Iterated trilinear fourier integrals with arbitrary symbols
Joeun Jung
Cornell University
ICM 2014, Satellite Conference in Harmonic Analysis,
Chosun University, Gwangju, Korea
August 6, 2014
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Motivation
the Coifman-Meyer theorem with classical paraproduct(1979)
Z
B(f1 , f2 )(x) :=
m(ξ1 , ξ2 )fb1 (ξ1 )fb2 (ξ2 )e2πix(ξ1 +ξ2 ) dξ1 dξ2
IR2
where m(ξ1 , ξ2 ) satisfies |∂ α (m(ξ))| ≤
multi-indices α.
1
|ξ||α|
for sufficiently many
the bilinear Hilbert transform (by Lacey and Thiele, 1997)
Z
fb1 (ξ1 )fb2 (ξ2 )e2πix(ξ1 +ξ2 ) dξ1 dξ2
BHT(f1 , f2 )(x) :=
ξ1 <ξ2
iterated trilinear Fourier integrals (called as Biest II) (by Muscalu, Tao
and Thiele, 2001)
Z
fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )e2πix(ξ1 +ξ2 +ξ3 ) dξ1 dξ2 dξ3
T(f1 , f2 , f3 )(x) :=
ξ1 <ξ2 <ξ3
Z
=
IR3
χξ1 <ξ2 · χξ2 <ξ3 fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )e2πix(ξ1 +ξ2 +ξ3 ) dξ1 dξ2 dξ3
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Motivation
the Coifman-Meyer theorem (classical paraproduct)
Z
B(f1 , f2 )(x) :=
m(ξ1 , ξ2 )fb1 (ξ1 )fb2 (ξ2 )e2πix(ξ1 +ξ2 ) dξ1 dξ2
IR2
where m(ξ1 , ξ2 ) satisfies |∂ α (m(ξ))| ≤
multi-indices α.
1
|ξ||α|
for sufficiently many
Flag paraproduct (by Muscalu, 2007)
Tab (f1 , f2 , f3 )(x)
Z
=
a(ξ1 , ξ2 ) · b(ξ2 , ξ3 )fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )e2πix(ξ1 +ξ2 +ξ3 ) dξ1 dξ2 dξ3
IR3
where a, b satisfies the classical Marcinkiewicz-Mikhlin-Hörmander
condition.
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Motivation
the bilinear Hilbert transform (by Lacey and Thiele)
Z
BHT(f1 , f2 )(x) :=
χξ1 <ξ2 fb1 (ξ1 )fb2 (ξ2 )e2πix(ξ1 +ξ2 ) dξ1 dξ2
IR2
the Biest II(the Fourier case) (by Muscalu, Tao and Thiele)
Z
T(f1 , f2 , f3 )(x) =
IR3
χξ1 <ξ2 · χξ2 <ξ3 fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )e2πix(ξ1 +ξ2 +ξ3 ) dξ1 dξ2 dξ3
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Motivation
the bilinear Hilbert transform (by Lacey and Thiele)
Z
BHT(f1 , f2 )(x) :=
χξ1 <ξ2 fb1 (ξ1 )fb2 (ξ2 )e2πix(ξ1 +ξ2 ) dξ1 dξ2
IR2
Bilinear operator with more generic symbol having 1-dim. singularity
Z
Bm (f1 , f2 )(x) := m(ξ1 , ξ2 )fb1 (ξ1 )fb2 (ξ2 )e2πix(ξ1 +ξ2 ) dξ1 dξ2
where m(ξ1 , ξ2 ) is smooth away from the line Γ = {ξ1 = ξ2 } and
2
1
satisfying |∂ α (m(ξ))| ≤ dist(Γ,ξ)
|α| for every ξ ∈ IR \ Γ, sufficiently many
multi-indices α.
Remark :Bm has the same Lp estimates as BHT by applying the same
model operator (modulo nice decaying Fourier coefficient).
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Motivation
the Biest II(the Fourier case)
Z
T(f1 , f2 , f3 )(x) =
χξ1 <ξ2 · χξ2 <ξ3 fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )e2πix(ξ1 +ξ2 +ξ3 ) dξ1 dξ2 dξ3
IR3
Key idea : In the region {|ξ3 − ξ2 | |ξ2 − ξ1 |},
χξ1 <ξ2 · χξ2 <ξ3 = χξ1 <ξ2 · χ ξ1 +ξ2 <ξ
2
3
.
