Presentation, Joint CRM-ISAAC Conference on Fourier Analysis and Approximation Theory, Bellaterra, November 6, 2013.

1. Estimates for a class of
non-standard bilinear multipliers
Vjekoslav Kovaˇc (University of Zagreb)
Joint work with Fr´ed´eric Bernicot (Universit´e de Nantes)
and Christoph Thiele (Universit¨at Bonn)
Joint CRM-ISAAC Conference on
Fourier Analysis and Approximation Theory
Bellaterra, November 6, 2013

2. Talk outline |
Part 1 — Introduction
Motivation for multilinear estimates
Concrete examples and some results

3. Talk outline |
Part 1 — Introduction
Motivation for multilinear estimates
Concrete examples and some results
Part 2 — The “entangled” structure
The scope of our techniques

4. Talk outline |
Part 1 — Introduction
Motivation for multilinear estimates
Concrete examples and some results
Part 2 — The “entangled” structure
The scope of our techniques
Part 3 — Dyadic model operators
Formulating a T(1)-type theorem
Setting up the Bellman function scheme

5. Talk outline |
Part 1 — Introduction
Motivation for multilinear estimates
Concrete examples and some results
Part 2 — The “entangled” structure
The scope of our techniques
Part 3 — Dyadic model operators
Formulating a T(1)-type theorem
Setting up the Bellman function scheme
Part 4 — Transition to continuous-type operators
Back to bilinear multipliers
“Entangled” operators with continuous kernels

6. Part 1 — Multilinear estimates ||
T = a multilinear integral operator
T is “singular” in some sense

7. Part 1 — Multilinear estimates ||
T = a multilinear integral operator
T is “singular” in some sense
We are interested in Lp
estimates:
T(F1, F2, . . . , Fk) Lp(Rn) ≤ Cp,p1,...,pk
k
j=1
Fj L
pj (R
nj )
in a subrange of 0 < p, p1, . . . , pk < ∞
Possibly also some Sobolev norm estimates, etc.

8. Part 1 — Multilinear estimates, motivation |||
Motivation for multilinear estimates:

9. Part 1 — Multilinear estimates, motivation |||
Motivation for multilinear estimates:
paraproducts
J.-M. Bony (1981) — paradifferential operators
G. David and J.-L. Journ´e (1984) — T(1) theorem

10. Part 1 — Multilinear estimates, motivation |||
Motivation for multilinear estimates:
paraproducts
J.-M. Bony (1981) — paradifferential operators
G. David and J.-L. Journ´e (1984) — T(1) theorem
multilinear expansions of nonlinear/“curved” operators
A. Calder´on (1960s) — Cauchy integral on Lipschitz curves
M. Christ and A. Kiselev (2001) — Hausdorff-Young
inequalities for the Dirac scattering transform
J. Bourgain and L. Guth (2010) — restriction estimates,
oscillatory integrals

11. Part 1 — Multilinear estimates, motivation |||
Motivation for multilinear estimates:
paraproducts
J.-M. Bony (1981) — paradifferential operators
G. David and J.-L. Journ´e (1984) — T(1) theorem
multilinear expansions of nonlinear/“curved” operators
A. Calder´on (1960s) — Cauchy integral on Lipschitz curves
M. Christ and A. Kiselev (2001) — Hausdorff-Young
inequalities for the Dirac scattering transform
J. Bourgain and L. Guth (2010) — restriction estimates,
oscillatory integrals
recurrence in ergodic theory
J. Bourgain (1988) — return times theorem
C. Demeter, M. Lacey, T. Tao, and C. Thiele (2008) —
extending the exponent range

12. Part 1 — A basic example ||||
Bilinear case only (for simplicity)

13. Part 1 — A basic example ||||
Bilinear case only (for simplicity)
From the viewpoint of bilinear singular integrals:
T(F, G)(x) = p.v.
(Rn)2
K(s, t)F(x − s)G(x − t) ds dt
K = translation-invariant Calder´on-Zygmund kernel
Generalized by L. Grafakos and R. H. Torres (2002):
multilinear C-Z operators
Take m = K

14. Part 1 — A basic example ||||
Bilinear case only (for simplicity)
From the viewpoint of bilinear multipliers:
Coifman-Meyer multipliers, R. Coifman and Y. Meyer (1978)
T(F, G)(x) =
(Rn)2
m(ξ, η)e2πix·(ξ+η)
F(ξ)G(η)dξdη
m∈C∞
R2{(0, 0)}
∂α1
ξ ∂α2
η m(ξ, η) ≤ Cα1,α2,n(|ξ| + |η|)−α1−α2
Note: m(ξ, η) is singular only at the origin ξ = η = 0

