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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932versão On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dez. 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.
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
![H : f ↦→ Hf = f * k](/img/revistas/ruma/v49n2/2a131x.png)
![1- k(x) = p.v.x](/img/revistas/ruma/v49n2/2a132x.png)
![x ∈ ℝ](/img/revistas/ruma/v49n2/2a133x.png)
![𝕋](/img/revistas/ruma/v49n2/2a134x.png)
![ℝn, n > 1.](/img/revistas/ruma/v49n2/2a135x.png)
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
,
![H : Lp(μ) → Lp(ν), 1 < p < ∞](/img/revistas/ruma/v49n2/2a1310x.png)
![μ, ν](/img/revistas/ruma/v49n2/2a1311x.png)
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
![Hf = - i f1 + if2.](/img/revistas/ruma/v49n2/2a1317x.png)
The analytic projector , associated with the Hilbert tranform operator
, is defined as
![1 P f = P (f1 + f2) = f1, P = -(I + iH ). 2](/img/revistas/ruma/v49n2/2a1320x.png)
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:
![- 1 S W1 ⊂ W1 and S W2 ⊂ W2.](/img/revistas/ruma/v49n2/2a1326x.png)
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
![∥Hf ∥2 = ∥f∥2](/img/revistas/ruma/v49n2/2a1337x.png)
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
![(Hf )2 = f 2 + 2 H (f.Hf ) (*)](/img/revistas/ruma/v49n2/2a1350x.png)
![f,](/img/revistas/ruma/v49n2/2a1351x.png)
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
![T(f.T g + Tf.g) = T f.T g - f.g.](/img/revistas/ruma/v49n2/2a1363x.png)
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.
6.1. Helson-Szeg theorem (1960).
![u+ Hv ω = e ,](/img/revistas/ruma/v49n2/2a1370x.png)
![u, v](/img/revistas/ruma/v49n2/2a1371x.png)
![∥v∥ ∞ < π ∕2](/img/revistas/ruma/v49n2/2a1372x.png)
is equivalent to
![logω = u + Hv ∈ BM O](/img/revistas/ruma/v49n2/2a1373x.png)
(with a special norm).
6.2. Hunt-Muckenhoupt-Wheeden theorem (1973).
![ω ∈ A2](/img/revistas/ruma/v49n2/2a1375x.png)
is equivalent to
![∫ ∫ ( 1- ω )(-1- -1 ) ≤ C, ∀ I interval |I| I |I| Iω](/img/revistas/ruma/v49n2/2a1376x.png)
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
![∀ f ∈ X, ∥f ∥ ≤ 1, ∃ g ∈ X, g ≥ |f |, ∥g∥ + ∥T g∥ ≤ C.](/img/revistas/ruma/v49n2/2a1382x.png)
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
![|H w (x)| ≤ C w(x) a.e.](/img/revistas/ruma/v49n2/2a1390x.png)
(Here means
)
is equivalent to
![-1 Tω : f ↦→ ω H (ω f)](/img/revistas/ruma/v49n2/2a1393x.png)
is
![u](/img/revistas/ruma/v49n2/2a1394x.png)
![L∞](/img/revistas/ruma/v49n2/2a1395x.png)
8.1. Hunt-Muckenhoupt-Wheeden Theorem (1973).
![ω ∈ Ap](/img/revistas/ruma/v49n2/2a1398x.png)
![1 ∫ 1 ∫ ( --- ω)(--- ω- 1∕(p-1))p-1 ≤ C , ∀ I interval |I| I |I| I](/img/revistas/ruma/v49n2/2a1399x.png)
8.2. Cotlar-Sadosky Theorem (1982).
![ω ∈ Ap,](/img/revistas/ruma/v49n2/2a13100x.png)
defined by
![T ω = ω-2∕pH (ω2 ∕p f), p ≥ 2,](/img/revistas/ruma/v49n2/2a13101x.png)
![2∕p -2∕p T ω = ω H (ω f), p < 2,](/img/revistas/ruma/v49n2/2a13102x.png)
is -bounded in
where
The following are equivalent:
1. The double Hilbert transform is bounded in
2.
