Hl-boundedness of Riesz transforms and imaginary powers of the Laplacian on Riemannian manifolds

Ark. Mat., 41 (2003), 115-132

@ 2003 by Institut Mittag-Leffler. All rights reserved

Hl-boundedness of Riesz transforms and imaginary powers of the Laplacian on Riemannian manifolds

Michel Marias a n d E m m a m m l t/uss

Abstract. We prove that the linearized Riesz transforms and the imaginary powers of the Laplacian are Hi-bounded on complete Riemannian manifolds satisfying the doubling property' and the Poinca% inequality, where H 1 denotes the Hardy" space on M.

1. I n t r o d u c t i o n a n d s t a t e m e n t o f t h e r e s u l t s

Let M be a complete, n o n c o m p a c t R i e m a n n i a n manifold. We d e n o t e by d the geodesic distance on M , by

dx

the R i e m a n n i a n measure, by V the R i e m a n n i a n gradient and by A t h e Laplace B d t r a m i operator. For all

xEM

a n d all r > 0 . let

B(x, r)

be t h e o p e n geodesic ball of radius r centered at x and

V(x, r)

its volume.

Say t h a t M satisfies t h e doubling p r o p e r t y if there exists a positive c o n s t a n t C such t h a t

(1.1) V(x, 2r) _<

CV(x, r)

for all x E 3 I and r > 0.

If (1.1) holds, one easily sees t h a t there exist C. D > 0 such t h a t for all

xC3I,

all r > 0 a n d all 0 > 1 ,

(1.2) v(x, oF) <_ coDv(x. ~).

Say t h a t t h e u n i f o r m L 2 - P o i n c a % inequality holds on 3 I if there exists a posi- tive c o n s t a n t C such t h a t , for all

xEM

and r > 0 .

(1.3)

~(~,,.) lf(x)-fu(x,,.)l 2 dx<<_Cr 2 fu(:,..2,.)llVf(x)ll2 dx

for all

fcC~(B(x,

2r)), where

1 f f(y) dy.

f B ( x . ~ ) - V ( x .

r) JB(.,:.,-)

116 Michel Marias and Emmanuel Russ

tt is well known (see [18]) t h a t the conjunction of (1.1) and (t.3) implies the so- called N e u m a m l - P o i n c a r ~ inequality': there exists C > 0 such that. for all x r M and r > 0 ,

(1.4)

s If(x)--fB(x.~)12dx~Cr2/

IlVf(f)ll2 dx

(x:~9 g u(z.,.)

for all

fEC~(B(x,r)).

Since M satisfies the doubling p r o p e r t y (1.1), it is a space of homogeneous type. One m a y therefore consider the Hardy" space H 1(31) as defined in [9]. We briefly recall how

H i (A4)

is defined. Sa.v that a complex-valued function a on ~ I is an a t o m if it is s u p p o r t e d in a ball

B(yo, r)

and satisfies

1

/,

(1.5) Ilal]2_< V.(y0,~.)l/2 and I Cl(2c) d x = O .

A function f on M belongs to H 1 (My if there exist (A,,),~eN ~ 11 and a sequence of a t o m s (a~)~eN such t h a t

(1.6) f--

hEN

where the series converges in L l ( 3 l ) . T h e norm

]]f}lttl(M )

is the infimum of

~ r IA~I over all such decompositions.

A function u on k~/ is said to be harmonic if A u = 0 on 1f. For d = l , 2 , . . . , denote by ~td(M) the space of harmonic functions on M of growth at most d.

This means t h a t u E ~ a ( M ) if u is harmonic and there exist C > 0 and

poEM

so t h a t

lu(x)l<C(l+d(x,po)) d

for all

xCM.

Notice t h a t the celebrated conjecture of Yau, which states t h a t

Jtd(AI)

is finite dimensional, is solved by Li and T a m for d = l , [19], in the case when M has nonnegative Ricci curvature, and by Colding and Minicozzi for all d_> 1 on manifolds satisfying the doubling volume property' and the N e u m a n n - P o i n c a r 6 inequality, [10], T h e o r e m 0.7.