15. Part 1 — More singular examples, 1D ||||
1D example: bilinear Hilbert transform
Suggested by A. Calderon (Cauchy integral on Lipschitz curves)
Bounded by M. Lacey and C. Thiele (1997)

16. Part 1 — More singular examples, 1D ||||
1D example: bilinear Hilbert transform
Suggested by A. Calderon (Cauchy integral on Lipschitz curves)
Bounded by M. Lacey and C. Thiele (1997)
As a singular integral:
T(f , g)(x) = p.v.
R
f (x − t)g(x + t)
dt
t

17. Part 1 — More singular examples, 1D ||||
1D example: bilinear Hilbert transform
Suggested by A. Calderon (Cauchy integral on Lipschitz curves)
Bounded by M. Lacey and C. Thiele (1997)
As a singular integral:
T(f , g)(x) = p.v.
R
f (x − t)g(x + t)
dt
t
As a multiplier:
T(f , g)(x) =
R2
πi sgn(η − ξ)e2πix(ξ+η)
f (ξ)g(η)dξdη
Note: m(ξ, η) = πi sgn(η − ξ) is singular along the line ξ = η

18. Part 1 — More singular examples, 2D | ||||
2D example: a variant of the 2D bilinear Hilbert transform
Introduced by Demeter and Thiele and bounded for “most” cases
of A, B ∈ M2(R) (2008)

19. Part 1 — More singular examples, 2D | ||||
2D example: a variant of the 2D bilinear Hilbert transform
Introduced by Demeter and Thiele and bounded for “most” cases
of A, B ∈ M2(R) (2008)
As a singular integral:
T(F, G)(x, y) = p.v.
R2
K(s, t)F (x, y)−A(s, t)
G (x, y)−B(s, t) ds dt

20. Part 1 — More singular examples, 2D | ||||
2D example: a variant of the 2D bilinear Hilbert transform
Introduced by Demeter and Thiele and bounded for “most” cases
of A, B ∈ M2(R) (2008)
As a singular integral:
T(F, G)(x, y) = p.v.
R2
K(s, t)F (x, y)−A(s, t)
G (x, y)−B(s, t) ds dt
Essentially the only case that was left out:
A =
1 0
0 0
and B =
0 0
0 1

21. Part 1 — More singular examples, 2D | ||||
2D example: a variant of the 2D bilinear Hilbert transform
As a multiplier:
T(F, G)(x, y) =
R4
µ(ξ1, ξ2, η1, η2)e2πi(x(ξ1+η1)+y(ξ2+η2))
F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2
µ(ξ1, ξ2, η1, η2) = m Aτ (ξ1, ξ2) + Bτ (η1, η2) , m = K
m ∈ C∞
R2{(0, 0)}
∂α1
τ1
∂α2
τ2
m(τ1, τ2) ≤ Cα1,α2 (|τ1| + |τ2|)−α1−α2
Note: µ(ξ1, ξ2, η1, η2) is singular along the 2-plane
Aτ (ξ1, ξ2) + Bτ (η1, η2) = (0, 0)

22. Part 1 — Remaining case of 2D BHT || ||||
T(F, G)(x, y) =
R4
m(ξ1, η2)e2πi(x(ξ1+η1)+y(ξ2+η2))
F(ξ1, ξ2)G(η1, η2)dξ1dξ2dη1dη2

23. Part 1 — Remaining case of 2D BHT || ||||
T(F, G)(x, y) =
R4
m(ξ1, η2)e2πi(x(ξ1+η1)+y(ξ2+η2))
F(ξ1, ξ2)G(η1, η2)dξ1dξ2dη1dη2
Theorem. Lp
estimate — F. Bernicot (2010), V. K. (2010)
T(F, G) Lr ≤ Cp,q,r F Lp G Lq
for 1 < p, q < ∞, 0 < r < 2, 1
p + 1
q = 1
r .

24. Part 1 — Remaining case of 2D BHT || ||||
T(F, G)(x, y) =
R4
m(ξ1, η2)e2πi(x(ξ1+η1)+y(ξ2+η2))
F(ξ1, ξ2)G(η1, η2)dξ1dξ2dη1dη2
Theorem. Lp
estimate — F. Bernicot (2010), V. K. (2010)
T(F, G) Lr ≤ Cp,q,r F Lp G Lq
for 1 < p, q < ∞, 0 < r < 2, 1
p + 1
q = 1
r .
Theorem. Sobolev estimate — F. Bernicot and V. K. (2013)
If supp m ⊆ (ξ1, η2) : |ξ1| ≤ c |η2| , then
T(F, G) Lr
y (Ws,r
x ) ≤ Cp,q,r,s F Lp G Ws,q
for s ≥ 0, 1 < p, q < ∞, 1 < r < 2, 1
p + 1
q = 1
r .

