-norm of f by using W+(f). Before starting

the argument we shall obtain some basic properties of the Euclidean fractional

integral operators WRƒÊ and WRƒÊ on R+, which correspond to the case of ƒ¿=, ƒÀ= -1/2 and ƒÐ=1 in (14) and (92) respectively: For ƒÊ •¸ C and y>0,

(30)

and

(31)

where n=0 RƒÊ>0 and -n<RƒÊ<-n+1, n=0, 1, 2, •c , if RƒÊ•…0. Let

0•…a<1 and let f be an integrable function on R+. Then, by changing the order of integration and by using integration by parts, we see that WRƒÊ o W R-ƒÊ(f)=f if f'(x)x-ƒÊ is integrable, and WR-ƒÊo WRƒÊ(f)=f if f(x)xƒÊ-1 is integrable. Moreover, if f, g are integrable functions on R+ such that f(x)xƒÊ-1, g' are integrable and

g(0)= limx•¨•‡(x)=0, then

(32)

LEMMA 4.4. Let f, f' be integrable on x•†0 and 0•…i<1. (1) For ƒÁ<0,

where BƒÁ(x, s) is smooth on 0<x•…s and there exists a constant c>0 such that

(33)

(2) For ƒÁ>O,

where C(x, s) is smooth on 0<x•…s and there exists a constant c>0 such that

(34)

(3) Let g be a smooth function.

(17)

where Dg (x, s) is smooth on 0<x•…s and there exists a constant c>0 such that

Especially, if g is supported on [0, 1] or g is constant on [1, •‡), then D9 (x, s)=0 for t•…x•…s.

(4) For ƒÁ>0,

where C(x, s) is smooth on 0<x•…s and satisfies (34).

PROOF, (1): We note that

where

Since ƒÁ-1<0 and (tƒÁ-xƒÁ)/(t-x)•…ƒÁxƒÁ-1 for 0<x<t, it follows that

(18)

where

Therefore,

(3): Since

it follows that

(35)

where

Then the desired result follows.

(4): We put C(x, s)=eƒÁx(e-ƒÁs-e-ƒÁx)(s-x)-ƒÂ-1 in (35). Since

(19)

We shall deduce the local and global forms of the Weyl type fractional operator

W- in (19). In what follows, for simplicity, we put

(36)

Furthermore,

we denote

(37)

and

(38)

Clearly, from the explicit values of ƒ¿, ƒÀ (cf. [5, Table 1] in p. 265), we may suppose that ƒÂ=0. However, for the sake of the Fourier-Jacobi analysis (cf. [4], [11]) we dare not take ƒÂ=0.

PROPOSITION 4.5. (1) 1f F is W+-smooth and supported on 0<x•…1, then

where the sum is taken over 0<m•…n and ƒÌ •¸ D, and A1m,0 (x, s)•ß0 and

A1m.ƒÌ(x, s) satisfies

(39)

(2) If F is W+-smooth, supported on x•†1 and F, F' are integrable, then

| W-(F)(x)|

(20)

(39) and there exists a positive constant c such that for j=3, 4,

(40)

PROOF. We shall consider the case of 0<ƒÂ,ƒÂ'<1. Other cases easily fol low from the same argument. In the following we use the same letters BƒÁ(x, s) and

C(x, s) to denote different functions satisfying (33) and (34) respectively.

(1): Let F be differentiable and supported on [0, 1]. For ƒÐ•†0, Lemma 4.4 (1) yields

and for p •¸ N, Lemma 4.4 (3) gives

Therefore,

combining

these equations,

we have

Since ID-p(x,s)I

x<

_ cx_p-1, IB_s(x,s)I < cx-s-1 and

for 0<x•…s, it follows that

(41)

(21)

and

Since

for 0<x•…s (see [10, Lemma 3.1, Lemma 3.3]), it follows that

(42)

Now let ƒÁ=-(ƒ¿-ƒÀ)=-(n+ƒÂ) and ƒÂ'=-(ƒÀ+1/2)=-(n'+ƒÂ') (see

(36)). Since WƒÐ-1(F)(x)=dF/d(chƒÐx)•`F'(x)/x, it follows that WƒÐ-n(F)(x)•`

ƒ°nm=0 n_m)F(m)(X) ~

_-(and thus

(43)

(44)

Here we note that

(45)

Then, combining (43), (42) and (44) in this order, we finally obtain that

(22)

(46)

Since 0•…x•…1, we can replace xm and B-2sƒ¿+ƒÂ+m by xl and B-2sƒ¿+ƒÂ+l respec

tively, the desired result follows.

(2): Let F be differentiable and supported on [1, •‡). We keep the notations in (1). It follows from Lemma 4.4 (2) that

Then, by substituting F with e-ƒÌxF, Lemma 4.4 (4) yields that for ƒÌ>0,

(47)

Let fi(x) be an even C•‡ function on R such that 0•…„U(x)•…1, „U(x)•ß1 if

|x|•… 1/2, and „U(x)•ß0 if |x|•†1. We put

(48)

We apply WƒÐ'ƒÂ', to each term in (48). We first substitute (47) into (48) and, to each resultant term we apply Lemma 4.4 (3) and (47) with a, b, replaced by ƒÐ', ƒÂ,ƒÌ+ƒÂƒÐ:

(23)

where, if we denote the first term in the right hand side of (49) by Il(x)(1

then K1(x) is given by K1(x)=•çsxIi (s)D1-ƒ³(x, s)ds. Here, to deduce the last

line, we used Lemma 4.4 (3) and the fact that •çtxC(s, t)D1-ƒ³(x, s) ds satisfies (34). Similarly, we have

