Table Of ContentPSEUDO-LOCALIZATION OF SINGULAR INTEGRALS AND
NONCOMMUTATIVE LITTLEWOOD-PALEY INEQUALITIES
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0 TAOMEIANDJAVIERPARCET
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2
n
a
J Contents
6
2 Introduction 1
1. Martingale transforms 5
]
2. Caldero´n-Zygmundoperators 16
A
Appendix A. Hilbert space valued pseudo-localization 26
O
Appendix B. Background on noncommutative integration 32
.
h References 37
t
a
m
[ Introduction
3
Understood in a wide sense, square functions play a central role in classical
v
1 Littlewood-Paley theory. This entails for instance dyadic type decompositions of
7 Fourier series, Stein’s theory for symmetric diffusion semigroups or Burkholder’s
3 martingalesquarefunction. Allthesetopicsprovideadeeptechniquewhendealing
4
withquasi-orthogonalitymethods,sumsofindependentvariables,Fouriermultiplier
.
7 estimates... The historical survey [34] is an excellent exposition. In a completely
0 different setting, the rapid development of operator space theory and quantum
8
probability has given rise to noncommutative analogs of several classical results
0
in harmonic analysis. We find new results on Fourier/Schur multipliers, a settled
:
v theory of noncommutative martingale inequalities, an extension for semigroups on
i
X noncommutativeLpspacesoftheLittlewood-Paley-Steintheory,anoncommutative
ergodic theory and a germ for a noncommutative Caldero´n-Zygmund theory. We
r
a refer to [6, 8, 10, 11, 17, 24, 27] and the references therein.
The aim of this paper is to produce weak type inequalities for a large class
of noncommutative square functions. In conjunction with BMO type estimates
interpolation and duality, we will obtain the corresponding norm equivalences in
the whole L scale. Apart from the results themselves, perhaps the main novelty
p
relies on our approach. Indeed, emulating the classical theory, we shall develop a
row/column valued theoryof noncommutativemartingale transformsandoperator
valuedCaldero´n-Zygmundoperators. Thisseemstobenewinthenoncommutative
settingandmayberegardedasafirststeptowardsanoncommutativevector-valued
theory. To illustrate it, let us state our result for noncommutative martingales.
2000Mathematics SubjectClassification: 42B20,42B25,46L51, 46L52,46L53.
Keywords: Caldero´n-Zygmundoperator,almostorthogonality, noncommutative martingale.
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2 TAOMEIANDJAVIERPARCET
Theorem A1. Let ( ) stand for a weak dense increasing filtration in a
n n 1 ∗
M ≥
semifinite von Neumann algebra ( ,τ) equipped with a normal semifinite faithful
M
trace τ. Given f =(f ) an L ( ) martingale, let
n n 1 1
≥ M
∞ ∞
T f = ξ df with sup ξ 2 .1.
m km k km
| |
k 1
Xk=1 ≥ mX=1
Then, there exists a decomposition T f =A f +B f, satisfying
m m m
∞ (A f)(A f) 12 + ∞ (B f) (B f) 12 . sup f .
m m ∗ m ∗ m n 1
1, 1, n 1k k
(cid:13)(cid:16)mX=1 (cid:17) (cid:13) ∞ (cid:13)(cid:16)mX=1 (cid:17) (cid:13) ∞ ≥
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
In the statement above, df denotes the k-th martingale difference of f relative
k
tothefiltration( ) and refersto thenormonL ( ). Inthe result
n n 1 1, 1,
M ≥ kk ∞ ∞ M
below, we also need to use the norm on BMO( ) relative to our filtration as well
M
as the norm on L ( ;ℓ2). All these norms are standard in the noncommutative
p M rc
setting and we refer to Section 1 below for precise definitions. Moreover, in what
followsδ ande willstandforunitvectorsofsequencespacesandmatrixalgebras
k ij
respectively.
Theorem A2. Let us set = ¯ (ℓ ). Assumethat f is an L ( )martingale
2
R M⊗B ∞ M
relative to the filtration ( ) and define T f with coefficients ξ satisfying
n n 1 m km
M ≥
the same condition above. Then, we have
∞ ∞
T f e + T f e . sup f .
m 1m m m1 n
⊗ BMO( ) ⊗ BMO( ) n 1k k∞
(cid:13)mX=1 (cid:13) R (cid:13)mX=1 (cid:13) R ≥
(cid:13) (cid:13) (cid:13) (cid:13)
Therefo(cid:13)re, given 1<p<(cid:13) and f (cid:13)Lp( ), we ded(cid:13)uce
∞ ∈ M
∞
T f δ c f .
m m p p
(cid:13)mX=1 ⊗ (cid:13)Lp(M;ℓ2rc) ≤ k k
(cid:13) (cid:13)
Moreover, the reverse i(cid:13)nequality also h(cid:13)olds if m|ξkm|2 ∼1 uniformly on k.