25. Part 1 — A warning example ||| ||||
Bi-parameter bilinear Hilbert transform

26. Part 1 — A warning example ||| ||||
Bi-parameter bilinear Hilbert transform
T(F, G)(x, y) = p.v.
R2
F(x − s, y − t) G(x + s, y + t)
ds
s
dt
t

27. Part 1 — A warning example ||| ||||
Bi-parameter bilinear Hilbert transform
T(F, G)(x, y) = p.v.
R2
F(x − s, y − t) G(x + s, y + t)
ds
s
dt
t
T(F, G)(x, y) =
R4
π2
sgn(ξ1 + ξ2)sgn(η1 + η2)
e2πi(x(ξ1+η1)+y(ξ2+η2))
F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2

28. Part 1 — A warning example ||| ||||
Bi-parameter bilinear Hilbert transform
T(F, G)(x, y) = p.v.
R2
F(x − s, y − t) G(x + s, y + t)
ds
s
dt
t
T(F, G)(x, y) =
R4
π2
sgn(ξ1 + ξ2)sgn(η1 + η2)
e2πi(x(ξ1+η1)+y(ξ2+η2))
F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2
Satisfies no Lp
estimates!
C. Muscalu, J. Pipher, T. Tao, and C. Thiele (2004)
Note: the symbol is singular along the union of two 3-planes,
ξ1 + ξ2 = 0 and η1 + η2 = 0

29. Part 1 — Open problem #1 |||| ||||
Triangular Hilbert transform

30. Part 1 — Open problem #1 |||| ||||
Triangular Hilbert transform
T(F, G)(x, y) = p.v.
R
F(x−t, y)G(x, y −t)
dt
t

31. Part 1 — Open problem #1 |||| ||||
Triangular Hilbert transform
T(F, G)(x, y) = p.v.
R
F(x−t, y)G(x, y −t)
dt
t
T(F, G)(x, y) =
R4
−πisgn(ξ1 + η2) e2πi(x(ξ1+η1)+y(ξ2+η2))
F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2

32. Part 1 — Open problem #1 |||| ||||
Triangular Hilbert transform
T(F, G)(x, y) = p.v.
R
F(x−t, y)G(x, y −t)
dt
t
T(F, G)(x, y) =
R4
−πisgn(ξ1 + η2) e2πi(x(ξ1+η1)+y(ξ2+η2))
F(ξ1, ξ2)G(η1, η2) dξ1dξ2dη1dη2
Still no Lp
estimates are known
Note: the symbol is singular along the 3-plane ξ1 + η2 = 0
Probably not the right way of looking at the operator

33. Part 1 — Open problem #1 |||| ||||
Triangular Hilbert transform
A singular integral approach to bilinear ergodic averages
(suggested by C. Demeter and C. Thiele):
1
N
N−1
k=0
f (Sk
ω)g(Tk
ω), ω ∈ Ω
S, T : Ω → Ω are commuting measure preserving transformations
L2
norm convergence as N → ∞ was shown by J.-P. Conze and E.
Lesigne (1984)
a.e. convergence as N → ∞ is still an open problem

34. Part 1 — Open problem #2 |||| ||||
Trilinear Hilbert transform

35. Part 1 — Open problem #2 |||| ||||
Trilinear Hilbert transform
T(f , g, h)(x) = p.v.
R
f (x−t)g(x+t)h(x+2t)
dt
t

36. Part 1 — Open problem #2 |||| ||||
Trilinear Hilbert transform
T(f , g, h)(x) = p.v.
R
f (x−t)g(x+t)h(x+2t)
dt
t
T(f , g, h)(x) =
R3
πi sgn(−ξ + η + 2ζ)e2πix(ξ+η+ζ)
f (ξ)g(η)h(ζ)dξdηdζ

37. Part 1 — Open problem #2 |||| ||||
Trilinear Hilbert transform
T(f , g, h)(x) = p.v.
R
f (x−t)g(x+t)h(x+2t)
dt
t
T(f , g, h)(x) =
R3
πi sgn(−ξ + η + 2ζ)e2πix(ξ+η+ζ)
f (ξ)g(η)h(ζ)dξdηdζ
Note: the symbol is singular along the 2-plane −ξ + η + 2ζ = 0
A complete mystery!
Only some negative results are known: C. Demeter (2008)

