Calderón-Zygmund operators associated to matrix-valued kernels

0.32 MB·English

by  Guixiang Hong

CALDERÓN-ZYGMUND OPERATORS
ASSOCIATED TO MATRIX-VALUED KERNELS
GUIXIANGHONG,LUISDANIELLÓPEZ-SÁNCHEZ,
JOSÉMARÍAMARTELLANDJAVIERPARCET
2
1 Abstract. Calderón-Zygmund operators with noncommuting kernels may
0 failto beLp-bounded forp6=2, evenfor kernels withgood sizeandsmooth-
2 ness properties. Matrix-valued paraproducts, Fourier multipliers on group
vNa’s or noncommutative martingale transforms are frameworks where we
n
findsuchdifficulties. WeobtainweaktypeestimatesforperfectdyadicCZO’s
a
andcancellativeHaarshiftsassociatedtononcommutingkernelsintermsofa
J
row/columndecompositionofthefunction. ArbitraryCZO’ssatisfyH1→L1
0 typeestimates. InconjunctionwithL∞→BMO,wegetcertainrow/column
2 Lp estimates. Ourapproach alsoappliestononcommutative paraproducts or
martingale transforms with noncommuting symbols/coefficients. Our results
] complementrecent resultsofJunge,Mei,ParcetandRandrianantoanina.
A
C
Introduction
.
h
at A semicommutative CZO has the formal expression
m
Tf(x) k(x,y)(f(y))dy,
[ ∼ ZRn

1 where the kernel acts linearly on the matrix-valued function f =(fij) and satisfies
v standardsize/smoothnessCalderón-Zygmundtypeconditions. Thisistheoperator
1
modelforquiteanumberofproblemswhichhaveattractedsomeattentioninrecent
5
years, including matrix-valued paraproducts, operator-valued Calderón-Zygmund
3
4 theoryorFouriermultipliersongroupvonNeumannalgebras,see[9,10,21,25,27]
1. andthereferencestherein. Tobemoreprecise,letB(ℓ2)standforthematrixalgebra
of bounded linear operators on ℓ . Consider the algebra formed by essentially
0 2
2 bounded functions f : Rn (ℓ2). Its weak operator closure is a von Neumann
→ B
1 algebra andassuchwemayconstructnoncommutativeL spacesoverit. Letus
p
A
: highlight a few significant examples:
v
i
X Scalar kernels. k(x,y) C and
• ∈
r
a k(x,y)(f(y)) = k(x,y)fij(y) .
(cid:16) (cid:17)
Schur product actions. k(x,y) (ℓ ) and
2
• ∈B
k(x,y)(f(y)) = k (x,y)f (y) .
ij ij
(cid:16) (cid:17)
Fully noncommutative model. k(x,y) (ℓ )¯ (ℓ ) and
2 2
• ∈B ⊗B
k(x,y)(f(y)) = tr k′′(y)f(y) k′ (x) .
m m ij
m
(cid:16)X (cid:0) (cid:1) (cid:17)
Partial traces, noncommuting kernels. k(x,y) (ℓ ) and
2
• ∈B
k (x,y)f (y) ,
s is sj
k(x,y)(f(y)) =
 (cid:16) (cid:17)
 Psfis(y)ksj(x,y) .
(cid:16) (cid:17)
1 P
2 HONG,LÓPEZ-SÁNCHEZ,MARTELLANDPARCET
Scalar kernels required in [27] a matrix-valued Calderón-Zygmund decomposi-
tion in terms of noncommutative martingales and a pseudo-localization principle
to control the tails of Tf in the L -metric. Hilbert space valued kernels were later
2
considered in [23], see also [20, 31, 34] for previous related results. The second
case refers to the Schur matrix product k(x,y) f(y), considered for the first time
•
in [10] to analyze cross product extensions of classical CZO’s. It is instrumental
for Hörmander-Mihlin type theorems on Fourier multipliers associated to discrete
groups and for Schur multipliers with a Calderón-Zygmund behavior [10, 11]. In
the fully noncommutative model, we approximate k(x,y) by a sum of elementary
tensors k′ (x) k′′(y) and the action is given by
m m ⊗ m
P
Tf(x) (id tr) k(x,y) 1 f(y) dy.
