Paraproducts with flag singularities I. A case study

0.32 MB·English

by  Camil Muscalu

PARAPRODUCTS WITH FLAG SINGULARITIES I. A CASE STUDY
6
0
0 CAMIL MUSCALU
2

n Abstract. In this paper we proveLp estimates for a tri-linear operator,whose symbol
a
isgivenbytheproductoftwostandardsymbols,satisfyingthewellknownMarcinkiewicz-
J
Ho¨rmander-Mihlincondition. OurmainresultcontainsinparticulartheclassicalCoifman-
9
Meyertheorem. Thistri-linearoperatoristhesimplestexampleofalargeclassofmulti-
1
linear operators,which we called paraproducts with flag singularities.

]
A
C
. 1. Introduction
h
t
a The purpose of the present article is to start a systematic study of the Lp boundedness
m
properties of a new class of multi-linear operators which we named paraproducts with flag
[
singularities.

1 For any d ≥ 1 let us denote by M(IRd) the set of all bounded symbols m ∈ L∞(IRd),
v
smooth away from the origin and satisfying the Marcinkiewicz-H¨ormander-Mihlin condi-
4
7 tion 1
4
1
0 1
|∂αm(ξ)| . (1)
6 |ξ||α|
0
h/ for every ξ ∈ IRd \{0} and sufficiently many multi-indices α. We say that such a symbol
t m is trivial if and only if m(ξ) = 1 for every ξ ∈ IRd.
a
m If n ≥ 1 is a fixed integer, we also denote by M (IRn) the set of all symbols m given
flag
: by arbitrary products of the form
v
i
X
r m(ξ) := mS(ξS) (2)
a
S⊆{1,...,n}
Y
where m ∈ M(IRcard(S)), the vector ξ ∈ IRcard(S) is defined by ξ := (ξ ) , while
S S S i i∈S
ξ ∈ IRn is the vector ξ := (ξ )n .
i i=1
Every symbol m ∈M (IRn) defines an n-linear operator T by the formula
flag m
T (f ,...,f )(x) := m(ξ)f (ξ )...f (ξ )e2πix(ξ1+...+ξn)dξ (3)
m 1 n 1 1 n n
IRn
Z
where f ,...,f are Schwartz functions on the real line IR.
1 n b b
In the particular case when all the factors (m ) in (2) are trivial the expression
S S⊆{1,...,n}
T (f ,...,f )(x)becomestheproductofourfunctionsf (x)·...·f (x)andasaconsequence,
m 1 n 1 n
H¨older inequalities imply the fact that T maps Lp1 ×...×Lpn → Lp boundedly as long
m
as 1 < p ,...,p < ∞, 1/p + ... + 1/p = 1/p and 0 < p < ∞. Similar estimates hold
1 n 1 n
1A . B simply means that there exists a universal constant C > 1 so that A ≤ CB. We will also
sometime use the notation A∼B to denote the statement that A.B and B .A
1
2 CAMIL MUSCALU
in the situation when all the factors (m ) in (2) are trivial except for the one
S S⊆{1,...,n}
corresponding to the set {1,...,n}. This deep and important fact is a classical result
in harmonic analysis known as the Coifman-Meyer theorem [1], [2], [3]. Clearly, the
same conclusion is also true if we assume that the only non-trivial symbols are those
corresponding to mutually disjoint subsets of {1,...,n}, because this case can be factored
out as a combination of the previous two.
It is therefore natural to ask the following question.
Question 1.1. Is it true that T maps Lp1 × ... × Lpn → Lp boundedly as long as 1 <
m
p ,...,p < ∞, 1/p +...+1/p = 1/p and 0 < p < ∞ for any m ∈M (IRn) ?
1 n 1 n flag
The main goal of the present paper, is to give an affirmative answer to the above ques-
tion,inthesimplest casewhichgoesbeyondtheCoifman-Meyertheorem. Wewillconsider
the case of a tri-linear operator whose non-trivial factors in (2) are those corresponding
to the subsets {1,2} and {2,3}.