38. Part 2 — Entangled structure | |||| ||||
k-linear operator (k+1)-linear form
Object of study:
Multilinear singular integral forms with functions that partially
share variables

39. Part 2 — Entangled structure | |||| ||||
k-linear operator (k+1)-linear form
Object of study:
Multilinear singular integral forms with functions that partially
share variables
Schematically:
Λ(F1, F2, . . .) =
Rn
F1(x1, x2) F2(x1, x3) . . .
K(x1, . . . , xn) dx1dx2dx3 . . . dxn
K = singular kernel
F1, F2, . . . = functions on R2

40. Part 2 — Generalized modulation invariance || |||| ||||
An alternative viewpoint: generalized modulation invariances

41. Part 2 — Generalized modulation invariance || |||| ||||
An alternative viewpoint: generalized modulation invariances
Rn
F1(x1, x2) F2(x1, x3) . . .
K(x1, . . . , xn) dx1dx2dx3 . . . dxn

42. Part 2 — Generalized modulation invariance || |||| ||||
An alternative viewpoint: generalized modulation invariances
Rn
F1(x1, x2) F2(x1, x3) . . .
K(x1, . . . , xn) dx1dx2dx3 . . . dxn
=
Rn
e2πiax1
F1(x1, x2) e−2πiax1
F2(x1, x3) . . .
K(x1, . . . , xn) dx1dx2dx3 . . . dxn

43. Part 2 — Generalized modulation invariance || |||| ||||
An alternative viewpoint: generalized modulation invariances
Rn
F1(x1, x2) F2(x1, x3) . . .
K(x1, . . . , xn) dx1dx2dx3 . . . dxn
=
Rn
e2πiax1
F1(x1, x2) e−2πiax1
F2(x1, x3) . . .
K(x1, . . . , xn) dx1dx2dx3 . . . dxn
=
Rn
ϕ(x1)F1(x1, x2)
1
ϕ(x1)
F2(x1, x3) . . .
K(x1, . . . , xn) dx1dx2dx3 . . . dxn

44. Part 2 — Estimates ||| |||| ||||
Goal: Lp estimates
|Λ(F1, F2, . . . , Fk)| F1 Lp1 F2 Lp2 . . . Fk Lpk
in a nonempty open subrange of
1
p1
+
1
p2
+ . . . +
1
pk
= 1

45. Part 2 — Estimates ||| |||| ||||
Goal: Lp estimates
|Λ(F1, F2, . . . , Fk)| F1 Lp1 F2 Lp2 . . . Fk Lpk
in a nonempty open subrange of
1
p1
+
1
p2
+ . . . +
1
pk
= 1
Desired results: characterizations of Lp boundedness
T(1)-type theorems

46. Part 2 — Back to examples, rem. case of 2D BHT|||| |||| ||||
T(F, G)(x, y) =
R2
F(x − s, y) G(x, y − t) K(s, t) ds dt

47. Part 2 — Back to examples, rem. case of 2D BHT|||| |||| ||||
T(F, G)(x, y) =
R2
F(x − s, y) G(x, y − t) K(s, t) ds dt
Substitute u = x − s, v = y − t:
Λ(F, G, H) = T(F, G), H
=
R4
F(u, y)G(x, v)H(x, y)K(x − u, y − v) dudvdxdy

48. Part 2 — Back to examples, rem. case of 2D BHT|||| |||| ||||
T(F, G)(x, y) =
R2
F(x − s, y) G(x, y − t) K(s, t) ds dt
Substitute u = x − s, v = y − t:
Λ(F, G, H) = T(F, G), H
=
R4
F(u, y)G(x, v)H(x, y)K(x − u, y − v) dudvdxdy
Non-translation-invariant generalization:
Λ(F, G, H) =
R4
F(u, y)G(x, v)H(x, y)K(u, v, x, y) dudvdxdy

49. Part 2 — Back to examples, rem. case of 2D BHT|||| |||| ||||
Λ(F, G, H) =
R4
F(u, y)G(x, v)H(x, y)K(u, v, x, y) dudvdxdy
Graph associated with its structure:
x ◦
H
G
◦
F
y
v ◦ ◦ u

50. Part 2 — Back to examples, triangular HT |||| |||| ||||
T(F, G)(x, y) := p.v.
R
F(x + t, y)G(x, y + t)
dt
t

51. Part 2 — Back to examples, triangular HT |||| |||| ||||
T(F, G)(x, y) := p.v.
R
F(x + t, y)G(x, y + t)
dt
t
Substitute: z = −x − y − t,
F1(x, y) = H(x, y), F2(y, z) = F(−y −z, y), F3(z, x) = G(x, −x−z)
Λ(F, G, H) = T(F, G), H
=
R3
F1(x, y) F2(y, z) F3(z, x)
−1
x + y + z
dxdydz
We do not know how to proceed in this example