Iterated trilinear fourier integrals with arbitrary symbols
Z
Tm1 m2 (f1 , f2 , f3 )(x) =
m1 (ξ1 , ξ2 )m2 (ξ2 , ξ3 )fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )e2πix(ξ1 +ξ2 +ξ3 ) dξ1 dξ2 dξ3
IR3
where mi (ξi , ξi+1 ) is smooth away from the line Γi = {ξi = ξi+1 } for
i = 1, 2 and satisfying |∂ α (mi (ξ))| ≤ dist(Γ 1,ξ)|α| for every ξ ∈ IR2 \ Γi ,
i
sufficiently many multi-indices α.
Remark : We are no longer able to apply this key idea to Tm1 m2 .
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Theorem
Let
Z
Tm1 m2 (f1 , f2 , f3 )(x) =
IR3
m1 (ξ1 , ξ2 )m2 (ξ2 , ξ3 )fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )e2πix(ξ1 +ξ2 +ξ3 ) dξ1 dξ2 dξ3
where mi (ξi , ξi+1 ) is smooth away from the line Γi = {ξi = ξi+1 } for i = 1, 2
and satisfying |∂ α (mi (ξ))| ≤ dist(Γ 1,ξ)|α| for every ξ ∈ IR2 \ Γi , sufficiently many
i
multi-indices α.
Theorem
Let 1 < p1 , p2 , p3 ≤ ∞ and 0 < p04 < ∞ such that p11 +
Tm1 m2 maps
0
Tm1 m2 : Lp1 × Lp2 × Lp3 → Lp4
1
p2
+
1
p3
=
1
.
p04
Then
as long as (1/p1 , 1/p2 , 1/p3 , 1/p4 ) ∈ D.
Remark: By using duality, we show that the quadrilinear form Λ associated to
Tm1 m2 via the formula
Z
Λ(f1 , f2 , f3 , f4 ) :=
Tm1 m2 (f1 , f2 , f3 )(x)f4 (x)dx
IR
is bounded on Lp1 × Lp2 × Lp3 × Lp4 for 1 < p04 < ∞.
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
the restricted weak-type interpolation theorems
Let (1/p1 , 1/p2 , 1/p3 , 1/p4 ) := (α1 , α2 , α3 , α4 ) := α.
Definition
1
2
P
A tuple α = (α1 , α2 , α3 , α4 ) is called admissible, if 4i=1 αi = 1 with
αi < 1 for all 1 ≤ i ≤ 4, and (possibly) only one αj < 0.
A 4-linear form Λ is of restricted type α if for every sequence
E1 , E2 , E3 , E4 of subsets of IR with finite measure, there exists a major
subset Ej0 of Ej for αj < 0 (if exists) such that
Y
|Λ(f1 , f2 , f3 , f4 )| . |Ej0 |αj
|Ei |αi
i6=j
for all functions fi supported on Ei and such that kfi k∞ ≤ 1.
Theorem
For every vertex of D there exist admissible tuples α arbitrarily close to the
vertex such that the form Λ is of restricted type α.
Lemma
Let α be an admissible tuple such that α ∈ D. Then Λ is of restricted type α.
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Reduction to the discretized model
By a standard partition of unity, we obtain
X
X
m1 (ξ1 , ξ2 )φQ (ξ1 , ξ2 )
m2 (ξ2 , ξ3 )φQ0 (ξ2 , ξ3 )
m1 (ξ1 , ξ2 )m2 (ξ2 , ξ3 ) =
Q0
Q
X
=
X
+
|Q||Q0 |
|Q|∼|Q0 |
+
X
|Q||Q0 |
Then
X
m1 (ξ1 , ξ2 )φQ (ξ1 , ξ2 )m2 (ξ2 , ξ3 )φQ0 (ξ2 , ξ3 )
|Q||Q0 |
=
X
n∈Z 4
X
Cn
φQ1 (ξ1 )φQ2 (ξ2 )φQ01 (ξ2 )φQ02 (ξ3 )
|Q||Q0 |
eQ0 (ξ1 + ξ2 )?
