This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.

1. Multilinear Twisted Paraproducts
Vjekoslav Kovaˇc, University of Zagreb
UCLA, April 20, 2012

2. Classical case
Classical model paraproduct:
Λ1DP(f1, . . . , fn) =
I∈D
|I|1−n
2
n
i=1
fi , ϕ
(i)
I L2(R)
=
I∈D
|I|1−n
2
Rn
n
i=1
fi (xi )ϕ
(i)
I (xi )dxi
D = dyadic intervals
ϕ
(i)
I = smooth bump function adapted to I, at least two mean 0
Calder´on-Zygmund theory establishes:
|Λ1DP(f1, . . . , fn)| n,(pi )
n
i=1
fi Lpi (R)
when n
i=1
1
pi
= 1, 1 < pi < ∞

3. Higher dimensions — for simplicity R2
Dyadic model paraproduct:
Λ2DP(F1, . . . , Fn) =
I×J∈C
|I|2−n
i∈S
Fi , ψD
I ⊗ϕD
J L2(R2)
i∈Sc
Fi , ϕD
I ⊗ϕD
J L2(R2)
=
I×J∈C
|I|2−n
R2n
i∈S
ψD
I (xi )
i∈Sc
ϕD
I (xi )
n
i=1
Fi (xi , yi )ϕD
J (yi )dxi dyi
C = dyadic squares in R2
ϕD
I := |I|−1/21I , ψD
I := |I|−1/2(1Ileft
− 1Iright
)
S ⊆ {1, 2, . . . , n}, |S| ≥ 2

4. Complications
Twisted paraproduct (Demeter and Thiele, ≈ 2006):
ΛTP(F, G, H) =
R2
k∈Z
22k
R
F(x − s, y)ϕ(2k
s)ds
R
G(x, y − t)ψ(2k
t)dt H(x, y)dxdy
ϕ, ψ smooth bump functions, supp( ˆψ) ⊆ {ξ ∈ R : 1
2 ≤|ξ| ≤ 2}
Dyadic twisted paraproduct:
ΛDTP(F, G, H) =
I×J∈C R4
F(x2, y1)G(x1, y2)H(x1, y1)
ψD
I (x1)ψD
I (x2)ϕD
J (y1)ϕD
J (y2) dx1dx2dy1dy2

5. Unified setting — Bipartite graph
m, n ∈ N
E ⊆ {1, . . . , m} × {1, . . . , n}
S ⊆ {1, . . . , m}, T ⊆ {1, . . . , n}
We require |S| ≥ 2 or |T| ≥ 2
y
y
x
x
x
y
x
TS
E
.
..
..
.
n
2
1
m
3
2
1
bipartite graph with selections of vertices ≡ triple (E, S, T)

6. Unified setting — Paraproduct
triple (E, S, T) multilinear form Λ = ΛE,S,T on |E| functions
Λ (Fi,j )(i,j)∈E :=
I×J∈C
|I|2−m+n
2
Rm+n
(i,j)∈E
Fi,j (xi , yj )
i∈S
ψD
I (xi )
i∈Sc
ϕD
I (xi )
j∈T
ψD
J (yj )
j∈Tc
ϕD
J (yj )
dx1 . . . dxm dy1 . . . dyn
(1) each edge (xi , yj ) function Fi,j
(2) each vertex xi or yj “dyadic bump function” ϕD or ψD
(3) selected vertex “mean 0 dyadic bump function” ψD
(4) at least two selected vertices in either {x1, . . . , xm} or
{y1, . . . , yn}

7. Main boundedness result
E connected components
di,j = larger size of the two bipartition classes of the connected
component containing an edge (xi , yj )
Theorem (K. 2011)
The series I×J∈C defining Λ converges absolutely and Λ satisfies
the estimate
Λ (Fi,j )(i,j)∈E m,n,(pi,j )
(i,j)∈E
Fi,j L
pi,j (R2)
when (pi,j )(i,j)∈E are such that (i,j)∈E
1
pi,j
= 1, di,j <pi,j <∞

8. Special case — Classical 2D dyadic
m = n
E = (i, i) : i ∈ {1, . . . , n}
|S| ≥ 2, T = ∅
di,i = 1
y
y
y
y
x
x
x
x
n
3
2
1
n
3
2
1
..
..
..
|Λ2DP(F1, . . . , Fn)| n,(pi )
n
i=1 Fi Lpi (R2)
in the range n
i=1
1
pi
= 1, 1 < pi < ∞

