12

MARIA CRISTINA PEREYRA

1.4. Paraproducts and

BMO.

The paraproducts are cousins of the square

functions. As such, they represent a class of objects rather than one specific opera-

tor. Here we will discuss the continuous paraproduct, the dyadic paraproduct will

be discussed in the next section.

Let

¢,

'¢ be compactly supported 0

00

functions (or functions in the Schwartz

class), such that I'¢

=

0, and I¢

=

1. Introduce the operators Qd

=

'1/Jt

*

f

and Ptf =¢*f. As before, the Qt operator represents differences and the Pt is an

approximation of the identity, and therefore represents averagings. The paraproduct

is defined formally as a bilinear operator:

1

00

dt

n:(f,

b)=

Qt(Ptf Qtb)-.

0

t

Heuristically the para product can be thought as "half a product": b f "' n:(f,

b)

+

n:(b,

f).

In our case one could do a formal Fourier analysis argument, see

[Ch]

p. 41. This will be more evident in the case of the dyadic paraproduct. For a fixed

function b consider the ordinary product as an operator in LP, Mb/

=

bf.

EXERCISE 1.18. Show that Mb is bounded in LP if and only if bE L

00

•

The paraproduct will behave better in the sense that for fixed b we will have

boundedness properties in a space larger than L

00

•

That space is the so-called space

of bounded mean oscillation or

BMO.

A locally integrable function bE

BMO

if

llbiiBMO

=

s~p l~lhlb(x)-

mrbldx oo;

where mrb

=

1}

1

Ir b denotes the mean value over I of the function b. This means

that the average oscillation of b is uniformly bounded on every interval I.

The null elements in the

BMO

norm are the constants, so a function in

BMO

is defined only up to additive constants.

EXERCISE 1.19. Show that

BMO

(modulo constant functions) is a Banach

space. Show that

L

00

C

BMO.

Show that log

lxl

E

BMO,

hence

BMO

is larger

than L

00

•

EXERCISE 1.20. Show that log

IP(x)l

E

BMO

for any polynomial

P

on

JR.

In the third lecture we will show that the singularities allowed in

BMO

are

precisely like those of log

lxl.

It is the content of the celebrated John-Nirenberg

Inequality. A corollary of that inequality is that for each

1

p

oo,

1

llbiiBMO"'

(s~p l~lhlb(x)-

mrWdx)

"P

THEOREM 1.21. Given b E

BMO

then

1rb

is bounded in LP for 1 p oo.

Moreover

llnbfiiP:::; llbiiBMollfllp·

(Here n:bf

=

n:(f, b).)

The proof can be found in

[Ch].

It uses square functions and Carleson's Lemma

which we will introduce in its dyadic incarnation in the next section, and we will

discuss more deeply in the fifth lecture.

The paraproduct appeared naturally in non-linear differential equations in the

work of Bony, see

[Bo].

It turns out that the paraproduct can be thought as a

singular integral operator which is far from being translation invariant. Moreover,

what the acclaimed T(1) Theorem says is that a large class of singular integral