(50)

where, if we denote the first term in the right hand side of (50) by 12(x) (1 -ƒ³) then K2(x) is given by K2(x)=•çsx I2 (s)D1-ƒ³(x, s) ds. In this case, to deduce the

last line, we used Lemma 4.4 (3), (32) and the facts that WS a W R , (C) N C (see

[l0, Lemma 3.1]) and •çtxC(s, t)C(x, s) ds satisfies (34). Therefore, we see that

x

(51)

As for (W(exF)I), ~s we use (47) and note that W ~~ (e_ F) J~ is

supported on [0, 1]. Hence, (46) with ƒÁ, ƒÏ replaced by 0, ƒÌ ƒÂƒÐ respectively yields

(24)

(52)

(53)

where, if we denote the first term in the right hand side of (52) by I3(x)ƒ³, then K3 (x) is given by K3(x)=f ~ I3 (s)D~ (x, s) ds. Here, to deduce the last line, we used Lemma 4.4 (3), WS oW S, (C) N C and the fact that I f C(s, t)B_b' (x, s) ds) < cx-s' -1. Combining (51) and (53), we can finally deduce that

w, oWfs (eF)

(x)

We substitute this formula to (45) and note that

for x•†1. Then desired result follows from the same argument used in the previous case.

Now let f •¸ W-(M-sƒ¿H1(R)), where sƒ¿=ƒ¿+1/2=n+ƒÂ(see (37)).

Since the Fourier multiplier M-sƒ¿ satisfies the Hormander condition (cf. [16, •˜5

in Chap. 11]), it is bounded on H1(R) (cf. [16, Theorem 4.4 in Chap. 14]). In

particular, F=W+(f) belongs to H1 (R). We note that

(54)

(cf [13, (4.39•Œ)]) and thus, M; o WR , 0 < < sa, is the the Fourier multiplier

(25)

W R is bounded on H1(R). Hence, each WR-ƒÁ (F) also belongs to H1(R). Therefore,

the condition that f •¸ W-(M-sƒ¿H1(R)) guarantees that for 0•…Y•…sƒ¿,

Comparing MRƒÓ (WR-ƒÁ(F)) with MƒÓf, we have the following inequalities.

THEOREM 4.6. Let ƒÓ be as in •˜3 and M•†2. For f E W- (M-sƒ¿ H1(R)) we

put F=W+(f ). Then there exist c1, c2 such that for all 0•…ƒÁ•…sƒ¿,

Especially,

and thus, W-(M-sƒ¿Hl(R)) •¼ H1ƒÓ(G//K).

PROOF. We denote sƒÕ=W+(ƒÓt), where ƒÕt is not a dilation of , however,

as shown in the previous section, W+(ƒÓt) has the same properties as a dilation. Therefore, we use this notation to abbrebiate W+(ƒÓt) . Let ƒ³(x) be the same as in the proof of Proposition 4.5. We decompose f * ƒÓt as

(see (22)). We apply Proposition 4.5 (1) to F * ƒÕt•Eƒ³. Then Lemma 4.4 (3) and the same argument used in the proof of Proposition 4.5 yield that

(26)

Similarly, applying Proposition 4.5 (2) to F *ƒÕt•E(1-ƒ³), we have

We recall that Am e (x, s), j=3, 4, satisfy (40), and for j=1, 1,2 and 0<x•…s•…1, Am (x, s)o(x) •…m+-1 (see (39), (4)) and thus,

Therefore,

we can deduce

that

As for the first inequality

in the theorem

we recall that

(27)

(see (15), (17)). Hence,

We take the supremum over 0<t<•‡. Since 2ƒ¿-ƒÁ>2ƒ¿-sƒ¿=ƒ¿-1/2>-1, we can deduce that,

This completes the proof of the theorem. •

REMARK

4.7. (1) Since

(see [3, Theorem 2]), Theorem 4.6 means that

Theorem A in •˜1 follows from this relation and the one in Corollary 4.3.

(2) Since (thx)m+ƒÌ is bounded, it is easy to see that MRƒÓ in Theorem 4.6 can be replaced by Ml, l •¸ AN,2M (see Definitions 8.4 and 8.5, and cf. Theorem 8.6). Especially, H1ƒÓ(G//K) does not depend on an individual q. We shall skip writing q and denote simply as

We keep the notations in the proof of Theorem 4.6 and we shall suppose that t•†1. Since we can transfer the Fourier multiplier WR-ƒÁ as

(28)

where d=c(1+(4n-1)C).

We note that

(55)

Since t•†1, Lemma 3.5 implies that each WR (fit)"(a), 0<'y<sa c + 1/2,

belongs to AN,2M (see Definition 8.4). Moreover, if ƒÁ>0, then W(b)A) -~ 0

as ItAl -' 0, because it has the term (tƒÉ)ƒÁ in (55). Since ()(A) zit"= ~t(A + ip) satisfies Lemma 3.6 (2), we may suppose that for a sufficiently large C>0, each

£m+(t,

A)= W (m+e)

(zl)t)"

(A) Czb (a) also belongs

to AN,2M

and satisfies

the

assumption (90) in Theorem 8.6. Hence, Proposition 3.7 and Theorem 8.6 yield

that, if M•†2 in (24), then

(56)

(see •˜8.B for the definitions of Mom+~ (t,.) and Mm+ ). We here define a truncated maximal operator MlocƒÓ on G as follows.

