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