Question: φQ01 (ξ2 ) ∼ φ
1
By using a Taylor decomposition of φQ01 (ξ2 ), we obtain that
M
φQ01 (ξ2 ) = φQ01 (
X (`)
ξ1 + ξ2
)+
φ Q0
1
2
`=1
Joeun Jung
ξ1 + ξ2
2
ξ2 − ξ1
2
`
1
+ RM (ξ1 , ξ2 )
`!
Iterated trilinear fourier integrals with arbitrary symbols
For 1 ≤ ` ≤ M,
"
X
φQ1 (ξ1 )φQ2 (ξ2 )
|Q||Q0 |
=
X
#≥1000
(`)
φQ0
1
X
2−(#+1)`
ξ1 + ξ2
2
ξ2 − ξ1
2
`
#
1
φQ02 (ξ3 )
`!
φ̃Q1 (ξ1 )φ̃Q2 (ξ2 )φ̃Q001 ,` (ξ1 + ξ2 )φQ02 (ξ3 )
Q,Q0 ;
2# |Q|∼|Q0 |
by letting |Q| = 2k1 , |Q0 | = 2k2 for k1 , k2 ∈ Z, and # = k2 − k1 . We denote
X
m#,` :=
φ̃Q1 (ξ1 )φ̃Q2 (ξ2 )φ̃Q001 ,` (ξ1 + ξ2 )φQ02 (ξ3 ).
Q,Q0 ;
2# |Q|∼|Q0 |
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Then the quadrilinear form Λ#,` associated to Tm#,` via the formula
Z
Λ#,` (f1 , f2 , f3 , f4 ) :=
Tm#,` (f1 , f2 , f3 )(x)f4 (x)dx
IR
Z
=
m#,` (ξ1 , ξ2 , ξ3 )fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )fb4 (−ξ1 − ξ2 − ξ3 )dξ1 dξ2 dξ3
IR3
X Z
=
φ̃Q1 (ξ1 )φ̃Q2 (ξ2 )φQ3 (ξ1 + ξ2 )φ̃Q001 ,` (ξ1 + ξ2 ) ·
Q,Q0 ;
2# |Q|∼|Q0 |
ξ1 +ξ2 +ξ3 +ξ4 =0
φ̃Q02 (ξ3 )φQ03 (ξ1 + ξ2 + ξ3 )fb1 (ξ1 )fb2 (ξ2 )fb3 (ξ3 )fb4 (ξ4 )dξ1 dξ2 dξ3 dξ4
Z
X
=
Q Q0 ;
2# |Q|∼|Q0 |
ξ1 +ξ2 +ξ3 +ξ4 =0
00
f1\
∗ φ̌Q1 (ξ1 ) f2\
∗ φ̌Q2 (ξ2 ) φ̌Q\
3 ∗ φ̌Q1 ` (ξ1 + ξ2 )
f3\
∗ φ̌Q02 (ξ3 ) f4\
∗ φ̌Q03 (ξ4 ) dξ1 dξ2 dξ3 dξ4


=
XZ
Q0
X
IR


(f1 ∗ φ̌Q1 )(x)(f2 ∗ φ̌Q2 )(x) ∗ φ̌Q3  ∗ φ̌Q01 ,`  (x)
Q;2# |Q|∼|Q0 |
(f3 ∗ φ̌Q02 )(x)(f4 ∗ φ̌Q03 )(x)dx
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Define that a tri-tile ~P = (P1 , P2 , P3 ) where each i-tile Pi = IPi × ωPi and ωPi =
~ =
Qi and the IPi = I~P are independent of i. Similarly, we define a tri-tile Q
0
0
0
0
(Q1 , Q2 , Q3 ) with frequency cube Q = Q1 × Q2 × Q3 . Then for # ≥ 1000
X 1
hB# (f1 , f2 ), ΦP1 ihf3 , ΦP2 ihf4 , ΦP3 i
Λ~#
(f
1 , f2 , f3 , f4 ) :=
~
P,Q
|I~P |1/2 P1
~
P∈~
P
where
X
B#
P1 (f1 , f2 ) :=
~
Q∈~
Q:
ωQ ⊂ωP , 2# |ωQ |∼|ωP |
1
1
3
3
1
hf1 , ΦQ1 ihf2 , ΦQ2 iΦQ3
|I~Q |1/2
and
Λ~P,~Q (f1 , f2 , f3 , f4 ) :=
X
2−(#+1)` Λ~#
(f1 , f2 , f3 , f4 ).
