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.