Lp bounds for Riesz transforms and square roots associated to second order elliptic operators

Publ. Ma . 47 (2003), 497–515
LpBOUNDS FOR RIESZ TRANSFORMS AND SQUARE
ROOTS ASSOCIATED TO SECOND ORDER ELLIPTIC
OPERATORS
S e e Ho mann∗and Jos´
e Ma ´
ıa Ma ell†
Abs ac
We conside he Riesz ans o ms ∇L−1/2, whe e L≡− di A(x)∇,
and Ais an acc e i e, n×nma ix wi h bounded measu able
complex en ies, defined on Rn.Wees ablish boundedness o
hese ope a o s on Lp(Rn), o he ange pn<p≤2, whe e
pn=2n/(n+ 2), n≥2, and we ob ain a weak- ype es ima e
a he endpoin pn. The case p=2was al eady known: i is
equi alen o he solu ion o he squa e oo p oblem o T. Ka o.
1. In oduc ion
Le A=A(x)=aj,k(x)j,k be an n×nma ix whe e he coe -
ficien s aj,k a e complex- alued L∞(Rn) unc ions. We assume ha
his ma ix sa isfies he ollowing ellip ici y (o “acc e i i y”) condi ion:
he e exis 0 <λ≤Λ<∞such ha
λ|ξ|2≤Re Aξ·¯
ξand |Aξ·¯
ζ|≤Λ|ξ||ζ|,
o all ξ,ζ ∈Cn.Weha e used he no a ion ξ·ζ=ξ1ζ1+···+ξnζn
and he e o e ξ·¯
ζis he usual inne p oduc in Cn. No e ha hen
Aξ ·¯
ζ=j,k aj,k(x)ξk¯
ζj. Associa ed wi h his ma ix we define he
second o de di e gence o m ope a o
L =−di (A∇ ),
which is unde s ood in he s anda d weak sense by means o a sesquilin-
ea o m.
2000 Ma hema ics Subjec Classifica ion. 42B20, 35J15.
Key wo ds. Riesz ans o ms, squa e oo s o di e gence o m ellip ic ope a o s.
∗Pa ially suppo ed by NSF.
†Pa ially suppo ed by MCYT G an BFM2001-0189.
498 S. Ho mann, J. M. Ma ell
The acc e i i y condi ion s a ed be o e allows one o define he squa e
oo L1
2=√L. Recen ly P. Ausche , S. Ho mann, M. Lacey, A. McIn-
osh and P. Tchami chian ha e ob ained an affi ma i e answe o he
so-called Ka o squa e oo p oblem o ellip ic diffe en ial ope a o s
(see [AHLMT]). Namely, hey ha e shown he ollowing:
Theo em 1.1 ([AHLMT, Theo em 1.4]).Fo any ope a o as be o e
he domain o √Lcoincides wi h he Sobole space H1
(Rn)and √L 2∼
∇ 2.
In [AT], assuming he esul o Theo em 1.1, he au ho s ob ain op i-
mal Lpbounds o √Land o he associa ed Riesz ans o ms ∇L−1/2,
unde he ex a hypo hesis ha one has a Gaussian uppe bound o
he hea ke nel (along wi h “Nash- ype” local H¨olde con inui y). In
his pape , we conside he same p oblem wi hou his ex a hypo he-
sis, i.e., we s udy he Lpboundedness o ∇L−1/2and √L o gene al
second o de ellip ic ope a o s Las abo e. Ou esul s a e new only in
he case n≥3. Indeed, Gaussian bounds (and Nash’s es ima es) always
hold o he ke nel o he semig oup e− L in dimensions 1 and 2 [AMT],
in which case he esul o [AT] (o also ha o [DM]) applies. In ac ,