T h e Riesz t r a n s f o r m on M is the o p e r a t o r R = V A -1/2. For u E ~ l ( 3 l ) , we define, as in

[22],

the linearized Riesz transform R . by

(1.7)

H ~ ( f ) ( x ) =

(t~(f)(x), Vu(x))

= ( V A - 1 / 2 f ( x ) . V u ( x ) )

for all

fEC~(M)

and all

xCM,

where ( . . . > is the R i e m a n n i a n inner product on tile tangent space of M at x.

Our first result deals with the H I (M)-boundedness of R~.

Hl-boundedness of Riesz transforms and imaginary powers of the Laplacian 117 T h e o r e m 1. Let M be a complete noncompact Riemannian manifold satis- fying the doubling property (1.1) and the Poincard inequality (1.3). Then for any u C ~ I ( M ) , R~, extends to a bounded operator on H i ( M ) .

Our next result is a b o u t imaginary powers of the L a p l a c e - B e l t r a m i operator.

For a l l / 3 E R , the o p e r a t o r A i3 is defined via spectral theory, it is L2-bounded and one has

II/vsIl>,2

= 1.

T h e following s t a t e m e n t holds.

T h e o r e m 2. Assume that M is a complete noncompact Riemannian manifold satisfying the doubling property (1.1) and the Poincard inequality (1.3). Then, there exists C > 0 such that, for all 3 E R , A i3 is Hl(M)-bounded and

II ' IIH H1 ~ c(1+ ~x/~e<~l/2).

We first say a few words a b o u t the geometric context of these two results.

Assumptions (1.1) and

(1.3)

are satisfied when M has nonnegative Ricci curva- ture. Indeed, by the Bishop comparison theorem (see [5]). M satisfies the doubling property. Also, in [6], P. Buser showed t h a t these manifolds s a t i s ~ the Poincar~

inequality. Recall t h a t b o t h (1.1) and

(1.3)

remain valid if 3 I is quasiqsometric to a manifold with nonnegative Ricci curvature, or is a cocompact covering manifold whose deck transformation group has polynomial growth. [12]. Note that there exist manifolds satisfying (1.1) and

(1.3)

and whose Ricci curvature is not nonnegative.

First considered in R n, the issue of Riesz transforms on Riemannian manifolds has been raised in

[32].

Note t h a t Riesz transforms have been studied in vari- ous geometric contexts, such as Riemannian manifolds (see [20], [3]), Lie groups (see [21], [271, {1]), discrete groups (see [17]) and graphs (see [25], [26]). See [2]

for an extended bibliography on the subject. Here. we concentrate on the case of Riemannian manifolds. Under very weak assumptions on 3 I (namely, under (1.1) and an on-diagonal upper bound on the kernel of e-t~), it is proved in [11] t h a t [~7/~-1/2[ is LP-bounded for all 1<p_<2 and weak (1, 1). When M has nonnegative Pdcci curvature, the Riesz transform is LP-bounded for all l < p < + 2 c ([3]). Its H 1- L 1 boundedness is proved in [7] on Riemannian manitolds with nonnegative Ricci curvature, and in [26] under the assumptions of T h e o r e m 1.

If one looks for an H l - b o u n d e d n e s s s t a t e m e n t for Riesz transforms on mani- folds, a new difficulty appears. Indeed,

~7A-1/2f

^{is a} vector-valued function, and it is not clear how to define a vector-valued H 1 space in this context. To overcome this difficulty, we take the scalar product of the Riesz transform with the gradient of a function in 7-/1 (M). Thus, one obtains a scalar-valued operator, called the linearized

118 Michel Marias and E m m a n u e l R u s s

Riesz transform, which was first introduced in [22], where the H a - b o u n d e d n e s s of the linearized Riesz transform on Riemannian manifolds with nonnegative Ricci curvature is established.