DEFINITION

4.8. For f E Ll (G//K),

Then (56) implies that ||f||H'(G)•…||MloxƒÓf||+c||F||H1(R)•E Hence, Corollary 4.3

yields the following.

(29)

5. Atomic

Hardy

spaces

on G

In this section we shall introduce some K-bi-invariant atoms on G and define the corresponding atomic Hardy spaces on G, on which the radial maximal operator MƒÓ is bounded to L1(G).

For x •¸ R and ƒÁ>0, let R(x, r) denote the interval on R centered at x with radius r and |R(x,r)| its volume with respect to ƒ¢(x) dx:

(57)

For x •¸ G and r>0, we also denote the annulus {y E G; ~cr(x) °(y) I <r} = K{as; s E R(a (x), r)}K on G by the same notation R(x, r). Clearly, R(x, r)=

R(aƒÐ(x),r) for x e G. We put B(r)=R(e, r) if x=e. Obviously, if a(x)•†r,

then the volume |R(x, r)| of R(x, r) with respect to dg coincides with |R(ƒÐ(x), r)| (see (57)) . Moreover, if a(x)•…r, then R(x, r)=B(ƒÐ(x)+r).

DEFINITION 5.1. We say that a K-hi-invariant function a on G is a (1,•‡,0) atom provided that there exist x •¸ G and 0<r•…ƒÐ(x) such that

(i)

(ii)

₍₅₈₎

(iii)

For ƒÃ•†0, we say that a is a (1,•‡, 0, ƒÃ)-atom if we replace (ii) by

(ii

)ƒÃ

(59)

and a (1, •‡, +)-atom if we replace (iii) by

(iii)+

(60)

Moreover,

if x=0,

we call a a centered

atom.

We introduce

atomic

Hardy

spaces

on G as follows.

DEFINITION

5.2. We define

and ~IIMH'

0(G) = inf where the infimum is taken over all such represen

tations f= >Similarly, replacing (1, •‡, 0)-atoms by (1, •‡, 0, ƒÃ)-atoms and

(30)

DEFINITION 5.3. We define the small Hardy space h~ _,0(G//K) _on _G _by

restricting (1, oo, 0)-atoms in the above definition of H~,0(G//K) to ones with

radius •…1.

Clearly,

we have

In what follows we shall characterize the difference between h1•‡,0(G//K) and

Ham' o (G//K), and then we shall obtain a relation between H1,+•‡,0(G//K) •¿

W-(Hl (R)) and H1ƒÓ(G//K). For x0 •¸ G and r>0, let xxo,r denote the char

acteristic function of R(x0, r) on G. For x, y e G, we define

(61)

Then Ix0 (x, r, y) is K-bi-invariant with respect to x, y and, as a function of y, it

is supported on R(x, ƒÐ(x0)+r), where R(x,ƒÐ(x0)+r)=B(ƒÐ(x)+ƒÐ(x0)+r) if

ƒÐ(x) •…ƒÐ(x0)+r. For simplicity, when x0=e, we skip writing the suffices x0=e:

LEMMA 5.4. Let x, y •¸ G and ƒÐ(x)>r. Then

(1) I(x>r,y) ~ j(x,r,x),

(2) ~B(r) I I R(x,r) J-1 < I (x, r, x) < IB(2r)I I R(x,r) 1-1,

(3) I~o(x, r, y) C IB(2r)~ ~R(x,r)~-1 if r > 1, a(x) > 1, v(xo) > r + 1.

PROOF. We regard I (x, r, y) as a function on R+•~R+. For a fixed x,

as a function of y, it is supported on |x-y|•…r and I(x, r, y) is increasing on

x-r•…y•…x. Hence (1) is obvious. As for (2), let f be an arbitrary function in

L•‡(G//K). Then

Therefore, letting f •ß 1, we see from (1) that

(31)

and thus, I (x, r, y) < IIXrIIiIIfIooIR(X,

~

fx y ~)

( -1 < IB(r)I (R(x, Ix y I) E1. Hence

As for (3), we may suppose that x0, x, y •¸ A+ _??_+ R+ and we use the kernel form of the product of spherical functions:

(see [4, (4.2)]). Since x•†1, x0-r•†1, it follows that

We set

(62)

where ƒÔ1 is the characteristic function of B(1), and for each (not necessary K-bi invariant) function f on G, we define a K-bi-invariant function f•óx, x •¸ G, as

(63)

PROPOSITION 5.5. For f •¸ H1,+•‡,0(G//K) there exist f0 •¸ h1•‡,0 (G//K) and

xi E G, )i E R such that

where 1110

Mh0(G)

~

_,and

_{> Iii I are both bounded by III IIH0(G)•}

PROOF. By Definitions 5.2 and 5.3, it is enough to obtain the above decom position for a (1, •‡, +)-atom a on G with radius r•†1. Let B(aƒÐ(x),r), x •¸ R+, denote the support of a. We identify K-bi-invariant functions on G with functions on R+, so a is supported on R(x, r). Without loss of generality we may suppose

(32)

that x, r •¸ N. Actually, since r•†1, |R(x, r)| N IR([x], [r]) I uniformly on x•†0 and r>1. We decompose R(x, r) as

and set ak=lIkI'IR(x,r)I

r.

aII

Then, ak is supported on Ik, IlakII00

< (IkI-1

and

Here >1k /tk N 1. Let 8k = 9ax_r+k (0 < k < r), where ax-r+k is the element in A+

corresponding to x-r+k •¸ R+ (see (3), (62), (63)). Clearly, ƒÆk is supported on

Ik, fOk(g)dg= G fO(g)dg G=1, and II9k ~~0 < Ilk' I-by Lemma 5.4 (1), (2). Let

mk = fG ak(g) dg and bk = (ak mkOk)/2. Then ImkI•…1, bk is supported on Ik,

bk (I 0 < I 1k1', -and fbk(g)dg G=0. This means that bk is a (1, •‡, 0)-atom with

radius

1. Therefore,

letting

we have the desired decomposition of a (1, •‡, +)-atom a with r•†1. •

We set

Then

Proposition

5.5 means

that

(64)

Next we shall restrict our attention to centered atoms a and consider a linear combination of their averaged translations a•óx, x •¸ G (see (63)).