he a gumen o [DM] equi es only he Gaussian uppe bounds, and
no he H¨olde con inui y. We ema k ha in he p esence o Gaussian
bounds, he boundedness on Lq o q≤2o he Riesz ans o ms as-
socia ed o o he ope a o s ( o example, he Laplace-Bel ami ope a o
on a mani old) ha e been ea ed (see [CD]). Fo he ope a o s unde
conside a ion in he p esen pape , i is known ha he Gaussian bounds
may ail in dimensions n≥5[AT, pp. 32–33] (also [MNP], whe e he
example o igina es). Ou main esul says:
Theo em 1.2. Le pn=2n
n+2 . Then he Riesz ans o m ∇L−1
2is o
weak ype (pn,p
n)and hus bounded on Lq(Rn) o pn<q≤2.
Rema ks. We lea ned du ing he p epa a ion o his manusc ip ha he
Lqboundedness o he Riesz ans o ms ∇L−1
2on he same ange o q
has also been ob ained independen ly by S. Blunck and P. C. Kun s-
mann [BK1], by essen ially he same me hod as ou s. Mo eo e , hey
ha e applied his echnique o ela ed ma e s, including Lpes ima es o
Riesz ans o ms associa ed o highe o de ellip ic ope a o s [BK1], as
well as o he exis ence o an H∞ unc ional calculus [BK2]inLpspaces.
We a e g a e ul o Pascal Ausche and Alan McIn osh o b inging hei
wo k o ou a en ion.
LpBounds o Riesz T ans o ms 499
We should poin ou ha p
n=2
whe e 2=2n
n−2is he Sobole
exponen o 2. On he o he hand, we ecall ha , by a s anda d duali y
a gumen , he L2es ima e o he Riesz ans o m is equi alen o
√L∗ 2≤C∇ 2,
whe e L∗is he adjoin ope a o o L ha sa isfies i s same p ope ies.
In iew o his ac , he L2es ima e o he Riesz ans o m ollows om
Theo em 1.1 ob ained in [AHLMT]. Le us also no e ha he p oblem
is in a ian wi h espec o he aking o adjoin s, i.e., Land L∗a e
ope a o s o he same na u e and i one o hem sa isfies he equi ed
hypo heses hen he o he one also does.
By using a s anda d duali y a gumen , one has ha om he bound-
edness o ∇L−1
2on some Lq(Rn)i ollows an Lq(Rn) domina ion o he
squa e oo s by he g adien . Then, as a consequence o ou main esul
we ge he ollowing.
Co olla y 1.3. Le 2≤q< 2n
n−2. Then,
√L q≤C∇ q.
The pape is o ganized as ollows. In he nex sec ion we p o e some
echnical es ima es ha will be used in he sequel. In Sec ion 3 we p o e
ou main heo em.
2. L2off-diagonal es ima es
Gi en E,F wo subse s o Rn,wewill deno e by dis (E,F) he dis-
ance be ween hem. We will use he no a ion 
o n- uples o unc ions.
The iden i y ope a o will be w i en as I.Webegin by s a ing he un-
damen al a e age decay es ima es sa isfied by he hea ke nels associa ed
o ou ope a o L.
Lemma 2.1. Le Eand Fbe wo closed se s o Rn. Then, o all >0,