In the Euclidean setting, the operators A i3 are H~-bounded, (this is a con- sequence of the classical C a l d e r d n - Z y g m u n d theory, see e.g. [31]). W h e n M is a Riemannian manifold, a universal multiplier theorem of Stein ([30], Corollary 4.

p. 121) shows t h a t A i'3 is

LP(M)-bounded

for all l < p < + ~ c . The

H1-L 1

bound- edness of A i~ on Riemannian manifolds with nonnegative Ricci curvature is shown in [23]. For other geometric settings, see for example [8] and [29].

T h e proofs of T h e o r e m 1 and T h e o r e m 2 are similar. T h e y go through a duality argument, which we quickly explain for R~,. In fact. to prove the H l - b o u n d e d n e s s of R~, it is enough to show t h a t there exists C > 0 such that. for all a t o m s a and

O~Cc(M),

(1.8)

~ I R,,a(z)o(:c) doc <_

C[[O[[BMO,

(see Section 2.2 for the definition of B M O and further explanations).

To prove (1.8), one first introduces a truncated version R ... e > 0 . of

R,,

and proves that, for all atoms a and all g > 0 .

R~,.~aELI(3I)

and

R,,.~a

has integral 0 over M, which is a consequence of the harmonicity of u. Then. thanks to the L2-boundedness of R~.~, weighted L2-estimates for the gradient of the heat kernel (see Section 2.1) and some classical estimates for BMO fimctions, one proves (1.8) with R~,e instead of R~, with a constant C > 0 independent of ~. Letting s go to 0 yields (1.8).

T h e p a p e r is organized as follows. In Section 2. we recall some known facts about the heat kernel of ill (Subsection 2.1) and the B M O space on eli (Subsec- tion 2.2). In Section 3. we first prove t h a t

R,.~a

has integral 0 and then t h a t (1.8) holds true. Finally, T h e o r e m 2 is proved in Section 4.

T h r o u g h o u t this article the different constants will ahvays be denoted by the same letter C. W h e n their dependence or independence is significant, it will be clearly stated.

2. P r e l i m i n a r i e s 2 . 1 . H e a t k e r n e l e s t i m a t e s

In the sequel, we d e n o t e by Pt the heat kernel on 3 I . i.e. the kernel of e - t A . Moreover, if y and Y0 are two fixed points in 3 I . define, for all x E 3 l and all t > 0 ,

qt (x) = > (~, y) - > (x. yo).

Hl-boundedness of Riesz transforms and imaginary powers of the Laplacian 119

When _~I satisfies (1.1) and (1.3), it is proved m [28] (see also [16]) that there exist c~, C1, c2, C2 > 0 such that. for all z, y E 3 I and all t > 0.

Cl e-Cld2(x.y)/t

<_pt(x, 9) <

V ( c t ~ ) e -c2d2(Jcy)/t.

(2.1) V(z, ~ )

Moreover, the parabolic Harnack inequality holds on 3I (actually, it is proved in [28]

that the conjunction of (1.1) and (1.3) is equivalent to the parabolic Harnaek in- equality on g,I, which is itself equivalent to (2.1)). As a consequence of this inequal- ity, one easily obtains that Pt is H61der continuous (see [26]).

L e m m a 1. There exist c3, Ca>O and 2. E]0.1[ such that, for all x.g, yoE~J and all t > 0 satisfying d(y, Yo)<_,~,

( d(> yo) T

Iqt(z)l < v(z, x/t) \ ~/~ ]

As a consequence of Lemma 1, we have proved the following estimate in [26].

L e m m a 2. For any c~<2c3, there ezists C[~ >0 such that, for all y, goEM and all t > 0 satisfying d(y, Yo) <_ x/t,

< 1 ( d ( ~ t t 9))2~' C'~ .