DEFINITION

5.6. We define

and I If II w 0(G) = inf > I )\ , where the infimum is taken over all such representa tions I = ~i a2abxi . Similarly, replacing centered (1,•‡,0)-atoms ai by centered

(33)

Real Hardy spaces on rank 1 sernisimple Lie groups 313

(1, •‡, 0, ƒÃ)-atoms, centered (1, •‡, +)-atoms and centered atoms with radius•…1

respectively, we define H0(G//K), H~ o(G//K), h~,0(G//K).

In this definition each atom ai is K-hi-invariant (see Definition 5.1). We shall

consider non-K-hi-invariant cases. Let ai be a centered function on G satisfying

(58) to (60), not necessary K-hi-invariant. Even if ai is not K-hi-invariant, a?,x2 is K-hi-invariant. Hence, by using such ai, let H ~,0 (G//K) q denote the space of all f = >I ~2ab,x2 with x2 •¸ G and > X1 <00. Similarly, we define H }o(G//K)q,

H o (G//K) a and h~ _{,o (G//K)} _{a respectively} _as _in _Definition _5.6.

PROPOSITION 5.7. Let ƒÃ>0. We have the following inclusions:

PROOF.

The

horizontal

inclusions

are clear from

the definitions.

We first

prove that

each space in the first line contains

the bottom

one in the second

line.

LEMMA 5.8. Let a be a K-bi-invariant function on G supported on R(x, r)

and IIatI < fR(x, r)1-l. Let r0=2r if r•…1 and r0=r+1 if r>1. Then there exists a constant c>0 such that

where c is independent of x, r.

PROOF. If r•…1, then Lemma 5.4 implies that for |x-g|<r,

Similarly, if r>1, then I(x, r0, y)•†I(x-r,1,x-r)•†|B(1)||R(x-r,l)1-1

e-ZP~x-T> > ~B(r) IIR(x,r)I'. -Thereby, ~Ia/1(x,ro, ')H oo <cfB(r)I-i •

We recall

the relation

H0 (C) = HJbo

(G) obtained

in [9, Theorem

5.5].

Here H0(C) is nothing

but H~

o(G//K) in this paper and HO(G) contains

H~ o (G//K) by Lemma

5.8. Then, it is easy to see that each space

in the first

(34)

line contains the bottom one in the second line. Furthermore, since I (x, r, y)•… c R(x, r)f-1 if r < 1 by Lemma 5.4, it follows that h~

,o(G//K)a C

we shall prove that

each space

in the second

line contains

the bottom

one in the third

line.

H~

,o(G//K) C H~,o(G//K): Let a be a centered (1, •‡, 0)-atom on G sup

ported on B(r) and x •¸ G. Then a is supported on R(x, r), fG ax(g) dg =

fa(g)dg G = 0, and ~IaII x ~ < `R(x, r)1"1 IB(2r)I ~B(r)I -1 by (61), (63) and Lem ma 5.4 (2). Therefore, if r•…1, then ~~a~~~ < c IR(x, r) I'. This means that ca

is a (1, •‡, 0)-atom on G. Let r>1 and ƒÐ(x)•…2. We may regard that a•óx is

supported on B(ƒÐ(x)+r) and we note that IIaII x ~ < (B(r)1' < c JR(or(x) + r)~-1 Therefore, c-1 a•óx is a .(1, •‡,0)-atom on G. Let r>1 and ƒÐ(x)>2. In this case, since a is centered, a has an L1 non-increasing denominator IB(r)L'XB(r)(x)

(see [7, 4.4]). Then a can be decomposed as a=> a2 a2, where ~2 (~2 N 1

and each a2 is a (1, •‡,0)-atom supported on R(xi,1), xi •¸ N and 0<xi•…r (see the proof of [7, Theorem 4.5]). Hence ab,x is supported on R(x, x2 + 1) and IIa,II bx ~o <_ IR(x~~ 1)f-1 fB(2)J IR(x, 1)1-1 c fR(x, x2 + 1)1-1 by (61), (63) and

Lemma 5.4 (3). Therefore, each c-lab x2 is a (l, •‡, 0)-atom on G. These ob

servations imply that a e H 0 (G//K) and its norm is bounded by a constant

independent

of a. Hence the desired

inclusion

follows.

h1~ _,o(G//K) _{C h1~,o(G//K):} _This _{is clear} _from _the _case _of _r•…1 _{in the} _above

argument.

H

,o(G//K) C H 0(G//K): Let r>1 and a be a centered (1, •‡, 0, ƒÃ)-atom on G. We repeat the above argument. Since ||a||•‡ has an extra decay r-ƒÃ, JaJIx

in a(x) < 2 and in u(x)>2 also have the same extra decay. Since

r-E < c(a(x) + r)-E in cr(x) < 2 and r-E < xi < c(xi + 1)~E in or(x) > 2, it follows that ax and ab_{,x are} _{(1, •‡,} _{0, ƒÃ)-atoms} _on _{G up} _to _{a constant} _{multiplication.} _Hence

the desired

inclusion

follows.

