SciELO - Scientific Electronic Library Online

 
vol.49 issue2A survey on hyper-Kähler with torsion geometryErratum to "Some aspects of the history of applied mathematics in Argentina" author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Article

Indicators

  • Have no cited articlesCited by SciELO

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Bookmark


Revista de la Unión Matemática Argentina

On-line version ISSN 1669-9637

Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 2008

 

The Hilbert transform and scattering

Cora Sadosky

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

H : f ↦→ Hf = f * k
is given by convolution with the singular kernel
 1- k(x) = p.v.x
for x ∈ ℝ . (Analogues in 𝕋 and ℝn, n > 1. )

The basic result of Marcel Riesz (1927) is

H is bounded on  p L , 1 < p < ∞ .

Similar boundedness properties are valid in the "weighted" cases, both for H and for the iterated H = H1H2...Hn ,

H : Lp(μ) → Lp(ν), 1 < p < ∞
where μ, ν are general measures.

3. The scattering property of the analytic projector

Given  2 f ∈ L , we can decompose f as f1 + f2 , where f1 is analytic and f2 is antianalytic.

Under this decomposition the Hilbert transform can be written as

Hf = - i f1 + if2.

The analytic projector P , associated with the Hilbert tranform operator H , is defined as

 1 P f = P (f1 + f2) = f1, P = -(I + iH ). 2

The crucial observation is that P supports the shift operator S : f(x) ↦→ eix f(x) .

Then, the range of P is the set W1 of analytic functions, and its kernel is the set W2 of antianalytic functions:

 - 1 S W1 ⊂ W1 and S W2 ⊂ W2.

The "scattering property" of the 1 -dimensional Hilbert transform provides the framework for a theory of invariant forms in scattering systems, leading to two-weight  2 L -boundedness results for H .

The scattering properties are also essential to providing the two-weight L2 - boundedness of the product Hilbert transform in product spaces, where the analytic projectors supporting the n -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 n -dimensional Calderón-Zygmund singular integrals, which do not share the scattering property.

4. H is bounded on L2

In fact, H is an isometry on  2 L , since

∥Hf ∥2 = ∥f∥2

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 H in L2 . The same happens for p ⁄= 2 .

5. Two ways for  1 H : L to  p L , 1 < p < ∞

Start checking that H is weakly bounded in  1 L , and apply the Marcinkiewicz Interpolation Theorem between p = 1 and p = 2 , and then, by duality, pass from p = 2 to p < ∞ .

The "Magic Identity" for H, given by

(Hf )2 = f 2 + 2 H (f.Hf ) (*)
is valid for all functions f, smooth and with compact support.

Using extrapolation, since for  2 f ∈ L , (*) implies  2 Hf ∈ L , then f ∈ L4 implies Hf ∈ L4 , and  k f ∈ L2 implies  k Hf ∈ L2 , ∀k ≥ 1 .

The boundedness of H in Lp′, 1∕p + 1∕p ′ = 1, p = 2k , follows by duality, and interpolation gives the boundedness of H in Lp, 1 < p < ∞.

By polarization, the Magic Identity for an operator T becomes

T(f.T g + Tf.g) = T f.T g - f.g.

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  2 S , and deduced the theorem of Krein and Macaev in a way similar to the passage from  2 L to  p L 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. H is bounded in L2(ω ), 0 ≤ ω ∈ L1

6.1. Helson-Szeg~o theorem (1960).

 u+ Hv ω = e ,
where u, v are real-valued bounded functions, such that ∥v∥ ∞ < π ∕2

is equivalent to

logω = u + Hv ∈ BM O

(with a special BM O norm).

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

ω ∈ A2

is equivalent to

 ∫ ∫ ( 1- ω )(-1- -1 ) ≤ C, ∀ I interval |I| I |I| Iω

6.3. Remark.

Take note that, although both conditions are necessary and sufficient, the first one is good to produce such weights, while the second one is good at checking them.

7. Definition of u -boundedness

An operator T acting in a Banach lattice  0 X, T : X → L (Ω) is u-bounded if

∀ f ∈ X, ∥f ∥ ≤ 1, ∃ g ∈ X, g ≥ |f |, ∥g∥ + ∥T g∥ ≤ C.

The u -boundedness of operators is considerably weaker than boundedness. For example,

T is u -bounded on L ∞ if and only if T 1 ∈ L∞

.