∼ ZRn ⊗ h (cid:0) ⊗ (cid:1)i
In this case, we regard the space L ( ) = L (Rn;L ( (ℓ ))) as a whole. In other
p p p 2
A B
words, the noncommutative nature of L ( ) predominates and the presence of a
p
A
Euclidean subspace is ignored. That is what happens for purely noncommutative
CZO’s [12] and justifies the presence of id tr, to integrate overthe full algebra
⊗ A
and not just over the Euclidean part. The last case refers to matrix-valued kernels
actingonf byleft/rightmultiplication,k(x,y)f(y)andf(y)k(x,y). Matrix-valued
paraproducts are prominent examples [17, 21, 22, 25, 33]. This is the only case in
which the kernel does not commute with f, since the Schur product is abelian and
we find (id tr)[k(x,y)(1 f(y))]=(id tr)[(1 f(y))k(x,y)] by traciality.
⊗ ⊗ ⊗ ⊗
Our main goal is to obtain endpoint estimates for CZO’s with noncommuting
kernels, motivated by a recent estimate from [10] for semicommutative CZO’s. If
k(x,y) acts linearly on (ℓ ) and satisfies the Hörmander smoothness condition in
2
B
the norm of bounded linear maps on (ℓ ), the content of [10, Lemma 1.3] can be
2
B
summarized as follows
If T is L ( (ℓ );Lr(Rn))-bounded, then T :L ( ) BMO ( ),
• ∞ B 2 2 ∞ A → r A
If T is L ( (ℓ );Lc(Rn))-bounded, then T :L ( ) BMO ( ).
• ∞ B 2 2 ∞ A → c A
Here, the L (Lc)-boundedness assumption refers to
∞ 2
1 1
Tf(x)∗Tf(x)dx 2 . f(x)∗f(x)dx 2 ,
(cid:13)(cid:16)ZRn (cid:17) (cid:13)B(ℓ2) (cid:13)(cid:16)ZRn (cid:17) (cid:13)B(ℓ2)
(cid:13) (cid:13) (cid:13) (cid:13)
while the(cid:13)column-BMO norm of a m(cid:13)atrix-val(cid:13)ued function g is given(cid:13)by
1
sup g(x) g ∗ g(x) g dx 2 .
Q Q
Q cube(cid:13)(cid:13)(cid:16)Z−Q(cid:0) − (cid:1) (cid:0) − (cid:1) (cid:17) (cid:13)(cid:13)B(ℓ2)
Taking adjoints —so th(cid:13)at the switches everywhere from(cid:13)left to right— we find
∗
L (Lr)-boundedness and the row-BMO norm. The noncommutative BMO space
∞ 2
BMO( )=BMO ( ) BMO ( ) was introducedin [31]. According to [24] it has
r c
A A ∩ A
the expected interpolation behavior in the L scale. Thus, standard interpolation
p
and duality arguments show that T : L ( ) L ( ) for 1 < p < provided
p p
A → A ∞
the kernel is smooth enough in both variables and T is a normal self-adjoint map
satisfying the L (Lr) and L (Lc) boundedness assumptions. In other words, the
∞ 2 ∞ 2
row/columnboundednessconditionsessentiallyplaytheroleoftheL -boundedness
2
assumption in classical Calderón-Zygmundtheory.
CZO’S WITH MATRIX-VALUED KERNELS 3
Although this certainly works for non-scalar kernels —Schur product actions
were used e.g. in [10, Theorem B]— the boundedness assumptions impose nearly
commuting conditions on the kernel which are too strong for CZO’s associated to
noncommuting kernels. Namely, given k : R2n ∆ (ℓ ) smooth and given
2
\ → B
x / suppRnf, let us set formally the row/column CZO’s
∈
T f(x)= k(x,y)f(y)dy and T f(x)= f(y)k(x,y)dy.
c r
ZRn ZRn
It is not difficult to construct noncommuting kernels with
i) T and T are L ( )-bounded,
r c 2
A
ii) T and T are not L ( )-bounded for 1<p=2< ,
r c p
A 6 ∞
see e.g. [27, Section 6.1]for specific examples. Therefore,the L (Lr) andL (Lc)
∞ 2 ∞ 2
boundedness assumption is in general too restrictive when kernel and function do
not commute. Assume for what follows that T and T are L ( )-bounded. We
r c 2
A
are interested in weakened forms of L boundedness and endpoint estimates for
p
these CZO’s. A dyadic noncommuting CZO will be a L ( )-bounded pair (T ,T )
2 r c
A
associated to a noncommuting kernel satisfying one of the following conditions:
a) Perfect dyadic kernels
k(x,y) k(z,y) + k(y,x) k(y,z) =0
− B(ℓ2) − B(ℓ2)
whenev(cid:13)er x,z Q and y(cid:13) R for(cid:13)some disjoint dy(cid:13)adic cubes Q,R.