H o(G//K) C Ham' o(G//K): Let r>1 and a be a centered (1, •‡, +)-atom

on G. Then Proposition 5.5 means that a is decomposed as a= > aZai, where

> ~~12) N 1 and each a2 is a (1, •‡, +)-atom supported on R(xi,1), xi •¸ N and |xi|•… r. As in the previous case, since R(xi,1) has the radius 1, each x e G, is also a (1, •‡, +)-atom on G up to constant multiplication. Hence the desired inclusion follows. •

Let ƒÓ be the K-bi-invariant function on G introduced in •˜3 and MlocƒÓ C the truncated maximal operator in Definition 4.8. Since H 0(G) in [9] coincides with H~ o(G//K)a, it follows from [9, Theorem 5.3] that

(35)

PROPOSITION 5.9. Let M•†2. MƒÓ is bounded from H o (G//K) fl

W_ (H1(R)) to Ll (G//K), that is, there exists a constant c>0 such that

for all f E H~

a(GlIK)# n W_(H'(R)) and thus,

THEOREM 5.10. Let M•†2. MƒÓ is bounded from H~lo(G//K) to L1(G//K),

that is, there

exists a constant

c>0

such that

for all f •¸ H f0 (G//K) and thus,

PROOF. Since H~1o(G//K) C H~ a(G//K) (see Proposition 5.7), it is

enough to prove that H f 0 (G//K) C W_ (H1(R)) (see Proposition 5.9). We shall prove that, for all (1, •‡, 0,1)-atoms a on G, ||W+(a) ||H1(R)•…c, where c is inde pendent of a.

LEMMA 5.11. Let ƒÁ •¸ R and ƒÐ>0. If F is supported on R(x0, r) and

smooth if ƒÁ <0, then W,y (F)(Ixl) is also supported on R(x0, r).

PROOF. When ƒÁ=-n, n •¸ N, WƒÐ-n is a differential operator (d/dch ƒÐx)n

and thus, the desired result is obvious. Hence, it is enough to consider the case of

ƒÁ >0. We denote F0(x)=F(x+x0). Then it follows from (14) that

(36)

where

and G1(x) is defined by replacing sh s ds with ch s ds. Since F0 is supported on

B(r), clearly G0 is also supported on B(r). Therefore, WƒÐ-ƒÂ(F) is supported on

R(x0, r). •

Let a be a (1, •‡, 0, 1)-atom supported on R(x0, r). By (59) and (58), ||a||•‡

~R(x0, r)f~1(1-I-r)-1 and •çG a(g) dg=0. We put A=W+(a). Then A is supported

on R(x0, r) by Lemma 5.11 and

Moreover, we recall that |A(x)|<ceƒÏ1th(x0+r)2sƒ¿ ||a||•‡ (see (15), (17) and

cf. [10, Lemma 3.3]).

Case I: x0-r•†1. Since A is supported on R(x0, r) and

it follows that IA(x)I < ce2p(x0+r) (62Px0 sh r)-1(1 + r)-1 ~ cr-1. Case II: x0-r<1 and r•†1. Since x0+r•†1,

Therefore, as in Case I, we have IIAIIOO cr-1.

Case III: x0-r<1, r<1 and x0>2r. Since x0>2r, it follows that

x0+r•†3 and thus

Then, since (x0+r)/(x0-r)•…3, IA(x)I < cth(xo + r)sa ((xo r)sar)-1•…cr-1.

Case IV: x0-r<1, r<1 and x0•…2r. Since x0+r•…3r<3 and

IR(x0, r) J > IB(r)I I B(3r) I , we may suppose that a is a centered atom supported on B(3r). Then IA(x)I < c(th 3r)Sa IB(3r) I-1 < cr~"1

These four cases imply that cA is a (1, •‡, 0)-atom on R and c is independent of a. •

(37)

THEOREM 5.12. Let M•†2. Then MƒÓ is bounded from H1•‡

,0(G//K) to L1(G//K), that is, there exists a constant c>0 such that

for all f •¸ H1•‡,0(G//K) and thus,

PROOF. Since MƒÓ is sublinear and iIMfIIi ~ ~ < IIMf ~ lii, in order to obtain the H _,0-L' _boundedness _{of MƒÓ, it is enough} _to _show _that _there _exists _{a constant} c>0 such that (tM~aIIi<c for all centered (1, •‡, 0)-atoms a on G. Let B(r) denote the support of a. Here we recall Theorem 4.5 in [7] and the proof. Since I B(r) (-ixs(r) (x) is an L1 non-increasing denominator of a, a has a (1, •‡, 0)-atomic decomposition a=> Aiai on G such that > i a2 < c, where each atom ai has radius ri•…1. This means that ai is a (1, •‡, 0,1)-atom on G. Hence, it follows from Theorem 5.10 that IMajM ~< c and thus, ~IMaIIi ~< c, where c is independent of a.•

6. Atomic decomposition of H1 (G//K)

We shall prove that each function f in H1(G//K) has a (1, •‡, +)-atomic

decomposition on G. This means that H1(G//K) C H1,+•‡,0(G//K) and then our

main Theorem C in •˜1 follows. In the following, first we shall introduce a space W_ (H~ o (R) a) and give a (1, •‡, +)-atomic decomposition on G for this space.