More specifically, let a,b ∈ M(IR2) and denote by T the operator given by
ab
T (f ,f ,f )(x) := a(ξ ,ξ )b(ξ ,ξ )f (ξ )f (ξ )f (ξ )e2πix(ξ1+ξ2+ξ3)dξ dξ dξ .
ab 1 2 3 1 2 2 3 1 1 2 2 3 3 1 2 3
IR3
Z (4)
b b b
Our main theorem is the following.
Theorem 1.2. The operator T previously defined maps Lp1 ×Lp2 ×Lp3 → Lp as long
ab
as 1 < p ,p ,p < ∞, 1/p +1/p +1/p = 1/p and 0 < p < ∞.
1 2 3 1 2 3
Moreover, we will show that in this particular case there are also some L∞-estimates
available (in general, one cannot hope for any of them, as one can easily see by taking all
the factors in (2) to be trivial, except for the ones corresponding to subsets which have
cardinality 1 ). We believe however that the answer to our Question 1.1 is affirmative in
general, and that the Lp-estimates described above are satisfied by the operators T in
m
(3) for all the symbols m ∈M (IRn). We intend to address this general situation in a
flag
separate, future paper.
To motivate the introduction of these paraproducts with flag singularities, we should
mention that some particular examples appeared implicitly in connection with the so-
called bi-est and multi-est operators studied in [7], [8], [13].
The bi-est is the tri-linear operator T defined by the following formula
bi−est
T (f ,f ,f )(x) := f (ξ )f (ξ )f (ξ )e2πix(ξ1+ξ2+ξ3)dξ dξ dξ (5)
bi−est 1 2 3 1 1 2 2 3 3 1 2 3
Zξ1<ξ2<ξ3
and we know from [7] and [8] that it satbisfiesbmanybLp-estimates of the type described
above. Its symbol χ can be viewed as a product of two bi-linear Hilbert transform
ξ1<ξ2<ξ3
type symbols, namely χ and χ [4], [5]. If one replaces them both with smoother
ξ1<ξ2 ξ2<ξ3
symbols in the class M(IR2), then one obtains our tri-linear operator T in (4).
ab
As mentioned in [10], the interesting fact about such operators as T in (3), is that
m
they have a very special multi-parameter structure which seems to be new in harmonic
analysis. This structure is specific to the multi-linear analysis since only in this context
one can construct operators given by multi-parameter symbols which act on functions
defined on the real line.
PARAPRODUCTS WITH FLAG SINGULARITIES I. A CASE STUDY 3
Acknowledgements: The author has been partially supported by NSF and by an
Alfred P. Sloan Research Fellowship.
2. Adjoint operators and interpolation
The purpose of the present section is to recall the interpolation theory from [6], that
will allow us to reduce our desired estimates in Theorem 1.2 to some restrictead weak
type estimates, which are more convenient to handle.
To each generic tri-linear operator T we associate a four-linear form Λ defined by the
following formula
Λ(f ,f ,f ,f ) := T(f ,f ,f )(x)f (x)dx. (6)
1 2 3 4 1 2 3 4
IR
Z
There are also three adjoint operators T∗j, j = 1,2,3 attached to T, defined by duality
as follows
T∗1(f ,f ,f )(x)f (x)dx := Λ(f ,f ,f ,f ), (7)
2 3 4 1 1 2 3 4
IR
Z
T∗2(f ,f ,f )(x)f (x)dx := Λ(f ,f ,f ,f ), (8)
1 3 4 2 1 2 3 4
IR
Z
T∗3(f ,f ,f )(x)f (x)dx := Λ(f ,f ,f ,f ). (9)
1 2 4 3 1 2 3 4
IR
Z
For symmetry, we will also sometimes use the notation T∗4 := T.
The following definition has been introduced in [6].