(cid:13) ∈ (cid:13)∈ (cid:13) (cid:13)
b) Cancellative Haar shift operators
k(x,y)= αQ h (x)h (y),
RS R S
Q dyadic R,S dyadic⊂Q
X ℓ(R)=X2−rℓ(Q)
ℓ(S)=2−sℓ(S)
forsomefixed r,s Z wherethe αQ (ℓ )with αQ √|R||S|.
∈ + RS ∈B 2 k RSkB(ℓ2) ≤ |Q|
Here h refers to any of the 2n 1 Haar functions related to the cube Q.
Q
−
Perfect dyadic kernels were introduced in [1] and include Haar multipliers, as well
as paraproducts and their adjoints. If J and J denote the left/right halves of
− +
a dyadic interval in R, the standard model for Haar shifts is the dyadic Hilbert
transform with kernel (h (y) h (y))h (x). It appeared after Petermichl’s
J J− − J+ J
crucial result [30], showing the classical Hilbert transform as a certain average of
P
dyadicHilberttransforms. Hytönen’srepresentationtheorem[7]extendsthisresult
to arbitraryCZO’s. We will writegeneric noncommuting CZO for L ( )-bounded
2
A
pairs(T ,T )withanoncommutingkernelsatisfyingthestandardsmoothness. Our
r c
first significant result is the following.
Theorem A. The following inequalities hold:
i) Dyadic noncommuting CZO’s. Given f L ( )
1
∈ A
inf T f + T f . f .
f=fr+fc r r 1,∞ c c 1,∞ k k1
(cid:13) (cid:13) (cid:13) (cid:13)
ii) Generic noncommutin(cid:13)g CZ(cid:13)O’s. Gi(cid:13)ven f(cid:13) H ( )
1
∈ A
inf T f + T f . f .
f=fr+fc r r 1 c c 1 k kH1(A)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
4 HONG,LÓPEZ-SÁNCHEZ,MARTELLANDPARCET
The noncommutative forms of L and the Hardy space H are well-known
1,∞ 1
in the subject. Nevertheless, they will also be properly defined in the body of
the paper. Our main result is the inequality given in Theorem A i) and their
noncommutative generalizations in Theorem C below. As we shall explain in the
Appendix,theleft/rightmodularnatureofT /T isessentialfortheweaktype(1,1)
r c
estimates, see also Remark 2.5. The following result easily follows from Theorem
A by interpolation/duality and it can also be derived from [10]. Nevertheless, it is
worth mentioning the L inequalities that we find.
p
Theorem B. The following inequalities hold for generic noncommuting CZO’s:
i) If 1<p<2 and f L ( )
p
∈ A
inf T f + T f . f .
f=fr+fc r r p c c p k kp
(cid:13) (cid:13) (cid:13) (cid:13)
ii) If 2<p< and f L(cid:13)( )(cid:13) (cid:13) (cid:13)
p
∞ ∈ A
T f + T f . f .
r Hrp(A) c Hcp(A) k kp
(cid:13) (cid:13) (cid:13) (cid:13)
iii) Given f ∈L∞(A(cid:13)), w(cid:13)e also hav(cid:13)e kT(cid:13)rfkBMOr(A)+kTcfkBMOc(A) .kfk∞.
TheoremsAandBalsoholdforotheroperator-valuedfunctions,replacing (ℓ )
2
B
by any semifinite von Neumann algebra . Our proof will be written in this
M
framework. Let us now consider a weak- dense filtration Σ = ( ) of von
A n n≥1
∗ A
Neumann subalgebras of an arbitrary semifinite von Neumann algebra . In the
A
following result, we will consider two kind of operators in L ( ):
p
A
a) Noncommuting martingale transforms
Mrf = ∆ (f)ξ and Mcf = ξ ∆ (f).
