The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M....
-
Upload
jack-hardy -
Category
Documents
-
view
218 -
download
0
Transcript of The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M....
![Page 1: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/1.jpg)
The Hilbert transform along a one variable vector field
Christoph Thiele
(joint work with M. Bateman)
Conference in honor of Eli Stein,
Princeton, 2011
![Page 2: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/2.jpg)
Partial list of work by Eli Stein that had impact on this research
- Stein: Problems in harmonic analysis related to curvature and oscillatory integrals, Proc ICM 1986
- Phong, Stein: Hilbert integrals, singular integrals, and Radon transforms II, Invent. Math, 1986
- Christ, Nagel, Stein, Wainger: Singular and maximal Radon transforms. Annals of Math, 1999
- Stein, Wainger: Oscillatory integrals related to Carleson’s theorem, Math Research Lett, 2001
- Stein, Street: Multi-parameter singular Radon transforms, preprint, 2011
![Page 3: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/3.jpg)
Outline of lecture
1) Hilbert transform along vector fields with
a) regularity (analytic, Lipshitz) condition
b) one variable condition (main topic here)
2) Connection with Carleson’s theorem
3) Reduction to covering lemmas
4) Three different covering lemmas
![Page 4: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/4.jpg)
Vector Field in the Plane
22: RRv
/
![Page 5: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/5.jpg)
Hilbert Transform/Maximal Operator along Vector Field
Question: - bounds
tdttyxvyxfvpyxfHv /)),(),((..),(
pL
|/)),(),((|sup),(0
dttyxvyxfyxfM v
![Page 6: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/6.jpg)
First observations
Bounded by 1D result if vector field constant.
Value independent of length of v(x,y). May assume unit length vector field.
Alternatively, may assume v(x,y)=(1,u(x,y)) for scalar slope field u.
![Page 7: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/7.jpg)
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.
![Page 8: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/8.jpg)
Gravitation vector field
HT/MO of bump function asymptotically , only for p>2, weak type 2
x
c
pL
![Page 9: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/9.jpg)
Propose modification/conditions
Truncate integral at (normalize )
Demand slow rotation (Lipshitz: )1v
1vt
![Page 10: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/10.jpg)
Zygmund/Stein conjectures
Assume , , and truncate .
Conj.1:Truncated MO bounded .
Conj.2:Truncated HT bounded .
No bounds known except 1) if p is infinity.
1v 1
v
,22 LL
,22 LL
pL
t
![Page 11: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/11.jpg)
Analytic vector fields
If v is real analytic, then on a bounded domain:
Bourgain (1989):
MO is bounded in , p>1.
Christ, Nagel, Stein, Wainger (1999):
HT bounded in (assume no straight integral curves. Stein,Street announce without assumption)
pL
pL
![Page 12: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/12.jpg)
One variable (meas.) vector field
)0,(),( xvyxv
![Page 13: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/13.jpg)
Theorem (M. Bateman, C.T.)
Measurable, one variable vector field
(HT not truncated;
Related earlier work: Bateman; Lacey/Li)
pppv fCfH
p2/3
![Page 14: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/14.jpg)
Linear symmetry group
Isotropic dilations
Dilation of second variable
Shearing
TfHTfH vTvT
)()(1
![Page 15: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/15.jpg)
Constant along Lipschitz
Angle of to x axis less than
Angle of to x-axis less than or equal to
Conjecture:
Same bounds for
as in Bateman,CT
2
v v
vH
![Page 16: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/16.jpg)
Relation with Carleson’s theorem and time-frequency analysis
![Page 17: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/17.jpg)
Carleson’s operator
Carleson 1966, Hunt 1968:
Carleson’s operator is bounded in ,pL p1
)(
2)(ˆ)(x
ix defxfC
tdtetxfvp xit /)(.. )(
![Page 18: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/18.jpg)
Coifman’s argument
),(2
/))(,(yxLR
tdttxuytxf
),(
)(
2
/),(ˆ
yxLR
txiu
R
iy dtdtetxfe
),(
)(
2
/),(ˆ
xLR
txiu tdtetxf 2),(2
),(ˆ~ fxfxL
![Page 19: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/19.jpg)
The argument visualized
![Page 20: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/20.jpg)
A Littlewood Paley band
![Page 21: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/21.jpg)
Bound for supported on Littlewood Paley band
Lacey and Li: Bound on for
arbitrary two variable measurable vector field
Bateman: Bound on for
one variable vector field.
p2
21 p
f̂
fHv
fHv
![Page 22: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/22.jpg)
Vector valued inequality, reduction to covering arguments
![Page 23: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/23.jpg)
Littlewood Paley decomp.
