Carleson’s Theorem,
Variations and Applications
Christoph Thiele
Kiel, 2010

x
• Translation in horizontal direction
Ty f ( x)  f ( x  y)
• Dilation
D f ( x)  f ( x /  ) / 
1/ 2
• Rotation by 90 degrees
fˆ ( ) 

 f ( x )e
2ix
dx

• Translation in vertical direction
2ix
M f ( x)  f ( x)e
Carleson Operator

2ix
ˆ
C f ( x )  sup  f ( )e d


(    identity op,   0 Cauchy projection)
Translation/Dilation/Modulation symmetry.
Carleson-Hunt theorem (1966/1968): 1  p  
C f
p
 cp f
p
Multiplier Norm
M q - norm of a function f is the operator norm
of its Fourier multiplier operator acting on Lq (R)
1
g  F ( f  Fg)
M 2 - norm is the same as supremum norm
f
M2
 f

 sup f ( )

M q -Carleson operator

CM f ( x ) ||  fˆ ( )e2ix d ||M
q

q ( )
Theorem: (Oberlin, Seeger, Tao, T. Wright ’10)
CM f
q
provided
p
 c p ,q f
p
| 1 / p  1 / 2 | 1 / q
Redefine Carleson Operator

Cf ( x )  sup p.v.  f ( x  t )e
i ( x t )
dt / t

Truncated Carleson operator
C f ( x )  sup

f ( x  t )e
[   , ]c
i ( x t )
dt / t
Truncated Carleson as average
sup  f ( x  t )e
i ( x t )
 (t /  ) dt / t
R
 sup   f ( x  t )e
i ( x t )
it
ˆ ( )e d dt / t
R R
  ˆ ( ) sup  f ( x  t )e
R
R
i ( x t ) it
e dt / t d
Maximal Multiplier Norm

p
M -norm of a family f of functions is the
p
L
(R)
operator norm of the maximal operator on
1
g  sup F ( f  Fg)
No easy alternative description for M 2
M

2
-Carleson operator
CM * f ( x ) ||
2

f ( x  t )ei ( x t ) dt / t ||M * ( )
2
[   , ]
c
Theorem: (Demeter,Lacey,Tao,T. ’07)
CM * f
2
p
 cp f
p
Conjectured extension to M p , range of p ?
Non-singular variant with M p by Demeter 09’.
Birkhoff’s Ergodic Theorem
X: probability space (measure space of mass 1).
T: measure preserving transformation on X.
f : measurable function on X (say in L2 ( X ) ).
Then
1
lim
N  N
N
 f (T
n 1
exists for almost every x .
n
x)
Harmonic analysis with    …
Compare
1
lim
N  N
With max. operator
N
n
f
(
T
x)

n 1
1
sup
N N
With Hardy Littlewood
sup

With Lebesgue Differentiation
1
N
n
f
(
T
 x)
n 1

f ( x  t )dt


0
lim
 0
…and no Schwartz functions
1

f ( x  t )dt


0
Weighted Birkhoff
A weight sequence an is called “good” if the
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
2
preserving transformation on Y, g  L (Y ) .
n
Then an  g ( S y ) is good for almost
every y.
Extended to g  Lp (Y ) , 1<p<2 by Demeter,
Lacey,Tao,T. Transfer to harmonic analysis,
take Fourier transform in f, recognize CM 2* .
Hilbert Transform / Vector Fields
v : R 2  R 2 Lipshitz,
v( x)  v( y)  C x  y
1
H v f ( x )   f ( x  v( x )t )dt / t
1
Stein conjecture:
Hv f

C
f
v
2
2
Also of interest are a) values other than p=2,
b) maximal operator along vector field (Zygmund
conjecture) or maximal truncated singular integral
Coifman: VF depends on 1 vrbl
 f ( x  t, y  v( x)t )dt / t
R

iy
ˆ ( x  t , )eiv( x ) t dt / t d
e
f
 
R


R
L2 ( x , y )
R
fˆ ( x  t , )eiv( x ) t dt / t
 fˆ ( x, )
L2 ( x , )
L2 ( x , y )
L2 ( x , )
 f
2
Other values of p: Lacey-Li/ Bateman
Open: range of p near 1, maximal operator
Application of M p -Carleson
(C. Demeter)
Vector field v depends on one variable and f
is an elementary tensor f(x,y)=a(x)b(y), then
H v f ( x)
p
C f
p
in an open range of p around 2.
iv ( x ) t
ˆ
 e b( ) a( x  t )e dt / t d
iy
R
R
LP ( x , y )
*
p
Application of M Carleson
Maximal truncation of HT along vectorfield
H f ( x)  C f
*
v
p
p
Under same assumptions as before