Definition 2.1. Let (p ,p ,p ,p ) be a 4-tuple of real numbers so that 1 < p ,p ,p ≤ ∞,
1 2 3 4 1 2 3
1/p + 1/p + 1/p = 1/p and 0 < p < ∞. We say that the tri-linear operator T is
1 2 3 4 4
of restricted weak type (p ,p ,p ,p ), if and only if for any (E )4 measurable subsets of
1 2 3 4 i i=1
the real line IR with 0 < |E | < ∞ for i = 1,2,3,4, there exists a subset E′ ⊆ E with
i 4 4
|E′| ∼ |E | so that
4 4
| T(f ,f ,f )(x)f (x)dx| . |E |1/p1|E |1/p2|E |1/p3|E |1/p′4, (10)
1 2 3 4 1 2 3 4
IR
Z
for every f ∈ X(E ), i = 1,2,3 and f ∈ X(E′) where in general X(E) denotes the space
i i 4 4
of all measurable functions f supported on E with kfk ≤ 1 and p′ is the dual index of
∞ 4
p (note that since 1/p +1/p′ = 1, p′ can be negative if 0 < p < 1 ).
4 4 4 4 4
As in [7], [8] let us consider now the 3 - dimensional hyperspace S defined by
S := {(α ,α ,α ,α ) ∈ IR4 : α +α +α +α = 1}.
1 2 3 4 1 2 3 4
Denote by D the open interior of the convex hull of the 7 extremal points A , A , A ,
11 12 21
A , A , A and A in Figure 1. They all belong to S and have the following coor-
22 31 32 4
dinates: A (−1,1,1,0), A (−1,1,0,1), A (1,−1,1,0), A (0,0,0,1), A (1,1,−1,0),
11 12 21 22 31
A (0,1,−1,1) and A (1,1,1,−2).
32 4
4 CAMIL MUSCALU
Figure 1. Polytopes
Denote also by D the open interior of the convex hull of the 5 extremal points A ,
22
G , G , G and A where G , G and G have the coordinates (1,0,0,0), (0,1,0,0) and
1 2 3 4 1 2 3
(0,0,1,0) respectiveely. The following theorem will be proved directly in the following
sections.
Theorem 2.2. If a,b ∈ M(IR2) are as before then, the following statements about the
operator T hold:
ab
(a) There exist points (1/p ,1/p ,1/p ,1/p ) ∈ D arbitrarily close to A so that T is
1 2 3 4 4 ab
of restricted weak type (p ,p ,p ,p ).
1 2 3 4
(b) There exist points (1/pij,1/pij,1/pij,1/pij) ∈ D arbitrarily close to A so that T∗i
1 2 3 4 ij ab
is of restricted weak type (pij,pij,pij,pij) for i = 1,2,3 and j = 1,2.
1 2 3 4
PARAPRODUCTS WITH FLAG SINGULARITIES I. A CASE STUDY 5
If we assume the above result, our main Theorem 1.2 follows immediately from the
interpolation theory developed in [6]. As a consequence of that theory, if (p ,p ,p ,p) are
1 2 3
so that 1 < p ,p ,p ≤ ∞, 1/p + 1/p + 1/p = 1/p and 0 < p < ∞ then T maps
1 2 3 1 2 3 ab
Lp1×Lp2×Lp3 → Lp boundedly, as long as the point (1/p ,1/p ,1/p ,1/p) belongs to D.
1 2 3
And this is clearly true if (p ,p ,p ,p) satisfies the hypothesis of Theorem 1.2. In fact, in
1 2 3
this case, the corresponding points (1/p ,1/p ,1/p ,1/p) belong to D which is a subset
1 2 3
of D.
