8
MARIA
CRISTINA PEREYRA
We have borrowed the presentation
in this section from lectures presented by
S. Treil in the 2000 Spring Analysis School held in Paseky, Czeck
Republic. For
generalizations to the multivariate setting see
[TV2].
1.2. The Hardy-Littlewood Maximal Function.
A natural question for
locally integrable functions
f
in R is whether the averages on small intervals
Ix
containing a point x converge to the value of the function there, i.e.
lim -IIll
j
f(t) dt = f(x).
llx 1--0
X
fx
It is clear that if
the function
f
is continuous this is true, by the Fundamental
Theorem of
Calculus. The Lebesgue Differentiation Theorem says that for locally
integrable functions
this is true almost everywhere ( a.e.).
A natural object
to study, instead of the limit, is the supremum. In this exam-
ple it corresponds to the Hardy-Littlewod
maximal function, a sublinear operator
defined by
M f(x) =sup
-11
1
r
lf(t)l dt.;
xEI
I
./1
here I are
intervals containing x.
It turns
out that the boundedness properties of maximal operators imply con-
vergence
a.e. of corresponding limits. It should be clear that the maximal function
is bounded in £
00

What is less obvious
is that M is of strong type (p,p) for p 1
and of weak type (1, 1).
We
will show
both facts in the next lecture.
EXERCISE
1.9. Show that the maximal function M is not of strong type (1, 1).
1.2.1.
Approximations of the identity. The maximal function controls a large
class of so-called
approximations of the identity.
Given ¢ a real valued integrable
function in
R,
such that
J
¢ = 1. Define for
each
t
0
c/Jt (
x)
=
t
¢(
'J').
We say that the family { ¢t}tO is an approximation of
the identity. One can check that c/Jt converges as
t
---+
0 to the Dirac delta function
in the sense of distributions. In particular this implies that for nice
functions (in
the Schwartz class S): limt_,0 c/Jt
*
g(x) = g(x) for all x. The question then becomes:
When does lim c/Jt
*
f(x) = f(x) a.e.?
t--0
EXERCISE
1.10. Given an
approximation of the identity { ¢t}tO show that
lim llc/Jt
*
f-
fliP=
0,
\If
E
£P, 1::; P
00.
t--0
As a consequence there exists a subsequence c/Jtk
*
f(x)
that converges a.e. to
f ( x). Therefore if limt_,o c/Jt
*
f ( x) exists it must coincide with f ( x) a.e.
1.2.2. More on Fourier series - Summability methods. The classical examples
of
approximations of the identity arise in the study of Fourier integrals and con-
vergence of truncated Fourier integrals. We will remind you the analogue problems
in the setting of Fourier series and convergence of partial Fourier sums, where in-
stead of approximations of the identity, as defined in the previous paragraph, we
encounter summability kernels.
A summability kernel is a sequence { K
N}
of continuous 1-periodic functions
whose averages are 1, whose £
1
norms are uniformly bounded, and
for all 0
6
n,
(1.4) lim {
1
-
8
IKN(t)ldt
= 0.
N-oo ./8
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