P
Let us briefly analyze Theorems A1 and A2. Taking ξ to be the Dirac delta
km
on (k,m), we find T f = df and our results follow from the noncommutative
m m
Burkholder-Gundy inequalities [27, 30]. Moreover, taking ξ = 0 for m > 1 we
km
simplyobtainamartingaletransformwithscalarcoefficients andourresultsfollow
from [29]. Other known examples appear by considering (ξ ) of diagonal-like
km
shape. For instance, taking an arbitrary partition
1 if k Ω ,
N= Ω and ξ = ∈ m
m km
(0 otherwise.
m 1
[≥
It is apparent that ξ 2 =1 and Theorem A2 gives e.g. for 2 p<
m| km| ≤ ∞
∞ P df 2 21 + ∞ df 2 12 c f .
k p k∗ p ∼ pk kp
(cid:13)(cid:13)(cid:16)mX=1(cid:12)(cid:12)kX∈Ωm (cid:12)(cid:12) (cid:17) (cid:13)(cid:13) (cid:13)(cid:13)(cid:16)mX=1(cid:12)(cid:12)kX∈Ωm (cid:12)(cid:12) (cid:17) (cid:13)(cid:13)
Except fo(cid:13)r p = 1(cid:12), this follo(cid:12)ws fr(cid:13)om t(cid:13)he nonc(cid:12)ommutati(cid:12)ve K(cid:13)hintchine inequality in
conjunctionwiththeL boundednessofmartingaletransforms. The newexamples
p
appear when consideringmore generalmatrices (ξ ) and will be further analyzed
km
in the body of the paper.
NONCOMMUTATIVE LITTLEWOOD-PALEY INEQUALITIES 3
In the framework of Theorems A1 and A2, the arguments in [27, 29, 30] are no
longer valid. Instead, we think in our square functions as martingale transforms
with row/columnvalued coefficients
ξr
k
∞ (Tmf)(Tmf)∗ 21 ∞ dfk e1,1 ∞ ξk,m1 e1m ,
∼ ⊗ z }|M⊗ {
(cid:16)mX=1 (cid:17) Xk=1(cid:0) (cid:1)(cid:16)mX=1 (cid:17)
∞ (T f) (T f) 21 ∞ ∞ ξ 1 e df e ,
m ∗ m k,m m1 k 1,1
∼ M⊗ ⊗
(cid:16)mX=1 (cid:17) Xk=1(cid:16)mX=1 (cid:17)(cid:0) (cid:1)
ξc
k
where means to have the same L ( |) or L {(z ) norm}. Tensorizing with the
1, p
∼ ∞ M M
identity on (ℓ ), we have df e = d(f e ) and we find our row/column
2 k 1,1 1,1 k
B ⊗ ⊗
valued transforms. According to [29], we might expect
∞ ∞
ξrd(f e ) c sup ξr df e . c f
k ⊗ 1,1 k p ≤ p k 1k kkB(ℓ2) k⊗ 1,1 p pk kp
(cid:13)kX=1 (cid:13) ≥ (cid:13)Xk=1 (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
andthe(cid:13)sameestimatefort(cid:13)heξc’s. However,itis(cid:13)essentialin[29](cid:13)tohavecommuting
k
coefficients ξ , where = ¯ (ℓ ) in our setting. This is
k ∈ Rk−1 ∩ R′k Rn Mn⊗B 2
not the case. In fact, the inequality above is false in general (e.g. take again
ξ = δ with 1 < p < 2) and Theorems A1 and A2 might be regarded as the
km (k,m)
rightsubstitute. Thesamephenomenonwilloccuristhecontextofoperator-valued
Caldero´n-Zygmundoperators below.