Now we can translate another equivalence for the Helson-Szeg~o theorem for p=2:

There exist w ~ ω , real-valued functions, such that

|H w (x)| ≤ C w(x) a.e.

(Here w ~ ω means cω (t) ≤ w (t) ≤ C ω(t), ∀ t )

is equivalent to

 -1 Tω : f ↦→ ω H (ω f)

is u -bounded in L∞ .

8. H is bounded in  p L (𝕋; ω), 1 < p < ∞

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

ω ∈ Ap
is equivalent to
 1 ∫ 1 ∫ ( --- ω)(--- ω- 1∕(p-1))p-1 ≤ C , ∀ I interval |I| I |I| I

8.2. Cotlar-Sadosky Theorem (1982).

ω ∈ Ap,

defined by
T ω = ω-2∕pH (ω2 ∕p f), p ≥ 2,
 2∕p -2∕p T ω = ω H (ω f), p < 2,

is u -bounded in  * Lp where 1∕p* = |1 - 2∕p |.

9. H is bounded in weighted L2 (𝕋2;ω )

The following are equivalent:

1. The double Hilbert transform H = H1 H2 is bounded in  2 2 L (𝕋 ;ω )

2.

 u1+H1 v1 u2+H2 v2 ω = e = e , u1, u2, v1, v2,
are real-valued bounded functions, ∥vi∥∞ < π∕2, i = 1,2

3. log ω ∈ bmo (with a special bmo norm)

4. ∃ w1,w2, w1 ~ ω ~ w2 , such that

|H w (x)| ≤ C w (x), |H w (x)| ≤ C w (x ) a.e. 1 1 1 2 2 2

5.

T1 : f ↦→ ω- 1H1 (ω f )
and
T2 : f ↦→ ω- 1H2 (ω f )

are simultaneously u -bounded in L∞ (𝕋2)

6.

ω ∈ A * 2

10. H bounded in weighted Lp (𝕋2;ω), 1 < p < ∞

1. The double Hilbert transform H = H1 H2 is bounded in  p 2 L (𝕋 ;ω )

2. T1 and T2 are simultaneously u -bounded in  * Lp , where for i = 1,2,

 -2∕p 2∕p Ti : f ↦→ ω Hi (ω f), if 2 ≤ p < ∞
Ti : f ↦→ ω2∕pHi (ω -2∕pf), if 2 ≤ p < ∞
are simultaneously u -bounded in  p* 2 L (𝕋 ) .

3.

 * ω ∈ A p

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

Let V be a vector space, and σ be a linear isomorphism in V.

The subspaces of V , W+, and W - , are linear subspaces satisfying

 -1 σ W+ ⊂ W+, σ W - ⊂ W - .

Let B1,B2 : V × V → ℂ , be positive σ -invariant forms, such that

∀B0 : W+ × W - → ℂ ∋ ∀ (x,y) ∈ W+ × W -

 - 1 B0(σ x,y) = B0 (x,σ y),
|B (x,y)| ≤ B (x, x)1∕2 B (y,y )1∕2 0 1 2

Then,

∃B ′ : V × V → ℂ ∋ ∀(x,y ) ∈ V × V
such that
 ′ ′ B (σ x,σ y) = B (x, y),
or, equivalently,
 ′ ′ -1 B (σ x,y) = B (x, σ y).
Furthermore,
|B ′(x, y)| ≤ B1(x, x)1∕2 B2(y,y)1∕2
such that
 ′ B (x,y) = B0 (x,y), ∀(x, y) ∈ W+ × W - .

12. The Lifting Theorem for trigonometric polynomials

Assume now that V = P is the set of trigonometric polynomials on 𝕋, and P1, P2 are the sets of analytic and antianalytic polynomials in P, where σ = S is the shift operator.

The Herglotz-Bochner theorem then translates to

B is positive and S -invariant in P × P if and only if it exists a measure μ ≥ 0, such that

 ∫ B(f,g ) = f ¯gd μ ∀f, g ∈ P.

Since P = P +P 1 2 , the domain of B splits in four pieces P × P i j for i,j = 1,2 .

A weaker concept of S -invariance is

B(Sf, Sg ) = B (f, g),
for all (f, g) in each quarter Pi × Pj, i,j = 1,2 .

