4 MARIA CRISTINA PEREYRA

Notice that it is also given by convolution with the distributional kernel

k(x)

=

p.v.

1rx.

1

If the kernel were an integrable function, then the integral operator:

Tf(x)

=

k* f(x)

=

J

k(x- y)f(y)dy

would be automatically bounded in

LP

for all

1 ::;

p::;

oo, by Young's inequality:

II!*

kiiP ::;

llkii1IIJIIP·

Unfortunately the Hilbert

transform kernel is not integrable, nevertheless the Hilbert transform is bounded

in LP(R) for all1

p

oo; and although it is not bounded at the endpoints

p

=

1

and

p

=

oo; there are appropriate substitutes.

In particular one can compute the Fourier transform of the Hilbert transform,

at least when applied to very smooth and compactly supported functions, and

obtain:

(H

!)"(~)

=

L

H j(x)e- 21rixedx

=

-i

sgn(~)}(~);

here we define

sgn(~)

=

-1

when~

0,

sgn(~)

=

1

when~

0, and sgn(O)

=

0.

This automatically shows that the Hilbert transform is an isometry

2

on a dense

subset of

L

2

(R),

and can then be extended by continuity as an isometry to

L

2

(R).

Notice also that from the above identity we conclude that

H

2

=-I.

In the next lecture we will give alternative proofs, based on Cotlar's and Schur's

lemmas, of the boundedness in L

2

of the Hilbert transform. We will also present

the original proof by M. Riesz of the boundedness in

LP

for 1

p

oo.

EXERCISE

1.1. Show that the Hilbert transform is not bounded in L 1 nor in

L

00

by explicitly calculating its action on the characteristic function of the interval

[0,

1], which is a function in

L

1

n

L

00

•

When

p

=

1, one can get away with a weaker notion of boundedness. Notice

that if an operator is bounded in

LP(X), (X, f.L)

a measure space, p

~

1, then the

following inequality is an immediate consequence of Tchebychev's inequality:

(1.1)

f.L({x

EX:

ITJ(x)

A})::;

c

e!II~(X)

)p'

C~l.

EXERCISE

1.2. Check the above inequality for

C

=

1 and

T

a bounded operator

in

LP(X).

An operator that satisfies (1.1) is said to be of weak type (p,p). An operator

that is bounded in

LP

is said to be of strong type (p, p). We have just shown that

strong (p, p) implies weak (p, p); but the converse, in general, is false.

We will show that the Hilbert transform is of weak type (1,1) in the next

lecture. As for bounded functions they are mapped into a larger space bounded

mean oscillation, BMO, to be defined later. This behaviour is shared by a large

class of very important operators the so-called Calder6n-Zygmund singular integral

operators. The departure point of the Calder6n-Zygmund theory is an a priori L

2

estimate; everything else unfolds from there. Having means other than Fourier

analysis to obtain such L

2

estimate is crucial, that is the content of the celebrated

T(1) Theorem of David and Journe which we will discuss in our fourth lecture.

Why did mathematicians get interested in the Hilbert transform? Here are a

few classical problems where the Hilbert transform appeared naturally.

2

IIHJII2 = II(HJ)AII2 = llfll = llfll2,

where Plancherel identity has been used twice