# Carleson's theorem

**Carleson's theorem** is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of *L*^{2} 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 *L*^{p} 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*L*^{p}periodic function for some*p*∈ (1, ∞), with Fourier coefficients . Then

- for almost every
*x*.

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

- Let
*ƒ*∈*L*^{p}(**R**) for some*p*∈ (1, ∞) have Fourier transform . Then

- 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 *L*^{2} function converges to it in *L*^{2} 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 *L*^{2} 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 *L*^{1} 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 *s*_{n} of the Fourier series of a function in *L*^{p} was

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 *L*^{1} 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 *L*^{2} 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 *L*^{p} 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 *L*log_{+}(*L*)log_{+}log_{+}(*L*) and by Antonov (1996) to the space *L*log_{+}(*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 *L*log_{+}(*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 *L*log_{+}(*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 *L*^{2}.

## The Carleson operator

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

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

## See also

## References

- Antonov, N. Yu. (1996), "Convergence of Fourier series",
*East Journal on Approximations*,**2**(2): 187–196, MR 1407066 - Arias de Reyna, Juan (2002),
*Pointwise convergence of Fourier series*, Lecture Notes in Mathematics,**1785**, Berlin, New York: Springer-Verlag, doi:10.1007/b83346, ISBN 978-3-540-43270-8, MR 1906800 - Carleson, Lennart (1966), "On convergence and growth of partial sums of Fourier series",
*Acta Mathematica*,**116**(1): 135–157, doi:10.1007/BF02392815, MR 0199631 - Fefferman, Charles (1973), "Pointwise convergence of Fourier series",
*Annals of Mathematics. Second Series*,**98**(3): 551–571, doi:10.2307/1970917, MR 0340926 - Fremlin, David H. (2003),
*Measure theory*,**2**, Torres Fremlin, Colchester, ISBN 978-0-9538129-2-9, MR 2462280 - Grafakos, Loukas (2014).
*Classical Fourier analysis*. Graduate Texts in Mathematics.**249**(Third ed.). New York: Springer-Verlag. doi:10.1007/978-1-4939-1194-3. ISBN 978-1-4939-1193-6. MR 3243734. - Grafakos, Loukas (2014).
*Modern Fourier analysis*. Graduate Texts in Mathematics.**250**(Third ed.). New York: Springer-Verlag. doi:10.1007/978-1-4939-1230-8. ISBN 978-1-4939-1229-2. MR 3243741. - Hunt, Richard A. (1968), "On the convergence of Fourier series",
*Orthogonal Expansions and their Continuous Analogues Proc. Conf., Edwardsville, Ill., 1967*, Carbondale, Ill.: Southern Illinois Univ. Press, pp. 235–255, MR 0238019 - Jørsboe, Ole G.; Mejlbro, Leif (1982),
*The Carleson-Hunt theorem on Fourier series*, Lecture Notes in Mathematics,**911**, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094072, ISBN 978-3-540-11198-6, MR 653477 - Kahane, Jean-Pierre (1995), "Sommes partielles des séries de Fourier (d'après L. Carleson)",
*Séminaire Bourbaki*,**9**, Paris: Société Mathématique de France, pp. 491–507, MR 1610981 - Katznelson, Yitzhak (1966), "Sur les ensembles de divergence des séries trigonométriques",
*Studia Mathematica*,**26**: 301–304, MR 0199632 - Kolmogorov, Andrey Nikolaevich (1923), "Une série de Fourier–Lebesgue divergente presque partout",
*Polska Akademia Nauk. Fundamenta Mathematicae*,**4**: 324–328 - Konyagin, S. V. (2000), "On the divergence everywhere of trigonometric Fourier series",
*Rossiĭskaya Akademiya Nauk. Matematicheskii Sbornik*,**191**(1): 103–126, doi:10.1070/sm2000v191n01abeh000449, MR 1753494 - Lacey, Michael T. (2004), "Carleson's theorem: proof, complements, variations",
*Publicacions Matemàtiques*,**48**(2): 251–307, arXiv:math/0307008, doi:10.5565/publmat_48204_01, MR 2091007 - Lacey, Michael; Thiele, Christoph (2000), "A proof of boundedness of the Carleson operator",
*Mathematical Research Letters*,**7**(4): 361–370, doi:10.4310/mrl.2000.v7.n4.a1, MR 1783613, archived from the original on 2008-07-05 - Luzin, N.N. (1915),
*The integral and trigonometric series (In Russian)*, Moscow-Leningrad (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212) - Mozzochi, Charles J. (1971),
*On the pointwise convergence of Fourier series*, Lecture Notes in Mathematics, Vol. 199,**199**, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0061167, MR 0445205 "This monograph is a detailed and essentially self-contained treatment of the work of Carleson and Hunt." - Raussen, Martin; Skau, Christian (2007), "Interview with Abel Prize recipient Lennart Carleson" (PDF),
*Notices of the American Mathematical Society*,**54**(2): 223–229, MR 2285126 - Sjölin, Per (1971), "Convergence almost everywhere of certain singular integrals and multiple Fourier series",
*Arkiv för Matematik*,**9**(1-2): 65–90, doi:10.1007/BF02383638, MR 0336222 - Telyakovskii, S.A. (2001), "c/c020450", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Zygmund, A. (2002) [1935],
*Trigonometric series. Vol. I, II*, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 978-0-521-89053-3, MR 1963498