14

MARIA CRISTINA PEREYRA

1.5.1.

Dyadic maximal function.

The

dyadic maximal function

is defined as

the ordinary maximal function, except that the supremum is taken over the dyadic

intervals:

Md f(x)

= sup

III 1

j

if(t)i dt

=sup

Enf(x).

xEIED I nEZl

Md

is bounded in

L

00

,

we will show that it is of weak type (1, 1) and by inter-

polation it will be of strong type

(p,p)

for all1

p

oo. The weak type property

will be an immediate consequence of the Calder6n-Zygmund decomposition to be

discussed in the next lecture. The interpolation theorem is also discussed in the

next lecture.

It is clear that

M

is pointwise larger than

Md, Md f(x) :::; Mf(x)

for all

x.

Therefore boundedness properties of

Md

are deduced from those of

M.

One can

actually reverse the process. See

[Duo]

for such an approach.

by

1.5.2.

Dyadic square function.

The

dyadic square function

is defined formally

1

sd

f(x)

=

(I:

l~nf(x)l

2

)

2

nEZl

EXERCISE 1.26. Show that

IISd

fll2

=

llfll2

(this is a consequence ofPlancherel).

THEOREM 1.27.

Let

1

p

oo,

then f

E

£P(R) if and only if Sdf

E

£P(R).

Moreover

II

flip

rv

IISd flip·

We will followS. Buckley

[Bul]

in his proof of this fact, by showing that

Sd

f

is

bounded in L

2

(w)

for all wE A2

,

see (1.3); and then a beautiful result of Rubio de

Francia, the Extrapolation Theorem, will give boundedness in

LP

for all1

p

oo.

We will discuss the extrapolation theorem as well as the proof of Theorem 1.27 in

the next lecture.

By Exercise 1.24,

~nf(x)

=

(!, hi)hi(x),

where

x

E IE

Vn,

therefore

(Sd

!)2(x) =

L

1(!,

hi)I2

xEIED

III

From here it is now easy to see that,

COROLLARY 1.28.

{hi hED is an unconditional basis in LP(R), for

1

p

oo.

1.5.3.

Dyadic paraproducts.

Formally the

dyadic paraproduct

is a bilinear op-

erator

1rd(b,J)

=

1rgf,

given by

7rgf(x)

=

L

Enf(x)~nb(x)

=

L

mif(b, hi)hi(x).

nEZl lED

EXERCISE 1.29. Check formally that

bf

=

Lj

~jb[Ejf

+(I- Ej)f]

=

1rgf

+

1rjb

+

Lj

~jb~jf,

so that the paraproduct can be thought as "half a product".

As mentioned before, the paraproduct will behave better than the ordinary

product, in the sense that we do not need b to be bounded to obtain boundedness

in

LP.

The substitute for

L

00

in this dyadic world will be

dyadic BMOd.

A locally

integrable function

b

is in

BMOd

if