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Paraproducts via H∞-functional calculus and a T(1)-theorem for non-integral operators [Elektronische Ressource] / von Dorothee Frey

160 pages
Ajouté le : 01 janvier 2011
Lecture(s) : 0
Signaler un abus

T
or
eines
rey
t:
ak
hnologie
f
on
non-integral
T
F
Erlangung
eer
a
T
hen
genehmigte
Dr.
TION
DER
Dorothee
-functional
F
opera
der
Prüfung:
Zur
2011
and
Dr.
des
Kunstmann
Lutz
t:
ademisc
W
T(1)-Theorem
(KIT)
Grades
DISSER
oducts
A
DOKTORS
v
or
F
arapr
via
calculus
tors
ündlic
NA
aus
TUR
reudenstadt
WISSENSCHAFTEN
ag
v
m
o
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n
09.
der
ebruar
F
Referen
akultät
HDoz.
für
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Mathematik
Christian
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∞Ha
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pro
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of
notes.
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others
Thank
down
ou
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those
co-examining
ab
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harmonics
con
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to
`intervals'
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ve
discussions
and
questions.
sound,
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to
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empirical
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sup
while.
ervisor
W.
p
Ov
.
.
op
.
.
.
.
.
.
op
.
th
.
.
.
.
op
.
.
Carleson
.
.
.
.
.
Carleson
.
spaces
.
81
.
.
3.4
.
.
.
.
.
12
.
.
.
.
.
.
.
.
.
.
.
.
.
59
Application
6.8
and
.
.
.
.
.
.
.
.
in
.
.
.
.
.
erators
.
5.2
.
.
.
Boundedness
12
.
.
u
.
.
so
tiabilit
.
.
BMO
6
.
erator
4.2
.
.
and
.
.
2.6
Necessary
of
.
.
108
.
.
.
.
to
.
.
.
.
w
19
.
.
.
.
24
a
.
.
.
.
.
.
.
.
.
.
3.2
.
.
.
.
.
.
.
.
.
.
.
.
.
Leb
apro
3.3
7
.
asso
vies-Ganey
.
.
.
.
ducts
.
.
.
.
.
.
.
on
.
.
.
.
.
.
.
of
homogeneous
.
and
.
Holomorphic
.
er
.
thesis
.
theory
.
.
.
.
-Theorem
.
Assumptions
.
.
.
.
ciated
.
.
Denition
.
.
.
.
.
.
.
.
.
.
sp
.
.
.
.
.
.
inequalities
.
.
.
.
.
.
BMO
6.5
.
.
.
.
.
.
.
.
.
second
.
.
A
.
measure
.
and
.
.
.
.
er
.
erators
spaces
Da
o
.
.
o-diagonal
.
.
.
.
.
.
.
.
.
.
.
.
71
.
revisited
.
.
.
.
on
.
.
.
.
.
.
.
.
4.8
.
.
.
.
.
olation
.
.
.
.
.
.
15
.
.
.
.
P
.
via
dieren
calculus
erties
Denition
of
rapro
satisfying
to
orem
.
.
.
.
.
.
.
.
of
.
.
.
.
estimates
.
.
.
.
.
.
.
.
87
.
parapro
.
.
.
.
.
.
.
.
.
.
Spaces
.
.
5.4
.
prop
.
ducts
2
.
4
.
16
.
spaces
.
yp
.
to
5.5
calculus
prop
42
.
tro
.
of
.
.
.
Hardy
.
.
.
.
.
.
.
.
non-in
Notation
102
.
the
.
.
.
.
.
.
spaces
.
.
.
op
.
.
102
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
103
en
.
4.3
.
.
.
spaces
.
ces
.
.
.
.
.
.
.
.
P
.
.
.
.
.
.
.
.
.
.
.
.
47
.
.
.
asso
theorem
.
.
erators
.
.
.
.
.
.
.
.
.
.
.
.
.
12
6.6
.
ersion
.
er
.
.
.
.
A
.
O-diagonal
.
.
124
eraging
parapro
.
.
ssumptions
.
.
.
.
op
.
op
.
ator
.
.
to
3.1
.
.
w
vies-Ganey
T
.
.
other
.
.
.
estimates
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4.7
.
measures
.
.
.
.
24
.
.
.
Assumptions
.
.
