carleson’s theorem, variations and applications christoph thiele kiel, 2010
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Carleson’s Theorem,Variations and Applications
Christoph Thiele
Kiel, 2010
x
• Translation in horizontal direction
• Dilation
• Rotation by 90 degrees
• Translation in vertical direction
)()( yxfxfTy
2/1/)/()( xfxfD
dxexff ix 2)()(ˆ
ixexfxfM 2)()(
Carleson Operator
( identity op, Cauchy projection)
Translation/Dilation/Modulation symmetry.
Carleson-Hunt theorem (1966/1968):
defxfC ix2)(ˆsup)(
pppfcfC
p1
0
Multiplier Norm
- norm of a function f is the operator normof its Fourier multiplier operator acting on
- norm is the same as supremum norm
qM
)(1 FgfFg
)(RLq
)(sup2
fff
M
2M
-Carleson operator
Theorem: (Oberlin, Seeger, Tao, T. Wright ’10)
provided
qM
)(
2 ||)(ˆ||)(
qq M
xi
MdefxfC
pqpp
MfcfC
q,
qp /1|2/1/1|
Redefine Carleson Operator
Truncated Carleson operator
tdtetxfvpxCf txi /)(..sup)( )(
tdtetxfxfCc
txi /)(sup)(],[
)(
Truncated Carleson as average
tdttetxfR
txi /)/()(sup )(
tdtdeetxfR R
titxi /)(ˆ)(sup )(
dtdteetxf
R R
titxi /)(sup)(ˆ )(
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 FgfFg
)(RLp
pM f
2M
-Carleson operator
Theorem: (Demeter,Lacey,Tao,T. ’07)
Conjectured extension to , range of p ?
Non-singular variant with by Demeter 09’.
2M
)(],[
)(*2
*2
||/)(||)(
M
txi
MtdtetxfxfC
c
pppM
fcfC *2
pMpM
Birkhoff’s Ergodic Theorem
X: probability space (measure space of mass 1).
T: measure preserving transformation on X.
: measurable function on X (say in ).
Then
exists for almost every x .
)(2 XL
)(1
lim1
xTfN
N
n
n
N
f
Harmonic analysis with …
Compare
With max. operator
With Hardy Littlewood
With Lebesgue Differentiation
…and no Schwartz functions
)(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 the
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, . Then is good for almost every y.
Extended to , 1<p<2 by Demeter, Lacey,Tao,T. Transfer to harmonic analysis, take Fourier transform in f, recognize .
)(2 YLg)( ySga n
n
)(YLg p
*2M
C
Hilbert Transform / Vector Fields
Lipshitz,
Stein conjecture:
Also of interest are a) values other than p=2, b) maximal operator along vector field (Zygmund
conjecture) or maximal truncated singular integral
1
1
/))(()( tdttxvxfxfHv
22: RRv yxCyvxv )()(
22fCfH vv
Coifman: VF depends on 1 vrbl
Other values of p: Lacey-Li/ Bateman
Open: range of p near 1, maximal operator
),(2
/))(,(yxLR
tdttxvytxf
),(
)(
2
/),(ˆ
yxLR
txiv
R
iy dtdtetxfe
2),(),(
)(2
2
),(ˆ/),(ˆ fxftdtetxfxL
xLR
txiv
Application of -Carleson (C. Demeter)
Vector field v depends on one variable and f
is an elementary tensor f(x,y)=a(x)b(y), then
in an open range of p around 2.
pM
ppv fCxfH )(
),(
)( /)()(ˆ
yxLR
txiv
R
iy
P
dtdtetxabe
Application of Carleson
Maximal truncation of HT along vectorfield
Under same assumptions as before
Carried out for Hardy Littlewood maximal operatoralong vector field by Demeter.
*pM
ppv fCxfH )(*
tdttxvxfxfHv /))((sup)(
1
*
Variation Norm
rrnn
N
nxxxNV
xfxffN
r/1
11,...,,,
)|)()(|(sup||||10
Another strengthening of supremum norm
Variation Norm Carleson
Thm. (Oberlin, Seeger, Tao, T. Wright, ‘09)
)(
2)(ˆ)(
r
r
V
ix
VdefxfC
ppVfCfC r
,1 p )',2max( pr
Rubio de Francia’s inequality
Rubio de Francia’s square function, p>2,
Variational Carleson, p>2
p
xL
ixN
nNfCdef
p
n
nN
)(
)2/(122
1,...,,,)|)(ˆ|(sup
110
p
xL
ixN
nNfCdef
p
n
nN
)(
2/122
1,...,,,)|)(ˆ|(sup
110
Coifman, R.d.F, Semmes
Application of Rubio de Francia’s inequality:Variation norm controls multiplier norm
Provided
Hence variational Carleson implies - Carleson
rp VM
mCm
rp /1|/12/1|
pM
Nonlinear theory
Fourier sums as products (via exponential fct)
dxexfygy
ix
2)(exp)(
)()()(' 2 ygexfyg ix
1)( g
))(ˆexp()( fg
Non-commutative theory
Nonlinear Fourier transform, other choices of matrices lead to other models, AKNS systems
)(0)(
)(0)('
2
2
yGexf
exfyG
ix
ix
10
01)(G
)()( fG
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 factsNonlinear Plancherel (a = first entry of G)
Nonlinear Hausdorff-Young (Christ-Kiselev)
Nonlinear Riemann-Lebesgue (Gronwall)
2)(2|)(|log fcaL
ppL
fcap
)('
|)(|log 21 p
1)(|)(|log fcaL
Conjectured analogues
Nonlinear Carleson
Uniform nonlinear Hausdorff Young
Both OK in Walsh case, WNLUHY by Vjeko Kovac
2)(2
|)(|logsup
fcyaLy
ppfca
'|)(|log 21 p
Picard iteration, exp series
Scalar case: symmetrize, integrate over cubes
idGxGxMxG )(),()()('
xy
dyyMidxGidxG )()(,)( 21
...)(!2
1)(1)(
2
xyxy
dyyMdyyMxG
...)()()()(12
1221 xyyxy
dydyyMyMdyyMidxG
Terry Lyons’ theory
Etc. … If for one value of r>1 one controls all with n<r, then bounds for n>r follow automatically as well as a bound for the series.
rr
xyx
N
nxxxNr dyymV
nnN
/1
1,...,,,1, )|)(|(sup
110
rr
xyyx
N
nxxxNr dydyymymV
nnN
/22/1221
1,...,,,2, )|)()(|(sup
12110
nrV ,
Lyons for AKNS, r<2, n=1
For 1<p<2 we obtain by interpolation between a trivial estimate ( ) and variational Carleson ( )
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
pp
L
pp
xyx
iyN
nxxxNfcdyeyf
pnnN
)(
)/(12
1,...,,,'1
10
)|)(|(sup
2L1L
22 r
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:
Appears to work fine when . This puts an algebraic condition on AKNS which unfortunately is violated by NLFT as introduced above.
rr
xyyx
yyiN
nxxxNdyeyfyf
nnN
/22/)(221
1,...,,,)|)()(|(sup
121
2211
10
021
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