![u1+H1 v1 u2+H2 v2 ω = e = e , u1, u2, v1, v2,](/img/revistas/ruma/v49n2/2a13110x.png)
![∥vi∥∞ < π∕2, i = 1,2](/img/revistas/ruma/v49n2/2a13111x.png)
3. (with a special
norm)
4. , such that
![|H w (x)| ≤ C w (x), |H w (x)| ≤ C w (x ) a.e. 1 1 1 2 2 2](/img/revistas/ruma/v49n2/2a13115x.png)
5.
![T1 : f ↦→ ω- 1H1 (ω f )](/img/revistas/ruma/v49n2/2a13116x.png)
![T2 : f ↦→ ω- 1H2 (ω f )](/img/revistas/ruma/v49n2/2a13117x.png)
are simultaneously -bounded in
6.
![ω ∈ A * 2](/img/revistas/ruma/v49n2/2a13120x.png)
1. The double Hilbert transform is bounded in
2. and
are simultaneously
-bounded in
, where for
![-2∕p 2∕p Ti : f ↦→ ω Hi (ω f), if 2 ≤ p < ∞](/img/revistas/ruma/v49n2/2a13130x.png)
![Ti : f ↦→ ω2∕pHi (ω -2∕pf), if 2 ≤ p < ∞](/img/revistas/ruma/v49n2/2a13131x.png)
![u](/img/revistas/ruma/v49n2/2a13132x.png)
![p* 2 L (𝕋 )](/img/revistas/ruma/v49n2/2a13133x.png)
3.
![* ω ∈ A p](/img/revistas/ruma/v49n2/2a13134x.png)
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
![-1 σ W+ ⊂ W+, σ W - ⊂ W - .](/img/revistas/ruma/v49n2/2a13141x.png)
Let , be positive
-invariant forms, such that
![- 1 B0(σ x,y) = B0 (x,σ y),](/img/revistas/ruma/v49n2/2a13145x.png)
![|B (x,y)| ≤ B (x, x)1∕2 B (y,y )1∕2 0 1 2](/img/revistas/ruma/v49n2/2a13146x.png)
Then,
![∃B ′ : V × V → ℂ ∋ ∀(x,y ) ∈ V × V](/img/revistas/ruma/v49n2/2a13147x.png)
![′ ′ B (σ x,σ y) = B (x, y),](/img/revistas/ruma/v49n2/2a13148x.png)
![′ ′ -1 B (σ x,y) = B (x, σ y).](/img/revistas/ruma/v49n2/2a13149x.png)
![|B ′(x, y)| ≤ B1(x, x)1∕2 B2(y,y)1∕2](/img/revistas/ruma/v49n2/2a13150x.png)
![′ B (x,y) = B0 (x,y), ∀(x, y) ∈ W+ × W - .](/img/revistas/ruma/v49n2/2a13151x.png)
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
![∫ B(f,g ) = f ¯gd μ ∀f, g ∈ P.](/img/revistas/ruma/v49n2/2a13161x.png)
Since , the domain of
splits in four pieces
for
.
A weaker concept of -invariance is
![B(Sf, Sg ) = B (f, g),](/img/revistas/ruma/v49n2/2a13167x.png)
![(f, g)](/img/revistas/ruma/v49n2/2a13168x.png)
![Pi × Pj, i,j = 1,2](/img/revistas/ruma/v49n2/2a13169x.png)
Then the Lifting theorem asserts that is positive in
and
-invariant in each quarter if and only if there exists
such that, for all
![∑ ∫ B (f1 + f2, g1 + g2) = fi ¯gj d μij. i,j=1,2](/img/revistas/ruma/v49n2/2a13175x.png)
Here means that the (complex) measures satisfy
and
![2 |μ12(D )| ≤ μ11(D ) μ22(D ), ∀ D ⊂ 𝕋.](/img/revistas/ruma/v49n2/2a13178x.png)
Let be an
-invariant in
.