.
the
.
Structure
.
erator
.
.
.
.
.
.
.
.
.
.
80
.
The
.
.
.
.
.
.
.
and
.
terp
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2.4
.
.
.
.
5
.
ar
esgue
ducts
32
.
.
-functional
Prop
8
tiation
5.1
of
of
In
a
erators
ducts
e
ciated
Da
op
.
.
estimates
.
.
.
.
.
.
.
.
.
.
.
.
87
.
Boundedness
.
parapro
.
on
34
.
.
.
Quadratic
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5.3
.
of
.
ducts
.
.
.
.
the
.
.
.
.
.
.
.
2.2
.
.
.
.
.
.
.
11
.
.
.
.
90
.
F
of
rther
.
erties
.
parapro
.
.
.
.
40
.
.
.
Hardy
.
t
.
BMO
.
2.5
.
as
.
Preliminaries
.
ciated
.
functional
95
op
Dieren
e
y
ators
erties
.
.
4.1
.
ts
.
erview
.
.
.
the
.
.
.
of
.
.
.
and
.
2.1
.
spaces
.
.
.
.
99
.
A
.
.
.
for
.
tegral
.
erators
.
6.1
.
on
.
op
.
.
.
.
.
.
42
.
.
.
Hardy
.
.
.
asso
.
.
.
to
.
.
.
erators
.
.
.
.
6.2
.
of
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
17
.
.
.
.
.
.
.
T
.
.
6.3
.
conditions
43
.
t
.
Characterizations
.
.
.
Hardy
.
a
.
.
.
.
.
.
.
and
.
.
.
duction
.
.
.
measures
.
.
.
.
6.4
.
oincaré
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4.4
.
.
.
spaces
113
.
Main
ciated
.
.
.
op
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
116
.
A
.
v
.
with
2.3
eak
.
assumptions
.
.
.
.
.
.
4.5
.
3
.
Carleson
.
v
.
estimate
.
estimates
.
.
6.7
.
to
.
ducts
a
.
.
.
and
.
.
.
on
.
.
.
.
.
.
ten
1
1
.
.
.
maximal
the
.
.
.
.
.
.
.
.
.
.
.
127
.
Extension
.
Hardy
.
.
.
-Theorem
.
ards
.
for
.
6
.
6.9
68
.
4.6
.
Dualit
.
y
.
of
.
Hardy
.
and
.
BMO
.
spaces
134
.
134
.
Con
p
H (X)L
∞H
2L (X)
pL (X)
T(1)
∗T(1) T (1)
pH (X) p = 2L
T(b).
.
o
.
constan
.
.
Comparison
.
Bernicot
.
role
.
ts
7.1
.
-Theorem
.
remarks
.
f
.
with
.
.
.
.
.
References
146
.
Concluding
144
a
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
144
.
7.2
.
The
149
of
7
T(1)of
app
In
e
semigroup
ys
and
e
en
a
for
a
tro
to
ciated
there
theory
asso
e
holo-
relev
results
to
of
the
ab
measures
space
called
of
urprising,
ss
the
com
rst
analysis
a
rs
space
space
W
-functional
of
only
ounded
main
Under
text
here
tegral
the
p
,
of
,
a
for
The
w
w
dev
via
yp
estima
op
satises
Coifman
estimate
spaces
and
is
estima
constructed
eraging
h
assumptions
erators.
tly
scop
spaces
ds
of
comparison
at
settin
op
v
for
new
sev
is
n
v
y
it
set
b
where
namely
on
thesis
kno
inequalities.
Moreo
elop
eerman-Stein
Besides,
measures
discussion
erator
i
repro
and
the
erties:
three
The
The
olom
theory
calculus
extends
o
recen
of
is
sets
non-in
du
homogeneous
alcu
ed
fact
as
the
du
due
b
this
and
e
W
some
b
t
[CW71].
erator
ss
earance
w
for
e
erator.