Moreover, since we also observe that all the points of the form (0,eα,β,γ), (α,0,β,γ),
˜ ˜ ˜
(α,β,0,γ) with α,β > 0, α+β +γ = 1 and (0,α˜,0,β) with α˜,β > 0 α˜ +β = 1 belong
to D, we deduce that in addition T maps L∞ × Lp × Lq → Lr, Lp × L∞ × Lq → Lr,
ab
Lp × Lq × L∞ → Lr and L∞ × Ls × L∞ → Ls boundedly, as long as 1 < p,q,s < ∞,
0 < r < ∞ and 1/p+1/q = 1/r.
The only L∞ estimates that do not follow from such interpolation argumens are those
of the form L∞ ×L∞ ×Ls → Ls and Ls ×L∞ × L∞ → Ls, because points of the form
(1/s,0,0,1/s′) and (0,0,1/s,1/s′) only belong to the boundary of D. But this is not
surprising since such estimates are false in general, as one can easily see by taking f ≡ 1
2
in (4).
In conclusion, to have a complete understanding of the boundedness properties of our
operator T , it is enough to prove Theorem 2.2.
ab
3. Discrete model operators
In this section we introduce some discrete model operators and state a general theorem
about them. Roughly speaking, this theorem says that they satisfy the desired restricted
weaktypeestimates inTheorem2.2. Lateron, inSection4,wewillprove thattheanalysis
of the operator T can in fact be reduced to the analysis of these discrete models. We
ab
start with some notations.
An interval I on the real line IR is called dyadic if it is of the form I = [2kn,2k(n+1)]
for some k,n ∈ Z. We denote by D the set of all such dyadic intervals. If J ∈ D is fixed,
we say that a smooth function Φ is a bump adapted to J if and only if the following
J
inequalities hold
1 1
(l)
|Φ (x)| ≤ C (11)
J l,α|J|l(1+ dist(x,J))α
|J|
for every integer α ∈ IN and sufficiently many derivatives l ∈ IN, where |J| is the length
of J. If Φ is a bump adapted to J, we say that |J|−1/pΦ is an Lp- normalized bump
J J
adapted to J, for 1 ≤ p ≤ ∞. We will also sometimes use the notation χ for the
J
approximate cutoff function defined by
e
dist(x,J)
χ (x) := (1+ )−10. (12)
J
|J|
Definition 3.1. A sequence of L2- normalized bumps (Φ ) adapted to dyadic intervals
e I I∈D
I ∈ D is called a non-lacunary sequence if and only if for each I ∈ D there exists
an interval ω (= ω ) symmetric with respect to the origin so that suppΦ ⊆ ω and
I |I| I I
|ω | ∼ |I|−1.
I
c
6 CAMIL MUSCALU
Definition 3.2. A sequence of L2- normalized bumps (Φ ) adapted to dyadic intervals
I I∈D
I ∈ D is called a lacunary sequence if and only if for each I ∈ D there exists an interval
ω (= ω ) so that suppΦ ⊆ ω , |ω | ∼ |I|−1 ∼ dist(0,ω ) and 0 ∈/ 5ω .
I |I| I I I I I
Let now consider I ,J ⊆ D two finite families of dyadic intervals. Let also (Φj) for
1 c1 I I∈I1
j = 1,2,3 be three sequences of L2- normalized bumps so that (Φ2) is non-lacunary
I I∈I1
while (Φj) for j 6= 2 are both lacunary in the sense of the above definitions.