![Page 24: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/24.jpg)
Vector valued inequality
Since LP projection commutes with HT
(vector field constant in vertical direction)
,
Enough to prove for any sequence
p
kpff
2/12 p
kvpv fHfH2/12
kf
p
kp
kkv fCfH2/122/12
![Page 25: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/25.jpg)
Weak type interpolation
Enough to prove for
Whenever
p3
2
pp
pGkkv GHCfH/11/12/12
1,
Hkf 12
![Page 26: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/26.jpg)
Cauchy Schwarz
Enough to prove
Which follows from
pp
pGkkv GHCfH/21/22
1,
2/21
21,
k
p
pGkkv fH
GCfH
![Page 27: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/27.jpg)
Single band operator estimate
By interpolation enough for
Lacey-Li (p>2) /Bateman (p<2) proved
Note: F,E depend on k, while G,H do not
2
/12/1
2/)1(1 fHGCfH
p
HkvG
2/12/1/12/1/1,1 EFHGCH
p
pEFkv
qEFkv EFCH/11/1
1,1
HFGE ,
![Page 28: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/28.jpg)
Induction on .
If p<2 and
Find such that and
the desired estimate holds (prove!) for .
Apply induction hypothesis on (gain )
GH GGexc 2/GGexc
)/(log2 HG
excG
excGG \2
![Page 29: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/29.jpg)
Induction on .
If p>2 and
Find such that and
And the desired estimate holds for
Apply induction hypothesis on
HG HH exc 2/HH exc
)/(log2 GH
excH
excHH \
![Page 30: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/30.jpg)
Finding exceptional sets. Covering arguments
![Page 31: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/31.jpg)
Parallelograms
![Page 32: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/32.jpg)
Kakeya example
Try union of all parallelograms R
with for appropriate .
Bad control on , example of Kakeya set.
RGR
excH
excH
![Page 33: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/33.jpg)
Vector field comes to aid
Restrict attention to U( R), set of points in R where the direction of the vector field is in angular interval E( R) of uncertainty of R
![Page 34: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/34.jpg)
1st covering lemma
The union of all parallelograms with
has measure bounded by for q>1
(vector field measurable, no other assumption)
RRURG )(
GC qq
excH
![Page 35: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/35.jpg)
Outline of argument
Find good subset of set of
parallelograms with large density, such that
1.
2.
Then:
'
'RR
RCR
'
'
')(1
R
q
RRU RC
RRR
GRGRUR/1
'/1
'
1
'
1
'
)(
![Page 36: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/36.jpg)
Greedy selection
Iteratively select R for with maximal
shadow such that for previously selected R’
Here 7R means stretch in vertical direction.
'
100/'770)'()(','
RRRRURUR
![Page 37: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/37.jpg)
Vertical maximal function
The non-selected parallelograms are all inside
which has acceptable size.
10000/1),)(1(:),('
yxMyxR
RV
![Page 38: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/38.jpg)
Additional property
R’ selected prior to R; U(R’) intersect U(R).
Then
RSR R 7'7
![Page 39: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/39.jpg)
argument
Assume in order of selection and
' ,...,
1)(
21
)(...)()1(R RRR
nn
RU
n
RURUC
0)()( 1 ii RURU
nR RRSC 7...7 11
11 7...71 nR RRSC
17... RC
nL
![Page 40: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/40.jpg)
2nd covering lemma (Lacey-Li)
The union of all parallelograms with
and
Has measure bounded by .
Use vector field Lipshitz in vertical direction. -power responsible for .
RRH
HC 21
RRUR )(
excG
2
3p
![Page 41: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/41.jpg)
Outline of argument
From set of parallelograms with large
densities select as before. Using as before
obtain:
To prove:
Power of 2 here responsible for -power
'
'RR
RCR
'
1
2
'
1RR
R RC
VM
![Page 42: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/42.jpg)
Expansion of .
Sum over all pairs with R selected before R’.
Case 1
Case 2
Second case has aligned directions, as before.
'RR
)'(100)( RERE
)'(100)( RERE
![Page 43: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/43.jpg)
Second selection in Case 1
Fix R, consider R’ selected later with Case 1.
Prove
Select so that each selected R’ has
Projection of U(R’) onto x-axis disjoint from
Projection of U(R’’) for previously selected R’’
RCRR 1'
'''
![Page 44: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/44.jpg)
Removing δ
Use disjointness of projections of U(R’) and
density δ to reduce to showing for fixed R’
Summing over those R’’ in which where
not selected for because of prior selection
of R’ . All R’’ have similar angle as R’.
(use vector field Lipschitz in vertical direction.)
''' RSRCRR '
''
![Page 45: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/45.jpg)
Picture of situation
![Page 46: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/46.jpg)
Back to maximal function
1) Hard case is when all rectangles are thin.
2) Intersection with R is only fraction α (depending on angle) of R
3) If too much overlap, then vertical maximal function becomes too large in extended rectangle (1/α) R.
4) Two effects of α cancel.
![Page 47: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/47.jpg)
3rd covering lemma (Bateman)
One variable vf, parallelograms of fixed
height h. Union of all parallelograms R with
has measure bounded by
,RRH RRUR )(
HC )1(1
![Page 48: The Hilbert transform along a one variable vector field Christoph Thiele (joint work with M. Bateman) Conference in honor of Eli Stein, Princeton, 2011.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649f2b5503460f94c46560/html5/thumbnails/48.jpg)
Difference to previous situation
Single height and constant vf in vertical
direction causes approximately constant slope
for all R’’ and thus avoids overlap.