Then, we shall prove that H1(G//K) C W_(H~ p(R)a) and obtain the desired

(1, •‡, +)-atomic decomposition for H1(G//K). We set

(65)

and define H1,+•‡,0(R)ƒ¿ as the space of all F=> a2 A2 such that > ~ a2<•‡ and each Ai satisfies

(i)

(ii)

₍₆₆₎

(iii)

(38)

PROPOSITION 6.2. Functions in W-(H~+o (R) a) have (1, •‡, +)-atomic de

compositions, that is, W-(H~ o(R)a) C H~ o(G//K).

PROOF. Let f •¸ W- (H~ o(R)a) and F=W+(f)=> i ) AZ the decompo

sition of F given by (66). By using the same argument as in the proof of Proposi tion 5.5 (see (64)), we may suppose that ri•…1 in (66). More precisely, when ri>1, we decompose the support of Ai by using a smooth decomposition of 1, where each

piece is supported in the interval with radius •…1. Then Ai=ƒ°jAij and each Aid

satisfies (66) with radius •…1 by Lemma 4.4 (3). Hence, we may suppose ri•…1 to

begin with.

When lxii < 2ri, Ai is supported on B (r'i), r'i=lxii + ri < 3ri, and satisfies

(ii) and (iii) with r'i<3, because

Hence, we may suppose that xi=0 with ri<3 or |xi|>2ri. We recall that

e-ƒÏxW+(f)(x)=F0f(x) is an even function on R (cf. [11, (3.6)]). Therefore, in the

decomposition e-ƒÏxF=>11i aie_PxAi, the counterparts eƒÏxAi(-x) must appear in

the decomposition. When xi=0 with ri•…3, we may suppose that e-ƒÏxAi(x) is

even. Actually, we let

Then Bi is supported on B (r) and e-ƒÏx Bi (x) is even. Moreover, if ri<1, then f Bi(x)dx = 0, because Ai satisfies (iii) originally and the even property

of e-ƒÏxF(x) implies that fCC e2PxAi(-x) dx=0. Since ri•…3, Bi also satisfies

(ii) and 2(1+e2ƒÏri)•`1. Therefore, replacing the left hand side with the right one,

we may suppose that e-ƒÏxAi(x) is even if xi=0 with ri•…3. Thereby, we can

rearrange the decomposition of F as

where each Ai satisfies (i), (ii) with xi=0, ri•…3, f A(x) dx=0; each Bj satisfies (i) to (iii) with ix3 I > 2r3, r3 < 1; each Ek satisfies (i), (ii) with |xk|•† 2rk, rk•†1, and moreover, > + > I,a I + > h'k I<•‡. Since F is W+-smooth, finally, we

have

(67)

where ai=W_ (Ai), bj=W_ (Bj) and ek=W-(Ek) . Lemma 5.11 implies that

each ai, bj, ek have the same supports as Ai, Bj, Ek respectively.

(39)

Now we apply fractional calculus in [10] to estimate each ai, bj, ek. For simplic ity, we skip writing the suffices i, j, k and denote the supports of a, b, e by R(x0, r). Without loss of generality, we may suppose that x0•†0.

As for e, since e is supported on R(x0,1) and x0•†2, it follows that x0-1•†1

and thus, da(x0,1)•`1. Thereby, (ii) and [10, Lemma 3.3]1 imply that on the

support of e

This means that c-ie is a (l, •‡, +)-atom on G.

As for b, we recall that

x0=0

or x0>2r.

Case I. x0-r•†1: Since x0-r•†1, dƒ¿(x0,r)•`r. Thereby, (ii) and [10,

Lemma 3.3] imply that on the support of b

This means that c-1 b is a (1, •‡, 0)-atom on G.

Case II. x0-r<1: Since r<1 and x0>2r, it follows that x0<r+1<2,

x0-r>x0/2, and x0 +r<3x0/2<3. Therefore, da(x0, r)•…c(x0-r)sƒ¿r and

thus, on the support of b

Since (x0+r)/(x0-r)•…3, it follows that

Therefore, |b(x)|•…c (R(x0, r)-1 on the support. This means that c-1b is a

(1, •‡, 0)-atom on G.

As for a, since x0=0 and r<1, it follows that da (0, r) •` rsƒ¿+1 and

(68)

We put

Clearly, |a(x)|•…a+(x) and a+ is a non-increasing function on R+ with finite

L1-norm:

1 If f(x)=O(x2r) around x=0, then WƒÐ-ƒÊ(f)(x)=O(x2(ƒÑ-RƒÊ)). T-However, under the assumption (ii) of (66), it follows that WƒÐ-ƒÊ(f)(x)=O(x-RƒÊ).

(40)

Since a is supported on B(r) and •çGa(g)dg=•ç•‡-•‡ A(x)dx=0, it follows that IB(s)1' -f80° a(x)a(x) dx is also supported on B(r) and

(69)

Here we used (68) and |B(s)|•`ƒ¢(s)s if s•…r•…1 (see (4)). Hence,

(70)

This means that ca+ is an L1 non-increasing denominator of a satisfying (70). Then

[7, Theorem 4.5] yields that a has a centered (1, •‡, 0)-atomic decomposition a=

> ~y3a1 on G such that > I~~ (< c IIa+IIL1(Z) < cc0. Especially, a •¸ H1•‡,0 (G//K) and ~Ia!IH1 0(G) cc0.

These three cases imply that all ai, bj, ek in (67), and thus f also, belong to H~ o(G//K). This completes the proof of the proposition. •

THEOREM

6.3. Let notations

be as above.

Then

PROOF. Because of Proposition 5.9 and Corollary 4.3, it is enough to prove

that H'(G//K) C Ham' o(C//K). First we shall consider the case that ƒÂ=ƒÂ'=0, that is, sƒ¿ is an integer (see (37)).