9. Special case — Dyadic twisted
m = n = 2
E = (1, 1), (1, 2), (2, 1)
S = {1, 2}, T = ∅
d1,1 = d1,2 = d2,1 = 2
y
y
x
x
2
1
2
1
|ΛDTP(F, G, H)| p,q,r F Lp(R2) G Lq(R2) H Lr (R2)
in the range 1
p + 1
q + 1
r = 1, 2<p, q, r <∞
Wider range: 1<p, q <∞, 2<r ≤∞ (Bernicot 2010)

10. Averages — Useful notation
[f ]I = [f (x)]x∈I :=
1
|I| I
f (x)dx
f I = f (x) x∈I :=
1
|I| Ileft
f (x)dx −
Iright
f (x)dx
[F(x, y)]x∈I =
1
|I| I
F(x, y)dx
F(x, y) x∈I =
1
|I| Ileft
F(x, y)dx −
Iright
F(x, y)dx
Φ(x, x , y) x,x ,y = Φ(x, x , y) x∈I x ∈I y∈J
Φ(x1, x2, . . .) xi ∈I for 1≤i≤n
= . . . Φ(x1, x2, . . .) x1∈I x2∈I
. . . xn∈I
[F]Q :=
1
|Q| Q
F(x, y) dxdy = F(x, y) x∈I, y∈J

11. Trees of dyadic squares
tree T with root (tree-top) QT ∈ T
= a collection of dyadic squares satisfying Q ⊆ QT
T is convex if (Q1 ⊆ Q2 ⊆ Q3) & (Q1, Q3 ∈ T ) ⇒ (Q2 ∈ T )
L(T ) = leaves of T
= squares that are not contained in T but their parents are

12. Bellman functions in multilinear setting
Bellman functions in harmonic analysis
Invented by Burkholder (1980s)
Developed by Nazarov, Treil, Volberg, etc. (1990s)
We primarily keep just the “induction on scales” idea
A broad class of interesting (dyadic) objects can be reduced to
ΛT (F1, . . . , F ) :=
Q∈T
|Q| AQ(F1, . . . , F )
T = a finite convex tree of dyadic squares
AQ(F1, . . . , F ) = some “scale-invariant” quantity depending on
F1, . . . , F and Q ∈ T

13. Calculus of finite differences
B = BQ(F1, . . . , F )
First order difference of B: B = BQ(F1, . . . , F )
BI×J(F1, . . . , F ) :=
1
4
BIleft×Jleft
(F1, . . . , F ) +
1
4
BIleft×Jright
(F1, . . . , F )
+
1
4
BIright×Jleft
(F1, . . . , F ) +
1
4
BIright×Jright
(F1, . . . , F )
− BI×J(F1, . . . , F )
B B
[F(x, y)]3
x y
3 [F(x, y)]x F(x, y) 2
x y
[F(x, y)]2
x,y [ F(x, y) x ]2
y + [F(x, y)]x
2
y + F(x, y) 2
x,y
[F(x, y)G(x , y)]2
y x,x
F(x, y)G(x , y) 2
y x,x
+ [F(x, y)G(x , y)]2
y x,x
+ F(x, y)G(x , y) 2
y x,x

14. Calculus of finite differences
Suppose |A| ≤ B, i.e.
|AQ(F1, . . . , F )| ≤ BQ(F1, . . . , F )
for all Q ∈ T and nonnegative bounded measurable F1, . . . , F .
|Q| |AQ(F1, . . . , F )| ≤
Q is a child of Q
|Q| BQ
(F1, . . . , F )
− |Q| BQ(F1, . . . , F ).
|ΛT (F1, . . . , F )| ≤
Q∈L(T )
|Q| BQ(F1, . . . , F )
− |QT | BQT
(F1, . . . , F )
B is called a Bellman function for ΛT

15. Paraproduct-type terms
Paraproduct-type term = a quantity A = AI×J(F1, . . . , Fl ) that
takes the form
A = . . . Φ(x1, . . . , xm, y1, . . . , yn)
x1∈I
. . .
yn∈J
where each pair (·) is replaced by either [·] or ·
Averaging paraproduct-type term contains only brackets of type [·]
Paraproduct-type expressions = linear combinations of
paraproduct-type terms
Example: Gowers box norm
F (I×J) = F(x1, y1)F(x1, y2)F(x2, y1)F(x2, y2)
1/4
x1,x2∈I, y1,y2∈J

16. Computing B
Lemma 1
The first order difference of the averaging paraproduct-type term
B = Φ(x1, . . . , xm, y1, . . . , yn)
x1,...,xm∈I, y1,...,yn∈J
is a paraproduct-type expression given by the formula
B =
S⊆{1,...,m}, T⊆{1,...,n}
|S|,|T| even, (S,T)=(∅,∅)
Φ(x1, . . . , yn) xi ∈I for i∈S
yj ∈J for j∈T
xi ∈I for i∈Sc
yj ∈J for j∈Tc
(1) The number of replacements corresponding to variables in I is even.
(2) The number of replacements corresponding to variables in J is even.
(3) At least two replacements are made, i.e. the derived terms are not
averaging.