OurmaintoolstoovercomeitwillbethenoncommutativeformsofGundy’sand
Caldero´n-Zygmunddecompositions[24,26]formartingalestransformsandsingular
integral operators respectively. As it was justified in [24], there exists nevertheless
a substantial difference between both settings. Namely, martingale transforms are
local operators while Caldero´n-Zygmund operators are only pseudo-local. In this
paper we will illustrate this point by means of Rota’s dilation theorem [33]. The
pseudo-localization estimate that we need in this setting, to pass from martingale
transforms to Caldero´n-Zygmund operators, is a Hilbert space valued version of
that given in [24] and will be sketched in Appendix A.
Now we formulate our results for Caldero´n-Zygmund operators. Let ∆ denote
the diagonalofRn Rn andfixa Hilbertspace . We willwriteinwhatfollowsT
to denote an integr×aloperatorassociatedto a keHrnelk :R2n ∆ . This means
\ →H
that for any smooth test function f with compact support, we have
Tf(x)= k(x,y)f(y)dy for all x / suppf.
ZRn ∈
Given two points x,y Rn, the distance x y between x and y will be taken
∈ | − |
for convenience with respect to the ℓ (n) metric. As usual, we impose size and
∞
smoothness conditions on the kernel:
a) If x,y Rn, we have
∈
1
k(x,y) . .
x y n
H | − |
(cid:13) (cid:13)
(cid:13) (cid:13)
4 TAOMEIANDJAVIERPARCET
b) There exists 0<γ 1 such that
≤
x x γ 1
k(x,y)−k(x′,y) . x| −y n′|+γ if |x−x′|≤ 2|x−y|,
H | − |
(cid:13) (cid:13) y y γ 1
(cid:13) (cid:13) ′
k(x,y)−k(x,y′) . x| −y n|+γ if |y−y′|≤ 2|x−y|.
H | − |
(cid:13) (cid:13)
We will r(cid:13)efer to this γ as t(cid:13)he Lipschitz parameter of the kernel. The statement
of our results below requires to consider appropriate -valued noncommutative
H
functionspacesasin[10]. Letusfirstconsiderthealgebra ofessentiallybounded
B
A
functions with values in
M
= f :Rn f strongly measurable s.t. ess sup f(x) < ,
B
A →M x Rn k kM ∞
n (cid:12) ∈ o
equipped with the n.s.f. t(cid:12)race ϕ(f) = τ(f(x))dx. The weak-operator closure
Rn
of is a von Neumann algebra. Given a norm 1 element e , take p to be
B e
A A R ∈H
the orthogonal projection onto the one-dimensional subspace generated by e and
define
L ( ; ) = (1 p )L ( ¯ ( )),
p r e p
A H A⊗ A⊗B H
L ( ; ) = L ( ¯ ( ))(1 p ).
p c p e
A H A⊗B H A⊗
Thisdefinitionisessentiallyindependentofthechoiceofe. Indeed,givenafunction
f L ( ; )wemayregarditasanelementofL ( ¯ ( )),sothattheproduct
p r p
∈ A H A⊗B H
ff belongs to (1 p )L ( ¯ ( ))(1 p ) which may be identified with
∗ e p/2 e
A ⊗ A⊗B H A ⊗
L ( ). When f L ( ; ) the same holds for f f and we conclude
p/2 p c ∗
A ∈ A H
kfkLp(A;Hr) = (ff∗)12 Lp(A) and kfkLp(A;Hc) = (f∗f)21 Lp(A).
Arguing as in [10, Cha(cid:13)pter 2],(cid:13)we may use these identities to(cid:13)regard(cid:13)L ( ) as
(cid:13) (cid:13) (cid:13) (cid:13) p A ⊗H
a dense subspace of L ( ; ) and L ( ; ). More specifically, given a function
p r p c
A H A H
f = g v L ( ) , we have
k k⊗ k ∈ p A ⊗H
1
P f = v ,v g g 2 ,
k kLp(A;Hr) (cid:13)(cid:16)Xi,j h i ji i j∗(cid:17) (cid:13)Lp(A)
(cid:13) (cid:13)
f = (cid:13) v ,v g g 21(cid:13) .
k kLp(A;Hc) (cid:13)(cid:16)Xi,j h i ji i∗ j(cid:17) (cid:13)Lp(A)
(cid:13) (cid:13)
This procedure may also be used to(cid:13)define the spaces (cid:13)
L ( ; ) and L ( ; ).
1, r 1, c
∞ A H ∞ A H
It is clear that L ( ; )=L ( ; ) and we will denote it by L ( ; ).