e− L 
L2(F)≤Ce
−dis (E,F)2
c  L2(E),supp ⊂E

 Le
− L 
L2(F)≤Ce
−dis (E,F)2
c  L2(E),supp ⊂E

 1
2∇e− L 
L2(F)≤Ce
−dis (E,F)2
c  L2(E),supp ⊂E

 1
2e− Ldi 

L2(F)≤Ce
−dis (E,F)2
c 
L2(E),supp 
⊂E
whe e c>0depends only on λand Λ, and C>0on n,λand Λ.
500 S. Ho mann, J. M. Ma ell
An analogous esul o esol en ke nels (I+ 2L)−1is p o ed in
[AHLMT, Lemma 2.1], ia an in eg a ion by pa s a gumen simila
o he p oo o Cacciopoli’s inequali y. The p esen lemma ollows om
he one o esol en ke nels by unc ional calculus. Al e na i ely, one
can simply modi y he in eg a ion by pa s a gumen o [AHLMT],
o ob ain a di ec p oo o he p esen lemma, as in he p oo o he
pa abolic Cacciopoli inequali y. We omi he de ails.
Lemma 2.2. Le m≥1be an in ege (e en ually, we shall choose m
depending only upon n). Le Eand Fbe wo closed se s o Rn. Then,


 1
2∇L−1
2I−e− Lmdi 


L2(F)≤Cdis (E,F)2
−(m+1
2)

L2(E),
supp 
⊂E
and


 1
2∇∇L−1
2I−e− Lm∗


L2(F)≤Cdis (E,F)2
−(m+1
2)

L2(E),
supp 
⊂E
whe e C>0depends only on n,m,λ,Λ, and
∇L−1
2I−e− Lm∗
=−L−1
2I−e− Lm∗di 
.(1)
To p o e his esul we need o es ablish an auxilia y lemma ha says
ha he composi ion o wo ope a o s ha sa is y wo L2off-diagonal
es ima es as in Lemma 2.1 e ifies a simila inequali y.
Lemma 2.3. Le {A } >0and {B } >0 wo amilies o linea ope a o s.
Assume ha o all closed se s E,F, o all such ha supp ⊂Eand
o all >0we ha e he ollowing es ima es
A L2(F)≤Ce
−dis (E,F)2
c  L2(E)
and
B L2(F)≤Ce
−dis (E,F)2
c  L2(E).
Then, o all , s > 0we ha e
A Bs L2(F)≤Ce
−dis (E,F)2
cmax{ ,s} L2(E).
LpBounds o Riesz T ans o ms 501
P oo : Se ρ= dis (E,F) and G={x: dis (x, F)<ρ/2}. Then, i is
clea ha dis (E,G)≥ρ/2 whe e we ha e w i en G o he opological
closu e o he se G. Besides, by defini ion dis (Rn G, F)≥ρ/2. Then,

A Bs ·χG
L2(F)≤
A Bs ·χG
L2(Rn)
≤C
Bs 
L2(G)
≤Ce
−dis (G,E)2
cs  L2(E)
≤Ce
−ρ2
cs L2(E).
No e ha in he second inequali y we ha e used ha A is uni o mly
bounded on L2(Rn), ac ha ollows om he hypo heses on A by
aking E=F=Rn. The hi d inequali y is jus he L2off-diagonal
es ima e assumed on Bs.On he o he hand,

A Bs ·χRn G
L2(F)≤Ce
−dis (Rn G,F )2
c Bs L2(Rn G)
≤Ce
−ρ2
c Bs L2(Rn)
≤Ce
−ρ2
c  L2(E),
whe e he fi s inequali y is a consequence o he L2off-diagonal es ima e
o A and he las one holds because, as be o e, Bsis uni o mly bounded
on L2(Rn). I we combine he wo es ima es we ge :
A Bs L2(F)≤
A Bs ·χG
L2(F)+
A Bs ·χRn G
L2(F)
≤Ce−ρ2
c +e−ρ2
cs L2(E)
≤Ce
−dis (E,F)2
cmax{ ,s} L2(E).
Now we can gi e a p oo o Lemma 2.2.
P oo o Lemma 2.2: We p o e he fi s es ima e since he o he one
ollows by duali y. We use he ollowing ep esen a ion o he Riesz
ans o m:
∇L−1
2h=1
2√π∞
0∇e−sLhds
√s=√m+2
2√π∞
0∇e−(m+2) sLh√sds
s

502 S. Ho mann, J. M. Ma ell
which leads o
∇L−1
2I−e− Lmdi 
=C∞
0∇e−sLe−msLI−e− Lm
e−sLdi 
√sds
s
=C
0···+∞
···
=C(I +II ).
No ice ha we ha e used he commu a ion p ope y o he semig oup.
Now we s udy each ope a o sepa a ely. Fo he fi s one we ha e
I−e− Lm=
m

k=0 m
j(−1)ke−k L =I+
m

k=1
cke−k L
and hen
I =
0∇e−sLe−msLe−sLdi 
√sds
s
+
m

k=1
ck
0∇e−k
2Le−(m+2) sLe−k
2Ldi 
√sds
s
=I ,0+
m

k=1
ckI ,k.
Then,
I ,0L2(F)≤
0
∇e−sLe−msLe−sLdi 

L2(F)√sds
s
=
0
s1
2∇e−sL◦e−msL◦s1
2e−sLdi )

