Carleson's theorem

Not to be confused with Carlson's theorem.

Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L2 functions, proved by Lennart Carleson (1966). The name is also often used to refer to the extension of the result by Richard Hunt (1968) to Lp functions for p ∈ (1, ∞) (also known as the Carleson–Hunt theorem) and the analogous results for pointwise almost everywhere convergence of Fourier integrals, which can be shown to be equivalent by transference methods.

Statement of the theorem

The result, in the form of its extension by Hunt, can be formally stated as follows:

Let ƒ be an Lp periodic function for some p ∈ (1, ∞), with Fourier coefficients \hat{f}(n). Then
\lim_{N \rightarrow \infty} \sum_{|n| \leq N} \hat{f}(n) e^{inx} = f(x)
for almost every x.

The analogous result for Fourier integrals can be formally stated as follows:

Let ƒ ∈ Lp(R) for some p ∈ (1, ∞) have Fourier transform \hat{f}(\xi). Then
\lim_{R \rightarrow \infty} \int_{|\xi| \leq R} \hat{f}(\xi) e^{2 \pi i x \xi} \, d\xi = f(x)
for almost every x ∈ R.

History

A fundamental question about Fourier series, asked by Fourier himself at the beginning of the 19th century, is whether the Fourier series of a continuous function converges pointwise to the function.

By strengthening the continuity assumption slightly one can easily show that the Fourier series converges everywhere. For example, if a function has bounded variation then its Fourier series converges everywhere to the local average of the function. In particular, if a function is continuously differentiable then its Fourier series converges to it everywhere. This was proved by Dirichlet, who expressed his belief that he would soon be able to extend his result to cover all continuous functions. Another way to obtain convergence everywhere is to change the summation method. For example, Fejér's theorem shows that if one replaces ordinary summation by Cesàro summation then the Fourier series of any continuous function converges uniformly to the function. Further, it is easy to show that the Fourier series of any L2 function converges to it in L2 norm.

After Dirichlet's result, several experts, including Dirichlet, Riemann, Weierstrass and Dedekind, stated their belief that the Fourier series of any continuous function would converge everywhere. This was disproved by Paul du Bois-Reymond, who showed in 1876 that there is a continuous function whose Fourier series diverges at one point.

The almost-everywhere convergence of Fourier series for L2 functions was conjectured by N. N. Luzin (1915), and the problem was known as Luzin's conjecture (up until its proof by Carleson (1966)). Kolmogorov (1923) showed that the analogue of Carleson's result for L1 is false by finding such a function whose Fourier series diverges almost everywhere (improved slightly in 1926 to diverging everywhere). Before Carleson's result, the best known estimate for the partial sums sn of the Fourier series of a function in Lp was

s_n(x)=o( \log (n)^{1/p})\text{ almost everywhere}, \,

proved by Kolmogorov–Seliverstov–Plessner for p = 2, by G. H. Hardy for p = 1, and by Littlewood–Paley for p > 1 (Zygmund 2002). This result had not been improved for several decades, leading some experts to suspect that it was the best possible and that Luzin's conjecture was false. Kolmogorov's counterexample in L1 was unbounded in any interval, but it was thought to be only a matter of time before a continuous counterexample was found. Carleson said in an interview with Raussen & Skau (2007) that he started by trying to find a continuous counterexample and at one point thought he had a method that would construct one, but realized eventually that his approach could not work. He then tried instead to prove Luzin's conjecture since the failure of his counterexample convinced him that it was probably true.

Carleson's original proof is exceptionally hard to read, and although several authors have simplified the argument there are still no easy proofs of his theorem. Expositions of the original paper Carleson (1966) include Kahane (1995), Mozzochi (1971), Jørsboe & Mejlbro (1982), and Arias de Reyna (2002). Charles Fefferman (1973) published a new proof of Hunt's extension which proceeded by bounding a maximal operator. This, in turn, inspired a much simplified proof of the L2 result by Michael Lacey and Christoph Thiele (2000), explained in more detail in Lacey (2004). The books Fremlin (2003) and Grafakos (2009) also give proofs of Carleson's theorem.

Katznelson (1966) showed that for any set of measure 0 there is a continuous periodic function whose Fourier series diverges at all points of the set (and possibly elsewhere). When combined with Carleson's theorem this shows that there is a continuous function whose Fourier series diverges at all points of a given set of reals if and only if the set has measure 0.

The extension of Carleson's theorem to Lp for p > 1 was stated to be a "rather obvious" extension of the case p = 2 in Carleson's paper, and was proved by Hunt (1968). Carleson's result was improved further by Sjölin (1971) to the space Llog+(L)log+log+(L) and by Antonov (1996) to the space Llog+(L)log+log+log+(L). (Here log+(L) is log(L) if L>1 and 0 otherwise, and if φ is a function then φ(L) stands for the space of functions f such that φ(f(x)) is integrable.)

Konyagin (2000) improved Kolmogorov's counterexample by finding functions with everywhere-divergent Fourier series in a space slightly larger than Llog+(L)1/2. One can ask if there is in some sense a largest natural space of functions whose Fourier series converge almost everywhere. The simplest candidate for such a space that is consistent with the results of Antonov and Konyagin is Llog+(L).

The extension of Carleson's theorem to Fourier series and integrals in several variables is made more complicated as there are many different ways in which one can sum the coefficients; for example, one can sum over increasing balls, or increasing rectangles. Convergence of rectangular partial sums (and indeed general polygonal partial sums) follows from the one-dimensional case, but the spherical summation problem is still open for L2.

The Carleson operator

The Carleson operator C is a non-linear operator defined by

Cf(x) = \sup_N\left|\int_{-N}^N \hat f(y)e^{2\pi i xy} \, dy\right|

A fundamental property of the Carleson operator is that it is a bounded (non-linear) map from Lp(R) to itself for 1 < p < ∞. The Carleson–Hunt theorem follows easily from this (and in fact from slightly weaker estimates).

See also

References