Symmetries in Analysis on R
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Transcript Symmetries in Analysis on R
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
2ix
dx
• Translation in vertical direction
2ix
M f ( x) f ( x)e
Carleson Operator
2ix
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 ( )e2ix 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 )
it
ˆ ( )e d dt / t
R R
ˆ ( ) sup f ( x t )e
R
R
i ( x t ) it
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 ( )e2ix 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 ( )e2ix d |2 )1/ 2
n 1 n 1
C f
p
Lp ( x )
Variational Carleson, p>2
N
sup
n
(| fˆ ( )e2ix 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
2ix
g ( y ) exp f ( x )e
dx
g' ( y) f ( x)e2ix g ( y)
g () 1
g () exp( fˆ ( ))
Non-commutative theory
0
G' ( y )
2ix
f
(
x
)
e
f ( x )e
0
2ix
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 )e2iy 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
2i ( 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.