2 r 2 c 2 oh
M H M H M H
Theorem B1. Given f L ( ), define formally
1
∈ A
Tf(x)= k(x,y)f(y)dy
ZRn
where the kernel k :R2n ∆ satisfies the size/smoothness conditions imposed
\ →H
above. Assume further that T defines a bounded map L ( ) L ( ; ). Then
2 2 oh
A → A H
we may find a decomposition Tf =Af +Bf, satisfying
Af + Bf . f .
L1,∞(A;Hr) L1,∞(A;Hc) k k1
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
NONCOMMUTATIVE LITTLEWOOD-PALEY INEQUALITIES 5
To state the following result, we need to define the corresponding -valued
H
BMO norm. Assume for simplicity that is separable and fix an orthonormal
H
basis (v ) in . Then, given a norm 1 element e , we may identify (as
k k 1
≥ H ∈ H
above) f = g v BMO( ) with
k k⊗ k ∈ A ⊗H
fP= g (e v ) = (1 p ) g (e v ) ,
e k k⊗ ⊗ k A⊗ e k k⊗ ⊗ k
fe = Pkgk⊗(vk⊗e) = kgk⊗(cid:16)(Pe⊗vk) (1A⊗pe(cid:17)),
(cid:16) (cid:17)
wheree v isundePrstoodastherank1operPatorξ v ,ξ eandv estands
k k k
⊗ ∈H7→h i ⊗
for ξ e,ξ v . Then we define the spaces BMO( ; ) and BMO( ; ) as
k r c
∈H7→h i A H A H
the closure of BMO( ) with respect to the norms
A ⊗H
f = f and f = f .
k kBMO(A;Hr) ke kBMO(A⊗B(H)) k kBMO(A;Hc) k ekBMO(A⊗B(H))
In the following result, we also use the standard terminology
L ( ; )+L ( ; ) 1 p 2,
Lp(A;Hrc)= Lpp(A;Hrr) Lpp(A;Hcc) 2≤p≤ .
(cid:26) A H ∩ A H ≤ ≤∞
Theorem B2. If f L ( ), we also have
∈ ∞ A
Tf + Tf . f .
BMO(A;Hr) BMO(A;Hc) k k∞
Therefore, given 1<(cid:13)p<(cid:13) and f L(cid:13)( (cid:13)), we deduce
(cid:13) (cid:13) ∞ ∈ (cid:13)p A(cid:13)
Tf c f .
Lp(A;Hrc) ≤ pk kp
Moreover, the reverse inequa(cid:13)lity (cid:13)holds whenever Tf = f .
(cid:13) (cid:13) k kL2(A;Hoh) k kL2(A)
In Section1 we proveTheoremsA1 andA2. Then we study anspecific example
on ergodic averages as in [35]. In conjunction with Rota’s theorem, this shows
therelevanceofpseudo-localizationintheCaldero´n-Zygmundsetting. Wealsofind
somemultilinearandoperator-valuedformsofourresults. TheoremsB1andB2are
proved in Section 2. The proof requires a Hilbert space valued pseudo-localization
estimate adapted from [24] in Appendix A. After the proof, we list some examples
and applications. Although most of the examples are semicommutative, we find
several new square functions not considered in [10, 19] and find an application in
the fully noncommutative setting which will be explored in [12]. Finally, follow-
ing a referee’s suggestion, we also include an additional Appendix B with some
backgroundonnoncommutativeL spaces,noncommutativemartingalesandafew
p
examples for nonexpert readers.
1. Martingale transforms
In this section, we prove Theorems A1 and A2. As a preliminary, we recall
the definition of some noncommutative function spaces and the statement of some
auxiliaryresults. We shallassumethatthe readerisfamiliarwithnoncommutative
L spaces. Given ( ,τ) a semifinite von Neumann algebra equipped with a n.s.f.
p
M
trace, the noncommutative weak L -space L ( ) is defined as the set of all
1 1,
∞ M
τ-measurable operators f for which the quasi-norm
f =sup λτ f >λ
k k1, | |
∞ λ>0
n o
6 TAOMEIANDJAVIERPARCET
is finite. In this case, we write τ f > λ to denote the trace of the spectral
| |
projection of f associated to the interval (λ, ). We find this terminology more
| | (cid:8) (cid:9) ∞
intuitive, since itis reminiscentofthe classicalone. The space L ( )satisfies a
1,
∞ M
quasi-triangle inequality that will be used below with no further reference
λτ f +f >λ λτ f >λ/2 +λτ f >λ/2 .