Let f •¸ H1 (C//K) and put F=W+(f). Then it follows from Theorem 4.6

that (I M o W R (F) (x) (th x) S« IL1(R) <•‡. We here recall the constructive proof of the atomic decomposition of H1 (R): For example, we refer to [12]. Then, mea

suring the volume of the dyadic cubes (intervals in the one-dimensional case) Qkj

appeared in the Whitney decomposition theorem by (65) and keeping the convolu

tion and the dilation, we can easily deduce that W R a (F) has a (1, •‡, sƒ¿ )-atomic decomposition with respect to I |th x|3ƒ¿ dx:

where Bi is supported on R(xi, ri), f°° B2(x)x' dx = 0, 0 < < s~, IIB2II~ <_ da(xZ, r2)-1, and > IAZI<•‡. We set

Since sƒ¿ is an integer and each Bi satisfies the sƒ¿-th moment condition, it follows

(41)

| Bi||•‡•…dƒ¿(xi, ri)-1. Therefore, Ai satisfies (66) and thus, F •¸ H~ o (R)ƒ¿. Hence Proposition 6.2 implies that f •¸ H~ o(G//K).

We shall remove the assumption that ƒÂ=ƒÂ'=0 and consider a general case

of 0•… ƒÂ,ƒÂ'<1. We recall the Fourier-Jacobi analysis (cf. [4], [11]) and note that all results obtained in the previous sections can be extended to the Fourier-Jacobi analysis, that is, for arbitrary ƒ¿, ƒÀ•†-1/2. Especially, we may denote W•}=Wƒ¿,ƒÀ•} and replace "G//K" as "(a,ƒÀ)" such as

For f •¸ H1 (ƒ¿,ƒÀ), we set

where ƒÏ=a+ƒÀ+1 and

We note that

Theorem

4.6 means

that

and thus, (f) belongs to H1(n, n'). Then, since n, n' are integers (see (36)), S(f) belongs to H~ o (n, n') and it has a (1, •‡, +)-atomic decomposition with respect to (n, n'):

where > i Pi I <00. Here

and each Ai satisfies (66) with respect to (n, n'). We here recall that F is orig

inally W+-smooth. Hence, from a constructive proof of atomic decompositions

W (n+n,) (F)=>i ) Bi (cf. [6, B in Chap. 3]) it follows that AZ satisfies (66) with

respect to (ƒ¿, ƒÀ). We let bi=W- (W+n,n'+ (ai)). Each bi has the same support

as ai by Lemma 5.11 and •çG bi(g)dg=A•`i(0)=0 if ri<1. Moreover, we see

(42)

If ||bi||F < c fR(x, iri)I -1. Thereby, we see that f=> is a (1, •‡, +)-atomic decomposition of f. •

THEOREM 6.4. Let ƒÃ•†0. Then H~Eo(G//K) fl W_(Hl(R)) is dense in

W-(Hl(R)). Especially, Hl(G//K) is dense in W-(Hl(R)).

PROOF. Let f •¸ W- (Hl (R)). We approximate f with a rapidly decreasing

function in W-(H1(R)). For example, let us take f1=f *ƒÓ t (see •˜3). Since W+(f1)

=W+(f) * W+(ƒÓt) and W+(ƒÓt) satisfies Lemma 3.6, it is easy to see that F1=

W+(fl) •¸ H1(R) and ||f-f1||H1(G)=||F-F1||H1(R)=ffF-F*W+(ct)lfHl(R) -~ 0

as t -* 0 (cf. [15, Chap. 3, 5.1 (d)]). We fix a sufficiently small t>0 and let

F1=W+(f * ƒÓt)=>iEI ~12A2 denote a (1, •‡, 0)-atomic decomposition of F1. For

a sufficiently small k> 0 and a finite large index set J •¼ I, we put

(71)

where we take K and J for which e-ƒÏxF0(x) is even. We here recall a constructive proof of the atomic decomposition of H1(R) (cf. [12] and [6, B in Chap. 3]). Since F, is smooth, each Ai is also smooth and if i •¸ J0, ri•…1, then

and if i•¸J0, ri•†l,then

(72)

In particular,

IIWR

_aAilloo

_{< R(xi,ri)-1 if supp(Ai)}

_{= R(xi,ri). Hence,}

_{each Ai,}

i E Jo, ri < 1, satisfies

(66)

and thus,

Fo E H~ o(R)a. Then,

by Proposition

6.2,

we

can define

f o E H~ o (G//K) such

that W+

(f o) = Fo. Since

Fo E H1(R),

it follows

that fo E H~+o(G//K)

fl W(H1(R)) and Iff, folf

Hl(G)

=IIF, FoffHl(R)

* 0

if K -~ 0 and J --~

I. This means

that H~ o (G//K) fl W_

(H1(R))

is dense

in

W_

(H1(R)).

Since

(see Proposition 5.9 and Corollary 4.3), in order to obtain the desired density it is enough to show that f o belongs to Ho (G//K).

We keep the atomic decomposition of F0 in (71). Let Ii=R(xi, ri) denote the

support of Ai. As in the proof of Proposition 6.2, we may suppose that xi=0 or

xi >2ri. We recall that e-ƒÏxF0 is even and Ii •¿-Ii=ƒÓ if 0 _??_ Ii. When xi=0, we put

(43)

Then Bi is a (1, •‡, 0)-atom on R and e-ƒÏxBj(x) is even. Since the index set Jo is finite, changing the coefficient X by 2Aie2pri and replacing the left hand side with the right one, we may suppose that e-ƒÏxAi(x) is even if xi=0. Since each Ai is

piecewise W+-smooth, we can define a K-bi-invariant function ai=W-(Ai) ion G

such that

Case I. xi>2ri and r2<1: As in the proof of Proposition 6.2, each ai is a (1, •‡, 0)-atom on G up to a constant multiplication.