ξ k k−1 ξ k−1 k
k≥1 k≥1
X X
b) Paraproducts with noncommuting symbol
Πr(f)= E (f)∆ (ρ) and Πc(f)= ∆ (ρ)E (f).
ρ k−1 k ρ k k−1
k≥1 k≥1
X X
Here ∆ denotes the martingale difference operator E E and ξ is an
k k k−1 k k
− ∈ A
adaptedsequence. Ofcourse,thesymbolsξ andρdonotnecessarilycommutewith
the function. Randrianantoanina considered in [34] noncommutative martingale
transforms with commuting coefficients. As for paraproducts with noncommuting
symbols, Mei studied the L -boundedness for p > 2 and regular filtrations in [21]
p
and also analyzed in [22] the case p < 2 in the dyadic matrix-valued case under
a strong BMO condition of the symbol. Our theorem below goes beyond these
results, see also [23] for related results.
Theorem C. Consider the pairs:
i) Martingale transforms (Mr,Mc), with sup ξ < .
ξ ξ kk kkM ∞
ii) Martingale paraproducts (Πr,Πc), with Πr/c L ( )-bounded.
ρ ρ ρ 2 A
If Σ is regular, we obtain weak type (1,1) inequalities like in Theorem Ai) for
A
martingale transforms and paraproducts . The estimates in Theorems Aii) and B
alsoholdforbothfamiliesandforarbitraryfiltrationsΣ . Moreover,themartingale
A
paraproducts Πr and Πc are L -bounded for 2<p< and L BMO.
ρ ρ p ∞ ∞ →
CZO’S WITH MATRIX-VALUED KERNELS 5
In the case of martingale transforms, there are also examples of noncommuting
kernelsfailingL -boundednessforp=2. Hence,ourresultsrecoverthosein[34,35]
p
6
and are in some sense sharp, providing appropriate substitutes for noncommuting
coefficients. Our result for paraproducts goes beyond [21, Theorem 1.2] in two
aspects. First, our estimates for p > 2 hold for arbitrary martingales, not just
for regular ones. Second, we give a partial answer to Mei’s question in [21] after
the proof of Theorem 1.2 for the case p < 2 and also for the weak type (1,1)
estimates. The paper is organized following the order in the Introduction. We
include an Appendix at the end with further comments and open problems. Along
thepaperweshallassumesomefamiliaritywithbasicnotionsfromnoncommutative
integration. The content of[27, Section 1]is enoughfor our purposes, more canbe
found in [16, 32, 37].
1. Calderón-Zygmund decomposition
Let be a semifinite von Neumann algebra equipped with a normal semifinite
M
faithful trace τ. Consider the algebra of essentially bounded functions Rn
→ M
equipped with the n.s.f. trace
ϕ(f)= τ(f(x))dx.
ZRn
Its weak-operator closure is a von Neumann algebra . If 1 p , we write
A ≤ ≤ ∞
L ( )andL ( )forthenoncommutativeL spacesassociatedtothepairs( ,τ)
p p p
andM( ,ϕ). TheAlattices of projections are written and , while 1 anMd 1
π π M A
A M A
standfor the unit elements. The set ofdyadic cubes in Rn is denotedby andwe
Q
use for the k-th generation,formed by cubes Q with side length ℓ(Q)=2−k. If
k
Q
f :Rn is integrable on Q , we set the average
→M ∈Q
1
f = f(y)dy.
Q
Q
| |ZQ
Let us write (Ek)k∈Z for the family of conditional expectations associated to the
classicaldyadicfiltrationonRn. E willalsostandforthetensorproductE id
k k M
⊗
acting on . If 1 p and f L ( )
p
A ≤ ≤∞ ∈ A
E (f) = f = f 1 ,
k k Q Q
QX∈Qk
∆k(f) = dfk = fQ−fQb 1Q,
QX∈Qk(cid:0) (cid:1)
where Q denotes the dyadic parent of Q. We will write ( k)k∈Z for the filtration
=E ( ). The noncommutative weak L -space, denoteAd by L ( ), is the set
k k 1 1,∞
A A A
of all ϕb-measurable operators f for which f =sup λϕ f >λ < , see
k k1,∞ λ>0 {| | } ∞
[5] for a more in depth discussion. In this case, we write ϕ f >λ to denote the
{| | }
traceofthespectralprojectionof f associatedtotheinterval(λ, ). Wefindthis
| | ∞
terminology more intuitive, since it is reminiscent of the classical one. The space
L ( ) is a quasi-Banach space and satisfies the quasi-triangle inequality below
1,∞
A
which will be used with no further reference
λϕ f +f >λ λϕ f >λ/2 +λϕ f >λ/2 .