I I∈I1
We also consider (Φj) for j = 1,2,3 three sequences of L2- normalized bumps so
J J∈J1
that at least two of them are lacunary. Then, define the discrete model operator T by
1
the formula
1
T (f ,f ,f )(x) := hf ,Φ1ihB1(f ,f ),Φ2iΦ3 (13)
1 1 2 3 |I|1/2 1 I I 2 3 I I
IX∈I1
where
1
B1(f ,f )(x) := hf ,Φ1ihf ,Φ2iΦ3. (14)
I 2 3 |J|1/2 2 J 3 J J
J∈J1;|ωJ3|≤X|ωI2|;ωJ3∩ωI26=∅
If k is a strictly positive integer, define also the operator T by
0 1,k0
1
T (f ,f ,f )(x) := hf ,Φ1ihB1 (f ,f ),Φ2iΦ3 (15)
1,k0 1 2 3 |I|1/2 1 I I,k0 2 3 I I
IX∈I1
where
1
B1 (f ,f )(x) := hf ,Φ1ihf ,Φ2iΦ3. (16)
I,k0 2 3 |J|1/2 2 J 3 J J
J∈J1;2k0|ωJ3X|∼|ωI2|;ωJ3∩ωI26=∅
Similarly, let us now consider two other finite families of dyadic intervals I ,J ⊆ D. As
2 2
before, we also consider sequences (Φj) , (Φj) for j = 1,2,3 of L2- normalized
I I∈I2 J J∈J2
bumps, where this time we assume that (Φ1) is non-lacunary while (Φj) are both
I I∈I2 I I∈I2
lacunary for j 6= 1 and at least two of the sequences (Φj) are lacunary. Using them,
J J∈J2
we define the operator T by the formula
2
1
T (f ,f ,f )(x) := hB2(f ,f ),Φ1ihf ,Φ2iΦ3 (17)
2 1 2 3 |I|1/2 I 1 2 I 3 I I
IX∈I2
where
1
B2(f ,f )(x) := hf ,Φ1ihf ,Φ2iΦ3. (18)
I 1 2 |J|1/2 1 J 2 J J
J∈J2;|ωJ3|≤X|ωI1|;ωJ3∩ωI16=∅
And finally, as before, for any strictly positive integer k define also the operator T by
0 2,k0
PARAPRODUCTS WITH FLAG SINGULARITIES I. A CASE STUDY 7
1
T (f ,f ,f )(x) := hB2 (f ,f ),Φ1ihf ,Φ2iΦ3 (19)
2,k0 1 2 3 |I|1/2 I,k0 1 2 I 3 I I
IX∈I2
where
1
B2(f ,f )(x) := hf ,Φ1ihf ,Φ2iΦ3. (20)
I 1 2 |J|1/2 1 J 2 J J
J∈J2;2k0|ωJ3X|∼|ωI1|;ωJ3∩ωI16=∅
The following theorem about these operators will be proved carefully in the forthcoming
sections.
Theorem 3.3. Our previous Theorem 2.2 holds also for all the operators T , T , T ,
1 2 1,k0
T with bounds which are independent on k and the cardinalities of the sets I , I , J ,
2,k0 0 1 2 1
J . Moreover, the subsets (E′)4 which appear implicitly due to Definition 2.1, can be
2 j j=1
chosen independently on the L2- normalized families considered above.
4. Reduction to the model operators
As we promised, the aim of the present section is to show that the analysis of our
operator T can be indeed reduced to the analysis of the model operators defined in the
ab
previous section. To achieve this, we will decompose the multipliers a(ξ ,ξ ) and b(ξ ,ξ )
1 2 2 3
separetely and after that we will study their interactions.
Fix M > 0 a big integer. For j = 1,2,...,M consider Schwartz functions Ψ so that
j
suppΨ ⊆ 10[j − 1,j], Ψ = 1 on [j − 1,j] and for j = −M,−M + 1,...,−1 consider
j 9 j
Schwartz functions Ψ so that suppΨ ⊆ 10[j,j +1] and Ψ = 1 on [j,j +1]. 2
j j 9 j
If λcis a positive real number and Ψ is a Schwartz function, we denote by
c
DpΨ(x) := λ−1/pΨ(λ−1x)
λ
the dilation operator which preserves the Lp norm of Ψ, for 1 ≤ p ≤ ∞.