L2(F)s−1
2ds
s
≤C
L2(E)
0
e−dis (E,F)2
cms s−1
2ds
s
=C
L2(E) −1
2∞
1
e−dis (E,F)2
c ss1
2ds
s,
whe e we ha e used Lemma 2.3 and no ed ha e e y ope a o sa isfies
he co esponding L2off-diagonal inequali y due o Lemma 2.1. Now we
LpBounds o Riesz T ans o ms 503
bound he in eg al:
∞
1
e−dis (E,F)2
c ss1
2ds
s≤C∞
1dis (E,F)2
c s−(m+1
2)
s1
2ds
s
=Cdis (E,F)2
−(m+1
2)
since m≥1. Then we ha e ob ained
I ,0L2(F)≤C
−1
2dis (E,F)2
−(m+1
2)

L2(E).
Now, fix 1 ≤k≤m, hen
I ,kL2(F)≤
0
∇e−k
2Le−(m+2) sLe−k
2Ldi 

L2(F)√sds
s
=
0



k
2∇e−k
2L◦e−(m+2) sL
◦k
2e−k
2Ldi 




L2(F)
2
k √sds
s
≤C
L2(E) −1
0
e−dis (E,F)2
cmax{k /2,(m+2) s}√sds
s
≤C
L2(E) −1e−dis (E,F)2
c 
0
√sds
s
≤C
−1
2e−dis (E,F)2
c 
L2(E)
≤C
−1
2dis (E,F)2
−(m+1
2)

L2(E).
Le us obse e ha because o Lemma 2.1 in he composi ion o he
ope a o s abo e each o hem e ifies an L2off-diagonal es ima e. This
ac allowed us o employ Lemma 2.3. We also used ha 1 ≤k≤m.
504 S. Ho mann, J. M. Ma ell
Collec ing his es ima e and he one p o ed o I ,0we ge
I L2(F)≤I ,0L2(F)+
m

k=1 |ck|I ,kL2(F)
≤C
−1
2dis (E,F)2
−(m+1
2)

L2(E).
Nex , we p oceed wi h he es ima e o II .
II L2(F)
≤C∞

√s∇e−sL◦e−sL
−e−(s+ )Lm
◦√se
−sLdi 

L2(F)s−1
2ds
s.
We know ha he fi s and he las ope a o s sa is y an L2off-diagonal
es ima e. We a e going o show no only ha he one in he middle
e ifies a simila es ima e bu also ha we gain some ex a decay ( /s)m.
Namely, le E,Fbe wo closed se s and gsuch ha supp g⊂E, hen

e−sL −e−(s+ )Lg
L2(F)=


−
0
d
d e−(s+ )Lgd



L2(F)
≤
0
(s+ )Le
−(s+ )Lg
L2(F)
d
s+
≤CgL2(E)
0
e−dis (E,F)2
c(s+ )d
s+
≤C
se−dis (E,F)2
cs gL2(E),
whe e in he las s ep we used ha ≤s≤s+ ≤s+ ≤2s, and
Lemma 2.1. Then we ha e p o ed ha



s
e−sL −e−(s+ )Lg

L2(F)≤Ce
−dis (E,F)2
cs gL2(E)
uni o mly on . Then his ope a o composed wi h i sel m imes will
sa is y he same inequali y in iew o Lemma 2.3. We can use his
LpBounds o Riesz T ans o ms 505
es ima e o bound II L2(F)as ollows:
II L2(F)≤C
L2(E)∞
e−dis (E,F)2
cs 
sm
s−1
2ds
s
=C
L2(E) −1
21
0
e−dis (E,F)2
c ssm+1
2ds
s
≤C
L2(E) −1
2∞
0
e−dis (E,F)2
c ssm+1
2ds
s
=C
L2(E) −1
2dis (E,F)2
−(m+1
2)∞
0
e−ssm+1
2ds
s
≤C
−1
2dis (E,F)2
−(m+1
2)