1 2 1 2
| | ≤ | | | |
n o n o n o
6 HONG,LÓPEZ-SÁNCHEZ,MARTELLANDPARCET
Let us consider the dense subspace
=L ( ) f :Rn f , supp f is compact L+( ).
Ac,+ 1 A ∩ →M ∈A+ Rn ⊂ 1 A
Here suppRn meansnthe support of(cid:12)(cid:12)f as a vector-valued function ion Rn. In other
words,we have suppRnf =supp f M. We employ this terminologyto distinguish
k k
fromsuppf,thesupportoff asanoperatorin . Anyfunctionf givesrise
c,+
A ∈A
to a martingale (fk)k∈Z with respect to the dyadic filtration. Moreover, it is clear
that given f and λ>0, there must exist m (f) Z so that 0 f λ for
c,+ λ k
∈A ∈ ≤ ≤
allk m (f). The noncommutativeanalogueofthe weaktype (1,1)boundedness
λ
≤
of Doob’s maximal function is due to Cuculescu. Here we state it in the context of
operator-valuedfunctions from .
A
Cuculescu’s construction [4]. Let f and consider the corresponding
c,+
∈ A
martingale (fk)k∈Z relative to the filtration ( k)k∈Z. Given λ R+, there exists a
A ∈
decreasing sequence of projections (qk(λ))k∈Z in satisfying
A
i) q (λ) commutes with q (λ)f q (λ) for each k,
k k−1 k k−1
ii) q (λ) belongs to for each k and q (λ)f q (λ) λq (λ),
k k k k k k
A ≤
iii) The following estimate holds
1 1
ϕ 1 q (λ) sup f = f .
A k k 1 1
(cid:16) −k^∈Z (cid:17) ≤ λ k∈Zk k λk k
Explicitly, take q (λ)=χ (q (λ)f q (λ)) with q (λ)=1 for k m (f).
k (0,λ] k−1 k k−1 k A λ
≤
Given f c,+, consider the Cuculescu’s sequence (qk(λ))k∈Z associated to
∈ A
(f,λ) for a given λ>0. Since λ will be fixed most of the time, we will shorten the
notation by qk and only write qk(λ) when needed. Define the sequence (pk)k∈Z of
disjoint projections p =q q , so that
k k−1 k
−
p =1 q with q = q .
k A k
−
k∈Z k∈Z
X ^
Calderón-Zygmund decomposition [27]. Given f and λ > 0, we may
c,+
∈ A
decompose f = g +g +b +b as the sum of four operators defined in terms
d off d off
of the Cuculescu’s construction as follows
g = qfq+ p f p ,
d k k k
k∈Z
X
b = p (f f )p ,
d k k k
−
k∈Z
X
b = p (f f )p ,
off i i∨j j
−
i6=j
X
g = p f p + qf(1 q)+(1 q)fq.
off i i∨j j A A
− −
i6=j
X
Moreover, we have the diagonal estimates
2
qfq+ p f p 2nλ f and p (f f )p 2 f .
k k k 2 ≤ k k1 k − k k 1 ≤ k k1
(cid:13)(cid:13)(cid:13) Xk∈Z (cid:13)(cid:13)(cid:13) Xk∈Z(cid:13)(cid:13) (cid:13)(cid:13)
CZO’S WITH MATRIX-VALUED KERNELS 7
The expression below for g will be also instrumental
off
∞ ∞ ∞ ∞ ∞
g = p df q +q df p = g = g .
off k k+s k+s−1 k+s−1 k+s k k,s (s)
Xs=1k=Xmλ+1 Xs=1k=Xmλ+1 Xs=1
2. Proof of Theorems A and B
The key result of this paper is Theorem A, since the remaining theorems follow
fromitorbyusinganalogideas. Webeginwiththeproofoftheweaktypeestimates
forperfectdyadicCZO’sandthenmakethenecessaryadjustmentstomakeitwork
for Haar shift operators. The proof of Theorem Aii) will require to recall some
recent results on square function and atomic Hardy spaces.
