On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 2008
To the memory of Mischa Cotlar, my teacher and my friend
Abstract. Through the prism of abstract scattering, and the invariant forms acting in them, we discuss the Hilbert transform in weighted Lp spaces in one and several dimensions.
2000 Mathematics Subject Classification. Primary: 42B30, Secondary: 47B35, 32A37.
It all started with the study of the Hilbert transform in terms of scattering...
In the late seventies, Cotlar and I began a systematic study of algebraic scattering systems, and the invariant forms acting on them.
In the late eighties we started working in multidimensional scattering-although many did not consider such approach as relevant.
In the late nineties our outlook was finally vindicated. Multidimensional abstract scattering systems appeared as couterparts of the input-output conservative linear systems.
The Hilbert transform operator
The basic result of Marcel Riesz (1927) is
is bounded on .
Similar boundedness properties are valid in the "weighted" cases, both for and for the iterated ,
Given we can decompose as , where is analytic and is antianalytic.
Under this decomposition the Hilbert transform can be written as
The analytic projector , associated with the Hilbert tranform operator , is defined as
The crucial observation is that supports the shift operator .
Then, the range of is the set of analytic functions, and its kernel is the set of antianalytic functions:
The "scattering property" of the -dimensional Hilbert transform provides the framework for a theory of invariant forms in scattering systems, leading to two-weight -boundedness results for .
The scattering properties are also essential to providing the two-weight - boundedness of the product Hilbert transform in product spaces, where the analytic projectors supporting the -dimensional shifts are at the basis of the lifting theorems in abstract scattering structures.
Notice that this fact, valid for the product Hilbert transforms, is not valid for the -dimensional Calderón-Zygmund singular integrals, which do not share the scattering property.
In fact, is an isometry on , since
and this follows easily from the Plancherel Theorem for the Fourier transform.
The result can also be obtained through the Cotlar Lemma on Almost Orthogonality, which extends the Hilbert transforms into ergodic systems.
These are two different ways to deal with the boundedness of in . The same happens for .
Start checking that is weakly bounded in , and apply the Marcinkiewicz Interpolation Theorem between and , and then, by duality, pass from to .
The "Magic Identity" for given by
Using extrapolation, since for implies , then implies , and implies .
The boundedness of in , follows by duality, and interpolation gives the boundedness of in
By polarization, the Magic Identity for an operator becomes
The Magic Identity, and similar ones have been used extensively in harmonic analysis, in particular by Coifman and Meyer. Cotlar and Sadosky, and Rubio de Francia, used the Identity in dealing with the weighted Hilbert transform in Banach lattices.
Gian-Carlo Rota used three different "magic indentities" in his work in combinatorics, and his school encompassed all particular cases in a general inequality.
Gohberg and Krein showed that the polarized Magic Inequality holds in the space , and deduced the theorem of Krein and Macaev in a way similar to the passage from to described before.
The "magic indentities" hold in a variety of non-commutative situations, starting with the non-commutative Hilbert transforms in von Neumann algebras, and that theory has been developed in the last years.
is equivalent to
(with a special norm).
is equivalent to
Take note that, although both conditions are necessary and sufficient, the first one is good to such weights, while the second one is good at them.
An operator acting in a Banach lattice is u-bounded if
The -boundedness of operators is considerably weaker than boundedness. For example,
is -bounded on if and only if
Now we can translate another equivalence for the Helson-Szeg theorem for p=2:
There exist , real-valued functions, such that
(Here means )
is equivalent to
is -bounded in .
is -bounded in where
The following are equivalent:
1. The double Hilbert transform is bounded in
3. (with a special norm)
4. , such that
are simultaneously -bounded in
1. The double Hilbert transform is bounded in
2. and are simultaneously -bounded in , where for
Let be a vector space, and be a linear isomorphism in
The subspaces of , and are linear subspaces satisfying
Let , be positive -invariant forms, such that
Assume now that is the set of trigonometric polynomials on and are the sets of analytic and antianalytic polynomials in where is the shift operator.
The Herglotz-Bochner theorem then translates to
is positive and -invariant in if and only if it exists a measure such that
Since , the domain of splits in four pieces for .
A weaker concept of -invariance is
Then the Lifting theorem asserts that is positive in and -invariant in each quarter if and only if there exists such that, for all
Here means that the (complex) measures satisfy and
Let be an -invariant in .
Then are positive, while is not positive but bounded,
Let the operator be defined in as
Defining by , is -invariant in each quarter and is also non-negative.
By the Lifting theorem, there exists such that
By the F. and M. Riesz theorem,
Then, the necessary and sufficient condition for boundedness in with norm , is that for all Borel sets,
In particular, is an absolutely continuous measure, for some .
When we return to the previous case:
The Hilbert transform is a bounded operator in with norm
if and only if
for which is the source to all the equivalences we mentioned above.
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Department of Mathematics
Washington, DC, USA
Recibido: 8 de abril de 2008
Aceptado: 20 de diciembre de 2008