Define the new symbol a(ξ ,ξ ) by the formula
1 2
e
a(ξ1,ξ2) := IRD2∞λ′Ψj1′(ξ1)D2∞λ′Ψj2′(ξ2)dλ′. (21)
max(|j′|,|j′|)=MZ
X1 2
c c
Clearly, by constreuction, a belongs to the class M(IR2). Also, things can be arranged so
that |a(ξ ,ξ )| ≥ c > 0 for every (ξ ,ξ ) ∈ IR2, where c is a universal constant. Roughly
1 2 0 1 2 0
speaking, this a should be understood as being essentially a decomposition of unity in
e
frequency space into a series of smooth functions, supported on rectangular annuli. Then,
e
we write a(ξ ,ξ ) as
1 e2
a(ξ ,ξ )
1 2
a(ξ ,ξ ) = ·a(ξ ,ξ ) := a(ξ ,ξ )·a(ξ ,ξ ) =
1 2 1 2 1 2 1 2
a(ξ ,ξ )
1 2
2If I is an interval, we denote by cI(c>0)ethe interval weeith the samee center as I and whose length is
c times the length of I e
8 CAMIL MUSCALU
IR(a(ξ1,ξ2)D2∞λ′Ψj1′(ξ1)D2∞λ′Ψj2′(ξ2))dλ′ (22)
max(|j′|,|j′|)=MZ
X1 2
e c c
e
and observe that a(ξ ,ξ ) has the same properties as a(ξ ,ξ ).
1 2 1 2
Fix now j′,j′ with max(|j′|,|j′|) = M and λ′ ∈ IR. By taking advantage of the fact
1 2 1 2
that a ∈ M(IR2),eeone can write it on the support of D2∞λ′Ψj1′ ⊗D2∞λ′Ψj2′ as a double Fourier
series and this allows us to decompose the inner term in (22) as
e c c
e
C(a)jλ1′′,,jn2′′1,n′2 D2∞λ′Ψj1′(ξ1)e2πin′11902−λ′ξ1 D2∞λ′Ψj2′(ξ2)e2πin′21902−λ′ξ2 :=
n′1X,n′2∈Z (cid:16) (cid:17)(cid:16) (cid:17)
c c
C(a)j1′,j2′ D∞ Ψn′1(ξ ) D∞ Ψn′2(ξ )
λ′,n′1,n′2 2λ′ j1′ 1 2λ′ j2′ 2
n′1X,n′2∈Z (cid:18) d (cid:19)(cid:18) d (cid:19)
n′ n′
where we denoted by Ψ 1 and Ψ 2 the functions defined by
j′ j′
1 2
Ψnj1′′1(ξ1) := Ψj1′(ξ1)e2πin′1190ξ1
and
d
c
Ψnj2′′2(ξ2) := Ψj2′(ξ2)e2πin′2190ξ2
j′,j′
and the corresponding constantds C(a) 1 2 satisfy the inequalities
λ′,n′,nc′
1 2
j′,j′ 1 1
|C(a) 1 2 | . , (23)
λ′,n′1,n′2 (1+|n′|)1000(1+|n′|)1000
1 2
for every n′,n′ ∈ Z, uniformly in λ′ ∈ IR.
1 2
In particular, the symbol a(ξ ,ξ ) can be written as
1 2
a(ξ ,ξ ) =
1 2
1
C(a)j1′,j2′ D∞ Ψn′1(ξ ) D∞ Ψn′2(ξ ) dκ′.