Case II. xi>2ri and ri•†1: Since xi>2ri, it follows that xi+ri>3 and xi-ri•†1. Thereby, |R(xi, ri)| •`e2ƒÏ(xi+ri). Since Jo is finite, we may replace

as

Then, similarly as for (68), we can deduce from (72) that

This means that a is a (1, •‡, 0, ƒÃ)-atom on G up to a constant multiplication.

Case III. xi=0 and Ti•…1: As in the proof of Proposition 6.2, each ai has a (1, •‡, 0)-atomic decomposition.

Case IV. xi=0 and ri>1: Since J0 is finite, we may replace ƒÉiai as

Then, similarly as for (68), we can deduce from (72) that

(73)

We set b+(x)=ƒ¢(x)-1r-1i. As in (69), using the moment condition, we see that

(74)

Therefore, b+ is an L1 non-increasing denominator of bi satisfying (74). Hence, bi has a (1, •‡, 0)-atomic decomposition (see [7, Theorem 4.5]).

These four cases yield that fo E H~~o (G//K). •

(44)

for 0 < ~}'

< a + 1/2, 0 < ry' < ,Q + 1/2. Clearly, S0,0 is the identity operator and

S+,/2@+1/2

a=

W

. Then it follows that

When 'y = a + 1/2 and y' = Q + 1/2, this relation is nothing but the one in

Corollary 4.3, because H1(-l/2, -1/2)=H1(R).

7. Other

operators

We shall consider (Hl, L1)-boundedness of singular integrals, heat and Poisson

maximal operators MH and MP, and the Riesz transform R on G.

Let TRm=Mm be a Fourier multiplier corresponding to a bounded function m

and K the distribution kernel of Mm, that is, KN(A) = m(a) and Mm(F)=K*F in

a distribution sense (cf. [15, Chapter 1, •˜6]). We put k=W-(K) and define an oper ator TGm on G as T(f)=k* f . Then it is easy to see from (22), Definition 4.2, The orem 4.6 that IIT(f)Mw m

_ H1 R = IIW+(k * f)IIH1(R) = IT(W+(f))H1(R) IRII and

If T (f)1 H1 G < clf MsaW+(T~(f))IfH1(R) = IfMSa oTR(W+(f))IIH1(R). There

fore, we have the following.

THEOREM 7.1. Let notation be as above. If TR is bounded on Hl (R), then

TGm is bounded on W-(H1 (R)). Moreover, if Msƒ¿ o TRm is bounded on H1 (R),

then Tm is bounded from W- (Hl (R)) to H1(G//K), especially, it is bounded on

H1(G//K).

Let ht (g) and pt (g), g •¸ G, t>0, denote the heat and Poisson kernels on

G. They are K-bi-invariant functions on G whose spherical Fourier transforms are

respectively given by

DEFINITION 7.2. For ƒÃ>0 we define the modified heat and Poisson maximal operators MƒÃH and MƒÃP on G as follows.

For simplicity,

we denote

M0H and M0P by MH and Mp respectively.

It is well-known that MH and Mp satisfy the maximal theorem (see [14,

p. 73 and p. 48], [1, •˜3 and •˜6]). Moreover, if we define their truncated maximal operators MlocH and MlocP by restricting the range of the supremum as 0<t<1

(45)

(cf. Definition 4.8), then they are bounded from H1,+•‡,0(G//K)# to Ll(G//K) (see

[9, Theorem 6.1, Remark 6.5]).

We here introduce a modified atomic Hardy space H1N (R), E>0, N=

0, 1, 2, •c, on R by replacing the norm condition ||A||•…r-1 of a (1,•‡, N)-atom

A on R. with

(75)

(cf. Definition 5.1 on G).

DEFINITION

7.3. We define

and

We give their norms by ||W+(f)||IL' (R) and W+ (1)11

H' (R) respectively.

Clearly,

(see (20) and Definition

5.6). Since

H~~N(R)

C H~,N(R) = H1(R) (cf. [6, (3.30)]),

it follows

that W_

(H~~N

(R)) C W (H1(R)) C W_ (L1(R)) and hence, Proposi

tion 5.7 and Theorem

6.3 imply that

(76)

THEOREM 7.4. Let notations be as above. Then MƒÃH is bounded from

to Ll(G//K).

PROOF. Since MlocH is bounded from H1,+•‡,0(G//K) to Ll(G//K), we may

suppose

that t>1 in the definition

of MH.

Let MHR,

'y E R, denote

the maximal

operator on R associated to the Fourier multiplier corresponding to t-E ht (A +

(46)

For f •¸ Ll(G//K) we put F=W+(f). Then the same argument as that used in the proof of Theorem 4.6 and (54) yield that

(77)

if the both

sides exist.

Let Ht denote

the heat kernel on R:

(78)

Since H~ ()) = e-tae, it follows that

(79)

Hence, (77) can be rewritten as

(so)

LEMMA 7.5. Let t•†1 and x, ƒÊ•†0. Then for each ƒÁ•†0, there exists a

constant c such that

(81)

and

PROOF. Let0•…ƒÂ<l and x•†0. We note that

Then the desired result for Ht follows. Since t•†1, by replacing x/\/ by x/t in