Carleson’s Theorem, Variations and Applications Christoph Thiele Colloquium, Amsterdam, 2011.
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Transcript of Carleson’s Theorem, Variations and Applications Christoph Thiele Colloquium, Amsterdam, 2011.
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
Nn
inxn
Nefxf 2ˆlim)(
1
0
2)(ˆ dxexff inxn
Hilbert space methods
The Functions with form an
orthonormal basis of a Hilbert space with
inner product
inxe 2
n
nn gfdxxgxfgf ˆˆ)()(,1
0
Zn
Carleson’s theorem
For f continuous or piecewise continuous,
converges to f(x) for almost every x in [0,1] .
N
Nn
inxn
Nef 2ˆlim
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
defxfC ix2)(ˆsup)(
dxexff ix
2)()(ˆ
)(
2)(ˆ)(x
ix defxfC
Carleson-Hunt Theorem
Carleson 1966, Hunt 1968 (1<p):
Carleson operator is bounded in .
pppfconstfC
dxxffpp
p
)(:
pL
Cauchy projection
An orthogonal projection, hence a bounded operator in Hilbert space .
0
2)(ˆ)( defxCf ix
2L
Symmetries
• Translation
• Dilation
)()( yxfxfTy
)/()( xfxfD
Invariance of Cauchy projection
Cauchy projection and identity operator span
the unique two dimensional space of linear
operator with these symmetries.
CDCDCTCT yy ,
Other operators in this space
• Hilbert transform
• Operator mapping real to imaginary part of functions on the real line with holomorphic extension to upper half plane.
tdttxfvpxHf /)(..)(
Wavelets
From a carefully chosen generating function
with integral zero generate the discrete
(n,k integers) collection
Can be orthonormal basis.
nkn TD k2,
Wavelets
Properties of wavelets prove boundedness of
Cauchy projection not only in Hilbert space
but in Banach space .
They encode much of singular integral theory.
For effective computations, choice of
generating function is an art.
pL
2L
Modulation
Amounts to translation in Fourier space
ixexfxfM 2)()(
fTf ˆˆ
Modulated Cauchy projection
Carleson’s operator has translation, dilation,
and modulation symmetry. Larger symmetry
group than Cauchy projection (sublinear op.).
def ix2)(ˆ
tdtetxfvp it /)(..
Wave packets
From a carefully chosen generating function
generate the collection (n,k,l integers)
Cannot be orthonormal basis.
nlkln TMD k2,,
Quadratic Carleson operator
Victor Lie’s result, 1<p<2
tdtetxfvpxQf tiit /)(..sup)(2
,
pppfconstQf
Vector Fields
Lipshitz,22: RRv yxcyvxv )()(
/
Hilbert Transform along Vector Fields
Stein conjecture:
(Real analytic vf: Christ,Nagel,Stein,Wainger 99)
1
1
/))((..)( tdttxvxfvpxfHv
22fCfH vv
Zygmund conjecture
Real analytic vector field: Bourgain (89)
22fCfM vv
/))((sup)(
10dttxvxfxfM v
One Variable Vector Field
R
tdttxvytxfvp /))(,(..
Coifman’s argument
),(2
/))(,(yxLR
tdttxvytxf
),(
)(
2
/),(ˆ
yxLR
txiv
R
iy dtdtetxfe
),(
)(
2
/),(ˆ
xLR
txiv tdtetxf 2),(2
),(ˆ fxfxL
Theorem with Michael Bateman
Measurable, one variable vector field
Prior work by Bateman, and Lacey,Li
pvpv fCfH
p2/3
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
mmm
M
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
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 .),(
)/(
Again 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 ),(),(),(..
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 fcaL
ppL
fcap
)('
|)(|log
21 p
1)(|)(|log fcaL
Conjectured analogues
Nonlinear Carleson
Uniform nonlinear Hausdorff Young
2)(2
|)(|logsup
fcyaLy
ppfca
'|)(|log 21 p
Couldn’t prove that….
But found a really interesting lemma.
THANK YOU!