k′+κ′,n′1,n′2 2k′+κ′ j1′ 1 2k′+κ′ j2′ 2
max(|jX1′|,|j2′|)=MZ0 n′1X,n′2∈ZkX′∈Z (cid:18) d (cid:19)(cid:18) d (cid:19) (24)
Similarly, the symbol b(ξ ,ξ ) can also be decomposed as
2 3
b(ξ ,ξ ) =
2 3
1
C(b)j1′′,j2′′ D∞ Ψn′1′(ξ ) D∞ Ψn′2′(ξ ) dκ′′
k′′+κ′′,n′1′,n′2′ 2k′′+κ′′ j1′′ 2 2k′′+κ′′ j2′′ 3
max(|j1X′′|,|j2′′|)=MZ0 n′1′X,n′2′∈ZkX′′∈Z (cid:18) d (cid:19)(cid:18) d (cid:19) (25)
j′′,j′′
where as before, the constants C(b) 1 2 satisfy the inequalities
k′′+κ′′,n′′,n′′
1 2
PARAPRODUCTS WITH FLAG SINGULARITIES I. A CASE STUDY 9
j′′,j′′ 1 1
|C(b) 1 2 | . , (26)
k′′+κ′′,n′1′,n′2′ (1+|n′′|)1000(1+|n′′|)1000
1 2
for every n′′,n′′ ∈ Z, uniformly in k′′ and κ′′.
1 2
As a consequence, their product a(ξ ,ξ )·b(ξ ,ξ ) becomes
1 2 2 3
a(ξ ,ξ )·b(ξ ,ξ ) =
1 2 2 3
·
max(|jX1′|,|j2′|)=Mmax(|j1X′′|,|j2′′|)=Mn′1X,n′2∈Zn′1′X,n′2′∈Z
1 1
j′,j′ j′′,j′′
C(a) 1 2 ·C(b) 1 2 ·
k′+κ′,n′,n′ k′′+κ′′,n′′,n′′
1 2 1 2
Z0 Z0 k′,Xk′′∈Z
D∞ Ψn′1(ξ ) D∞ Ψn′2(ξ ) D∞ Ψn′1′(ξ ) D∞ Ψn′2′(ξ ) dκ′dκ′′.
2k′+κ′ j1′ 1 2k′+κ′ j2′ 2 2k′′+κ′′ j1′′ 2 2k′′+κ′′ j2′′ 3
(cid:20)(cid:18) (cid:19)(cid:18) (cid:19)(cid:21)(cid:20)(cid:18) (cid:19)(cid:18) (cid:19)(cid:21) (27)
d d d d
Clearly, one has to have
supp(D∞ Ψn′2)∩supp(D∞ Ψn′1′) 6= ∅ (28)
2k′+κ′ j2′ 2k′′+κ′′ j1′′
otherwise, the expression in (27) vandishes. d
Let now # be a positive integer, much bigger than logM. If k′ and k′′ are two integers
as in the sum above then, there are three possibilities: either k′ ≥ k′′+# or k′′ ≥ k′+#
or |k′ −k′′| ≤ #. As a consequence, the multiplier a(ξ ,ξ )·b(ξ ,ξ ) can be decomposed
1 2 2 3
accordingly as
a(ξ ,ξ )·b(ξ ,ξ ) = m (ξ ,ξ ,ξ )+m (ξ ,ξ ,ξ )+m (ξ ,ξ ,ξ ).
1 2 2 3 1 1 2 3 2 1 2 3 3 1 2 3
Since it is not difficult to see that m ∈ M(IR2), the desired estimates for the tri-linear
3
operator T follow from the classical Coifman-Meyer theorem quoted before. It is there-
m3
fore enough to concentrate our attention on the remaining operators T and T . Since
m1 m2
their definitions are symmetric, we will only study the case of T where the summation
m1
in (27) runs over those k′,k′′ ∈ Z having the property that k′ ≥ k′′+#. We then observe
that since # is big in comparison to logM, we have to have j′ = −1 or j′ = 1 in order
2 2
for (28) to hold. In particular, this simplies that the intervals supp(D∞ Ψn′2) are
2k′+κ′ j2′ k′∈Z
all intersecting each other.
d
At this moment, let us also remind ourselves that in order to prove restricted weak type
estimates for T , we would need to understand expressions of the form
m1
T (f ,f ,f )(x)f (x)dx =
m1 1 2 3 4
IR
(cid:12)Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
m (ξ ,ξ ,ξ )f (ξ )f (ξ )f (ξ )f (ξ )dξ (29)
1 1 2 3 1 1 2 2 3 3 4 4
(cid:12)Zξ1+ξ2+ξ3+ξ4=0 (cid:12)
(cid:12) (cid:12)
(cid:12) b b b b (cid:12)
(cid:12) (cid:12)
10 CAMIL MUSCALU
andasaconsequence, fromnowon,wewillthinkofourtri-dimensionalvectors(ξ ,ξ ,ξ ) ∈
1 2 3
IR3 as being part of 4- dimensional ones (ξ ,ξ ,ξ ,ξ ) ∈ IR4 for which ξ +ξ +ξ +ξ = 0.