52. Part 2 — Back to examples, triangular HT |||| |||| ||||
Λ(F1, F2, F3) =
R3
F1(x, y) F2(y, z) F3(z, x)
−1
x + y + z
dxdydz
Associated graph:
x
◦
F3F1
y ◦
F2
◦ z
Note: this graph is not bipartite

53. Part 2 — A manageable modification | |||| |||| ||||
Quadrilinear variant:
Λ(F1, F2, F3, F4)
=
R4
F1(u, v)F2(u, y)F3(x, y)F4(x, v) K(u, v, x, y) dudvdxdy

54. Part 2 — A manageable modification | |||| |||| ||||
Quadrilinear variant:
Λ(F1, F2, F3, F4)
=
R4
F1(u, v)F2(u, y)F3(x, y)F4(x, v) K(u, v, x, y) dudvdxdy
Associated graph:
x ◦
F3
F4
◦
F2
y
v ◦
F1
◦ u

55. Part 2 — A manageable modification | |||| |||| ||||
Quadrilinear variant:
Λ(F1, F2, F3, F4)
=
R4
F1(u, v)F2(u, y)F3(x, y)F4(x, v) K(u, v, x, y) dudvdxdy
Associated graph:
x ◦
F3
F4
◦
F2
y
v ◦
F1
◦ u
x ◦
F3
F2
◦
F4
y
u ◦
F1
◦ v
Note: this graph is bipartite

56. Part 3 — Dyadic model operators || |||| |||| ||||
Scope of our techniques

57. Part 3 — Dyadic model operators || |||| |||| ||||
Scope of our techniques
We specialize to:
bipartite graphs
multilinear Calder´on-Zygmund kernels K
“perfect” dyadic models

58. Part 3 — Perfect dyadic conditions ||| |||| |||| ||||
m, n = positive integers
D := (x, . . . , x
m
, y, . . . , y
n
) : x, y ∈ R
the “diagonal” in Rm+n

59. Part 3 — Perfect dyadic conditions ||| |||| |||| ||||
m, n = positive integers
D := (x, . . . , x
m
, y, . . . , y
n
) : x, y ∈ R
the “diagonal” in Rm+n
Perfect dyadic Calder´on-Zygmund kernel K : Rm+n → C,
Auscher, Hofmann, Muscalu, Tao, Thiele (2002):
|K(x1, . . . , xm, y1, . . . , yn)|
i1<i2
|xi1 − xi2 | + j1<j2
|yj1 − yj2 |
2−m−n
K is constant on (m+n)-dimensional dyadic cubes disjoint
from D
K is bounded and compactly supported

60. Part 3 — Bipartite structure |||| |||| |||| ||||
E ⊆ {1, . . . , m}×{1, . . . , n}
G = simple bipartite undirected graph on
{x1, . . . , xm} and {y1, . . . , yn}
xi —yj ⇔ (i, j) ∈ E

61. Part 3 — Bipartite structure |||| |||| |||| ||||
E ⊆ {1, . . . , m}×{1, . . . , n}
G = simple bipartite undirected graph on
{x1, . . . , xm} and {y1, . . . , yn}
xi —yj ⇔ (i, j) ∈ E
|E|-linear singular form:
Λ (Fi,j )(i,j)∈E :=
Rm+n
K(x1, . . . , xm, y1, . . . , yn)
(i,j)∈E
Fi,j (xi , yj ) dx1 . . . dxmdy1 . . . dyn
Assume: there are no isolated vertices in G
avoids degeneracy

62. Part 3 — Adjoints |||| |||| |||| ||||
There are |E| mutually adjoint (|E|−1)-linear operators Tu,v ,
(u, v) ∈ E:
Λ (Fi,j )(i,j)∈E =
R2
Tu,v (Fi,j )(i,j)=(u,v) Fu,v

63. Part 3 — Adjoints |||| |||| |||| ||||
There are |E| mutually adjoint (|E|−1)-linear operators Tu,v ,
(u, v) ∈ E:
Λ (Fi,j )(i,j)∈E =
R2
Tu,v (Fi,j )(i,j)=(u,v) Fu,v
Explicitly:
Tu,v (Fi,j )(i,j)∈E{(u,v)} (xu, yv )
=
Rm+n−2
K(x1, . . . , xm, y1, . . . , yn)
(i,j)∈E{(u,v)}
Fi,j (xi , yj )
i=u
dxi
j=v
dyj

