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