Recall (see [11]) that, as a consequence of the upper estimate of Pt in (2.1), one also has the following estimate.

L e m m a 3. There exists d > 0 such that, for all IlEM and all t > 0 . I IV*Pt(a:'Y)tl2eaa~(x"J)/t da: <_ tV(g. x/l)"

2 . 2 . A f e w f a c t s a b o u t B M O

Say that a locally square integrable function o on 3 I is in BMO(M) if

(2.2)

II011gx~o = sup V - 7 ~

,2 1 s

I o ( x ) - o B dz < +~c,

12

where V ( B ) is the volume of the ball B and the suprenmm is taken over all the balls of M. Since gJ satisfies the doubling property (1.1), M is a space of homogeneous

120 M i c h e l M a r i a s a n d E m m a n u e l R u s s

type and the general theory of BMO developed in [9] holds. Using (2.2), one proves the classical inequality

IOB-o=BI <_ CIIollBvo.

which yields, as in [14], p. 142, that there exists C > 0 such that. for all oEBMO(III), all k >_ 1 and all balls B C M,

(2.3)

i ~ io(x)_OzBlZdx<Ok211oll~x,o

Define VMO(M) as the closure in BMO(3I) of Cc(3I), the space of continuous functions on M with compact support. In the sequel, we also use the fact that the dual of

Hi(M)

is BMO(M) ([9], Theorem B. p. 593) and that. as a consequence,

H~(M)

itself is the dual of VMO(M) ([9], Theorem 4.1). The duality implies the following characterization of H I ( M ) :

fEH~(3I),

if

fEL~(M)

and if there exists C > 0 such that, for all functions

OEC~(M),

J(~I f(X)O(X) dx <_

CIIollBvo.

Furthermore, in this situation,

Ilfllm(.~/) ~ KC.

where K > 0 only depends on M.

3. H l - b o u n d e d n e s s o f R~

For all c>0, define the truncated operator R~.~ by

l f l/~ dt

R~,ef(x) = ~ (Vxe-tnf(x),

Vu(x)} 7 ' The following holds.

L e m m a 4.

For all fEC~:(M),

lira

R~,.~ f = R~, f in

L2(3I).

r

f E C ~ (:I).

H L b o u n d e d n e s s of Riesz transforms and imaginary powers of the Laplacian 121

Proof.

The proof relies on H :~ calculus for A (see [24] and [331). Fix p r 1 and set

F , = {z E C : [ arg z[ < p } . For all z E F , and all e > 0 , define

1 /1/~ e-t'~z 1/2 dt

For any function gETP(A 1/2) and all e > 0 . define

1 fl/~e_tAA1/2g dt

so that Observe that

lim ~,~ (z) = 1

r

uniformly on all compact subsets of F , . By H ~ functional calculus for A one therefore has

lira I I A 1 / 2 ~ - A 1 / 2 g l I 2 = 0.

~--+0

T h e L2-boundedness of

VA -1/2

yields

lim [IVu~ -VgI[2 = ! ~ ] ] V A - 1 / 2 A 1 / 2 U s - V A - 1 / 2 A 1 / 2 g [ [ 2 s-+o

< C lira

NA1/2u~ -A1/2gt12 = O.

s-+O

Applying this with

g=A-1/2f,

one obtains

/ ^1/~ VA-1/2f 2

lira 1 V

e_t& f dt

_{= o .}

which proves the claim. []

Observe that the kernel of R~.~ is given by

I f l/~ dt

v ~ <Vxp,(x.y).W,(.~)> ~ .

As in [26], the L2-boundedness of R~,.~, Lemma 3 and Lemma 2 show that, for all r and all atoms

a, R~,~aELI(5.I).

The computations are analogous to those in [26], except that the integrals with respect to t are computed on ]~, l / c [ instead of ]0, +oc[.

Using the fact that u is harmonic, we prove the following result.