H v* f ( x )  sup  f ( x  v( x )t )dt / t
 1 
Carried out for Hardy Littlewood maximal operator
along vector field by Demeter.
Variation Norm
|| f ||V r 
N
sup
( | f ( xn )  f ( xn 1 ) |r )1/ r
N , x0 , x1 ,...,x N n 1
Another strengthening of supremum norm
Variation Norm Carleson

CV r f ( x ) 

ˆf ( )e2ix d
V r ( )
Thm. (Oberlin, Seeger, Tao, T. Wright, ‘09)
CV r f
1  p  ,
p
C f
p
r  max(2, p' )
Rubio de Francia’s inequality
Rubio de Francia’s square function, p>2,
N
sup
N ,0 ,1 ,..., N
n
(  |  fˆ ( )e2ix d |2 )1/ 2
n 1 n 1
C f
p
Lp ( x )
Variational Carleson, p>2
N
sup
n
(|  fˆ ( )e2ix d |2 )1/(2 )
N ,0 ,1 ,..., N n 1
n 1
C f
Lp ( x )
p
Coifman, R.d.F, Semmes
Application of Rubio de Francia’s inequality:
Variation norm controls multiplier norm
m
Mp
 C m Vr
Provided
| 1 / 2  1 / p | 1 / r
Hence variational Carleson implies
M p - Carleson
Nonlinear theory
Fourier sums as products (via exponential fct)
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

0
G' ( y )  
2ix
f
(
x
)
e

f ( x )e
0
2ix

G ( y )

 1 0
G ( )  

 0 1

G (  )  f ( )
Nonlinear Fourier transform, other choices of
matrices lead to other models, AKNS systems
Incarnations of NLFT
• (One dimensional) Scattering theory
• Integrable systems, KdV, NLS, inverse
scattering method.
• Riemann-Hilbert problems
• Orthogonal polynomials
• Schur algorithm
• Random matrix theory
Analogues of classical facts
Nonlinear Plancherel (a = first entry of G)
log | a () |
L ( )
2
c f
2
Nonlinear Hausdorff-Young (Christ-Kiselev)
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
Both OK in Walsh case, WNLUHY by Vjeko Kovac
Picard iteration, exp series
G' ( x)  M ( x)G( x),
G()  id
G1 ( x )  id , G2 ( x )  id 
 M ( y )dy
y x
G( x )  id 
 M ( y )dy   M ( y ) M ( y )dy dy  ...
1
y x
2
2
1
y2  y1  x
Scalar case: symmetrize, integrate over cubes
2

1 
  ...
G( x )  1   M ( y )dy 
M
(
y
)
dy
 

2
!
y x
 y x

Terry Lyons’ theory
Vr ,1 
Vr ,2 
sup
N
sup
( |
r 1/ r
m
(
y
)
dy
|
)

N , x0 , x1 ,...,xN n 1
xn 1  y  xn
N
( |
 m( y )m( y )dy dy |
N , x0 , x1 ,...,xN n 1
xn 1  y2  y1  xn
r /2 2/ r
1
2
2
1
)
Etc. …
If for one value of r>1 one controls
all Vr ,n with n<r, then bounds for n>r follow
automatically as well as a bound for the series.
Lyons for AKNS, r<2, n=1
For 1<p<2 we obtain by interpolation between a
trivial estimate ( L1 ) and variational Carleson ( L2 )
N
sup
(|

f ( y )e2iy dy |p  )1/( p )
N , x0 , x1 ,...,x N n 1
xn 1  y  xn
 cp f
Lp ' (  )
This implies nonlinear Hausdorff Young as well as
variational and maximal versions of nonlinear HY.
Barely fails to prove the nonlinear Carleson theorem
because cannot choose
2  r  2
p
Lyons for AKNS, 2<r<3, n=1,2
Now estimate for n=1 is fine by variational Carleson.
Work in progress with C.Muscalu and Yen Do:
N
sup
( |
 f ( y ) f ( y )e
N , x0 , x1 ,...,xN n 1
xn 1  y2  y1  xn
1
2
2i ( 1 y1 2 y2 )
r /2 2/ r
dy | )
Appears to work fine when 1  2  0. This puts an
algebraic condition on AKNS which unfortunately
is violated by NLFT as introduced above.