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
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