122 Michel Marias and Emmanuel Russ P r o p o s i t i o n

1. For any atom a and any

^z>O,

(3.1)

J~M R, ~a(x) dx

^{= 0 .}

Proof.

The proof follows tim lines of the one of the corresponding statement m [221, Section 4, and we give it for the sake of completeness. Let us assmne that a is supported in

B(yo,r).

For K_>2, let

OK(X)

be a C ~ function on M with 0_<0K_<I, which is equal to 1 in

B(yo.K-1)

and to 0 outside

B(yo, K+l),

^and

satisfying II V0K II ~ -< C, where C > 0 is an absolute constant. Define

I= ~s ,a(y), fl/e~l~ ~

[(V,,pt(x. y). Va,(x))[

IoK(.r)l dx ~ dy.

By the boundedness of rVu] and the Le-estimate of

V~p~(x, y)

given in Lemma 3.

one has, for all

yEM

and all t>0.

p(v~>

(~,

dx

_< cIIIV~lll~ [%.p,(x, ~)12

^dz V1/2 (~/0, /s

X)

(y0,K+l)

\i/2

< ClllWlll~ ___(/~(~o.K+~)~dl~"~)~/r '~)[~ &) V'/2(Y~ K + 1)

_< ClllW Ill ~ ( '~/<~-~t ~) ) 1/2

This estimate and the fact that

Ilalll<i

yield

L /U~(V(g~ ~/2dt

I < CltlVulll~ - /~o.,-) la(Y)l v ( > ~ ) V dy

(3.2)

-

dt

_ < C m a x { l ,

( K+~]+r) D},I, Vu,,,~ /B(uo.,')

,a(y), ~1/=

T dy < +::~c"

By the harmonicity of u and the Fnbini theorem, applicable by (3.2). one obtains

i f 1:.'-

(3.3) = 1

a(y) dx ~ dy

I 1/~ dt

-- ; ~ia(lj).~ 'jf3i(VoK()j).V~l(2t))p,(gc.~l)dz ~d~.

Hl-boundedness of Pdesz transforms and iinaginary powers of the Laplacian

Proposition 1 will therefore be a consequence of

(3.4)

lira

[ la(y)l I(VxOK(X).Vu(x))lpt(x,y)dx~dy=O.

K - + + :,c J M 9 I

Indeed, if (3.4) is proved, then, since

R,~:aELI(3I). (3.3)

and (3.4) yield

~ R~.~a(x) d:c =

I

To prove (3.4), set

lim

~

O K ( X ) R u . ~ o ( x ) d x = O .

K--++~c 1

123

IK = la(y)l I(VxO~(x). W(X)> IP,(~, Y) dx ~ dy.

I . I

By the boundedness of [Vu t and [VoK[. the fact t h a t ot< is equal to 1 on

B(yo, K-

¹⁾

and the upper bound in (2.1), one has

IK <C ]a(y)[ p~(x.g)dx dt

- f (~jo.K+l)\u(~,io.K- t)

~ dg

<_C/i I ,a(y), ~ 1/~-~(K-l ")2/tV(y~ t-7-~.~)) Vr-z'dY

CIIclll I e - c ( K - I - r ) 2 / t Hlax 1, ~k ~ ] J V '

which goes to zero when K goes to +~c by the dominated convergence theorem. []

Proof of Theorem

1. Let a be an a t o m s u p p o r t e d in B = B ( y 0 , r) and consider

CECc(M)

such t h a t

It011BMO_<I.

Proposition 1 yields, ior all c > 0 ,

~ R,,.~a(x)(@r)dx= ~ R~,.~a(x)(o(x)-o2B)dx.

Decompose 0 - 0 2 B as

r162 =

(e--,O2B)X2a+(O--O~S)X(2e)c = ol +02.

and write

(3.5)

~,, Ru,ea(x),(x)dx= ~i Ru.ea(x)ol(x)dx+ fdM R.,.,a(x)o2(x)dx= El+E2.