Then
![B ≥ 0 ⇐ ⇒ μ ≥ 0,](/img/revistas/ruma/v49n2/2a13183x.png)
![B](/img/revistas/ruma/v49n2/2a13184x.png)
![S](/img/revistas/ruma/v49n2/2a13185x.png)
![Pi × Pj](/img/revistas/ruma/v49n2/2a13186x.png)
![∫ B ≥ 0 ⇐ ⇒ ∑ f ¯f dμ ≥ 0 i j ij i,j](/img/revistas/ruma/v49n2/2a13187x.png)
![f1 ∈ P1, f2 ∈ P2](/img/revistas/ruma/v49n2/2a13188x.png)
![∑ ∫ μ ≥ 0 ⇐ ⇒ fi ¯fj dμij ≥ 0 i,j](/img/revistas/ruma/v49n2/2a13189x.png)
![f1, f2 ∈ P](/img/revistas/ruma/v49n2/2a13190x.png)
Let .
Then are positive, while
is not positive but bounded,
![|B0 (f1,f2)| ≤ B1(f1,f1)1∕2B2 (f2,f2)1∕2.](/img/revistas/ruma/v49n2/2a13194x.png)
Since ,
![|B0 (f1,f2)| ≤ ∥f1∥L2(μ11)∥f2∥L2(μ22).](/img/revistas/ruma/v49n2/2a13196x.png)
![B0](/img/revistas/ruma/v49n2/2a13197x.png)
![∫ B0(f1,f2) = f1f¯2 dμ12](/img/revistas/ruma/v49n2/2a13198x.png)
![f1 ∈ P1, f2 ∈ P2.](/img/revistas/ruma/v49n2/2a13199x.png)
And since
![2 |μ12(D )| ≤ μ11(D ) μ22(D ), ∀ D ⊂ 𝕋,](/img/revistas/ruma/v49n2/2a13200x.png)
![∫ ′ ¯ B (f1,f2) := f1 f2dμ12, ∀f1, f2 ∈ P](/img/revistas/ruma/v49n2/2a13201x.png)
![′ ∥B ∥ = ∥B0 ∥](/img/revistas/ruma/v49n2/2a13202x.png)
Let the operator be defined in
as
![H (f1 + f2) = - if1 + if2.](/img/revistas/ruma/v49n2/2a13207x.png)
![H](/img/revistas/ruma/v49n2/2a13208x.png)
![∫ ∫ |Hf |2dμ ≤ M 2 |f|2dν 𝕋 𝕋](/img/revistas/ruma/v49n2/2a13209x.png)
![∑ ∫ (⋆) = fi ¯fj dρij ≥ 0, f1 ∈ P1, f2 ∈ P2 i,j=1,2 𝕋](/img/revistas/ruma/v49n2/2a13210x.png)
![2 --- 2 ρ11 = ρ22 = M ν - μ, ρ12 = ρ21 = M ν + μ.](/img/revistas/ruma/v49n2/2a13211x.png)
Defining by ,
is
-invariant in each quarter and is also non-negative.
By the Lifting theorem, there exists such that
![ˆρii(n) = ˆμii(n),∀n ∈ ℤ,](/img/revistas/ruma/v49n2/2a13216x.png)
![ˆρ12(n) = ˆμ12(n),](/img/revistas/ruma/v49n2/2a13217x.png)
![n < 0](/img/revistas/ruma/v49n2/2a13218x.png)
By the F. and M. Riesz theorem,
![2 ---- 2 μ11 = μ22 = M ν - μ, μ12 = μ21 = M ν + μ - h,](/img/revistas/ruma/v49n2/2a13219x.png)
![h ∈ H1 (𝕋 ).](/img/revistas/ruma/v49n2/2a13220x.png)
Then, the necessary and sufficient condition for boundedness in with norm
, is that for all
Borel sets,
![∫ |(M 2ν + μ)(D ) - h dt| ≤ (M 2ν - μ )(D ) D](/img/revistas/ruma/v49n2/2a13224x.png)
![h ∈ H1 (𝕋)](/img/revistas/ruma/v49n2/2a13225x.png)
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
![2 2 |(M + 1)ω (t) - h (t)| ≤ (M - 1)ω (t), a.e.𝕋,](/img/revistas/ruma/v49n2/2a13233x.png)
for which is the source to all the equivalences we mentioned above.
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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