Then the Lifting theorem asserts that B is positive in P × P and S -invariant in each quarter if and only if there exists μ = (μij) ≥ 0 such that, for all f1, f2 ∈ P ∞ × P∈,

 ∑ ∫ B (f1 + f2, g1 + g2) = fi ¯gj d μij. i,j=1,2

Here (μij) ≥ 0 means that the (complex) measures satisfy  ---- μ11 ≥ 0, μ22 ≥ 0, μ21 = μ12 and

 2 |μ12(D )| ≤ μ11(D ) μ22(D ), ∀ D ⊂ 𝕋.

13. The Lifting Theorem in P × P

Let B be an S -invariant in P × P .

Then

B ≥ 0 ⇐ ⇒ μ ≥ 0,
but, when B is S -invariant in each Pi × Pj ,
 ∫ B ≥ 0 ⇐ ⇒ ∑ f ¯f dμ ≥ 0 i j ij i,j
only for f1 ∈ P1, f2 ∈ P2 , which is far less than
 ∑ ∫ μ ≥ 0 ⇐ ⇒ fi ¯fj dμij ≥ 0 i,j
for f1, f2 ∈ P .

Let B1 = B |P1 × P1, B2 = B |P2 × P2, B0 = B|P1 × P2 .

Then B1, B2 are positive, while B0 is not positive but bounded,

|B0 (f1,f2)| ≤ B1(f1,f1)1∕2B2 (f2,f2)1∕2.

Since μ11 ≥ 0, μ22 ≥ 0 ,

|B0 (f1,f2)| ≤ ∥f1∥L2(μ11)∥f2∥L2(μ22).
By the definition of B0 it follows
 ∫ B0(f1,f2) = f1f¯2 dμ12
only for f1 ∈ P1, f2 ∈ P2.

And since

 2 |μ12(D )| ≤ μ11(D ) μ22(D ), ∀ D ⊂ 𝕋,
defining
 ∫ ′ ¯ B (f1,f2) := f1 f2dμ12, ∀f1, f2 ∈ P
it is  ′ ∥B ∥ = ∥B0 ∥ .

13.1. H is bounded in  2 L (𝕋; ν,μ) .

Let the operator H be defined in P as

H (f1 + f2) = - if1 + if2.
The two-weight inequality for H is
∫ ∫ |Hf |2dμ ≤ M 2 |f|2dν 𝕋 𝕋
or, equivalently,
 ∑ ∫ (⋆) = fi ¯fj dρij ≥ 0, f1 ∈ P1, f2 ∈ P2 i,j=1,2 𝕋
where
 2 --- 2 ρ11 = ρ22 = M ν - μ, ρ12 = ρ21 = M ν + μ.

Defining by B(f,g ) = B (f1 + f2, g1 + g2) = (⋆) , B is S -invariant in each quarter and is also non-negative.

By the Lifting theorem, there exists (μij), i,j = 1,2, such that

ˆρii(n) = ˆμii(n),∀n ∈ ℤ,
while ˆρ12(n) = ˆμ12(n), is only for n < 0 .

By the F. and M. Riesz theorem,

 2 ---- 2 μ11 = μ22 = M ν - μ, μ12 = μ21 = M ν + μ - h,
and h ∈ H1 (𝕋 ).

Then, the necessary and sufficient condition for boundedness in  2 L (𝕋;ν, μ), with norm M , is that for all 0 ⊂ 𝕋 Borel sets,

 ∫ |(M 2ν + μ)(D ) - h dt| ≤ (M 2ν - μ )(D ) D
where h ∈ H1 (𝕋) .

In particular, μ is an absolutely continuous measure, dμ = w dt for some 0 ≤ w ∈ L1 .

When μ = ν = ω dt we return to the previous case:

The Hilbert transform H is a bounded operator in L2(ω ) with norm M

if and only if

 2 2 |(M + 1)ω (t) - h (t)| ≤ (M - 1)ω (t), a.e.𝕋,

for  1 h ∈ H (𝕋), 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  2 L 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  p L versions of the Helson-Szeg~o 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 Lp 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, Hp spaces of several variables, Acta Math. 129(1972), 137-193.        [ Links ]

[HS]    H. Helson and G. Szeg~o , 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 ]

 

Cora Sadosky
Department of Mathematics
Howard University
Washington, DC, USA
csadosky@howard.edu

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