1 2 1 2
| | ≤ | | | |
n o n o n o
We refer the reader to [4, 28] for a more in depth discussion on these notions.
Let us now define the space BMO( ). Let L ( ) stand for the -algebra of
0
M M ∗
τ-measurable operators affiliated to and fix a filtration ( ) . Let us write
n n 1
E : for the correspondMing conditional expectaMtion. ≥Then we define
n n
BMOMr a→ndMBMOc as the spaces of operators f L ( ) with norm (modulo
0
multipMles of 1 ) M ∈ M
M
1
kfkBMOrM = nsup1 En (f −En−1(f))(f −En−1(f))∗ 2 ,
≥ (cid:13) (cid:16) (cid:17) (cid:13)M
(cid:13) 1(cid:13)
kfkBMOcM = nsup1(cid:13)En (f −En−1(f))∗(f −En−1(f)) 2(cid:13) .
≥ (cid:13) (cid:16) (cid:17) (cid:13)M
(cid:13) (cid:13)
It is easily checked that we hav(cid:13)e the identities (cid:13)
1
kfkBMOrM = sup En dfkdfk∗ 2 ,
n 1
≥ (cid:13)(cid:13) (cid:16)kX≥n (cid:17) (cid:13)(cid:13)M
kfkBMOcM = sup(cid:13)En dfk∗dfk 12(cid:13) .
n 1
≥ (cid:13)(cid:13) (cid:16)kX≥n (cid:17) (cid:13)(cid:13)M
We define BMO( )=BMOr BMOc(cid:13) with norm given(cid:13)by
M M∩ M
kfkBMO(M) =max kfkBMOrM,kfkBMOcM .
n o
Finally, the space L ( ;ℓ2) was already defined in the Introduction.
p M rc
A key tool in proving weak type inequalities for noncommutative martingales
is due to Cuculescu. It can be viewed as a noncommutative analogue of the weak
type (1,1) boundedness of Doob’s maximal function.
Cuculescu’s construction [2]. Let f = (f ,f ,...) be a positive L martingale
1 2 1
relative to the filtration ( ) and let λ be a positive number. Then there exists
n n 1
M ≥
a decreasing sequence of projections
q(λ) ,q(λ) ,q(λ) ,...
1 2 3
in satisfying the following properties
M
i) q(λ) commutes with q(λ) f q(λ) for each n 1.
n n 1 n n 1
− − ≥
ii) q(λ) belongs to for each n 1 and q(λ) f q(λ) λq(λ) .
n n n n n n
M ≥ ≤
iii) The following estimate holds
1
τ 1 q(λ) sup f .
n n 1
M− ≤ λ n 1k k
(cid:16) n^≥1 (cid:17) ≥
Explicitly, we set q(λ) =1 and define q(λ) =χ (q(λ) f q(λ) ).
0 n (0,λ] n 1 n n 1
M − −
NONCOMMUTATIVE LITTLEWOOD-PALEY INEQUALITIES 7
AnotherkeytoolforwhatfollowsisGundy’sdecompositionfornoncommutative
martingales. We need a weak notion of support which is quite useful when dealing
withweaktypeinequalities. Foranon-necessarilyself-adjointf ,thetwosided
∈M
null projection of f is the greatest projection q in satisfying qfq =0. Then we
M
define the weak support projection of f as
supp f =1 q.
∗
A−
Itisclearthatsupp f =suppf when isabelian. Moreover,thisnotionisweaker
∗
M
than the usual support projection in the sense that we have supp f suppf for
∗
≤
any self-adjoint f and supp f is a subprojection of both the left and right
∗
∈ M
supports in the non-self-adjoint case.
Gundy’s decomposition [26]. Let f = (f ,f ,...) be a positive L martingale
1 2 1
relative to the filtration ( ) and let λ be a positive number. Then f can be
n n 1
M ≥
decomposed f = α+β +γ as the sum of three martingales relative to the same
filtration and satisfying
1 ∞
max sup α 2, dβ , λτ supp dγ . sup f .