practice
Under
sectorial
w
rmoni
v
ning
as
widens
elop
us
of
of
some
to
y
applications
corresp
explain
and
the
deca
in
to
w
sector
and
giv
little
presen
thi
results,
in
c
short
space
there
functions
of
the
situation
the
of
of
es
and
and
are
t
on
of
as-
the
,
for
to
P
no
efore
b
e
er,
yp
a
op
describin
more
of
order
elemen
era
is
rou
duct
an
a
the
form
erator
ts
wing
ev
bac
and
estimates
v
Kato
der
ounded
to
erlying
e
ic
the
functional
to
un
the
e
that
of
for
elemen
sho
a
tly
e
a
stage
parapro
of
of
s
elop
functional
t
also
tegral
that
e
o-diagonal
theory
tes;
in
The
w
to
duced
is
Hardy
an
y
but
erators
s
and
o-diagonal
BMO
for
ei
b
cts
seem
in
migh
a
an
This
op
y
of
no
o-diagonal
o
te
ada
some
In
The
common
op
to
.
in
the
bi
t
a
o
op
on
c
,
via
w
and
recen
Let
dev
the
ed
metho
theory
e
Hardy

a
the
denotes
consider
,
for
and
t
a
in
onding
used
innit
to
zero
the
y
Euclidean
with
from
ciated
e
the
g
erator
at
.
th
e
ery
e
giv
unied
cost.
tation
tro
the
fact,
including
a
eral
th
haracterizations
study
the
thesis
a
o
o
is
morphic
calculus
and
one
dualit
erview
of
where
spaces
ounded
b
mak
consisting
the
the
a
from
t
tak
an
,
results
,
dierence,
.
theory
these
in
sumptions
the
ounded
con
the
dev
are
of
our
b
wledge
oincaré
w
non-in
stated
W
efore.
coming
v
consider
w
s
generalize
sectorial
F
a
criterion,
erator
g
op
connection
of
Carleson
detailed
and
.
ts
a
b
on
(1.1)
and
op
g
parapro
tors
and
sem
Calderón
explanation
ducing
with
ula
es
elemen
follo
of
the
ery
prop
for
the
on
on
assumptions
kground.
e
has
o
op
the
b
.
und
basic
h
ol
problem
th
orph
whole
space
is
functional
b
holomorphic
o
for
In
duction
thesis,
of
-Theorem
with
a
calculus
ounded
n
.
ernel
conn
k
cti
y
n
an
Carleson
ose
and
;
ts
imp
t,
The
th
semigroup
sho
not
W
do
the
e
for
generated
denition
b
parapro
y
ct
w
constructed
satises
holomorphic
Da
c
vies-Ganey
lus.
estimates,
1
a
1
∞H
T(1)
pL
(X,d,μ)
nR
2L 2m L (X)
2• L L (X)
−tL• e L
2L
−tL p 2• e L −L 1<p< 2
2 qL −L 2<q<∞
L
pH (X) BMO (X) LLL
1 1H (X) H (X) BMO ∗(X)LL L
L
1BMO (X) H (X)L L
BMO ∗(X)L
L
BMO (X)L
L b ∈ BMO (X)L
Z ∞
2m dt2m 2m −t L˜Π :f 7→ ψ(t L)[ψ(t L)b·A (e f)]b t
t0
2 ˜L (X) ψ ψ Ψ
At
At
−tLe
ppΠ L (X) H (X)b Lnon-
tion
e
that
let
the
ounde
with
ab
the
tegral
b
v
tion
thesis
y
eak
ma
a
the
where
a
to
hosen
that
The
o
zero
e
as
Littlew
uniformly
oincaré
erties
t
conser-
if
and
lar
second
e
consideration
ws:
y
der
e
t
op
on
as
a
the
op
so-called
for
pro
functions.
s
t
all
setting,
supp
g
Euclidean
dieren
y
e
aley-Stein
h
inequalit
is
y
a
e
erties
itrary
W
space
the
on
complete
all
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2
and
∞p∈ (2,∞) L (X) BMO (X)L
2L
2 ∗ ∗T :D(L)∩R(L)→L (X) T :D(L )∩loc
∗ 2R(L ) → L (X) ψ ,ψ ∈ Ψ1 2loc
−γ2mdist(B ,B )1 2
kTψ (tL)fk ≤C 1+ kfk2 21 L (B ) L (B )2 1t
−γ2mdist(B ,B )1 2∗ ∗kT ψ (tL )fk ≤C 1+ kfk2 2 2L (B ) L (B )2 1t
1/2m 2γ > 0 t > 0 B ,B r = t f ∈ L (X)1 2
B1
nR GLZ 1/2∞ 22m dt