2.1. Perfect dyadic CZO’s. To the best of our knowledge, the notion of perfect
dyadic Calderón-Zygmund operator was rigorously defined for the first time in [1]
by Auscher, Hofmann, Muscalu, Tao and Thiele. Accordingly, we define a perfect
dyadic CZO with noncommuting kernel as a pair (T ,T ) formally given by
r c
T f(x) k(x,y)f(y)dy,
r
∼ ZRn
T f(x) f(y)k(x,y)dy,
c
∼ ZRn
with an -valued kernel satisfying the perfect dyadic conditions
M
k(x,y) k(z,y) + k(y,x) k(y,z) =0
− M − M
wheneverx,z Q(cid:13)andy Rforso(cid:13)medis(cid:13)jointdyadiccube(cid:13)sQ,R. Alternatively,we
may think of p∈erf(cid:13)ect dya∈dic kernel(cid:13)s k :R(cid:13)2n ∆ as t(cid:13)hose which are constant
\ →M
on 2n-cubes of the form Q R, where Q,R are distinct dyadic cubes in Rn with
×
the same side length and sharing the same dyadic parent. Classical perfect dyadic
CZO’s include Haar multipliers/martingale transforms and dyadic paraproducts.
In other words, operators of the following form
ξ(Q)
Hξf(x) = ZRn(cid:16)QX∈Q |Qb| 1Q(x)(1Q−2−n1Qb)(y)(cid:17)f(y)dy,
1
Πρf(x) = ZRn(cid:16)QX∈Q|Q|(ρQ−ρQb)1Q(x)2−n1Qb(y)(cid:17)f(y)dy,
with sup ξ(Q) < and ρ : Rn C in dyadic BMO. Adjoints of paraproducts
Q| | ∞ →
arealsoperfectdyadic. Inthenoncommutingsetting, the coefficientsξ(Q)andthe
symbolρ become operatorsin and an -valuedfunction respectively which do
M M
not commute a priori with f L ( ). Nevertheless, the perfect dyadic condition
p
∈ A
for the kernel is still satisfied in these cases.
Proof of Theorem Ai) — Perfect dyadic CZO’s. Splitting f as a sum of
four positive operators and by density of in the positive cone of L ( ), we
c,+ 1
A A
may clearly assume that f . A well-known lack of Cuculescu’s construction
c,+
∈A
is that we do not necessarily have q (λ ) q (λ ) for λ λ . This is typically
k 1 k 2 1 2
≤ ≤
solved restricting our attention to lacunary values for λ. Define
π = q (2s) q (2s) for j,k Z.
j,k k k
− ∈
s≥j s≥j−1
^ ^
8 HONG,LÓPEZ-SÁNCHEZ,MARTELLANDPARCET
We have π S=OT1 ψ , where
j j,k A− k
P ψ = q (2s).
k k
s∈Z
^
Observe that ψ df =df ψ =0 for k Z. Indeed, we have
k k k k
∈
1 1 1 1
ψ df ψ f2 f 2 + ψ f2 f 2
k k kkA ≤ k k k kAk kkA k k k−1kAk k−1kA
= kψkfkψkkA21kfkkA12 +kψkfk−1ψkkA21kfk−1kA21 ≤ s→lim−∞21+2skfkA21.
In particular,we find f = (1 ψ )df (1 ψ ) and set f =f +f with
k A− k−1 k A− k−1 r c
f = LPT (df ) = π df π ,
r k−1 k i,k−1 k j,k−1
Xk∈Z Xk∈Z(cid:16)Xi>j (cid:17)
f = UT (df ) = π df π .
c k−1 k i,k−1 k j,k−1
Xk∈Z Xk∈Z(cid:16)Xi≤j (cid:17)
This is the decomposition we will use for any perfect dyadic CZO. Given such an
operator T = (T ,T ) and λ > 0, the goal is to show that there exists an absolute
r c
constant c so that λϕ T f > λ +λϕ T f > λ c f for any f
0 r r c c 0 1 c,+
{| | } {| | } ≤ k k ∈ A
andanyλ>0. Bysymmetryin the argument,wewilljust provethe inequalityfor
T f . Moreover,replacingc by2c wemayalsoassumethatλ=2ℓ forsomeℓ Z.
