Carleson’s Theorem,
Variations and Applications
Christoph Thiele
Colloquium, Amsterdam, 2011
Lennart Carleson
• Born 1928
• Real/complex
Analysis, PDE,
Dynamical systems
• Convergence of
Fourier series 1968
• Abel Prize 2006
Fourier Series
N
2inx
ˆ
f ( x )  lim  f n e
N 
1
n N
ˆf  f ( x )e  2inx dx
n

0
Hilbert space methods
2inx
The Functions e
with n  Z form an
orthonormal basis of a Hilbert space with
inner product
1
f , g   f ( x ) g ( x )dx 
0

ˆ gˆ
f
 n n
n  
Carleson’s theorem
For f continuous or piecewise continuous,
N
2inx
ˆ
lim  f n e
N 
n N
converges to f(x) for almost every x in [0,1] .
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 Operator

C f ( x )  sup



C f ( x ) 
ˆf ( ) e2ix d


ˆf ( ) e2ix d
( x)
fˆ ( ) 

 f ( x) e

 2ix
dx
Carleson-Hunt Theorem
Carleson 1966, Hunt 1968 (1<p):
C f
p
 const p f
p

f
:

p
p

p
f ( x ) dx

Carleson operator is bounded in
p
L
.
Cauchy projection

Cf ( x )  
ˆf ( ) e2ix d
0
An orthogonal projection, hence a bounded
2
operator in Hilbert space L .
Symmetries
• Translation
Ty f ( x)  f ( x  y)
• Dilation
D f ( x)  f ( x /  )
Invariance of Cauchy projection
TyC  CTy ,
DC  CD
Cauchy projection and identity operator span
the unique two dimensional space of linear
operator with these symmetries.
Other operators in this space
• Hilbert transform

Hf ( x )  p.v.  f ( x  t ) dt / t

• Operator mapping real to imaginary part of
functions on the real line with holomorphic
extension to upper half plane.
Wavelets
From a carefully chosen generating function
 with integral zero generate the discrete
(n,k integers) collection
n,k  D2 Tn
k
Can be orthonormal basis.
Wavelets
Properties of wavelets prove boundedness of
2
Cauchy projection not only in Hilbert space L
p
but in Banach space L .
They encode much of singular integral theory.
For effective computations, choice of
generating function is an art.
Modulation
2ix
M f ( x)  f ( x)e
Amounts to translation in Fourier space
fˆ  T fˆ
Modulated Cauchy projection


ˆf ( ) e 2ix d

p.v.  f ( x  t )e dt / t

it
Carleson’s operator has translation, dilation,
and modulation symmetry. Larger symmetry
group than Cauchy projection (sublinear op.).
Wave packets
From a carefully chosen generating function
 generate the collection (n,k,l integers)
n,l ,k  D2 M lTn
k
Cannot be orthonormal basis.
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
Vector Fields
v : R 2  R 2 Lipshitz,
v( x)  v( y)  c x  y
/
Hilbert Transform along
Vector Fields
1
H v f ( x )  p.v.  f ( x  v( x )t )dt / t
1
Stein conjecture:
Hv f

C
f
v
2
2
(Real analytic vf: Christ,Nagel,Stein,Wainger 99)
Zygmund conjecture

M v f ( x )  sup
0 1
Mv f
 f ( x  v( x)t )dt / 


C
f
v
2
2
Real analytic vector field: Bourgain (89)
One Variable Vector Field
p.v. f ( x  t , y  v ( x )t )dt / t
R
Coifman’s argument
 f ( x  t, y  v( x)t )dt / t
R

e 
iy
R


R
L2 ( x , y )
ˆf ( x  t , )eiv( x ) t dt / t d
L2 ( x , y )
R
fˆ ( x  t , )eiv( x ) t dt / t

L ( x , )
2
fˆ ( x, )
L ( x , )
2
 f
2
Theorem with Michael Bateman
Measurable, one variable vector field
3/ 2  p  
Hv f
p
 Cv f
p
Prior work by Bateman, and Lacey,Li
Variation Norm
|| f ||V r 
N
sup
( | f ( xn )  f ( xn 1 ) |r )1/ r
N , x0 , x1 ,...,x N n 1
sup f ( x )  f
x
Vr
Variation Norm Carleson

CV r f ( x ) 

fˆ ( ) e2ix d
V r ( )
Oberlin, Seeger, Tao, T. Wright, ’09: If r>2,
CV r f
2
C f
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 Lq (R)
1
g  F (m  Fg)
M 2 - norm is the same as supremum norm
m M  m   sup m( )
2

Coifman, Rubio de Francia,
Semmes
Variation norm controls multiplier norm
m
Provided
Mp
 C m Vr
|1/ 2  1/ p |  1/ r
Hence
Vr -Carleson implies M p - Carleson
Maximal Multiplier Norm

p
M -norm of a family m of functions is the
p
L
(R)
operator norm of the maximal operator on
1
g  sup F (m  Fg)
No easy alternative description for M 2
Truncated Carleson Operator
C f ( x )  sup
 f ( x  t )e
[   , ]c
it
dt / t
M

2
-Carleson operator

CM * f ( x ) ||
2
f ( x  t )eit dt / t ||M * ( )
2
[   , ]
c
Theorem: (Demeter,Lacey,Tao,T. ’07) If 1<p<2
CM * f
2
p
 cp f

M
Conjectured extension to
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
1
lim
N  N
N
 f (T
n 1
exists for almost every x .
n
x)
Harmonic analysis with  
.
Compare
With max. operator
With Hardy Littlewood
1
lim
N  N
N
n
f
(
T
x)

n 1
1
sup
N N
sup

N
n
f
(
T
 x)
1
n 1

f ( x  t )dt


0
1

f ( x  t )dt

With Lebesgue Differentiation lim
 0 
0
Weighted Birkhoff
A weight sequence an is called “good” if
2
f

L
(X )
weighted Birkhoff holds: For all X,T,
lim N 
1
N
N
a
n 1
n
n
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
p
g

L
(Y ) , 1<p<2 by D.L.T.T,
Extended to
q
f

L
( X ) by Demeter 09,
Further extension
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
1
lim
N  N
N
 f (T
n
n
x) g ( S x)
n 1
exist for almost every x ? (Yes for S  T a .)
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)
Again 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.
Nonlinear theory
Exponentiate Fourier integrals
y

 2ix
g ( y )  exp  f ( x )e
dx 
 

g' ( y)  f ( x)e2ix g ( y)
g ()  1
g ()  exp( fˆ ( ))
Non-commutative theory
The same matrix valued…

0
G' ( y )  
2ix
 f ( x )e
 1 0
G ( )  

 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
p
1 p  2
Nonlinear Riemann-Lebesgue (Gronwall)
log | a () |

L ( )
c f
1
Conjectured analogues
Nonlinear Carleson
 c f 2
sup log | a ( y ) |
y
L2 (  )
Uniform nonlinear Hausdorff Young
log | a( ) |
p'
c f
p
1 p  2
Couldn’t prove that….
But found a really interesting lemma.
THANK YOU!