64. Part 3 — A T(1)-type theorem | |||| |||| |||| ||||
Theorem. “Entangled” T(1) — V. K. and C. Thiele (2013)
(a) For m, n ≥ 2 and a graph G there exist positive integers di,j
such that (i,j)∈E
1
di,j
> 1 and the following holds. If
|Λ(1Q, . . . , 1Q)| |Q|, Q dyadic square,
Tu,v (1R2 , . . . , 1R2 ) BMO(R2) 1, (u, v) ∈ E,
then
Λ (Fi,j )(i,j)∈E
(i,j)∈E
Fi,j L
pi,j (R2)
for exponents pi,j s.t. (i,j)∈E
1
pi,j
= 1, di,j < pi,j ≤ ∞.

65. Part 3 — A T(1)-type theorem | |||| |||| |||| ||||
Theorem. “Entangled” T(1) — V. K. and C. Thiele (2013)
(a) For m, n ≥ 2 and a graph G there exist positive integers di,j
such that (i,j)∈E
1
di,j
> 1 and the following holds. If
|Λ(1Q, . . . , 1Q)| |Q|, Q dyadic square,
Tu,v (1R2 , . . . , 1R2 ) BMO(R2) 1, (u, v) ∈ E,
then
Λ (Fi,j )(i,j)∈E
(i,j)∈E
Fi,j L
pi,j (R2)
for exponents pi,j s.t. (i,j)∈E
1
pi,j
= 1, di,j < pi,j ≤ ∞.
(b) Conversely, the estimate for some choice of exponents implies
the conditions.

66. Part 3 — A T(1)-type theorem, reformulation|| |||| |||| |||| ||||
Theorem. “Entangled” T(1) — V. K. and C. Thiele (2013)
For m, n ≥ 2 and a graph G there exist positive integers di,j such
that (i,j)∈E
1
di,j
> 1 and the following holds. If
Tu,v (1Q, . . . , 1Q) L1
(Q)
|Q|, Q dyadic square, (u, v) ∈ E,
then
Λ (Fi,j )(i,j)∈E
(i,j)∈E
Fi,j L
pi,j (R2)
for exponents pi,j s.t. (i,j)∈E
1
pi,j
= 1, di,j < pi,j ≤ ∞.

67. Part 3 — Proof outline ||| |||| |||| |||| ||||
The only nonstandard part — sufficiency of the testing conditions

68. Part 3 — Proof outline ||| |||| |||| |||| ||||
The only nonstandard part — sufficiency of the testing conditions
Scheme of the proof:

69. Part 3 — Proof outline ||| |||| |||| |||| ||||
The only nonstandard part — sufficiency of the testing conditions
Scheme of the proof:
decomposition into paraproducts

70. Part 3 — Proof outline ||| |||| |||| |||| ||||
The only nonstandard part — sufficiency of the testing conditions
Scheme of the proof:
decomposition into paraproducts
a stopping time argument for reducing global estimates to
local estimates

71. Part 3 — Proof outline ||| |||| |||| |||| ||||
The only nonstandard part — sufficiency of the testing conditions
Scheme of the proof:
decomposition into paraproducts
a stopping time argument for reducing global estimates to
local estimates
cancellative paraproducts with ∞ coefficients
“most” cases of graphs G
di,j related to sizes of connected components of G
stuctural induction + Bellman function technique
exceptional cases of graphs G

72. Part 3 — Proof outline ||| |||| |||| |||| ||||
The only nonstandard part — sufficiency of the testing conditions
Scheme of the proof:
decomposition into paraproducts
a stopping time argument for reducing global estimates to
local estimates
cancellative paraproducts with ∞ coefficients
“most” cases of graphs G
di,j related to sizes of connected components of G
stuctural induction + Bellman function technique
exceptional cases of graphs G
non-cancellative paraproducts with BMO coefficients
reduction to cancellative paraproducts

73. Part 3 — Proof outline ||| |||| |||| |||| ||||
The only nonstandard part — sufficiency of the testing conditions
Scheme of the proof:
decomposition into paraproducts
a stopping time argument for reducing global estimates to
local estimates
cancellative paraproducts with ∞ coefficients
“most” cases of graphs G
di,j related to sizes of connected components of G
stuctural induction + Bellman function technique
exceptional cases of graphs G
non-cancellative paraproducts with BMO coefficients
reduction to cancellative paraproducts
counterexample for m = 1 or n = 1

74. Part 3 — Multilinear Bellman functions |||| |||| |||| |||| ||||
Bellman functions in harmonic analysis
Invented by Burkholder (1980s)
Developed by Nazarov, Treil, Volberg, etc. (1990s)
We only keep the “induction on scales” idea

