Carleson's theorem

Short description: 1966 result in mathematical analysis

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 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 f be an L^p periodic function for some p ∈ (1, ∞], with Fourier coefficients [math]\displaystyle{ \hat{f}(n) }[/math]. Then [math]\displaystyle{ \lim_{N \to \infty} \sum_{|n| \leq N} \hat{f}(n) e^{inx} = f(x) }[/math] for almost every x.

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

Let f ∈ L^p(R) for some p ∈ (1, 2] have Fourier transform [math]\displaystyle{ \hat{f}(\xi) }[/math]. Then [math]\displaystyle{ \lim_{R \to \infty} \int_{|\xi| \leq R} \hat{f}(\xi) e^{2 \pi i x \xi} \, d\xi = f(x) }[/math] 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 proven 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 postulated 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 s_n of the Fourier series of a function in L^p was [math]\displaystyle{ s_n(x)=o( \log (n)^{1/p})\text{ almost everywhere}. }[/math] In other words, the function s_n(x) can still grow to infinity at any given point x as one takes more and more terms of the Fourier series into account, though the growth must be quite slow (slower than the logarithm of n to a small power). This result 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 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) 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), 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 2014) 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 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 the non-linear operator defined by [math]\displaystyle{ Cf(x) = \sup_N\left|\int_{-N}^N \hat f(y)e^{2\pi i xy} \, dy\right| }[/math]

It is relatively easy to show that the Carleson–Hunt theorem follows from the boundedness of the Carleson operator from L^p(R) to itself for 1 < p < ∞. However, proving that it is bounded is difficult, and this was actually what Carleson proved.

See also

• Convergence of Fourier series

References

• Antonov, N. Yu. (1996), "Convergence of Fourier series", East Journal on Approximations 2 (2): 187–196 
• 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 
• Carleson, Lennart (1966), "On convergence and growth of partial sums of Fourier series", Acta Mathematica 116 (1): 135–157, doi:10.1007/BF02392815 
• Fefferman, Charles (1973), "Pointwise convergence of Fourier series", Annals of Mathematics, Second Series 98 (3): 551–571, doi:10.2307/1970917 
• Fremlin, David H. (2003), Measure theory, 2, Torres Fremlin, Colchester, ISBN 978-0-9538129-2-9, http://www.essex.ac.uk/maths/people/fremlin/mt.htm, retrieved 2010-09-09 
• 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. 
• 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 
• 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 
• 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, http://www.numdam.org/item?id=SB_1964-1966__9__491_0 
• Katznelson, Yitzhak (1966), "Sur les ensembles de divergence des séries trigonométriques", Studia Mathematica 26 (3): 301–304, doi:10.4064/sm-26-3-305-306, http://matwbn.icm.edu.pl/tresc.php?wyd=2&tom=26 
• Kolmogorov, Andrey Nikolaevich (1923), "Une série de Fourier–Lebesgue divergente presque partout", Fundamenta Mathematicae 4: 324–328, doi:10.4064/fm-4-1-324-328, https://eudml.org/doc/213617 
• 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, Bibcode: 2000SbMat.191...97K 
• Lacey, Michael T. (2004), "Carleson's theorem: proof, complements, variations", Publicacions Matemàtiques 48 (2): 251–307, doi:10.5565/publmat_48204_01 
• 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 
• 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, ISBN 978-3-540-05475-7  "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", Notices of the American Mathematical Society 54 (2): 223–229, https://www.ams.org/notices/200702/comm-carleson.pdf 
• 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, Bibcode: 1971ArM.....9...65S 
• Hazewinkel, Michiel, ed. (2001), "Carleson theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Zygmund, A. (2002), Trigonometric Series. Vol. I, II, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 978-0-521-89053-3