λ k nk2 k kk1 ∗ k k nk1
n 1 n 1
n ≥ kX=1 (cid:16)k_≥1 (cid:17)o ≥
We may write α,β and γ in terms of their martingale differences
dα = q (λ)df q (λ) E q (λ)df q (λ) ,
k k k k k 1 k k k
− −
dβk = qk 1(λ)dfkqk 1(λ) q(cid:0)k(λ)dfkqk(λ)+(cid:1)Ek 1 qk(λ)dfkqk(λ) ,
− − − −
dγ = df q (λ)df q (λ).
k k k 1 k k 1 (cid:0) (cid:1)
− − −
1.1. Weak type (1,1) boundedness. Here we prove Theorem A1. Let us begin
with some harmless assumptions. First, we shall assume that is a finite von
M
Neumann algebra with a normalized trace τ. The passage to the semifinite case
is just technical. Moreover, we shall sketch it in Section 2 since the von Neumann
algebra we shall work with can not be finite. Second, we may assume that the
A
martingale f is positive and finite, so that we may use Cuculescu’s construction
and Gundy’s decomposition for f and moreover we do not have to worry about
convergence issues.
Now we provide the decomposition T f = A f +B f. If (q (λ)) denotes
m m m n n 1
≥
the Cuculescu’s projections associated to (f,λ), let us write in what follows q(λ)
for the projection
q(λ)= q (λ).
n
n 1
^≥
Then we define the projections
π = q(2s) and π = q(2s) q(2s)
0 k
−
s 0 s k s k 1
^≥ ^≥ ≥^−
for k 1. Since π =1 , we may write
≥ k≥0 k M
dfP= π df π + π df π =∆ (df )+∆ (df ).
k i k j i k j r k c k
i j i<j
X≥ X
Then, our decomposition T f =A f +B f is given by
m m m
∞ ∞
A f = ξ ∆ (df ) and B f = ξ ∆ (df ).
m km r k m km c k
k=1 k=1
X X
8 TAOMEIANDJAVIERPARCET
Since both terms can be handled in a similar way, we shall only prove that
supλτ ∞ (A f)(A f) 12 >λ . sup f .
m m ∗ 1
k k
λ>0 n 1
n(cid:16)mX=1 (cid:17) o ≥
By homogeneity, we may assume that the right hand side equals 1. This means in
particular that we may also assume that λ 1 since, by the finiteness of , the
≥ M
left hand side is bounded above by 1 for 0 < λ < 1. Moreover, up to a constant
2 it suffices to prove the result for λ being a nonnegative power of 2. Let us fix a
nonnegative integer ℓ, so that λ=2ℓ for the rest of the proof.
Let us define
w = q(2s).
ℓ
s ℓ
^≥
By the quasi-triangle inequality, we are reduced to estimate
λτ wℓ ∞ (Amf)(Amf)∗ wℓ >λ2 +λτ(1 wℓ)=A1+A2.
M−
n (cid:16)mX=1 (cid:17) o
According to Cuculescu’s theorem, A is dominated by
2
1
λ τ(1 q(2s)) 2ℓ sup f 2 sup f .
M− ≤ 2s n 1 k nk1 ≤ n 1 k nk1
Xs≥ℓ (cid:16)Xs≥ℓ (cid:17) ≥ ≥
LetusnowproceedwiththetermA . Wefirstnoticethatw π =π w =0forany
1 ℓ k k ℓ
integer k > ℓ. Therefore, we find w ∆ (df ) = w ( π df π ) = w ∆ (df ).
ℓ r k ℓ j i ℓ i k j ℓ rℓ k
Similarly, we have ∆ (df ) w =∆ (df ) w and letti≤ng≤
r k ∗ ℓ rℓ k ∗ ℓ
P
∞
A f = ξ ∆ (df ),
mℓ km rℓ k
k=1
X
we conclude
A =λτ w ∞ (A f)(A f) w >λ2 .
1 ℓ mℓ mℓ ∗ ℓ
n (cid:16)mX=1 (cid:17) o
Moreover,usingthefactthatthespectralprojectionsχ (xx )andχ (x x)
(λ, ) ∗ (λ, ) ∗
are Murray-von Neumann equivalent, we may kill the p∞rojection w ab∞ove and
ℓ
obtain the inequality A λτ (A f)(A f) > λ2 . Now we use Gundy’s
1 ≤ m mℓ mℓ ∗
decomposition for (f,λ) and quasi-triangle inequality to get
(cid:8)P (cid:9)
A . λτ ∞ (A α)(A α) >λ2
1 mℓ mℓ ∗
nmX=1 o
∞
+ λτ (Amℓβ)(Amℓβ)∗ >λ2
nmX=1 o
+ λτ ∞ (A γ)(A γ) >λ2 = A +A +A .
