LECTURE NOTES ON DYADIC HARMONIC ANALYSIS
15
EXERCISE
1.30.
Show that forb E
BMOd,
2
1"'"'
2
\\b\\BMOd
=sup
-\I\
~
\(b,hJ)\ .
lED JED(I)
EXERCISE
1.31.
Show that
BMOd
is strictly larger than
BMO.
THEOREM
1.32.
Given
bE
BMOd, then
7rb
is bounded in
£P(1R),
for
1
p
oo.
Moreover, \\1rgf\\P:::; Cp\\b\\BMOd \\f\\p·
For p = 2 this theorem is an immediate consequence of Carleson's Lemma that
we will prove in the fifth lecture.
A positive sequence {
Ar} lED
is a
Carleson sequence
if there exists a constant
C
0 such that for all IE
V,
L
AJ:::; C\I\.
JED
(I)
LEMMA
1.33
(Carleson'sLemma).
Let{.r}IED, be
aCarlesonsequence.
Given
any positive sequence {ar
}rED,
let a*(x)
= supxEIED
ar; then
(1.5)
Given bE
BMOd,
let
Ar
= \(b,hr)\
2
,
and
ar
=
m7J,
then the sequence {)..1
}
is Carleson with constant
C
=
\\b\\1Mod,
and
a*(x)
=
(Mdf(x))
2
.
By Carleson's
Lemma and the boundedness in £
2
of the dyadic maximal function we obtain the
boundedness in £
2
of the dyadic paraproduct,
\\1rgf\\~
=
L
m7f b7 :S \\b\\1Mod \\Md
f\\~
:S \\b\\1Mod
\\f\\~.
lED
For p
"/=-
2 one can use Littlewood-Paley theory (square functions) plus Car-
leson's Lemma. Instead we will show in the last lecture that
1rg
is bounded in
£
2
(
w)
for all
w
E A2 and invoke the extrapolation theorem, see Section
6.2.2.
Al-
ternatively we will prove that
1rg
is of weak type (1, 1); by interpolation this time
we can show that it is of strong type
(p, p)
for 1
p
2. If we could show the
same for its adjoint, then a duality argument will give us the range 2 p oo.
EXERCISE
1.34.
Show that the adjoint of
7rb
is given by
* "'"'
xr(x)
1rbg(x)
=
~
(g, hr)br-\-I\-.
lED
Both the paraproduct and its adjoint are linear operators which are bounded
in £
2
,
and such that for every dyadic interval
I,
the image under either of them is
supported on I, we will see in the next lecture that this implies that they are of
weak type
(1, 1),
see Lemma
2.10.
EXERCISE
1.35.
Check formally that
1rg1
=band that
(7rg)*1
= 0.
1.5.4.
Haar multipliers and the Hilbert transform.
The operators in this section
do not have a priori a continuous analogue. We will see that averages over random
dyadic grids of appropriate Haar multipliers will give back the Hilbert transform.
A
Haar multiplier
is an operator defined formally by
Tf(x)
=
L
wr(x)(f, hr)hr(x);
lED
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