LECTURE NOTES ON DYADIC HARMONIC ANALYSIS 5
1.1.1. Connection to complex analysis. Consider a real valued function f E
L
2
(R)
and let F(z) be twice its analytic extension to the upper half planeR!
=
{z
=
x +it : t
0},
suitably normalized. F(z) can be explicitly computed by
means of the well known Cauchy integral formula:
F(z)
=
~
{
f(y) dy, z
E
R!.
27rZ }R
Z-
y
Notice the resemblance with the Hilbert transform. No principal value is needed
here since the singularity is never achieved. By separating the real and imaginary
parts of the kernel, one can obtain explicit formulae for the real and imaginary
parts of F(z)
=
u(z) + iv(z) in terms of convolutions with the so-called Poisson
and conjugate Poisson kernels: u(x +it)
=
f
*
Pt(x), v(x +it)
=
f
*
Qt(x). The
function u is the harmonic extension off to the upper-half plane, and the function
v is its harmonic conjugate.
EXERCISE
1.3. Show that the Poisson kernel is given by Pt(x)
=
~ x
2
~t
2
,
and
the conjugate Poisson kernel by Qt(x)
=
~ x
2
~t
2
Show that for each t 0,
Qt(~)
=
-i sgn(~)e_ 2 ,.1te1, therefore as
t ___. 0,
Qt(~) approaches -isgn(~), the
Fourier multiplier corresponding to the Hilbert transform.
The Poisson kernel is an example of an approximation of the identity that we
will discuss more deeply in the next section. As such, the limit as
t ___.
0 of
u
=
Pt
*
f
is fin the L
2
sense and almost everywhere. On the other hand, as
t ___.
0, v
=
Qt
*
f
approaches the Hilbert transform H
f
in L
2
.
1.1.2. Connection to Fourier series. For functions integrable on TI'
=
[0, 1],
the
n-th Fourier coefficient is well defined by the formula
](n)
=
11
J(x)e-2,.inxdx.
Since
L
2
(TI')
c
L
1
(TI'), this is also well defined for square integrable functions. It is
well known that the trigonometric system {
e2,.inx}nEZl
is an orthonormal complete
system in L
2
(TI'); therefore the following reconstruction and isometry formulae hold
in L
2
:
f(x)
=
L
](n)e2,.inx,
2 ""
A
2
ll!ll£2'1rl
=
~
lf(n)l .
nEZl nEZl
The N-th partial sum is given by
SNJ(x)
=
L
](n)e2,.inx.
lni:SN
In the XIX century mathematicians asked for which 21r-periodic functions
f
would it be true that limN__,oo SNJ(x)
=
f(x) at a given point x
E
TI'? Some
partial answers were given, more than continuity at the point was always required
(eg. Dini's condition). In 1889, Du Bois Raymond found a continuous function
whose partial Fourier sum diverges at a point. With the advent of measure theory
and
LP
spaces, new questions were formulated: Is there convergence a. e.? Is there
convergence in the
LP
sense? The second question is answered positively for 1
p oo and it is a consequence of the boundedness of the Hilbert transform in such
LP's.
The first question is much more difficult, for
p
=
2 the positive answer was
given by L. Carleson in a celebrated paper published in 1965, see
[Car]
(settling the
question for periodic continuous functions which had remained open until then);
two years later, R. Hunt extended the result for the remaining p's, 1
p
oo, see
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