## Articulo

• Similares en SciELO

## versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637

### Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dic. 2008

The Hilbert transform and scattering

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.

1. Introduction

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.

2. The Hilbert transform

The Hilbert transform operator

is given by convolution with the singular kernel
for . (Analogues in and )

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 ,

where are general measures.

3. The scattering property of the analytic projector

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.

4. is bounded on

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 .

5. Two ways for to

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

is valid for all functions smooth and with compact support.

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.

6. is bounded in

6.1. Helson-Szeg theorem (1960).

where are real-valued bounded functions, such that

is equivalent to

(with a special norm).

6.2. Hunt-Muckenhoupt-Wheeden theorem (1973).

is equivalent to

6.3. Remark.

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.

7. Definition of -boundedness

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 .

8. is bounded in

8.1. Hunt-Muckenhoupt-Wheeden Theorem (1973).

is equivalent to

defined by

is -bounded in where

9. is bounded in weighted

The following are equivalent:

1. The double Hilbert transform is bounded in

2.

are real-valued bounded functions,

3. (with a special norm)

4. , such that

5.

and

are simultaneously -bounded in

6.

10. bounded in weighted

1. The double Hilbert transform is bounded in

2. and are simultaneously -bounded in , where for

are simultaneously -bounded in .

3.

11. The Lifting Theorem for invariant forms in algebraic scattering structures

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

Then,

such that
or, equivalently,
Furthermore,
such that

12. The Lifting Theorem for trigonometric polynomials

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

for all in each quarter .

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

13. The Lifting Theorem in

Let be an -invariant in .

Then

but, when is -invariant in each ,
only for , which is far less than
for .

Let .

Then are positive, while is not positive but bounded,

Since ,

By the definition of it follows
only for

And since

defining
it is .

13.1. is bounded in .

Let the operator be defined in as

The two-weight inequality for is
or, equivalently,
where

Defining by , is -invariant in each quarter and is also non-negative.

By the Lifting theorem, there exists such that

while is only for .

By the F. and M. Riesz theorem,

and

Then, the necessary and sufficient condition for boundedness in with norm , is that for all Borel sets,

where .

In particular, is an absolutely continuous measure, for some .

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.

References

[BSV]    J. A. Ball, C. Sadosky, V. Vinnikov, Scattering systems with several evolutions and multidimensional input/state/output systems, Int.Eqs.Op.Theory 52 (2005), 323-393.        [ Links ]

[C1]    M. Cotlar, A combinatorial inequality and its applications to spaces, Rev. Mat. Cuyana 1 (1955), 41-55.        [ Links ]

[C2]    M. Cotlar, A general interpolation for linear operations, Rev. Mat. Cuyana 2 (1955), 57-84.        [ Links ]

[C3]    M. Cotlar, Some generalizations of Hardy-Littlewood maximal theorem, Rev. Mat. Cuyana 3 (1955), 85-104.        [ Links ]

[C4]    M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems. Rev. Mat. Cuyana 1 (1955), 105-167.        [ Links ]

[CF]    R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250.        [ Links ]

[CM]    R. R. Coifman and Y. Meyer, Calderón-Zygmund and multilinear operators, Cambridge University Press, Cambridge, 1997.        [ Links ]

[ChS]    D.-C. Chang and C. Sadosky, Functions of bounded mean oscillation, Taiwanese J. Math. 10(2006), 573-601.        [ Links ]

[CRW]    R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635.        [ Links ]

[CS1]    M. Cotlar and C. Sadosky, On some versions of the Helson-Szeg theorem, Conference on harmonic analysis in honor of Antoni Zygmund, Vol I, II (Chicago, IL, 1981), 306-317, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983.        [ Links ]

[CS2]    M. Cotlar and C. Sadosky, Integral representations of bounded Hankel forms defined in scattering systems with a multidimensional evolution group, in Contributions to Operator Theory and its Applications (Mesa, AZ, 1987) (Ed. I. Gohberg, J.W. Helton and L. Rodman), pp. 357-375, OT35 Birkhäuser, Basel-Boston, 1988.        [ Links ]

[CS3]    M. Cotlar and C. Sadosky, Generalized Bochner Theorem in algebraic scattering systems, in Analysis at Urbana vol.II, London Math. Soc. Lecture Notes Ser. 138, Cambridge Univ. Press, Cambridge, 1989, pp. 144-169.        [ Links ]

[CS4]    M. Cotlar and C. Sadosky, The Helson-Szegö theorem in of the bidimensional torus, Contemp. Math. 107 (1990), 19-37.        [ Links ]

[CS5]    M. Cotlar and C. Sadosky, Transference of metrics induced by unitary couplings, a Sarason theorem for the bidimensional torus, and a Sz.Nagy-Foias theorem for two pairs of dilations, J. Funct. Anal. 111(1993), 473-488.        [ Links ]

[F]    C. Fefferman, Characterization of bounded mean oscillation, Bull. Amer. Math. Soc. 77(1971), 587-588.        [ Links ]

[FS]    C. Fefferman and E.M. Stein, spaces of several variables, Acta Math. 129(1972), 137-193.        [ Links ]

[HS]    H. Helson and G. Szeg, A problem in prediction theory, Ann. Math. Pura Appl. 51 (1960), 107-138.        [ Links ]

[HMW]    R. Hunt, B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for the conjugate function and the Hilbert transform, Trans. Amer. Math. Soc.  176 (1973), 227-251.        [ Links ]

[JN]    F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426.        [ Links ]

[LP]    P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Math. Vol. 26, Academic Press, Boston, 1989, (second revised edition, first edition published in 1967).        [ Links ]

[S]    C. Sadosky, Liftings of kernels shift-invariant in scattering systems, in Holomorphic Spaces (Ed. S. Axler, J. E. McCarthy and D. Sarason), Mathematical Sciences Research Institute Publications, Vol. 33, Cambridge University Press, 1998, pp. 303-336.        [ Links ]

[Sa]    D. Sarason, On spectral sets having connected complement, Acta Sci. Math. (Szeged), 26 (1965), 289-299.        [ Links ]

Department of Mathematics
Howard University
Washington, DC, USA

Recibido: 8 de abril de 2008
Aceptado: 20 de diciembre de 2008

Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons