L^p estimates for a singular integral operator motivated by Calderón's second commutator

0.3 MB·English

by  Eyvindur Palsson

Lp ESTIMATES FOR A SINGULAR INTEGRAL OPERATOR
MOTIVATED BY CALDERO´N’S SECOND COMMUTATOR
EYVINDURARI PALSSON
Abstract. WeproveawiderangeofLpestimatesforatrilinearsingularintegraloperator
2 motivated bydroppingoneaverageinCaldero´n’s secondcommutator. Forcomparison by
1 dropping two averages in Caldero´n’s second commutator one faces the trilinear Hilbert
0 transform. The novelty in this paper is that in order to avoid difficulty of the level of the
2 trilinearHilberttransform,wechoosetoviewthesymboloftheoperatorasanon-standard
symbol. The methods used come from time-frequency analysis but must be adapted to
n
a the fact that oursymbol is non-standard.
J

9
1
1. Introduction

]
A
1.1. History. The k-th Calder´on commutator, k ∈ {1,2,3,...}, is given by
C
k
1 A(x)−A(y)
. (k)
h C f(x)= p.v. f(y)dy
A x−y x−y
at ZR (cid:18) (cid:19)
m where A is Lipschitz and A′ ∈L∞(R). Calder´on studied these operators in connection with
an algebra of pseudo-differential operators. He was also motivated by possible applications
[
to operators of the type

3
v
0 1 A(x)−A(y)
(1.1) p.v. F f(y) dy
6
x−y x−y
1 Z (cid:18) (cid:19)
R
5
. where F is an analytic function. The Cauchy integral on Lipschitz curves and double layer
4
potentials are examples of the previous operator. In 1965 Calder´on showed
0
1
1 C(k) : Lp → Lp for 1< p < ∞
: A
v for k = 1 [2]. Coifman and Meyer extended his result in 1975 to k = 2,3,... [4]. The
i
X estimates obtained did not clearly indicate how the boundedness constant depended on
r k. Building on the work of Coifman and Meyer, Calder´on was able to prove the above
a
estimates with a boundednessconstant that depended on k exponentially. This way he was
able to prove bounds for operators of the type (1.1), as long as the Lipschitz constant was
small. Finally, in 1982 Coifman, McIntosh and Meyer showed the above estimates with a
boundedness constant that depended on k polynomially [5] and were thus able to show a
wide range of Lp estimates for operators of the type (1.1).
1.2. Motivation. Calder´on observed that one can write the following as an average
A(x)−A(y) 1
= A′(x+α(y−x))dα.
x−y
Z0
Using this trick and a substitution he rewrote his first commutator as
Date: January 20, 2013.
2010 Mathematics Subject Classification. Primary 42B20, 47H60; Secondary 45P05.
Key words and phrases. Fourier analysis, multilinear operators.
1
2 EYVINDURARIPALSSON
1 1
C(1)f(x)= A′(x+αt)f(x+t) dtdα.
A t
Z0 ZR
He then asked if one dropped the average and fixed α whether Lp estimates could be found
for the resulting operator, uniformly in α. This motivated the definition of the bilinear
Hilbert transform
1
BHT (f ,f )(x) = p.v. f (x+αt)f (x+t) dt.
α 1 2 1 2
t
Z
R
In two papers from 1997 and 1999, Lacey and Thiele showed that the bilinear Hilbert
transform BHT maps Lp×Lq into Lr when 1+ 1 = 1, 1 < p,q ≤ ∞ and 2 < r < ∞ with
α p q r 3
a bound depending on α [9, 10]. Uniform boundedness of these Lp estimates was resolved
later [7, 17]. Note that r only goes down to 2, not 1 as one would expect from H¨older type
3 2
estimates. It is still an open problem whether r can be pushed all the way down to 1.
2
In a similar fashion then one can rewrite the second Calder´on commutator with two
averages. Droppingbothaverages motivatesthedefinitionofthetrilinearHilberttransform.
1
THT (f ,f ,f )(x) = p.v. f (x+α t)f (x+α t)f (x+t) dt
α~ 1 2 3 1 1 2 2 3
t
Z
R
IncontrasttothebilinearHilberttransformthennoLp estimatesareknownforthetrilinear
Hilbert transform.
(2)
In this paper we will study a trilinear operator motivated by C in a similar fashion as
A
THT , except we drop one average, not two. Define
α~
1 1
(1.2) T (f ,f ,f )(x) := p.v. f (x+αt)dα f (x+βt)f (x+t) dt.
β 1 2 3 1 2 3
t
Z (cid:18)Z0 (cid:19)
R
1.3. Known estimates. Benyi, Demeter, Nahmod,Thiele, TorresandVillarroya obtained
a modulation invariant bilinear T(1) theorem [1]. If one fixes f ∈L∞(R) and looks at the
1
bilinear operator
1
f (x+αt)dα
0 1
(f ,f ) 7→ p.v. f (x+βt)f (x+t)dt,
2 3 (cid:16)R t (cid:17) 2 3
Z
R
one can apply their theorem to obtain the following Lp estimates for T
β
T : L∞×Lp1 ×Lp2 → Lp
β
1 1 1
for β ∈/ {0,1} if + = , 1 < p ,p ≤ ∞ and 2 < p < ∞. These are the only known
p p p 1 2 3
1 2
Lp estimates for T .
β
1.4. Result. The main theorem of this paper establishes the following wide range of Lp
estimates for T .
β
Theorem 1.1. Let β ∈/ {0,1}, 1< p ,p ,p ≤ ∞,
1 2 3
1 p p p 2 p p
< p := 1 2 3 < ∞ and < 2 3 ≤ ∞.
2 p p +p p +p p 3 p +p
1 2 1 3 2 3 2 3
Lp ESTIMATES FOR A SINGULAR INTEGRAL OPERATOR 3
Then there exists a constant C such that
β,p1,p2,p3
kT (f ,f ,f )k ≤ C kf k kf k kf k
β 1 2 3 p β,p1,p2,p3 1 p1 2 p2 3 p3
for all f , f and f in S(R).
1 2 3
ThetheoremrecoversallknownLp estimates fortheoperator. KnownLp estimatesforboth
the bilinear Hilbert transform and for Calder´on’s first commutator follow as a corollary.
Compared to the theorem on the bilinear Hilbert transform, this theorem has an extra
condition.
2 p p
2 3
< ≤ ∞
3 p +p
2 3
This condition implies that we have not improved the previously known Lp estimates for
thebilinear Hilbert transform. We also requirethe condition 1 < p, which is notthe largest
2
possible range of Lp estimates expected. Based on the known estimates for the bilinear
Hilbert transform one would expect to be able to go all the way down to 2. This remains
5
an open problem.
Note that if β = 0,1 then we obtain trilinear operators that only involve multiplication
of functions and the first Calder´on commutator. The Lp-boundsof these operators are easy
to determine.
1.5. Approach. The standard way of understanding the boundedness of the Calder´on
commutators is to use the T(1) theorem. In order to use such an approach on T we would
β
need some sort of a trilinear T(1) theorem. Despite the existence of some multilinear T(1)
theorems[3,8]thenthereisnosuchappropriatetheoremforT . Theothercanonicalwayof
β
trying to understand T would be to establish uniform Lp estimates on the trilinear Hilbert
β
transform. Since no Lp estimates exist, uniform estimates are out of reach. The obvious
approaches to find Lp estimates fail so we need some novel ideas.
On the Fourier side it is equivalent to show Lp estimates for an operator T given by
β
1
(1.3) T (f ,f ,f )(x) = sgn(αξ +βξ +ξ )dα f (ξ )f (ξ )f (ξ )
β 1 2 3 1 2 3 1 1 2 2 3 3
Z (cid:20)Z0 (cid:21)
R3
b b b
e2πi(ξ1+ξ2+ξ3)xdξ dξ dξ .
1 2 3
where sgn is the usual sign function. The symbol 1sgn(αξ +βξ +ξ )dα has a singularity
0 1 2 3
around the line ξ = 0, βξ +ξ =0 in the sense that it is discontinuous. This is similar to
1 2 3
R
the bilinear Hilbert transform. Unlike standard symbols, which are assumed to be smooth
outside the set where they are singular, this symbol is continuous but not differentiable
on the planes ξ + βξ + ξ = 0 and ξ = 0 away from the previous line. We approach
1 2 3 1
the symbol as a rough non-standard symbol and use techniques in the spirit of the bilinear
Hilbert transform. An important ingredient in that approach are new proofs of the Lp
estimates for the Calder´on commutators by Muscalu [11]. The techniques and notation are
also heavily inspired by Muscalu, Tao and Thiele [12, 13].
There exist theorems that give immediate Lp estimates for operators with standard sym-
bols where the dimension of the singularity is strictly less than half the dimension of the
frequency space of the form associated to the operator [15]. Even if our symbol had been
standard outside the line then those kind of theorems would not have been applicable be-
cause the line is degenerate.
1.6. Acknowledgements. The author would like to thank his thesis adviser, Camil Mus-
calu, for his guidance and many helpful conversations about this problem.
4 EYVINDURARIPALSSON
2. Notation
We use A . B to denote the statement that A ≤ CB for some large constant C and
A≪ B to denote the statement that A≤ C−1B for some large constant C. Our constants
C shall always be independent of the tiles P~.
Given any interval I, let |I| denote the Lebesgue measure of I and let cI denote the
interval with the same center as I but c times the side-length. Also define the approximate
cutoff function χ˜ by
I
|x−x |
χ˜ (x) := (1+( I )2)−1/2
I
|I|
where x is the center of I.
I
Define hni:= 2+|n| for n∈ Z.
3. Symbol
The meaning of (1.2) is
1 1
(3.1) lim f (x+αt)dα f (x+βt)f (x+t) dt
1 2 3
ǫ→0+ Z (cid:18)Z0 (cid:19) t
|t|>ǫ
where the limit exists. Assume f , f and f are Schwartz functions on R. We will show
1 2 3
that (3.1) exists in that case and we will rewrite it in a convenient way.
Write (3.1) as
1
lim f (ξ )e2πiξ1(x+αt)dξ dα
1 1 1
Nǫ→→0∞+ ǫ<|tZ|<N (cid:20)Z0 ZR (cid:21)
b
1
f (ξ )e2πiξ2(x+βt)dξ f (ξ )e2πiξ3(x+t)dξ dt
2 2 2 3 3 3
t
ZR ZR
which is equal to
b b
1 1
lim e−2πi(−αξ1−βξ2−ξ3)dα f (ξ )f (ξ )f (ξ )e2πix(ξ1+ξ2+ξ3)dξ dξ dξ dt
1 1 2 2 3 3 1 2 3
Nǫ→→0∞+ ǫ<|tZ|<N ZR3(cid:20)Z0 t (cid:21)
b b b
The function being integrated, viewed as depending on ξ , ξ , ξ and t is clearly absolutely
1 2 3
integrable on R4 and by applying Fubini’s theorem together with dominated convergence
we see that the formula becomes equivalent to
1
(3.2) sgn(−αξ −βξ −ξ )dα f (ξ )f (ξ )f (ξ )e2πi(ξ1+ξ2+ξ3)xdξ dξ dξ
1 2 3 1 1 2 2 3 3 1 2 3
Z (cid:20)Z0 (cid:21)
R3
b b b
which clearly exists since f , f and f are also Schwartz functions.
1 2 3
A product of three functions satisfies a H¨older type inequality as we obtain in Theorem
1.1. Since the product canbbebwrittenbas
(3.3) f (ξ )f (ξ )f (ξ )e2πi(ξ1+ξ2+ξ3)xdξ dξ dξ
1 1 2 2 3 3 1 2 3
Z
R3
b b b
Lp ESTIMATES FOR A SINGULAR INTEGRAL OPERATOR 5
and using sgn(−x) = −sgn(x) it becomes clear by subtracting (3.2) from (3.3) that it is
enough to consider Lp estimates for
1
(3.4) T˜ (f ,f ,f )(x) := 1 (αξ +βξ +ξ )dα f (ξ )f (ξ )f (ξ )
β 1 2 3 R+ 1 2 3 1 1 2 2 3 3
Z (cid:20)Z0 (cid:21)
R3
b b b
e2πi(ξ1+ξ2+ξ3)xdξ dξ dξ .
1 2 3
where 1 is the characteristic function for the positive real axis.
R+
Similar to what was mentioned in the introduction then the symbol
1
1 (αξ +βξ +ξ )dα
R+ 1 2 3
Z0
is not continuous around the line ξ = 0, βξ +ξ = 0, continuous but not differentiable
1 2 3
around the planes ξ +βξ +ξ = 0 and βξ +ξ = 0, away from the previous line, but
1 2 3 2 3
smooth everywhere else. It is tempting to view the symbol as a trilinear symbol of the
variables ξ , ξ , ξ . That would however result in a problem of the same difficulty as the
1 2 3
trilinear Hilbert transform. We choose thus instead to view it as a non-standard bilinear
symbol of the variables ξ and βξ +ξ .
1 2 3
4. Discretization
We will now come up with a ”discretized” variant of the ”continuous” form associated
to (3.4). We start by reviewing some standard definitions and comments [13].
Definition 4.1. Letn ≥ 1 and σ ∈ {0, 1,2}n. Wedefine the shifted n-dyadic mesh D = Dn
3 3 σ
to be the collection of cubes of the form
Dn := {2j(k+(0,1)n +(−1)jσ)|j ∈ Z, k ∈ Zn}
σ
We define a shifted dyadic cube to be any member of a shifted n-dyadic mesh.
Observe that for every cube Q, there exists a shifted dyadic cube Q′ such that Q ⊆ 7 Q′
10
and |Q′|∼ |Q|; this is best seen by first verifying the n = 1 case.
Definition 4.2. A subset D′ of a shifted n-dyadic grid D is called sparse, if for any two
cubes Q, Q′ in D with Q 6= Q′ we have |Q| < |Q′| implies |109Q| < |Q′| and |Q| = |Q′|
implies 109Q∩109Q′ = ∅.
Observe that any subset of a shifted n-dyadic grid (with n ≤ 4 say), can be split into
O(1) sparse subsets.
Definition 4.3. Let σ = (σ ,σ ,σ ,σ ) ∈ {0, 1,2}4, and let 1 ≤ i ≤ 4. An i-tile with shift
1 2 3 4 3 3
σ is a rectangle P = I ×ω with area 1 and with I ∈ D1, ω ∈ D1 . A quadtile with
i P P P 0 P σi
shift σ is a 4-tuble P~ = (P ,P ,P ,P ) such that each P is an i-tile with shift σ , and the
1 2 3 4 i i
I = I are independent of i. The frequency cube Q of a quadtile is defined to be Π4 ω
Pi P~ P~ i=1 Pi
We sometimes refer to i-tiles with shift σ just as i-tiles, or even as tiles, if the parameters
σ, i are unimportant.
Definition 4.4. A set P~ of quadtiles is called sparse, if all quadtiles in P~ have the same
shift and the set {Q : P~ ∈ P~} is sparse.
P~
Again, any set of quadtiles can be split into O(1) sparse subsets.
6 EYVINDURARIPALSSON
Definition 4.5. Let P and P′ be tiles. We write P′ < P if IP′ ( IP and 5ωP ⊆ 5ωP′, and
P′ ≤ P if P′ < P or P′ = P. We write P′ . P if I ⊆ I and 107ω ⊆ 107ω . We write
P′ P P P′
P′ .′ P if P′ . P and P′ (cid:2) P.
This ordering by Muscalu, Tao and Thiele [13] is in the spirit of that in Fefferman [6] or
Lacey and Thiele [9, 10]. The main difference from the previous orderings is that P′ and P
do not quite have to intersect which turns out to be convenient for technical purposes.
Definition 4.6. Let P be a tile. An Lp normalized wave packet on P, 1 ≤ p < ∞, is a
function φ which has Fourier support in 9 ω and obeys the estimates
P 10 P
|φ (x)| . |I |−1/pχ˜ (x)M
P P I
for all M > 0, with the implicit constant depending on M.
Heuristically, φ is Lp-normalized and is supported in P.
P
Now that we have the tools from Muscalu, Tao and Thiele [13] then let us start decom-
posing. We start with two standard Littlewood-Paley decompositions and write
1 (ξ ) = Ψ (ξ )
R 1 k1 1
Xk1
and d
1 (βξ +ξ ) = Ψ (βξ +ξ )
R 2 3 k2 2 3
Xk2
d
where as usual, Ψ (ξ ) and Ψ (βξ +ξ ) are bumps supported in the regions |ξ | ∼ 2k1
k1 1 k2 2 3 1
and |βξ +ξ | ∼ 2k2 respectively. In particular we get
2 3
d d
(4.1) 1 (ξ ,βξ +ξ ) = Ψ (ξ )Ψ (βξ +ξ )
R 1 2 3 k1 1 k2 2 3
kX1,k2
d d
By splitting (4.1) over the regions where k ≪ k , k ≪ k and k ∼ k we obtain the
1 2 2 1 1 2
decomposition
(4.2) 1 (ξ ,βξ +ξ ) = Ψ (ξ )Φ (βξ +ξ ) +
R 1 2 3 k 1 k 2 3
k
X
c c
(4.3) Φ (ξ )Ψ (βξ +ξ ) +
k 1 k 2 3
k
X
c c
(4.4) Ψ (ξ )Ψ (βξ +ξ ).
k1 1 k2 2 3
kX1∼k2
d d
whereΦ isabumpsupportedonaninterval, symmetricwithrespecttotheorigin oflength
k
∼ 2k.
Notecthat Φ (βξ +ξ )is supportedin R2 on astrip aroundthe line βξ +ξ = 0 of width
k 2 3 2 3
∼ 2k. We can cover that strip with shifted dyadic cubes with side-length ∼ 2k. Similarly
then Ψ (βξ +cξ ) is supported in R2 on two strips of width ∼ 2k but this time away from
k 2 3
βξ +ξ = 0. Again we can cover those strips with shifted dyadic cubes of a similar scale.
2 3
Thucs we come up with a decomposition
Lp ESTIMATES FOR A SINGULAR INTEGRAL OPERATOR 7
(4.5) a(ξ ,ξ ,ξ ) = φ[(ξ )φ[(ξ )φ[(ξ )
1 2 3 Q1,1 1 Q2,2 2 Q3,3 3
Q~X∈Q~
for each of the three cases (4.2), (4.3), (4.4) such that
1
< a(ξ ,ξ ,ξ ) < 10.
1 2 3
10
Here φ is an L1 normalized wave packet on a tile I ×Q for i = 1,2,3, where Q is a
Qi,i Q~ i i
shifted dyadic interval that depends on the decomposition in each of the three cases and I
Q~
is a dyadic interval such that |I | ∼ |Q |−1 for i= 1,2,3.
Q~ i
Sinceξ ∈ 9 Q ,ξ ∈ 9 Q andξ ∈ 9 Q itfollowsthatξ +ξ +ξ ∈ 9 Q + 9 Q + 9 Q
1 10 1 2 10 2 3 10 3 1 2 3 10 1 10 2 10 3
and as a consequence one can find a shifted dyadic interval Q with the property that
4
9 Q + 9 Q + 9 Q ⊆ − 7 Q and also satisfying |Q | = |Q |= |Q | ∼ |Q |. In particular
10 1 10 2 10 3 10 4 1 2 3 4
there exists an L1 normalized wave packet φ adapted to I ×Q such that φ[ ≡ 1 on
Q4,4 4 Q4,4
− 9 Q − 9 Q − 9 Q .
10 1 10 2 10 3
Thus (4.5) can be written as
(4.6) a(ξ ,ξ ,ξ )= φ[(ξ )φ[(ξ )φ[(ξ )φ[(−ξ −ξ −ξ )
1 2 3 Q1,1 1 Q2,2 2 Q3,3 3 Q4,4 1 2 3
Q~X∈Q~
where this time Q~ is a collection of shifted dyadic quasi-cubes in R4. Modulo a finite
refinement we can assume that a sum of the type
(4.7) φ[(ξ )φ[(ξ )φ[(ξ )φ[(−ξ −ξ −ξ )
Q1,1 1 Q2,2 2 Q3,3 3 Q4,4 1 2 3
Q~X∈Q~
runs over a sparse collection of tiles Q~. In such a sparse collection, then for every Q ∈ Q~
there exists a unique shifted cube Q˜ in R4 such that Q ⊆ 7 Q˜ and with the diameter of Q
10
similar to the diameter of Q˜. This allows us to assume that a sum of the type (4.7) runs
over a sparse collections of shifted dyadic cubes such that |Q | ∼ |Q | ∼ |Q | ∼ |Q |. Let
1 2 3 4
|Q~|∼ |Q |, i = 1,2,3,4, be the scale of the dyadic cube.
