Carleson’s Theorem,
Variations and Applications
Christoph Thiele
Santander, September 2014
Lennart Carleson
• Born 1928
• Real/complex
Analysis, PDE,
Dynamical systems
• Convergence of
Fourier series 1968
• Abel Prize 2006
Quote from Abel Prize
“The proof of this result is so difficult that for over
thirty years it stood mostly isolated from the rest
of harmonic analysis. It is only within the past
decade that mathematicians have understood the
general theory of operators into which this
theorem fits and have started to use his powerful
ideas in their own work.”
Carleson’s Operator
Closely related maximal operator

C  f ( x )  sup


2  ix 
fˆ (  ) e
d

Carleson-Hunt theorem (1966/1968):
C f
p
 cp f
1 p  
p
Can be thought as stepping stone to Carl. Thm
Other forms of Carleson operator

C* f ( x ) 

2  ix 
fˆ (  )e
d

L  ( )

it 
C˜ * f ( x )  p.v .  f ( x  t )e dt / t



L ( )

C f ( x ) 


ˆf (  )e 2  ix  d 
 (x )

it  ( x )
C˜  f ( x )  p.v .  f ( x  t )e
dt / t

Quadratic Carleson operator

Qf ( x )  sup
 ,
p .v .  f ( x  t ) e

Victor Lie’s result, 1<p<2
Qf
p
 const
p
f
p
it   i  t
2
dt / t
Directional Hilbert transform
In the plane:
H u f ( x )  p.v .  f ( x  t, y  ut ) dt / t
Rotate so that u=0, apply HT in first variable
and Fubini:
Hu f
p
 Cp f
p
Alternative description
1)Take the Fourier transform of f
2)Multiply by a certain function constant on
half planes determined by (1,u)
3)Take the inverse Fourier transform

Maximal directional Hilbert t.
In the plane:
sup
u
p.v .  f ( x  t, y  ut )dt / t
Turns out unbounded
Nikodym set example
Set E of null measure containing for each
(x,y) a line punctured at (x,y). If vector field
points in direction of this line then averages
of characteristic fct of set along vf are one.
“Half” max directional HT
Unbounded:
Bounded:
sup u H u f ( x, y )
sup
H u f ( x, y )
u
p
L (y )
p
L (y)
p
C
 p f
p
p
 Cp f
p
L (x )
L (x)

Bounded (3/2<p<infty)(Bateman, T. 2012)

sup
u
H u f ( x, y )
p
L (y)
p
L (x)
 Cp f
p
Direct.HT w.r.t Vector Field
H u f ( x )  p.v .  f ( x  t, y  u( x, y )t )dt / t
BT case: One Variable V.F
p.v .  f ( x  t, y  u( x ) t ) dt / t
R
L2:Coifman’s argument

f ( x  t, y  u( x ) t ) dt / t
2
R

e
R



R
iy 
L ( x ,y )

ˆf ( x  t,  )e iu( x )t dt / t d 
2
R
L ( x ,y )
iu( x )t 
fˆ ( x  t,  )e
dt / t

L ( x , )
2
fˆ ( x ,  )
L ( x , )
2
 f
2
Coifman’s argument visualized
A Littlewood Paley band
For Lp theory need Littlewood Paley instead FT.
Idea of Lacey and Li: Generalization of Carleson
Further generalization
Vector field constant along suitable family of
Lispchitz curves (tangents nearly
vertical,vector field nearly horizontal)
Shaming Guo 2014: HT bounded in L2/Lp
Lipschitz conjecture
Conjecture: The truncated Hilbert transform
(integral from -1 to 1) along (two variable) vector
field is bounded in L2 provided the vector field is
Lipschitz with small enough constant
Only known for real analytic vector fields.
Christ,Nagel,Stein,Wainger 99, Stein/Street 2013
Triangular Hilbert transform

T ( f , g )( x , y )  p . v .  f ( x  t , y ) g ( x , y  t ) dt / t

All non-degenerate triangles equivalent
Triangular Hilbert transform
Open problem: Do any bounds of type
T ( f ,g)
pq /( p  q )
 const . f
p
g
q
hold? (exponents as in Hölder’s inequality)
Symmetric dual trilinear form
 ( f , g, h ) 


p.v . 
r r
r r
r r
f ( x  1 t) g( x  2 t) h ( x  3 t )dt / t dx 1 dx 2

All non-degenerate triangles equivalent by
linear transformation. No parameters.
 ( f , g, h )  p.v .  f ( x, y )g( y, z) h ( z, x )
1
xyz
dxdydz
Stronger than Carleson:

p . v .  f ( x  t , y ) g ( x , y  t ) dt / t

Specify
f ( x, y )  f ( x)
g ( x, y )  e
 2  iN ( x ) y
Degenerate triangles
Bilinear Hilbert transform (one dimensional)

B ( f , g )( x )  p . v .  f ( x  t ) g ( x  at ) dt / t

Satisfies Hölder bounds. (Lacey, T. 96/99)
Uniform in a. (T. , Li, Grafakos, Oberlin)
Vjeko Kovac’s Twisted
Paraproduct (2010)
 
p .v . 