mℓ mℓ ∗ α β γ
nmX=1 o
We claim that A is identically 0. Indeed, note that
γ
∆ (dγ )= π df q (2ℓ)df q (2ℓ) π =0
rℓ k i k k 1 k k 1 j
− − −
j≤Xi≤ℓ (cid:16) (cid:17)
NONCOMMUTATIVE LITTLEWOOD-PALEY INEQUALITIES 9
since π q (2ℓ) = π and q (2ℓ)π = π for i,j ℓ. Therefore, it remains to
i k 1 i k 1 j j
control th−e terms A and A .−Let us begin with A ≤. Applying Fubini to the sum
α β α
defining it, we obtain
∞ ∞ ∞
(A α)(A α) = ξ ξ ∆ (dα )∆ (dα ) .
mℓ mℓ ∗ jm km rℓ j rℓ k ∗
mX=1 jX,k=1(cid:16)mX=1 (cid:17)
It will be more convenient, to write this as follows
∞
(A α)(A α)
mℓ mℓ ∗
m=1
X
∞ ∞ ∞ ∞
= ∆rℓ(dαj)e1j ξjmξkm ejk ∆rℓ(dαk)∗ek1
(cid:16)Xj=1 (cid:17)(cid:16)jX,k=1hmX=1 i (cid:17)(cid:16)kX=1 (cid:17)
∞ ∞ ∞ ∞
= ∆ (dα )e ξ e ξ e ∗ ∆ (dα ) e .
rℓ j 1j jk jk jk jk rℓ k ∗ k1
(cid:16)Xj=1 (cid:17)(cid:16)jX,k=1 (cid:17)(cid:16)jX,k=1 (cid:17) (cid:16)Xk=1 (cid:17)
In particular, Chebychev’s inequality gives
A 1 ∞ ∆ (dα )e ∞ ξ e 2
α rℓ j 1j jk jk
≤ λ(cid:13)(cid:16)Xj=1 (cid:17)(cid:16)jX,k=1 (cid:17)(cid:13)L2(M⊗¯B(ℓ2))
(cid:13) (cid:13)
1 (cid:13) ∞ ∞ (cid:13)2
= ∆ dα e ξ e ,
rℓ j 1j jk jk
λ(cid:13) h(cid:16)Xj=1 (cid:17)(cid:16)jX,k=1 (cid:17)i(cid:13)L2(M⊗¯B(ℓ2))
with∆ =∆ id (cid:13)(cid:13)datriangulartruncation,bounded(cid:13)(cid:13)onL ( ¯ (ℓ )). Thus,
rℓ rℓ⊗ B(ℓ2) 2 M⊗B 2
we get
d
A . 1 ∞ dα e ∞ ξ e 2
α j 1j jk jk
λ(cid:13)(cid:16)Xj=1 (cid:17)(cid:16)jX,k=1 (cid:17)(cid:13)L2(M⊗¯B(ℓ2))
(cid:13) (cid:13)
1 (cid:13)∞ ∞ (cid:13) ∞ 1 ∞
= ξ ξ τ(dα dα ) = sup ξ 2 τ(dα dα ).
λ jm km j ∗k | km| λ k ∗k
k 1
jX,k=1(cid:16)mX=1 (cid:17) (cid:16) ≥ mX=1 (cid:17) Xk=1
Therefore, the estimate for A follows from our hypothesis on the ξ ’s and from
α km
the estimate for the α-term in Gundy’s decomposition. Let us finally estimate the
term A . Arguing as above, we clearly have
β
A ∞ (A β)(A β) 12
β mℓ mℓ ∗
≤ 1,
(cid:13)(cid:16)mX=1 (cid:17) (cid:13) ∞
(cid:13) (cid:13)
(cid:13) ∞ ∞ (cid:13)
= ∆ dβ e ξ e .
rℓ j 1j jk jk
(cid:13) h(cid:16)Xj=1 (cid:17)(cid:16)jX,k=1 (cid:17)i(cid:13)L1,∞(M⊗¯B(ℓ2))
Then we use the wea(cid:13)(cid:13)kdtype (1,1) boundedness of triang(cid:13)(cid:13)ular truncations to get
A . ∞ ∞ ξ dβ e
β jk j 1k
(cid:13)Xk=1(cid:16)Xj=1 (cid:17)⊗ (cid:13)L1(M⊗¯B(ℓ2))
(cid:13) (cid:13)
(cid:13)sup ∞ ξ 2 21 ∞ d(cid:13)β . sup f .