L2(E).
Collec ing he es ima es o I and II we ge

 1
2∇L−1
2I−e− Lmdi 

L2(F)≤C 1
2I L2(F)+II L2(F)
≤Cdis (E,F)2
−(m+1
2)

L2(E).
3. P oo o Theo em 1.2
The p oo o his esul is inspi ed by [DM], whe e a gene al me hod
is de eloped o p o e weak- ype (1,1) es ima es o singula in eg al
ope a o s wi hou assuming explici egula i y on he space a iables
o he ke nel. Tha me hod can be applied o de i e he boundedness
o he Riesz ans o ms i one assumes sufficien poin wise decay o he
hea ke nel. I is o be no ed ha in [DM], he e is an ex a assump ion
in ol ing a poin wise es ima e o he g adien o he hea ke nel, bu
his hypo hesis is supe fluous, as obse ed in [CD]. He e, a u he
s ep is aken: we de i e he bounds o he Riesz ans o ms om off-
diagonal es ima es on he hea semig oup ha hold in an a e age, as
opposed o a poin wise, sense. These a e age off-diagonal es ima es hold
o all ope a o s o he ype unde conside a ion ( his is he con en o
Lemma 2.1), in con as o poin wise es ima es which may ail.
512 S. Ho mann, J. M. Ma ell
and we ge he desi ed es ima e o II3.Now we a e conce ned wi h II2.
By Chebyche ’s inequali y we ge
(II2)1
2=


x∈E∗:
j
Djbj(x)
>α



1
2
≤1
α





j
Djbj




L2(E∗)
=1
αsup

hRn
j
Djbj(x),
h(x)dx
,
(8)
whe e ·,· deno es he inne p oduc in Cnand he sup emum is aken
o e all Cn- alued unc ions 
h∈L2(E∗) wi h 
hL2(E∗)=1. Le us
fix such a unc ion 
h.Fo e e y j,wedefine as be o e 
h(l,j)(x)=

h(x)χS(l,j)(x). Le us ecall ha E∗=Rn ∪
jQ∗
j.In his way, and
since supp
h⊂E∗⊂(2 Qj)cwe ha e
Rn
j
Djbj(x),
h(x)dx
=
j
∞

l=1 Rn!Djbj(x),
h(l,j)(x)"dx
=
j
∞

l=1 Qj
bj(x)D∗
j
h(l,j)(x)dx
=
j
∞

l=1 Rn
bj(x)D∗
j
h(l,j)(x)−D∗
j
h(l,j)Qjdx
≤
j
∞

l=1 bjLp(Qj)

D∗
j
h(l,j)−D∗
j
h(l,j)Qj

Lp(Qj)
≤Cα
j
∞

l=1 |Qj|1
p

D∗
j
h(l,j)−D∗
j
h(l,j)Qj

Lp(Qj),
(9)

LpBounds o Riesz T ans o ms 513
whe e D∗
jis he ope a o gi en in (1) wi h = j, and we ha e used bo h
p ope ies in (3). We apply Poinca ´e-Sobole inequali y and Lemma 2.2
o ge


D∗
j
h(l,j)−D∗
j
h(l,j)Qj

Lp(Qj)
≤C

∇D∗
j
h(l,j)

L2(Qj)
=C

∇TI−e− jLm∗
h(l,j)

L2(Qj)
≤C
−1
2
jdis (S(l,j),Q
j)2
j−(m+1
2)

h(l,j)L2(S(l,j))
≤C(Qj)−12−2(m+1
2)l
hL2(S(l,j)),
since j=(Qj)2and o l≥1weha e dis (S(l, j),Q
j)≥2l−2(Qj).
We now plug his es ima e in o (9) and i ollows ha
Rn
j#Djbj(x),
h(x)$dx
≤Cα
j
∞

l=1 |Qj|1
p(Qj)−12−2(m+1
2)l
hL2(S(l,j))
≤Cα
j
∞

l=1 |Qj|1
22−2(m+1
2)l|2l+1Qj|1
21
|2l+1Qj|2l+1Qj|
h(y)|2dy1
2
≤Cα
j|Qj|ess in
y∈Qj
M|
h|2(y)1
2
∞