75. Part 3 — Multilinear Bellman functions |||| |||| |||| |||| ||||
Bellman functions in harmonic analysis
Invented by Burkholder (1980s)
Developed by Nazarov, Treil, Volberg, etc. (1990s)
We only keep the “induction on scales” idea
A broad class of interesting dyadic objects can be reduced to
bounding expressions of the form
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

76. Part 3 — Calculus of finite differences |||| |||| |||| |||| ||||
B = BQ(F1, . . . , F )
First order difference of B: B = BQ(F1, . . . , F )
BI×J := 1
4BIleft×Jleft
+ 1
4BIleft×Jright
+ 1
4BIright×Jleft
+ 1
4BIright×Jright
− BI×J

77. Part 3 — Calculus of finite differences |||| |||| |||| |||| ||||
B = BQ(F1, . . . , F )
First order difference of B: B = BQ(F1, . . . , F )
BI×J := 1
4BIleft×Jleft
+ 1
4BIleft×Jright
+ 1
4BIright×Jleft
+ 1
4BIright×Jright
− BI×J
Suppose: |A| ≤ B, i.e.
|AQ(F1, . . . , F )| ≤ BQ(F1, . . . , F )
for all Q ∈ T and nonnegative bounded measurable F1, . . . , F

78. Part 3 — Calculus of finite differences |||| |||| |||| |||| ||||
|AQ(F1, . . . , F )| ≤ BQ(F1, . . . , F )
|Q| |AQ(F1, . . . , F )| ≤
Q is a child of Q
|Q| BQ
(F1, . . . , F )
− |Q| BQ(F1, . . . , F )

79. Part 3 — Calculus of finite differences |||| |||| |||| |||| ||||
|AQ(F1, . . . , F )| ≤ BQ(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 = a Bellman function for ΛT

80. Part 4 — Ordinary paraproduct | |||| |||| |||| |||| ||||
Dyadic version
Td(f , g) :=
k∈Z
(Ekf )(∆kg)
Ekf := |I|=2−k
1
|I| I f 1I , ∆kg := Ek+1g − Ekg

81. Part 4 — Ordinary paraproduct | |||| |||| |||| |||| ||||
Dyadic version
Td(f , g) :=
k∈Z
(Ekf )(∆kg)
Ekf := |I|=2−k
1
|I| I f 1I , ∆kg := Ek+1g − Ekg
Continuous version
Tc(f , g) :=
k∈Z
(Pϕk
f )(Pψk
g)
Pϕk
f := f ∗ ϕk, Pψk
g := g ∗ ψk
ϕ, ψ Schwartz, supp( ˆψ) ⊆ {ξ ∈ R : 1
2 ≤|ξ| ≤ 2}
ϕk(t) := 2kϕ(2kt), ψk(t) := 2kψ(2kt)

82. Part 4 — Twisted paraproduct || |||| |||| |||| |||| ||||
Dyadic version
Td(F, G) :=
k∈Z
(E
(1)
k F)(∆
(2)
k G)
E
(1)
k martingale averages in the 1st variable
∆
(2)
k martingale differences in the 2nd variable

83. Part 4 — Twisted paraproduct || |||| |||| |||| |||| ||||
Dyadic version
Td(F, G) :=
k∈Z
(E
(1)
k F)(∆
(2)
k G)
E
(1)
k martingale averages in the 1st variable
∆
(2)
k martingale differences in the 2nd variable
Continuous version
Tc(F, G) :=
k∈Z
(P(1)
ϕk
F)(P
(2)
ψk
G)
P
(1)
ϕk , P
(2)
ψk
L-P projections in the 1st and the 2nd variable
(P
(1)
ϕk F)(x, y) := R F(x−t, y)ϕk(t)dt
(P
(2)
ψk
G)(x, y) := R G(x, y −t)ψk(t)dt

84. Part 4 — Twisted paraproduct || |||| |||| |||| |||| ||||
Dyadic version
Td(F, G) :=
k∈Z
(E
(1)
k F)(∆
(2)
k G)
Continuous version
Tc(F, G) :=
k∈Z
(P(1)
ϕk
F)(P
(2)
ψk
G)
Bilinear multipliers from our theorems reduce to these
using cone decomposition of the symbol:
m =
j
m[j]
from the Fourier series
m[j]
(ξ1, η2) =
k∈Z
ϕ
[j]
k (ξ1) ψ
[j]
k (η2)

