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,
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