km j 1 n 1
≤ | | k k k k
k 1 n 1
(cid:16) ≥ mX=1 (cid:17) Xj=1 ≥
The last inequality follows from our hypothesis and Gundy’s decomposition. (cid:3)
10 TAOMEIANDJAVIERPARCET
1.2. BMO estimate, interpolation and duality. In this paragraph we prove
TheoremA2. ThekeyfortheBMOestimateiscertaincommutationrelationwhich
can not be exploited in L for finite p. Namely, we have
p
∞ ∞ ∞ ∞
T f e = ξ 1 e df 1 = d ξr(f 1 )
m ⊗ 1m km M⊗ 1m k⊗ B(ℓ2) k ⊗ B(ℓ2) k
mX=1 Xk=1(cid:16)mX=1 (cid:17)(cid:0) (cid:1) kX=1 (cid:0) (cid:1)
ξr
k
where the last marting|ale diffe{rzence is }considered with respect to the filtration
= ¯ (ℓ ) of . Note that ξr commutes with df 1 and therefore we
Rn Mn⊗B 2 R k k⊗ B(ℓ2)
find that
(cid:13)(cid:13)mX∞=1Tmf ⊗e1m(cid:13)(cid:13)2BMOrR = (cid:13)(cid:13)snu≥p1kX≥nEn(cid:0)(dfk⊗1B(ℓ2))ξkrξkr∗(dfk⊗1B(ℓ2))∗(cid:1)(cid:13)(cid:13)R
(cid:13) (cid:13) (cid:13)sup ∞ ξ 2 f 1 2 (cid:13)
≤ k 1 | km| ⊗ B(ℓ2) BMOrR
(cid:16) ≥ mX=1 (cid:17)(cid:13) (cid:13)
(cid:13) (cid:13)
. f 1 2 f 2 f 2 .
⊗ B(ℓ2) BMOrR ≤ k kBMOrM ≤ k k∞
Similarly, k mTmf ⊗e1mkBMO(cid:13)(cid:13)cR .kfkBM(cid:13)(cid:13)OcM ≤kfk∞ so that
P ∞
T f e . f sup f .
m 1m BMO( ) n
⊗ BMO( ) k k M ≤n 1k k∞
(cid:13)mX=1 (cid:13) R ≥
(cid:13) (cid:13)
The estimate f(cid:13)or mTmf ⊗em(cid:13)1 is entirely analogous. This gives the L∞−BMO
estimate, or even better the BMO BMO one. Note that we make crucial use of
P −
the identity f 1 = f ! However, f 1 = f for p
k ⊗ B(ℓ2)k∞ k k∞ k ⊗ B(ℓ2)kLp(R) 6 k kLp(M)
finite. That is why we can not reduce the L estimate to the commutative case.
p
With the weak type (1,1) and the BMO estimates in hand, we may follow by
interpolation. Namely, since the case p = 2 is trivial, we interpolate for 1 <p < 2
following Randrianantoanina’s argument [30] and for 2 < p < following Junge
∞
and Musat [13, 21]. This gives rise to
∞
T f δ c f
m m p p
(cid:13)mX=1 ⊗ (cid:13)Lp(M;ℓ2rc) ≤ k k
(cid:13) (cid:13)
for all 1<p<∞. Assu(cid:13)ming further th(cid:13)at m|ξkm|2 =γk ∼1, we find
∞ P
f = sup df ,dg
p k k
k k
kgkp′≤1 Xk=1(cid:10) (cid:11)
∞ 1 ∞
= sup ξ df ,ξ dg
km k km k
γ
kgkp′≤1 Xk=1 k mX=1(cid:10) (cid:11)
∞ ∞ ∞ ξkm
= sup ξ df , dg
km k k
kgkp′≤1 mX=1DXk=1 Xk=1 γk E
∞ ∞ ∞ ξkm
T f δ sup dg δ .
m m k m
≤ (cid:13)mX=1 ⊗ (cid:13)Lp(M;ℓ2rc)kgkp′≤1 (cid:13)mX=1kX=1 γk ⊗ (cid:13)Lp′(M;ℓ2rc)
(cid:13) (cid:13) (cid:13) (cid:13)
Therefore, th(cid:13)e reverse inequa(cid:13)lity follows by dua(cid:13)lity whenever ξ (cid:13)2 1. (cid:3)
m| km| ∼
P