124 M i c h e l M a r i a s a n d E m m a n u e l R u s s

By Cauchy-Schwarz and (2.2),

j~ IR~,~a(x)l Ir d~ < IIR,,.~all~ll(o-o2BDx2~ll2

^<

IIR~,.~alI2V(2B) ~/2.

I

We now deal with the term involving 02 in (3.5). Observe that, since a has meat]

value zero, one has

~ a(y)<Vj:pt(x, y),

Vu(x)}

dy= ~ a(y)<Vzqt(x), Vu(x)) dy,

with

qt(z)=pt(z, y)-pt(z, Yo).

^Write

E2 = ~s R~,,~a(x)o2(x) dr

= k~> l f~.+l Bk2,.B R".~-a('v)O2('v) d:c

= 1 02(,)

(Vxp,(x, y),

W,(x)>~(y) @ ~ dx

k+lB\2kB ^_

+-~1 "+'B\2"B

02(*') ~ <V,qt(x),

Vu(x)>a(y) dy --~ dx

k_>l k > l

Fix k > l . When

yEB(yo, r)

^and

2kr<d(x, yo)<2~+lr,

^{one has}

2k-tr<d(x.y)<

2k+2r,

which implies

_< C ~ f

IIkl /2 _k+lB\2kB 102(x)l/

^{J B}

I<Vxpt(x,y),Vu(x))tla(~)l@-~dxdt

/B ~r2j~2k--lr~d(x.y)<2k+2r dt

_< cIIIWlll~ I~(y)l IV~pt(a, y)110~(x)l dx -~ @.

The Cauchy Schwarz inequality, Lemma 3, (2.3) and the doubling property yield

~ _,~<_a(x,~)<2,.+~, IVxpt( x, Y)I

Io~(x)l dx

,,1/2

~-- (~ '~Tzpt(x, ~l)'2edd2(x'y)/t dx) k-lr<d(x.g)<2k+ir

X (~22~. lr<d(x.y)<2k+2r'O2(X)[2r 1/2

Hl-boundedness of Riesz transforms and imaginary powers of the Laplacian 125

< C ( k + 2 )

V1/2 (~],

2k+2r)e_~22(~,-~),z/2 t

- ~/~v(~,, ~ )

C ( 2 k + 2 r \ D/2

--

) e--522(~-l)r~/2t

< ~ (k+2)\ ,fi

C' (k+2)e_222kr2/t

< _ ~ ^{, .}

Therefore,

1.2 F2

k--lr~_d(x,y)<2k+2 r -- t

~

^{+ ~ dt}

< C k e - 3 t - - < C k 2 - 2 k

- - 2 k t - -

which shows that

y-~ I_rkl _< C,

k > l

since Ilalll < 1.

T h e treatment of Jk is similar. One has

fB fl/~f IVq~(x)IIo2(x)Idx dt

IJ~l-<ClllWlll= la(y)lj?`2 J2k-'~<_d(x~)<2k+2?` ~dy.

When t_<22k+4r 2, use L e m m a 2; (2.3) and the doubling property to write, for an O~C3,

jf2k lr<_d(x,y)<2k+XrlVqt(x)l

^{tr ld3g}

c(k+2) (,-)= .,._,2/~ ,/~ k+~

_< v/tv( y, ~ ) ~ ~ - ~ k " v (~,2 )

D / 2 e _ o 2 2 ( k - , )r 2 / t

_< ~ (,~+2) ' 32~,-~/~

.

126 M i c h e l M a r i a s a n d E m m a n u e l R u s s

When t>22k+% 9, just write that V(y.2k+2r)_<V(y. ~/t) and the result still holds.