i
Further we know that in all three cases (4.2), (4.3) and (4.4) then the scale |Q~| fixes the
location of the tile Q . Also in the case (4.2) where we are close to the line βξ +ξ = 0
1 2 3
then the tiles Q and Q can be made to overlap while in the second two cases (4.3), (4.4),
2 3
when we are away from the line βξ +ξ = 0 then Q and Q can be made to be a couple
2 3 2 3
of units of length |Q~| away from another so they don’t overlap.
We will now study the quadlinear form associated to (3.4).
(4.8) T˜ (f ,f ,f )(x)f (x)dx
β 1 2 3 4
ZR
1
= 1 (αξ +βξ +ξ )dα f (ξ )f (ξ )f (ξ )f (ξ )dξ dξ dξ dξ
R+ 1 2 3 1 1 2 2 3 3 4 4 1 2 3 4
Z (cid:20)Z0 (cid:21)
ξ1+ξ2+ξ3+ξ4=0
b b b b
1
1 (αξ +βξ +ξ )dα
0 R+ 1 2 3
= φ[(ξ )φ[(ξ )φ[(ξ )φ[(−ξ −ξ −ξ )
hR a(ξ1,ξ2,ξ3) i Q1,1 1 Q2,2 2 Q3,3 3 Q4,4 1 2 3
Q~X∈Q~ ξ1+ξ2+Zξ3+ξ4=0
8 EYVINDURARIPALSSON
f (ξ )f (ξ )f (ξ )f (ξ )dξ dξ dξ dξ
1 1 2 2 3 3 4 4 1 2 3 4
1
1 (αξ +βξ +ξ )dα
0b R+ b 1 b 2 b3
= φ[(ξ )φ[(ξ )φ[(ξ )φ[(ξ )
hR a(ξ1,ξ2,ξ3) i Q1,1 1 Q2,2 2 Q3,3 3 Q4,4 4
Q~X∈Q~ ξ1+ξ2+Zξ3+ξ4=0
f (ξ )f (ξ )f (ξ )f (ξ )dξ dξ dξ dξ
1 1 2 2 3 3 4 4 1 2 3 4
b b b b
We can write
1
1 (αξ +βξ +ξ )dα
0 R+ 1 2 3
φ[(ξ )φ[(ξ )φ[(ξ )
hR a(ξ1,ξ2,ξ3) i Q1,1 1 Q2,2 2 Q3,3 3
as
1
1 (αξ +βξ +ξ )dα
0 R+ 1 2 3 φ˜[(ξ )φ˜[(ξ )φ˜[(ξ )φ[(ξ )φ[(ξ )φ[(ξ )
hR a(ξ1,ξ2,ξ3) i Q1,1 1 Q2,2 2 Q3,3 3 Q1,1 1 Q2,2 2 Q3,3 3
where φ˜[ ⊗φ˜[ ⊗φ˜[ is identically equal to 1 on the support of φ[ ⊗φ[ ⊗φ[.
Q1,1 Q2,2 Q3,3 Q1,1 Q2,2 Q3,3
Now split
1
1 (αξ +βξ +ξ )dα
0 R+ 1 2 3 φ˜[(ξ )φ˜[(ξ )φ˜[(ξ )
hR a(ξ1,ξ2,ξ3) i Q1,1 1 Q2,2 2 Q3,3 3
as a Fourier series
CQ~ e2πi|nQ~1|ξ1e2πi|nQ~2|ξ2e2πi|nQ~3|ξ3.
n1,n2,n3
n1X,n2,n3
Q~
The coefficient C is given by
n1,n2,n3
1
1 (αξ +βξ +ξ )dα
(4.9) CQ~ = 1 0 R+ 1 2 3 φ˜[(ξ )φ˜[(ξ )φ˜[(ξ )
n1,n2,n3 |Q~|4 ZR3 hR a(ξ1,ξ2,ξ3) i Q1,1 1 Q2,2 2 Q3,3 3
e−2πi|nQ~1|ξ1e−2πi|nQ~2|ξ2e−2πi|nQ~3|ξ3dξ1dξ2dξ3
Lemma 4.7.
|CQ~ |. C(n ,n ,n )
n1,n2,n3 1 2 3
where the implicit constant does not depend on Q~.