85. Part 4 — Twisted paraproduct, estimates ||| |||| |||| |||| |||| ||||
B( ), _
2
1 C( )_
2
1 , _
2
1
1
2
_,1
4
_ )(E
D( )_
2
1 ,
0
0,1
2
_ )(A
_
4
1
_1
q
p
1_
1
0
10
the shaded region – the
strong estimate
two solid sides of the square
– the weak estimate
two dashed sides of the
square – no estimates
the white region –
unresolved

86. Part 4 — Proof outline |||| |||| |||| |||| |||| ||||
B( ), _
2
1 C( )_
2
1 , _
2
1
1
2
_,1
4
_ )(E
D( )_
2
1 ,
0
0,1
2
_ )(A
_
4
1
_1
q
p
1_
1
0
10
Dyadic version Td
ABC – a very special case
of the technique in Part 3
the rest of the shaded region
– conditional proof,
F. Bernicot (2010)
dashed segments –
counterexamples
D, E – an alternative purely
Bellman function proof
Continuous version Tc
transition using the
Jones-Seeger-Wright square
function

87. Part 4 — Transition to cont. version |||| |||| |||| |||| |||| ||||
Assume: ψk = φk+1 − φk for some φ Schwartz, R φ = 1
The general case is then obtained by composing with a bounded
Fourier multiplier in the second variable

88. Part 4 — Transition to cont. version |||| |||| |||| |||| |||| ||||
Assume: ψk = φk+1 − φk for some φ Schwartz, R φ = 1
The general case is then obtained by composing with a bounded
Fourier multiplier in the second variable
A. Calder´on (1960s), R. L. Jones, A. Seeger and J. Wright (2008)
If ϕ is Schwartz and R ϕ = 1, then the square function
SF :=
k∈Z
Pϕk
F − EkF
2 1/2
satisfies SF Lp
(R) p F Lp
(R)
for 1 < p < ∞.

89. Part 4 — Transition to cont. version |||| |||| |||| |||| |||| ||||
Assume: ψk = φk+1 − φk for some φ Schwartz, R φ = 1
The general case is then obtained by composing with a bounded
Fourier multiplier in the second variable
A. Calder´on (1960s), R. L. Jones, A. Seeger and J. Wright (2008)
If ϕ is Schwartz and R ϕ = 1, then the square function
SF :=
k∈Z
Pϕk
F − EkF
2 1/2
satisfies SF Lp
(R) p F Lp
(R)
for 1 < p < ∞.
Proposition
Tc(F, G) − Td(F, G) Lpq/(p+q) p,q F Lp G Lq

90. Part 4 — “Entangled” + cont. kernel | |||| |||| |||| |||| |||| ||||
General bipartite graphs G
How to obtain boundedness of
Λ (Fi,j )(i,j)∈E :=
Rm+n
K(x1, . . . , xm, y1, . . . , yn)
(i,j)∈E
Fi,j (xi , yj ) dx1 . . . dxmdy1 . . . dyn
at least for some continuous singular kernels K?

91. Part 4 — “Entangled” + cont. kernel | |||| |||| |||| |||| |||| ||||
General bipartite graphs G
How to obtain boundedness of
Λ (Fi,j )(i,j)∈E :=
Rm+n
K(x1, . . . , xm, y1, . . . , yn)
(i,j)∈E
Fi,j (xi , yj ) dx1 . . . dxmdy1 . . . dyn
at least for some continuous singular kernels K?
We can average “entangled” dyadic operators from Part 3 over
translated, dilated, and rotated dyadic grids
Partial results: One can recover some very special kernels K
Possibly all sufficiently smooth translation-invariant kernels
This is still far from a complete T(1)-type theorem

92. Currently open problems || |||| |||| |||| |||| |||| ||||
Further directions:

93. Currently open problems || |||| |||| |||| |||| |||| ||||
Further directions:
Translating the results to the case of more general continuous
C-Z kernels K
Ultimately obtaining a “real” (i.e. non-dyadic) T(1)-type
theorem

94. Currently open problems || |||| |||| |||| |||| |||| ||||
Further directions:
Translating the results to the case of more general continuous
C-Z kernels K
Ultimately obtaining a “real” (i.e. non-dyadic) T(1)-type
theorem
Forms corresponding to non-bipartite graphs (such as odd
cycles, recall a triangle)

95. Currently open problems || |||| |||| |||| |||| |||| ||||
Further directions:
Translating the results to the case of more general continuous
C-Z kernels K
Ultimately obtaining a “real” (i.e. non-dyadic) T(1)-type
theorem
Forms corresponding to non-bipartite graphs (such as odd
cycles, recall a triangle)
More singular kernels K, like K(x, y, z) = 1
x+y+z

96. Thank you! ||| |||| |||| |||| |||| |||| ||||
Thank you!