As a consequence,

fl/~ f

2 ,]2 k l<d(x,y)<2k+2 r

IVq,(m)1102(x)ldx ~ < C ( # + 2 ) e _ 3 2 ~ / e

r dt

- 2 7 -

22~

_< C ( k + 2 ) 2 -~~ f

e-3~'v -'/'e-1 dv

.JO

<_ C ( k + 2 ) 2 -~,~'

j o e-3~'v ~/2-1 dr.

Thus,

~-~ la.I <_ c.

k.>l u s i n g the fact that Ilalll <_ 1 again.

Finally, we have proved that. for all functions

oCC~(3I),

(3.6) ~ I Ru.ea(x)O(x) dx, < litzlu.ea{{2V (2B) I/2IIO{{BMO § 0

for all ~>0. Since

R,,..=a

converges to

R~,a in L2(M)

when z goes to 0, (3.6) yields

Z~ R~,a(x)o(x) dx ~ IIR, alI2V(2B)I/211OIIB_x~O+CIIotlB.VO ~ CIIoIIB.x~O.

In the last inequality, we use the L2-boundedness of

R~,.

(1.1) and the fact that

[]aI[2<_V(B) -1/2.

Therefore, (1.8) and Theorem 1 are proved. []

4. I m a g i n a r y p o w e r s o f t h e L a p l a c e o p e r a t o r

We now prove Theorem 2. The argmnents are analogous to those used in the proof of Theorem 1.

For all ,gER, set T ~ = A ~9. For all ~>0. define 1

f l / s t-i3-]e -tA dt.

A

For all

,fEL2(AI), T2.ef

converges to T3f in L2(3I) by H x functional calculus.

Fix 3 c R . One first proves the following result.

H l - b o u n d e d n e s s of Riesz transforms and imaginary powers of the Laplacian 127

P r o p o s i t i o n 2.

For all

c > 0 .

T3.~ is HI(3I)-LI(M) bmmded.

Proof.

L e t a b e a n a t o m s u p p o r t e d in a b a l l

B=B(yo, 7").

T h e n . since e -tLx is a M a r k o v s e m i g r o u p , i.e.

(4.1)

fM Pt (z, y) dx = 1

for all y E M a n d all t > 0 (see [15]), one has 1

< c I,(y)l

p,(x..v) d.~. dy

- - I J a 31 Y

( 4 . 2 ) c ~/-~ 1

dt

< _ C / ' / ~ d t

: c ( ~ ) by the fact that Ila111<_1. []

We now state the following cancellation property (cf. Proposition 1).

P r o p o s i t i o n 3.

For all

~ > 0

and all atoms a

~ T3.~a(x) dx =0.

I

Proof.

L e t us a s s m n e t h a t a is s u p p o r t e d in a ball

B=B(yo. r).

F r o m (4.2) we have t h a t

,/s la(Y)' /U~ fA[ pt(x. y) dx d-f dY < + ~c.

T h i s allows us t o a p p l y F u b i n i a n d get 1

-

r(-i3) ~(y) dx d~ dt

- F ( - i ~ ) dt a ( y ) d y

~ 0 ,

since (4.1) h o l d s a n d a has m e a n value 0. []

128 Michel Marias and E m m a n u e l R u s s

Proof of Theorem

2. It is enough to show that there is a C > 0 , such that for all

CECc(M)

and all atoms a,

Lr~a(z)r dx

_{< - -}

c

_II011BMO-

- I F ( - i 3 ) I

Let a be an atom supported in

B=B(yo. r)

and

oEC,:(M).

Write r -~- ( ~ - - O 2 B )X2B-[-( O--O2B )X( 2B) c = O1 + 0 2 , Then, for all r

L T~3,~a(x)r LBT3.~a(x)Ol(x)dx+

f2B)~

T3.~a(x)o2(x)dx=El+E2.

The Cauehy Sehwarz inequality yields

]Eli <-- _..Le ITa.~a(x)ol(x)l dx < NT3.~aN21Ir

(4.3)

<_ IlZ~,~all2Vl/2(2B)llollBxlO.