17. More on the structure of B
Lemma 2
S⊆{1,...,m}
|S| even
Ψ(x1, x2, . . . , xm)
xi ∈I for i∈S xi ∈I for i∈Sc
=
1
2
Ψ(x1, . . . , xm) x1,...,xm∈Ileft
+
1
2
Ψ(x1, . . . , xm) x1,...,xm∈Iright
In particular, if Ψ(x1, x2, . . . , xm) ≥ 0, then the above sum will
also be nonnegative.
The sum is constructed by adding up all terms of the form
. . . Ψ(x1, x2, . . . , xm)
x1∈I x2∈I
. . .
xm∈I
where an even number of pairs of parentheses (·) is replaced with pairs of
brackets · and the remaining ones are replaced with pairs of brackets [·].

18. Single tree estimate
Λ = ΛE,S,T
Λ (Fi,j )(i,j)∈E =
Q∈C
|Q| AQ (Fi,j )(i,j)∈E
AI×J (Fi,j )(i,j)∈E =
(i,j)∈E
Fi,j (xi , yj ) xi ∈I for i∈S
yj ∈J for j∈T
xi ∈I for i∈Sc
yj ∈J for j∈Tc
Assume Fi,j ≥ 0
Proposition (Single tree estimate)
For a finite convex tree T we have
ΛT (Fi,j )(i,j)∈E m,n |QT |
(i,j)∈E
max
Q∈T ∪L(T )
F
di,j
i,j
1/di,j
Q
Normalize: |QT | = 1 and maxQ∈T ∪L(T ) F
di,j
i,j
1/di,j
Q
= 1

19. Single tree estimate
It is enough to consider complete bipartite graphs:
E = {1, . . . , m} × {1, . . . , n}
only one connected component
di,j = max{m, n}
A single scale estimate:
Lemma 3
For Gi,j ≥ 0 on a dyadic square I × J:
1≤i≤m
1≤j≤n
Gi,j (xi , yj )
x1,...,xm∈I
y1,...,yn∈J
≤
1≤i≤m
1≤j≤n
G
max{m,n}
i,j
1/ max{m,n}
I×J

20. Single tree estimate
Combinatorics of integer partitions
selective (m, n)-partition = a (2m + 2n)-tuple of integers
p = (a1, . . . , am; b1, . . . , bn; α1, . . . , αm; β1, . . . , βn)
satisfying:
(1) 0 ≤ αi ≤ ai for i = 1, . . . , m and 0 ≤ βj ≤ bj for j = 1, . . . , n,
(2) a1 + . . . + am = m and b1 + . . . + bn = n,
(3) α1 + . . . + αm and β1 + . . . + βn are even,
(4) α1 + . . . + αm = 0 or β1 + . . . + βn = 0.
Ωm,n = the set of all selective (m, n)-partitions

21. Single tree estimate
Integer partitions and paraproduct-type terms
To every p ∈ Ωm,n we associate a paraproduct-type term
A(p)
= A
(p)
I×J (Fi,j )(i,j)∈E
by
A
(p)
I×J :=
1≤i≤m
1≤j≤n
1≤µ≤ai
1≤ν≤bj
Fi,j (x
(µ)
i , y
(ν)
j ) x
(µ)
i ∈I for all (i,µ)
such that 1≤µ≤αi
y
(ν)
j ∈J for all (j,ν)
such that 1≤ν≤βj
x
(µ)
i ∈I for all (i,µ)
s. t. αi +1≤µ≤ai
y
(ν)
j ∈J for all (j,ν)
s. t. βj +1≤ν≤bj
For instance, to
p = (2, 0; 2, 0, 1; 0, 0; 1, 0, 1) ∈ Ω2,3
we associate
A(p)
= F1,1(x1, y1)F1,1(x1, y1)F1,3(x1, y3)
F1,1(x1, y1)F1,1(x1, y1)F1,3(x1, y3) y1,y3 x1,x1,y1

22. Single tree estimate
Combinatorics of integer partitions
p = (a1, . . . , am; b1, . . . , bn; α1, . . . , αm; β1, . . . , βn)
Composition type of p:
comp(p) := (a1, . . . , am; b1, . . . , bn)
Partition type of p and A(p):
part(p) := (a∗
1, . . . , a∗
m; b∗
1, . . . , b∗
n),
where a∗
1, . . . , a∗
m is the decreasing rearrangement of a1, . . . , am
and b∗
1, . . . , b∗
n is the decreasing rearrangement of b1, . . . , bn.
Ω∗
m,n = the set of all these partition types
Ω∗
m,n = p#(m)p#(n)

