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
Previous Page Next Page