f ( x  s , y ) g ( x , y  t ) K ( s , t ) dtds

Satisfies Hölder type bounds. K is a Calderon
Zygmund kernel, that is 2D analogue of 1/t.
Weaker than triangular Hilbert transform.
Variation Norm
N
|| f ||V r 
sup
N , x 0 , x1 ,..., x N
(  | f ( x n )  f ( x n 1 ) | )
r 1/ r
n 1
sup f ( x )  f
x
V
r
Variation Norm Carleson

CV r f ( x ) 

2  ix 
fˆ (  ) e
d

V ( )
r
Oberlin, Seeger, Tao, T. Wright, ’09: If r>2,
CV r f
C f
2
2
Quantitative convergence of Fourier series.
Multiplier Norm
M q - norm of a function m is the operator norm
of its Fourier multiplier operator acting on
g F
M
2
1
( m  Fg )
- norm is the same as supremum norm
m
M2
 m

 sup m ( )

q
L (R)
Coifman, Rubio de Francia,
Semmes
Variation norm controls multiplier norm
m
Provided
M
C m
p
V
r
|1 / 2  1 / p | 1 / r
Hence V r -Carleson implies M p - Carleson
Maximal Multiplier Norm

p
-norm of a family m  of functions is the
operator norm of the maximal operator on
M
g  sup

F
1
( m   Fg )
No easy alternative description for
M

2
p
L (R)
Truncated Carleson Operator
C  f ( x )  sup 

it 
f ( x  t ) e dt / t
[   , ]
c
M

2
-Carleson operator

C M * f ( x )  ||
2
it 
f ( x  t ) e dt / t || M * ( )
2
[   , ]
c
Theorem: (Demeter,Lacey,Tao,T. ’07) If 1<p<2
CM * f
2
p
 cp f
Conjectured extension to M

q
.
p
Birkhoff’s Ergodic Theorem
X: probability space (measure space of mass 1).
T: measure preserving transformation on X.
2
f: measurable function on X (say in L ( X ) ).
Then
lim
N 
1
N
N

n 1
exists for almost every x .
n
f (T x )
Harmonic analysis with  .
Compare
With max. operator
1
lim
N 
N
sup
With Hardy Littlewood
N

n
f (T x )
n 1
1
N
N

sup

With Lebesgue Differentiation
1

N

n
f (T x )
n 1


f ( x  t ) dt
0
lim
0
1



0
f ( x  t ) dt
Weighted Birkhoff
A weight sequence a n is called “good” if
2
f

L
(X )
weighted Birkhoff holds: For all X,T,
lim
1
N 
N
N
a
n
n
n 1
exists for almost every x.
f (T x )
Return Times Theorem
Bourgain (88)
Y: probability space
S: measure preserving transformation on Y.
2
g: measurable function on Y (say in L (Y ) ).
Then
n
an  g (S x )
Is a good sequence for almost every x .
Return Times Theorem
After transfer to harmonic analysis and one
partial Fourier transform, this can be
essentially reduced to * Carleson
M2
Extended to
g  L (Y )
p
Further extension
, 1<p<2 by D.L.T.T,
f  L (X )
q
by Demeter 09,
1/ p  1/ p  3/ 2
Two commuting transformations
X: probability space
T,S: commuting measure preserving
transformations on X
f.g: measurable functions on X (say in L2 ( X ) ).
Open question: Does
lim
N 
1
N
N

n
n
f (T x ) g ( S x )
n 1
exist for almost every x ? (Yes for S  T a .)
Nonlinear theory
Exponentiate Fourier integrals
 y
 2  ix 
g  ( y )  exp   f ( x ) e
dx

 
g ' ( y )  f ( x ) e
g  (  )  1
 2  ix 




g ( y )
g  (  )  exp( fˆ (  ))
Non-commutative theory
The same matrix valued…

0
G ' ( y )  
 f ( x ) e 2  ix 

1
G  (  )  
0
0

1
f ( x )e
0
 2  ix 

G  ( y )



G  (  )  f ( )
Communities talking NLFT
• (One dimensional) Scattering theory
• Integrable systems, KdV, NLS, inverse
scattering method.
• Riemann-Hilbert problems
• Orthogonal polynomials
• Schur algorithm
• Random matrix theory
Classical facts Fourier transform
Plancherel
fˆ
 f
2
2
Hausdorff-Young
fˆ
 f
p'
p
Riemann-Lebesgue
fˆ

 f
1
1  p  2 , p '  p /( p  1)
Analogues of classical facts
Nonlinear Plancherel (a = first entry of G)
log | a  (  ) |
L ( )
2
c f
2
Nonlinear Hausdorff-Young (Christ-Kiselev
‘99, alternative proof OSTTW ‘10)
log | a  (  ) |
L
p'
( )
 cp f
1 p  2
p
Nonlinear Riemann-Lebesgue (Gronwall)
log | a  (  ) |

L ( )
c f
1
Conjectured analogues
Nonlinear Carleson
sup
 c f 2
log | a  ( y ) |
y
L ( )
2
Uniform nonlinear Hausdorff Young
log | a (  ) |
 c f
p'
p
1 p  2
THANK YOU!