We now treat the term involving 02. %Ve write E2 =

~2B) c T,3,ea(x)O2(x) dx

t-i3-1 .i,~ pt(x, y)a(y) dy dt dx

1 o2(x) f ~

1

= 1 E(ik+jk)"

V ( - i ~ ) k>l

By the estimates (2.1) of

pt(x,y) we

obtain that for all k > l . all

yEB

and all O < t < r 2,

e-cd(x.y)2/t

_ c . I o 2 ( x ) l

dx

< c [ , _ l o ~ ( ~ ) l &

V(y

2 k+2 r ) 0 BMO

< C ( k + 2 ) ~

--c22~'r2/t

- / Y , )

< - c ' 2 ~-t r 2/t

_ C ( ] ~ -{- 2) e IIoll g.klO,

Hl-boundedness of Riesz transforms and imaginary powers of the Laplacian 129

where the last line follows from the doubling property. Thus

l&l < Ir(-i3~c l(k+2)lloliB.~.~ ~ la(y)l

^dy

_ -e-~:it

^{r /~ dt}

/ r2 1 c22~r2/t

< C ~ ( k-I- 2 II(~llB.~[O )llll _ _ { l

at

(4.4)

- I r ( - i ~ ) l ~ t

-< Ir(-ig)l

C (k+2)212kllOIIBM~

where C > 0 only depends on M.

Let us now deal with Jk. Since a has mean value 0. one has

When

d(y, Yo)<_ x/t,

L e m m a 1 yields

r 1 e_cd2(x.yo}/t"

[qt(x)[ < C -~

~(Y0, x / / ) 9 So, when r < x / t _<2kHr, one has

C y e -c22kr2/t

( ~" x.) l(2221")'2/t V(2~--~].B )

_< c(~-+l)l ~ )___ e V(yo, 7/)IIOIIBMo

Q r )h (2k+lr)D -c22kr2/t

_ < C ( k + l ) ~ \

~

e llollBxlo

) (@tt ~ el~:~kr~/~ll II

_< C ' ( k + l O B.xio.

When x/t _>2kHr, just write that

V(2kHB)<_V(yo, vq)

and the result still holds.

Therefore,

tJ~l< - -

C

Ir(-ig) l

< _ _ C

(k+l)ll011Bx.~olIall~j~ R ~ T

(k+l)2-k~:ll011BX~O,

130 Michel Marias and Emmanuel Russ

which, combined with (4.4), gives,

(4.5) ~ IZkl§ < o IIo11~:,~o

~"~(h2-21"§ < c

k > l

--Ir(-i3)~ -Ir(-i3)~ IIoIIB-',,O.

_ J,.>l

From (4.3) and (4.5), it follows that, tbr all ~>0.

fa~ T~%~,Xx)o(x)d~ < CllT~.~allCr*/2(2B)lloll

^C

Since T2,~a converges to T s a in L~(M) when ~ goes to 0,

IIrjII2-~2=1

^and

Ilalt25

V(B) 1/2, one obtains

fMTza(z)O(z)d.r <_C(1-~ IF(li3)I)IIOHBMO . which completes the proof of the H l - b o u n d e d n e s s of y~.

Finally, recall that, for 3 E R .

Ir(-i3)l

⁼ 3 s i n h ~ r 3 7r "

see [13]. Therefore,

IIAi~IIH1---~H 1 ~ C ( l + ~ e ~ ' ! ~ l / 2 ) . which colnpletes the proof of T h e o r e m 2. []

Acknowledgment. T h e authors would like to thank Thierry Coulhon for useful advice.

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Received December 3, 2001 Michel Marias

Aristotle University of Thessaloniki GR-54006 Thessaloniki

Greece

email: m a r i a s ~ a u t h . g r Emmanuel Russ

Universit4 d'Aix-Marseille 3 FR-13397 Marseille Cedex 20 France

emaih emmanuel.russ guniv.u-Smrs, fr