c c 0 0
∈
Havingfixedthevalueofλ,wemayconsidertheCalderón-Zygmunddecomposition
f =g +g +b +b and set
d off d off
gc = UT ∆ (g ) , gc = UT ∆ (g ) ,
d k−1 k d off k−1 k off
k∈Z k∈Z
X (cid:0) (cid:1) X (cid:0) (cid:1)
bc = UT ∆ (b ) , bc = UT ∆ (b ) .
d k−1 k d off k−1 k off
k∈Z k∈Z
X (cid:0) (cid:1) X (cid:0) (cid:1)
By the quasi-triangle inequality it suffices to show
λ ϕ T gc >λ +ϕ T bc >λ +ϕ T gc >λ +ϕ T bc >λ . f .
| c d| | c d| | c off| | c off| k k1
h n o n o n o n oi
The first term is first estimated by Chebychev’s inequality in
A
λϕ T gc >λ 1 T gc 2 . 1 gc 2.
| c d| ≤ λ c d 2 λk dk2
We use that UTk−1 ∆kn(gd) are inofact m(cid:13)(cid:13)arting(cid:13)(cid:13)ale differences, so that
1 gc 2 = (cid:0)1 U(cid:1)T ∆ (g ) 2 1 ∆ (g ) 2
λk dk2 λ k−1 k d 2 ≤ λ k k d k2
k∈Z k∈Z
X(cid:13) (cid:0) (cid:1)(cid:13) X
1 (cid:13) 2 1(cid:13) 2
= ∆ (g ) = qfq+ p f p 2n f .
k d k k k 1
λ 2 λ 2 ≤ k k
(cid:13)Xk∈Z (cid:13) (cid:13) Xk∈Z (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
Indeed, the first inequ(cid:13)ality above f(cid:13)ollows fro(cid:13)m the fact that tr(cid:13)iangular truncations
arecontractiveinL ( ) while the lastinequality arisefromthe diagonalestimates
2
A
in the noncommutative CZ decomposition stated above. To handle the remaining
terms, we introduce the projection
q = q(2s) = q (2s).
k
s≥ℓ s≥ℓk∈Z
^ ^ ^
b
CZO’S WITH MATRIX-VALUED KERNELS 9
According to Cuculescu’s construction, we find
1 2
ϕ 1 q ϕ 1 q(2s) f = f .
A− ≤ A− ≤ 2sk k1 λk k1
s≥ℓ s≥ℓ
(cid:0) (cid:1) X (cid:0) (cid:1) X
This reduces our probblem to show that
λ ϕ T (bc)q >λ +ϕ T (gc )q >λ +ϕ T (bc )q >λ . f .
c d c off c off k k1
Thehpernf(cid:12)ect dyad(cid:12)ic naoture onf(cid:12)T comes(cid:12)nowointo scne(cid:12)ne. Indee(cid:12)d, weocilaim that the
three term(cid:12) s T (bbc(cid:12))q,T (gc )q,(cid:12)Tc(bc )qb(cid:12)vanish when(cid:12)ever T ibs(cid:12)perfect dyadic. This
c d c off c off c
will be enough to conclude the proof. If Q (x) is the only cube in containing
k k
Q
x, we find a.e. x b b b
T (bc)(x)q(x) = T UT (∆ (b )) (x)q(x)
c d c k−1 k d
k∈Z
X (cid:0) (cid:1)
b = T UT (∆ (b ))1 b (x)q(x)
c k−1 k d Qk−1(x)
Xk∈Z (cid:16) (cid:17)
+ k(x,y)UT (∆ (bb))(y)dy q(x).
k−1 k d
Xk∈ZQ∈XQk−1(cid:16)ZQ (cid:17)
x∈/Q
b
The last term on the right vanishes since the term UT (∆ (b )) has mean 0 in
k−1 k d
anyQ ,sothatwemayreplacek(x,y)byk(x,y) k(x,c ),whichis0when
k−1 Q
∈Q −
x / Q by the perfect dyadic cancellation of the kernel. On the other hand, if we
∈
define the projection
q = q (2s),
k−1 k−1
s≥ℓ
^
we see that q(x)=qk−1(x)q(x)b=qk−1(y)q(x) for any y Qk−1(x). This gives
∈
T (bc)(x)q(x) = T UT (∆ (b ))q 1 (x)q(x).