P,~
Q
#≥1000
Let
Λ~#
(f1 , f2 , f3 , f4 ) =
P,~
Q
X
~
Q∈~
Q
where
(1)
aQ1
:=
(3),#
:=
aQ3
1
(1) (2) (3),#
a a a
|I~Q |1/2 Q1 Q2 Q3
(2)
hf1 , ΦQ1 i
X
aQ2 := hf2 , ΦQ2 i
~
P∈~
P:ωQ ⊆ωP
1
3
2# |ωQ |∼|ωP |
1
3
Joeun Jung
X
1
hf3 , ΦP2 ihf4 , ΦP3 ihΦP1 , ΦQ3 i.
|I~P |1/2
Iterated trilinear fourier integrals with arbitrary symbols
Definition (Tile norms)
~ be a finite collection of tri-tiles, j = 1, 2, 3, and let (aQj )~ ~ be a
Let Q
Q∈Q
sequence of complex numbers. We define the size of this sequence by
sizej ((aQj )~Q∈~Q ) := sup (
T⊂~
Q
1 X
|aQj |2 )1/2
|IT |
~
Q∈T
~ which are i-trees for some i 6= j.
where T ranges over all trees in Q
We also define the energy of the sequence by
X
energyj ((aQj )~Q∈~Q ) := sup sup 2n (
|IT |)1/2
n∈Z
T
T∈T
~ such that
where T ranges over all collections of strongly j-disjoint trees in Q
X
(
|aQj |2 )1/2 ≥ 2n |IT |1/2
~
Q∈T
for all T ∈ T, and
(
X
|aQj |2 )1/2 ≤ 2n+1 |IT 0 |1/2
~
Q∈T 0
0
for all sub-trees T ⊂ T ∈ T.
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Proposition
~ be a finite collection of tri-tiles, and for each Q
~ and j = 1, 2, 3 let a(j)
~ ∈Q
Let Q
Qj
be a complex number. Then
3
|
X
~
Q∈~
Q
Y
1
(j)
(j)
(1) (2) (3)
sizej ((aQj )~Q∈~Q )θj energyj ((aQj )~Q∈~Q )1−θj
aQ1 aQ2 aQ3 | .
1/2
|I~Q |
j=1
for any 0 ≤ θ1 , θ2 , θ3 < 1 with θ1 + θ2 + θ3 = 1, with the implicit constant
depending on the θj .
Let Ej be sets of finite measure and fj be functions supported on Ej and such
that kfj k∞ ≤ 1.
R χ̃MI~
(j)
sizej ((aQj )~Q∈~Q ) . sup~Q∈~Q Ej |I Q| for j = 1, 2
~
Q
(j)
energyj ((aQj )~Q∈~Q ) ≤ kfj k2 ≤ |Ej |1/2 for j = 1, 2
!1−θ
R χ̃MI~Q
R
(3),#
size3 ((aQ3 )~Q∈~Q ) . sup~Q∈~Q
E |I |
E
3
~
Q
4
R
(3),#
energy3 ((aQ3 )~Q∈~Q )
.2
#/2
(|E4 |
1/2
sup
~
P∈~
P
E3
χ̃M
I~P
|I~P |
!θ
χ̃M
I
~
Q
|I~
|
Q
R
)
1−θ
(|E3 |
1/2
sup
~
P∈~
P
E4
χ̃M
I~P
|I~P |
. 2#/2 |E4 |(1−θ)/2 |E3 |θ/2
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
)θ
With these tile norms and by Proposition, we obtain that
|Λ~#
(f1 , f2 , f3 , f4 )| . 2#/2 |E|α
P,~
Q
for given sequence E1 , E2 , E3 , E4 with finite measure and fi supported on Ei
and such that kfi k∞ ≤ 1. Hence, finally we have
X −(#+1)` #
|Λ~P,~Q (f1 , f2 , f3 , f4 )| ≤
2
|Λ~P,~Q (f1 , f2 , f3 , f4 )|
#≥1000
.
X
2−(#+1)` 2#/2 |E|α
#≥1000
≤ |E|α .
This complete the proof.
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols
Thank you!
Joeun Jung
Iterated trilinear fourier integrals with arbitrary symbols