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
=
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
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