This lemma is a consequence of lemma 6.2 that we prove in section 6. Themain point for
now is that the Fourier coefficient is bounded uniformly independently of the dyadic cube
Q~.
We can now majorize (4.8) by
C(n ,n ,n ) | φ[(ξ )φ[(ξ )φ[(ξ )φ[(ξ )
1 2 3 Q1,1 1 Q2,2 2 Q3,3 3 Q4,4 4
n1X,n2,n3 Q~X∈Q~ ξ1+ξ2+Zξ3+ξ4=0
f1(ξ1)f2(ξ2)f3(ξ3)f4(ξ4)e−2πi|nQ~1|ξ1e−2πi|nQ~2|ξ2e−2πi|nQ~3|ξ3dξ1dξ2dξ3dξ4|
b b b b
Lp ESTIMATES FOR A SINGULAR INTEGRAL OPERATOR 9
= C(n ,n ,n ) | f \∗φ (ξ )f \∗φ (ξ )f \∗φ (ξ )f \∗φ (ξ )
1 2 3 1 Q1,1 1 2 Q2,2 2 3 Q3,3 3 4 Q4,4 4
n1X,n2,n3 Q~X∈Q~ RZ5
e2πi(ξ1+ξ2+ξ3+ξ4)xdξ dξ dξ dξ dx|
1 2 3 4
= C(n ,n ,n ) | (f ∗φn1 )(x)(f ∗φn2 )(x)(f ∗φn3 )(x)(f ∗φ )(x)dx|
1 2 3 1 Q1,1 2 Q2,2 3 Q3,3 4 Q4,4
n1X,n2,n3 Q~X∈Q~ ZR
Here the meaning of φni is that if φ was an L1 normalized wave packet on I ×Q then
Qi,i Qi,i Q~ i
φni is an L1 normalized wave packet on Ini ×Q where Ini is a dyadic interval sitting n
Qi,i Q~ i Q~ i
units of length |I | away from I .
Q~ Q~
Split Q~ = Q~ where Q~ has cubes Q~ of scale |Q~| = 2k and thus |I |= 2−k.
k∈Z k k Q~
S
(4.10) | (f ∗φn1 )(x)(f ∗φn2 )(x)(f ∗φn3 )(x)(f ∗φ )(x)dx|
1 Q1,1 2 Q2,2 3 Q3,3 4 Q4,4
Q~X∈Q~ ZR
= |2−k (f ∗φn1 )(2−ky)(f ∗φn2 )(2−ky)(f ∗φn3 )(2−ky)(f ∗φ )(2−ky)dy|
1 Q1,1 2 Q2,2 3 Q3,3 4 Q4,4
Xk∈ZQ~X∈Q~k ZR
1
= |I | (f ∗φn1 )(2−km+2−kγ)(f ∗φn2 )(2−km+2−kγ)
Q~ 1 Q1,1 2 Q2,2
Xk∈ZQ~X∈Q~k(cid:12) Z0 mX∈Z
(cid:12)
(cid:12) (f ∗φn3 )(2−km+2−kγ)(f ∗φ )(2−km+2−kγ)dγ
3 Q3,3 4 Q4,4
(cid:12)
(cid:12)
Now observe that for i= 1,2,3,4 (where we take n4 = 0) (cid:12)
(f ∗φni )(2−km+2−kγ) = f (z)φni (2−km+2−kγ −z)dz
i Qi,i i Qi,i
ZR
1
= f (z)|I |1/2φni (2−km+2−kγ−z)dz
|I |1/2 i Q~ Qi,i
Q~ ZR
1
= hf ,φ˜ni i
|I |1/2 i Qi,i,m,γ
Q~
where φ˜ni is a wave packet translated from φni by m steps in time and then addi-
Qi,i,m,γ Qi,i
tionally shifted by γ steps. Note that φ˜ni is an L2 normalized wave packet since φni
Qi,i,m,γ Qi,i
was L1 normalized. Now (4.10) becomes
1 1 1 1
|I | hf ,φ˜n1 i hf ,φ˜n2 i hf ,φ˜n3 i
Q~ |I |1/2 1 Q1,1,m,γ |I |1/2 2 Q2,2,m,γ |I |1/2 3 Q3,3,m,γ
Z0 Q~X∈Q~ mX∈Z Q~ Q~ Q~
1
hf ,φ˜ idγ
|I |1/2 4 Q4,4,m,γ
Q~
1 1
= Z0 P~X∈P~ |IP~|hf1,φ˜P1n1,1,γihf2,φ˜P2n2,2,γihf3,φ˜P3n3,3,γihf4,φ˜P4,4,γidγ
10 EYVINDURARIPALSSON
where Pni denotes the tile Ini+m×Q where Ini+m is a dyadic interval such that |Ini+m| ∼
i Pi i Pi Pi
|Q |−1 for i = 1,2,3,4 (again we have n = 0). Again then Ini+m sits n +m units of length
i 4 Pi i
|I | away from I .