cbd b b k cb k−1b k d k−1 Qk−1(x)
The exact same argument aXpplies(cid:16)for gc and bc , so that it su(cid:17)ffices to prove
off off
b b b
UT (∆ (b ))q = 0,
k−1 k d k−1
UT (∆ (g ))q = 0,
k−1 k off k−1
UT (∆ (b ))qb = 0,
k−1 k off k−1
b
for all k Z. In all these cases we will be using the following two key identities
∈
b
q π =π q =0 for i,j >ℓ and k Z,
k−1 i,k−1 j,k−1 k−1
• ∈
π p =p π =0 for s 1, i,j ℓ and k Z.
i,k−1 k−s k−s j,k−1
• ≥ ≤ ∈
b b
The proof is straightforward and left to the reader. It only requires to apply the
monotonicity properties of q (2s), which increases in j and decreases in k. If
s≥j k
we apply the first identity to UT (∆ (γ))q for any γ, we get
k−1 k k−1
V
UT (∆ (γ))q = π dγ π q .
k−1 k k−1 i,k−1 k j,k−1 k−1
b
i≤j≤ℓ
X
Therefore, if we know that dγ b= A +B where the left subpport of A and the
k k k k
rightsupportofB aredominatedby p =1 q ,thenwededucethat
k s≥1 k−s A− k−1
UT (∆ (γ))q =0. In other words, it suffices to prove that
k−1 k k−1
P
q ∆ (γ)q = 0 for γ =b ,g ,b .
k−1 k k−1 d off off
b
10 HONG,LÓPEZ-SÁNCHEZ,MARTELLANDPARCET
We have
∆ (b ) = ∆ p (f f )p
k d k j j j
−
j
X (cid:0) (cid:1)
= p (f f )p p (f f )p
j k j j j k−1 j j
− − −
j<k j<k−1
X X
= p df p = (1 q )∆ (b )(1 q ).
j k j A k−1 k d A k−1
− −
j≤k−1
X
To calculate the martingale differences for g , we invoke the formula
off
∞
g = p df q +q df p
off j j+s j+s−1 j+s−1 j+s j
s=1j∈Z
XX
given in the statement of the Calderón-Zygmunddecomposition. Then we find
∞
∆ (g ) = p df q +q df p
k off k−s k k−1 k−1 k k−s
s=1
X
= (1 q )df q +q df (1 q ).
A k−1 k k−1 k−1 k A k−1
− −
Finally, it remains to consider the martingale differences of b
off
∞
∆ (b ) = ∆ p (f f )p +p (f f )p
k off k j j+s j+s j+s j+s j
− −
s=1j∈Z
XX (cid:0) (cid:1)
∞
= p (f f )p +p (f f )p
j k j+s j+s j+s k j+s j
− −
s=1 j<k−s
X X
∞
p (f f )p +p (f f )p
j k−1 j+s j+s j+s k−1 j+s j
− − −
s=1j<k−s−1
X X
∞ ∞
= p df p + p df p = A +B .
j k j+s j+s k j k k
s=1 j<k−s s=1 j<k−s
X X X X
So q A =B q =0 and q ∆ (γ)q =0 for γ=b ,g ,b as desired. (cid:3)
k−1 k k k−1 k−1 k k−1 d off off
2.2. Haar shift operators. The Haar system has the form
n
1
hεQ(x) = Q 1Ij−(xj)+εj1Ij+(xj)
| |jY=1(cid:0) (cid:1)
where Q = I I Ip and ε = (ε ,ε ,...,ε ) = (1,1,...,1) with
1 2 n 1 2 n
ε 1. We a×re usi×ng··I·−×and∈I+Qfor the left/right halves of6the intervals I . It
j ∈ ± j j j
yields an orthonormal system in L (Rn) composed of mean zero functions. If we
2
write h for any Haar function of the form hε , a noncommuting dyadic shift with
Q Q
complexity (r,s) has the form
X f(x) = A f = αQ f,h h (x),
α Q RS S R
Q∈Q Q∈Q R,S dyadic⊂Q
X X ℓ(R)=X2−rℓ(Q) (cid:10) (cid:11)
ℓ(S)=2−sℓ(Q)
where f,h = fh and αQ are operators in satisfying αQ √|R||S|.
h Si S RS M k RSkM ≤ |Q|
R