23. Single tree estimate
Combinatorics of integer partitions
We define a strict total order relation on Ω∗
m,n as the restriction
of the inverse of the lexicographical order on (m + n)-tuples of
integers.
Natural rank (i.e. order) function:
ord: Ω∗
m,n → {1, 2, . . . , p#(m)p#(n)}
Simply write ord(p) for ord part(p)
Example: The total order on Ω∗
2,3 and its rank function.
(2, 0; 3, 0, 0) (2, 0; 2, 1, 0) (2, 0; 1, 1, 1) · · · (1, 1; 1, 1, 1).
ord = 1 ord = 2 ord = 3 ord = 6

24. Single tree estimate
Our goal:
Dominate all terms A(p) by B for some averaging
paraproduct-type expression B = BQ (Fi,j )(i,j)∈E that is
controlled in the sense
max
Q∈T ∪L(T )
BQ (Fi,j )(i,j)∈E m,n 1
This expression B will be the desired Bellman function.
The goal will be achieved by mathematical induction on ord(p).

25. Single tree estimate
Lemma 4 (Reduction lemma)
For any p ∈ Ωm,n there exists a controlled averaging
paraproduct-type term B(p) = B
(p)
Q (Fi,j )(i,j)∈E such that for any
0 < δ < 1 we have the estimate
|A(p)
| ≤ B(p)
+ Cm,n δ−1
p∈Ωm,n
ord(p)<ord(p)
|A(p)
| + Cm,n δ
p∈Ωm,n
ord(p)≥ord(p)
|A(p)
|
with some constant Cm,n > 0.
Proof: Multiple applications of Lemma 1 and Lemma 2 together
with the Cauchy-Schwarz inequality.

26. Recursive control of part. types in Ω∗
2,4
(1, 1; 1, 1, 1, 1)
δ, δ−1
//
δ, δ−1
(2, 0; 1, 1, 1, 1)
δ, δ−1
(1, 1; 2, 1, 1, 0)
δ, δ−1
//
δ, δ−1
δ−1
''
δ
•
•
•
•
•
(2, 0; 2, 1, 1, 0)
δ, δ−1
δ−1
ww
δ
||
•
•
•
•
•
(1, 1; 2, 2, 0, 0)
δ, δ−1
//
δ, δ−1
•
•
•
•
•
(2, 0; 2, 2, 0, 0)
δ, δ−1
•
•
•
•
•
(1, 1; 3, 1, 0, 0)
δ, δ−1
//
δ−1
δ
YY•
•
•
•
•
(2, 0; 3, 1, 0, 0)
δ−1
δ
EE •
•
•
•
•
(1, 1; 4, 0, 0, 0)
δ, δ−1
// (2, 0; 4, 0, 0, 0)

27. Single tree estimate
Corollary
There exists a “universal” controlled averaging paraproduct-type
expression B(m,n) = B
(m,n)
I×J (Fi,j )(i,j)∈E satisfying
p∈Ωm,n
|A(p)
| ≤ B(m,n)
.

28. Proof of the main estimate
We prove the bound
Q∈C
|Q| AQ (Fi,j )(i,j)∈E m,n,(pi,j )
(i,j)∈E
Fi,j L
pi,j (R2)
For each |E|-tuple of integers k = (ki,j )(i,j)∈E ∈ Z|E| we denote
Pk := Q ∈ C : 2ki,j
≤ sup
Q ∈C, Q ⊇Q
F
di,j
i,j
1/di,j
Q
2ki,j +1
for every (i, j) ∈ E

29. Proof of the main estimate
Mk = the collection of maximal squares in Pk
For each Q ∈ Mk the family
TQ := Q ∈ Pk : Q ⊆ Q
is a finite convex tree with root Q.
For different squares Q ∈ Mk the corresponding trees cover
disjoint regions in the plane.
Single tree estimate gives
ΛTQ
(Fi,j )(i,j)∈E m,n |Q| 2 (i,j)∈E ki,j
Adding up the estimates over different trees is standard.

30. Graphs that are not bipartite
For instance, a multilinear form associated to a triangle, i.e. a cycle
of length 3:
Λ (F, G, H) =
I,J,K∈D
|I|=|J|=|K|
c(I,J,K)=0
|I|1/2
R3
F(x, y)G(y, z)H(z, x)
ψD
I (x)ψD
J (y)ψD
K (z) dxdydz,
where c(I, J, K) = 0 is some constraint.
Such forms seem to share many characteristics with the
two-dimensional “triangular” Hilbert transform
Λ HT(F, G, H) =
R2
p.v.
R
F(x−t, y) G(x, y−t)
dt
t
H(x, y)dxdy,
for which no Lp bounds are known.

31. Thank you!
Thank you!