1 2 3 4 1 2 3 4
Fix now the parameters j′, j′, j′′, j′′, n′, n′, n′′, n′′, k′, k′′, κ′, κ′′ so that k′ ≥ k′′ +#
1 2 1 2 1 2 1 2
and look at the corresponding inner term in (27). It can be rewritten as
\ \ \ \
n′ n′ n′′ n′′
Ψ 1 (ξ )Ψ 2 (ξ )Ψ 1 (ξ )Ψ 2 (ξ ) (30)
k′,κ′,j′ 1 k′,κ′,j′ 2 k′′,κ′′,j′′ 2 k′′,κ′′,j′′ 3
1 2 1 2
where
Ψn′1 := D1 Ψn′1,
k′,κ′,j1′ 2−k′−κ′ j1′
Ψn′2 := D1 Ψn′2,
k′,κ′,j2′ 2−k′−κ′ j2′
Ψn′1′ := D1 Ψn′1′
k′′,κ′′,j1′′ 2−k′′−κ′′ j1′′
and
Ψn′2′ := D1 Ψn′2′.
k′′,κ′′,j1′′ 2−k′′−κ′′ j2′′
\
Consider now Schwartz functions Ψk′,κ′,j′,j′ and Ψk′′,κ′′,j′′,j′′ so that Ψk′,κ′,j′,j′ is identi-
1 2 1 2 1 2
\ \
n′ n′
cally equal to 1 on the interval −2(supp(Ψ 1 ) + supp(Ψ 2 )) and is supported
k′,κ′,j′ k′,κ′,j′
1 2
\
on a 190 enlargement of it, while Ψk′′,κ′′,j1′′,j2′′ is identically equal to 1 on the interval
\ \
(supp(Ψn′1′ )+supp(Ψn′2′ )) and is also supported on a 10 enlargement of it. Since
k′′,κ′′,j′′ k′′,κ′′,j′′ 9
1 2
ξ + ξ + ξ + ξ = 0, one can clearly insert these two new functions into the previous
1 2 3 4
expression (30), which now becomes
\ \
n′ n′ \
Ψk′1,κ′,j1′(ξ1)Ψk′2,κ′,j2′(ξ2)Ψk′,κ′,j1′,j2′(ξ4) · (31)
(cid:20) (cid:21)
\ \
n′′ n′′ \
Ψk′1′,κ′′,j1′′(ξ2)Ψk′2′,κ′′,j2′′(ξ3)Ψk′′,κ′′,j1′′,j2′′(ξ2 +ξ3) .
(cid:20) (cid:21)
The following elementary lemmas will play animportant roleinour further decomposition
(see also [8]).
Lemma 4.1. Let η ,η ,η ,η ,η ,η be Schwartz functions. Then,
1 2 3 4 14 23
η (ξ )η (ξ )η (ξ )η (ξ )η (ξ +ξ )η (ξ +ξ )f (ξ )f (ξ )f (ξ )f (ξ )dξ =
1 1 2 2 3 3 4 4 14 1 4 23 2 3 1 1 2 2 3 3 4 4
Zξ1+ξ2+ξ3+ξ4=0
b b b b
b b b b c c
[(f ∗η )(f ∗η )]∗η ·[(f ∗η )(f ∗η )]∗η dx.
1 1 4 4 14 2 2 3 3 23
IR
Z