carleson’s theorem, variations and applications christoph thiele santander, september 2014
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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
Carleson-Hunt theorem (1966/1968):
Can be thought as stepping stone to Carl. Thm
defxfC ix2)(ˆsup)(
€
C∗ fp
≤ c p fp
p1
Other forms of Carleson operator
€
˜ C * f (x) = p.v. f (x − t)e itη
−∞
∞
∫ dt / tL∞ (η )
€
˜ C η f (x) = p.v. f (x − t)e itη (x )
−∞
∞
∫ dt / t
€
Cη f (x) = ˆ f (ξ)e2πixξ dξη (x )
∞
∫
€
C* f (x) = ˆ f (ξ)e2πixξ dξη
∞
∫L∞(η )
Quadratic Carleson operator
Victor Lie’s result, 1<p<2
tdtetxfvpxQf tiit /)(..sup)(2
,
pppfconstQf
Directional Hilbert transform
In the plane:
Rotate so that u=0, apply HT in first variable and Fubini:
€
Hu f (x) = p.v. f (x + t,y + ut)dt / t∫
€
Hu fp
≤ Cp fp
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:
Turns out unbounded
€
supu p.v. f (x + t, y + ut)dt / t∫
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:
Bounded (3/2<p<infty)(Bateman, T. 2012)
€
supu Hu f (x, y)Lp (y ) Lp (x )
/ ≤ Cp fp
€
supu Hu f (x, y)Lp (y ) Lp (x )
≤ Cp fp
€
supu Hu f (x,y)Lp (y ) Lp (x )
≤ Cp fp
Direct.HT w.r.t Vector Field
€
Hu 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 / tR
∫
L2:Coifman’s argument
€
f (x + t,y + u(x)t)dt / tR
∫L2 (x,y )
€
= e iyη
R
∫ ˆ f (x + t,η)e iu(x )tη dt / t dηR
∫L2 (x,y )
€
= ˆ f (x + t,η)e iu(x )tη dt / tR
∫L2 (x,η )
2),(2),(ˆ fxf
xL
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
All non-degenerate triangles equivalent
tdttyxgytxfvpyxgfT /),(),(..),)(,(
Triangular Hilbert transform
Open problem: Do any bounds of type
hold? (exponents as in Hölder’s inequality)
qpqppqgfconstgfT .),(
)/(
Symmetric dual trilinear form
All non-degenerate triangles equivalent by linear transformation. No parameters.
€
Λ( f ,g,h) = ∫∫ p.v. f (r x +
r β 1t)g(
r x +
r β 2t)h(
r x +
r β 3t)dt / t
−∞
∞
∫ dx1dx2
€
Λ( f ,g,h) = p.v. f (x,y)g(y,z)h(z, x)1
x + y + zdxdydz∫∫∫
Stronger than Carleson:
Specify
tdttyxgytxfvp /),(),(..
)(),( xfyxf
yxiNeyxg )(2),(
Degenerate triangles
Bilinear Hilbert transform (one dimensional)
Satisfies Hölder bounds. (Lacey, T. 96/99)
Uniform in a. (T. , Li, Grafakos, Oberlin)
tdtatxgtxfvpxgfB /)()(..))(,(
Vjeko Kovac’s Twisted Paraproduct (2010)
Satisfies Hölder type bounds. K is a Calderon
Zygmund kernel, that is 2D analogue of 1/t.
Weaker than triangular Hilbert transform.
dtdstsKtyxgysxfvp ),(),(),(..
Variation Norm
rrnn
N
nxxxNV
xfxffN
r/1
11,...,,,
)|)()(|(sup||||10
rVx
fxf )(sup
Variation Norm Carleson
Oberlin, Seeger, Tao, T. Wright, ’09: If r>2,
Quantitative convergence of Fourier series.
)(
2)(ˆ)(
r
r
V
ix
VdefxfC
22fCfC rV
Multiplier Norm
- norm of a function m is the operator normof its Fourier multiplier operator acting on
- norm is the same as supremum norm
qM
)(1 FgmFg
)(RLq
)(sup2
mmmM
2M
Coifman, Rubio de Francia, Semmes
Variation norm controls multiplier norm
Provided
Hence -Carleson implies - Carleson
rp VM
mCm
rp /1|/12/1|
pMrV
Maximal Multiplier Norm
-norm of a family of functions is the
operator norm of the maximal operator on
No easy alternative description for
)(sup 1 FgmFg
)(RLp
pM m
2M
Truncated Carleson Operator
tdtetxfxfCc
it /)(sup)(],[
-Carleson operator
Theorem: (Demeter,Lacey,Tao,T. ’07) If 1<p<2
Conjectured extension to .
2M
)(],[
*2
*2
||/)(||)(
M
it
MtdtetxfxfC
c
pppM
fcfC *2
qM
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 ).
Then
exists for almost every x .
)(2 XL
)(1
lim1
xTfN
N
n
n
N
Harmonic analysis with .
Compare
With max. operator
With Hardy Littlewood
With Lebesgue Differentiation
)(1
lim1
xTfN
N
n
n
N
)(1
sup1
xTfN
N
n
n
N
00
)(1
lim dttxf
0
)(1
sup dttxf
Weighted Birkhoff
A weight sequence is called “good” if
weighted Birkhoff holds: For all X,T,
exists for almost every x.
na
)(1
lim1
xTfaN
nN
nnN
)(2 XLf
Return Times Theorem
Bourgain (88)
Y: probability space
S: measure preserving transformation on Y.
g: measurable function on Y (say in ).
Then
Is a good sequence for almost every x .
)(2 YL
)( xSga nn
Return Times Theorem
After transfer to harmonic analysis and one
partial Fourier transform, this can be
essentially reduced to Carleson
Extended to , 1<p<2 by D.L.T.T,
Further extension by Demeter 09,
2/3/1/1 pp
)(YLg p
*2M
)(XLf q
Two commuting transformations
X: probability space
T,S: commuting measure preserving transformations on X
f.g: measurable functions on X (say in ).
Open question: Does
exist for almost every x ? (Yes for .)
)(2 XL
)()(1
lim1
xSgxTfN
nN
n
n
N
aTS
Nonlinear theory
Exponentiate Fourier integrals
dxexfygy
ix
2)(exp)(
)()()(' 2 ygexfyg ix
1)( g ))(ˆexp()( fg
Non-commutative theory
The same matrix valued…
)(0)(
)(0)('
2
2
yGexf
exfyG
ix
ix
10
01)(G
)()( fG
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 transformPlancherel
Hausdorff-Young
Riemann-Lebesgue
22
ˆ ff
ppff
'
ˆ )1/(',21 pppp
1ˆ ff
Analogues of classical factsNonlinear Plancherel (a = first entry of G)
Nonlinear Hausdorff-Young (Christ-Kiselev ‘99, alternative proof OSTTW ‘10)
Nonlinear Riemann-Lebesgue (Gronwall)
2)(2|)(|log fca
L
ppL
fcap
)('
|)(|log
21 p
1)(|)(|log fca
L
Conjectured analogues
Nonlinear Carleson
Uniform nonlinear Hausdorff Young
2)(2
|)(|logsup
fcyaLy
ppfca
'|)(|log 21 p
THANK YOU!