10 MARIA CRISTINA PEREYRA

Its proof can be found in

[Duo],

p. 37.

In the case of the approximations of the identity, let Ttf

= cPt

*f. The Schwartz

class S (or if you prefer

C0

)

is a subset of A0. S is dense in LP, therefore if A0

is closed, we conclude that A

0

=

LP. The fact that A

0

is closed can be shown by

the previous theorem, provided we can show that

T*

is of weak type

(p, p).

One

can control a large class of approximation of the identity kernels (in particular the

Fejer and Poisson kernels) by the Hardy-Littlewood maximal function; therefore all

we need to check is that M is of weak type

(p, p).

EXERCISE

1.14. Consider

¢

E

L1

such that

¢

is even and decreasing as a

function oft

=

lxl 0. Show that

T* f(x)

=

suptO lc/Jt

*

f(x)l :::;

llc/JII1MJ(x).

Show that if

M

is of weak type

(p, p)

so is

T*, p 2::

1.

1.2.4.

Lebesgue Differentiation Theorem.

The previous results can be used in

particular to prove the well known Lebesgue Differentiation Theorem. Namely

1

1x+t

f

E

L1(lR)

=}

lim-

f(s)ds

=

f(x)

a.e ..

t--0

2t

x-t

In higher dimensions one would like to consider similar problems. One could

use instead of intervals, cubes or rectangles, or more general sets. If one uses cubes

then the theory runs parallel to the 1-dim theory. If instead we use rectangles with

sides parallel to the axis then the corresponding maximal function is of strong type

(p, p)

for

p

1 but not of weak type ( 1, 1), moreover the Lebesgue differentiation

theorem is false for functions

f

E

L1(lRn),

although it works for functions such

that j(1 +log lfl)n-

1

E

Lfoc(lRn), in particular iff

E

Lfoc(lRn). If we allow all

possible rectangles then the corresponding maximal function is not even of strong

type

(p, p)

for any

p.

For this and much more on differentiability properties of basis

of rectangles, see the classical book by Miguel de Guzman

[Guz].

1.3. Square functions/Littlewood-Paley Theory.

The so-called

square

functions

are ubiquitous objects in harmonic analysis. This is not just one object

but several who share some properties. It is best to describe some of the most

classical examples to give the flavor of the so-called Littlewood-Paley Theory. This

theory has been a favorite tool for proving LP estimates.

1.3.1.

Trigonometric series and Littlewood-Paley square function.

Given a func-

tion

f

E

L

2

(1I'), denote the

N-th dyadic partial Fourier sum

by

PNf(x)

=

L

j(n)e21rinx

(

=

S2Nj(x)).

lni9N

PN

f

should be viewed as an approximation of

f

which gets better as N increases.

Consider the difference operators

tlof(x)

=

L

j(n)e2Kinx.

lnl9

Because the trigonometric system is an orthonormal basis in L

2

(1I') we have the

reconstruction formula

f =

tl

0

f

+ l:N

1

b.N

j,

and Plancherel's identity

11111~

=

l:nEZl

lf(n)

1

2

.

It is clear that if we

decid~

to change the sign of some of the Fourier

coefficients and utilize them to crate a new function

T f(x)

=

2:

±](n)e21finx,