l=1
2−l2(m+1
2)−n
2
≤Cα
{x∈Rn:M( p)(x)1
p>α}
1
2,
whe e in he las s ep we used ha m>n−2
4and so 2 (m+1
2)−n
2>0,
and we ha e p oceeded as in (7). Then we can use his es ima e, (8) and
514 S. Ho mann, J. M. Ma ell
ha Mis o weak ype (1,1) o ge
II2≤C{x∈Rn:M( p)(x)1
p>α}≤C
αpRn
(x)pdx.
The p oo o he ac ha Tis o weak ype (p, p)isnow comple ed by
collec ing he es ima es ha we ha e ob ained o I,II1,II2and II3.
No e added in p oo : P. Ausche has ecen ly obse ed ha he expo-
nen pnmay be eplaced by pn−,=(n, λ, Λ), n≥3[A].
Re e ences
[A] P. Ausche ,Onnecessa y and sufficien condi ions o Lp
es ima es o Riesz ans o ms associa ed o ellip ic ope a o s
on Rn:asu ey, P ep in (2003).
[AHLMT] P. Ausche , S. Ho mann, M. Lacey, A. McIn osh and
P. Tchami chian, The solu ion o he Ka o squa e oo
p oblem o second o de ellip ic ope a o s on Rn,Ann. o
Ma h. (2) 156(2) (2002), 633–654.
[AMT] P. Ausche , A. McIn osh and P. Tchami chian, Hea
ke nels o second o de complex ellip ic ope a o s and appli-
ca ions, J. Func . Anal. 152(1) (1998), 22–73.
[AT] P. Ausche and P. Tchami chian, Squa e oo p oblem
o di e gence ope a o s and ela ed opics, As ´e isque 249
(1998), 172 pp.
[BK1] S. Blunck and P. C. Kun smann,Weak- ype (p, p) es i-
ma es o Riesz ans o ms, P ep in .
[BK2] S. Blunck and P. C. Kun smann, Calde on-Zygmund
heo y o non-in eg al ope a o s and he H∞ unc ional cal-
culus, Re . Ma . Ibe oame icana ( o appea ).
[CD] T. Coulhon and X. T. Duong, Riesz ans o ms o 1 ≤
p≤2, T ans. Ame . Ma h. Soc. 351(3) (1999), 1151–1169.
[Du] J. Duoandikoe xea,“Fou ie analysis”,T ansla ed and e-
ised om he 1995 Spanish o iginal by Da id C uz-U ibe,
G adua e S udies in Ma hema ics 29, Ame ican Ma hema i-
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[DM] X. T. Duong and A. McIn osh, Singula in eg al ope a-
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[Ma1] J. M. Ma ell, Desigualdades con pesos en el An´alisis de
Fou ie : de los espacios de ipo homog´eneo a las medidas no
LpBounds o Riesz T ans o ms 515
doblan es, Ph. D. Thesis, Uni e sidad Au ´onoma de Mad id
(2001).
[Ma2] J. M. Ma ell, Sha p maximal unc ions associa ed wi h
app oxima ions o he iden i y in spaces o homogeneous ype
and applica ions, P ep in (2002).
[MNP] V. G. Maz’ya, S. A. Naza o and B. A. Plamene ski˘
ı,
Absence o a De Gio gi- ype heo em o s ongly ellip ic
equa ions wi h complex coefficien s, (Russian), Bounda y
alue p oblems o ma hema ical physics and ela ed ques ions
in he heo y o unc ions, 14, Zap. Nauchn. Sem. Lening ad.
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S e e Ho mann:
Depa men o Ma hema ics
Uni e si y o Missou i
Columbia, MO 65211
U.S.A.
E-mail add ess:[email p o ec ed]
Jos´eMa ´ıa Ma ell:
Depa amen o de Ma em´a icas, C-XV
Uni e sidad Au ´onoma de Mad id
28049 Mad id
Spain
E-mail add ess:[email p o ec ed]
and
Depa men o Ma hema ics
Uni e si y o Missou i
Columbia, MO 65211
U.S.A.
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 19 de desemb e de 2002,
da e a e si´o ebuda el 2 de maig de 2003.