LECTURE NOTES ON DYADIC HARMONIC ANALYSIS
11
then
Tf
E
L
2
(T),
moreover, Tis an isometry: IITflb
=
llfll2· This means that
the trigonometric system is an
unconditional basis
in
L2(TI').
The question then
becomes: Is the trigonometric system and unconditional basis in
LP ('IT)
for p -:/- 2?
The answer is NO, there is no way we can decide if a function is in
LP('IT)
just by
analyzing the absolute value of its Fourier coefficients, unless p
=
2. The closest
substitute is obtained analyzing the
Littlewood-Paley square function:
I
Sf(x)
=
(~
1.6-nf(xW)
2
THEOREM 1.15 (Littlewood-Paley).
Let
1
p
oo,
then f
E
LP(TI') if and
only if Sf
E
LP(TI'). Moreover
llfiiP"" liS flip·
Notice for p
=
2 this is an immediate consequence of Plancherel's identity. For
a proof for
p-:/-
2, see for example
[St2]
p.104, and Appendix D. As usual A"" B
means that there exist constants c,
C
0 such that
cB
s;
A
s;
C B.
1.3.2.
The g-function.
Let
1/J
be a compactly supported
C
00
(R)
function with
zero mean,
J
1/J
=
0. Let
1/Jt(x)
=
t'I/J(f).
Define the family of operators
Qtf
=
1/Jt
*
f,
which now play the role of the differences in the previous example. One
obtains a
reproducing formula: f(x)
=
c
1
(1j;)
J00
0
QU(x)slf-.
EXERCISE 1.16. Show that
c('I/J)
=
J00
0
l~(t)l2df- oo is the right constant in
the reproducing formula.
The analogue to the Littlewood-Paley square function is the
square function
defined by:
I
Sf(x)
=
(1
00
1Qtf(x)l
2
~t)
2
Same theorem holds,
THEOREM 1.17.
Let
1
p
oo,
then f
E
LP(R) if and only if Sf
E
LP(R).
Moreover
llfiiP"" IISfllp·
The case p
=
2 is an immediate consequence of Plancherel. The inequalities
for p -:/- 2 are more subtle. One can view square functions as ordinary singular
integrals, but now taking their values on a Hilbert space, a machinery similar to
the one necessary to handle the Hilbert transform and its siblings can be used to
prove this theorem, see
[Stl].
When the functions
1/Jt
arise as derivatives of the Poisson kernel, more precisely,
1/Jt
=
t
ft
Pt,
then the square function is called the
g-function:
I
g(f)(x)
=
(1
00 tl\i'u(x, t)!
2
dt)
2
,
where
u(x, t)
=
Pt
*
f(x)
is the harmonic extension of
f.
In this case one can use
Green's formula to show that llfll2
=
llg(f)ll2·
There is a third very illustrative example, the
dyadic square function.
We have
decided to present all dyadic analogues at the end of this first lecture. All these
square functions share the property that we go from a function of x to a function
of
(x, t), t
0, or of
(x, n), n 2':
0, ornE
7l (Qtf(x)
or
.6-nf(x)).
There is always
some reconstruction formula and the way the square function is constructed is by
taking an
L
2
(or
l2
)
norm on the new variable. The square function
Sf
has now
the same
LP
properties as the function
f.
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