Precise (mathematical) formulations of Carleson's results

Convergence of Fourier series

Let f be an integrable function defined in the interval [-π,π]. We define the mth Fourier coefficient of f as

The nth partial sum of the function f is given by

Theorem (Carleson)

If f(x) is quadratically integrable, then s_n(x) converges toward f(x) almost everywhere.

L. Carleson: On Convergence and Growth of Partial Sums of Fourier Series. Acta Mathematica, Volume 116, pp. 135-157.

The Corona problem

Theorem (Carleson)

Let B be the Banach algebra of bounded analytic functions in the open unit disk in the complex plane under the natural norm. Let f₁,…,f_n be given functions in B so that

for a real number δ. Then I(f₁,…,f_n) = B.

L. Carleson: Interpolations by Bounded Analytic Functions and the Corona Problem. Annals of Mathematics, Volume 76, No. 3, November 1962, pp. 547-559.

An alternative formulation:

Let M be the set of maximal ideals in B (defined above). Since the quotient B/m for a maximal ideal m is isomorphic to the complex numbers C, we can identify m in a natural way with a homomorphism φ_m: B→C so that m=ker(φ_m). This induces a natural map π: M→D of M into the closed unit disk in the complex plane, given by π(φ)=φ(z) where z is the identity map of the disk. The continuity of φ ensures that |φ(z)|≤|z|≤1. For each element ω in the interior D⁰ of the unit disk, we can form the maximal ideal m=m_ω consisting of all functions vanishing in ω. Since z-ω lies in m then

and we get a natural embedding of the open unit disk D⁰ into M. The complement of D⁰ is mapped by π into the unit circle |ω|=1, and for each complex number ω of absolute value 1, we can form the fibre M_ω=π^-1(ω). This fibre contains homomorphisms that correspond to "evaluation in ω", where the quotes remind us that the homomorphisms are actually not defined at the boundary. Now, however, we have the following result (taken from K. Hoffman: Banach spaces of Analytic functions, Prentice-Hall, 1962):

Let f be a function in B and let ω be a point on the unit circle. Let {λ_n} be a sequence of points in the open unit disk D⁰ so that λ_n →ω. Assume further that the limit ζ of the sequence f (λ_n) also exists. Then there is a complex homomorphism φ in the fibre M_ω so that φ(f)=ζ.

Thus, a priori there may be a set of “evaluation homomorphisms” on the unit circle, corresponding to various ways of approaching the point ω. It is this manifold that has given rise to the name “the Corona problem”. Carleson's result states that D⁰ is dense in M, i.e. that the whole corona is contained in the closure of the open disk.
In Carleson's proof of the Corona theorem, he introduces a particular measure, which has been given the name Carleson measure by posterity.

Definition

Let μ be a non-negative measure of the open unit disk D⁰ in the complex plane and assume that

for all sets S of the form

Then μ is called a Carleson measure.

Carleson needs this definition in order to prove the following result. We let H^P, 1<p<∞ be the Banach space of bounded analytic functions f defined in the interior of the complex unit disk D⁰ under the norm

Theorem (Carleson)

Let μ be a non-negative measure of D⁰. Then μ is a Carleson measure if and only if

for every f in the space H^P, 1<p<∞ and where C_p is a constant.

Existence of a strange attractor for the Hénon map

The Hénon map is defined as the discrete-time dynamic system given by

We call the map T. In Hénon's original example, a=1,4 og b=0,3.

Theorem (Benedicks, Carleson)

Let W^u be the unstable manifold of T at its fixed points in x,y > 0. Then for all c < log2, there exists a b₀ > 0 such that for all 0 < b < b₀ there exists a set E(b) of positive one-dimensional Lebesgue measures such that for every a in the set E(b):

(i) There exists an open set U=U(a,b) such that for every z in U,

(ii) There is a point z₀=z₀(a,b) such that

   a. is dense in

   b.

M. Benedicks, L. Carleson: The dynamics of the Hénon Map. Annals of Mathematics, Volume 133, No. 1, 1991, pp. 73-169.

In this case, DT stands for the Jacobi matrix of T.

The Hénon attractor is fractal, smooth in one direction and a Cantor set in another. Numerical estimates yield a correlation dimension of 1.42 ± 0.02 (Grassberger, 1983) and a Hausdorff dimension of 1.26 ± 0.003 (Russel, 1980) for the attractor of the canonical map.

Simply stated, Benedicks-Carleson's result says that the Hénon map has so-called strange attractors for a non-empty (up to and including of positive measure) set of parameter values.

HomeNews ArchiveCalendar      Editor: Anne Marie Astad  The Norwegian Academy of Science and Letters  E-mail: dnva@online.no