P~ Pi
If we now fix n ,n ,n ∈ Z and γ ∈ [0,1] then it is sufficient to study estimates for the
1 2 3
following discrete variant of (4.8)
1
P~X∈P~ |IP~|hf1,φP1n1,1ihf2,φP2n2,2ihf3,φP3n3,3ihf4,φP4,4i
Write
1
(4.11) ΛP~(f1,f2,f3,f4) := P~X∈P~ |IP~|hf1,φP1n1,1ihf2,φP2n2,2ihf3,φP3n3,3ihf4,φP4,4i
and define T (f ,f ,f ) with
P~ 1 2 3
hT (f ,f ,f ),f i = Λ (f ,f ,f ,f )
P~ 1 2 3 4 P~ 1 2 3 4
To compare our quadtiles with the tiles one faces in the bilinear Hilbert transform then
notice that if P~ = (P ,P ,P ,P ) then P is like a paraproduct tile, P and P might at
1 2 3 4 1 2 3
a first glance seem just as in the bilinear Hilbert transform and P is essentially as in the
4
bilinear Hilbert transform, just potentially translated a bit in frequency by P . Note that
1
the constant in the definition of .′ is 5 as opposed to 3 in [13]. We choose a bigger constant
to make up for this extra possible translation of P . In the next section we will see in which
4
cases weareessentially as inthebilinear Hilberttransformcase, and inwhich cases wehave
to be more careful.
5. Rank (1,0)
Recall a standard definition of rank [13].
Definition 5.1. A collection P~ of quadtiles is said to have rank 1 if one has the following
properties for all P~,P~′ ∈ P~:
• If P~ 6= P~′, then P 6= P′ for all j=1,2,3,4.
j j
• If P′ ≤ P for some j = 1,2,3,4, then P′ . P for all 1 ≤ i≤ 4.
j j i i
• If we further assume that 109|I |< |I |, then we have P′ .′ P for all i6= j.
P~′ P~ i i
This definition does not work for our collection of quadtiles because the paraproduct tile
P does not uniquely determine the other three tiles.
1
We only need a frequency or time interval from one of our tiles to determine P , while
1
we need a whole tile P , j = 2,3 or 4, to determine the other three. Motivated by this fact
j
and what ingredients are really important in a rank definition [15] we give the following
definition.
Definition 5.2. Let {i ,i ,i ,i } be some rearrangement of {1,2,3,4}. A collection P~ of
1 2 3 4
quadtiles is said to have rank (1,0) with respect to {{i ,i ,i },{i }} if one has the following
1 2 3 4
properties for all P~,P~′ ∈ P~:
• If P~ 6= P~′, then P 6= P′ for all j=1,2,3 and if I = I then P = P′ .
ij ij P~ P~′ i4 i4
• If P′ ≤ P for some j = 1,2,3, then P′ . P for all 1 ≤ k ≤ 4.
ij ij ik ik
• If we further assume that 109|I |< |I |, then there exist at least two indices
P~′ P~
τ (i ),τ (i ) ∈ {1,2,3,4}\{i }, τ (i ) 6= τ (i )
1 j 2 j j 1 j 2 j