The $L\log L$ Fourier-convergence conjecture

The explanation

Take a square wave. It equals 1 on the first half of a circle and −1 on the second half. Fourier's method rebuilds it from smooth waves of rising frequency. Adding the waves up to frequency N gives a partial sum. At the quarter point, where the true value is 1, the partial sums swing around it. Up to frequency 1 the sum is about 1.273. Up to 3 it is about 0.849, and up to 5 about 1.104. By 101 it is about 1.0062, and by 1001 about 1.0006. The sums close in on the true value.

Does this always happen? Here "always" can only mean almost everywhere. That means at every point except a set of total length zero. A few stray points may misbehave. The question is which functions allow more than that. The paper uses the ordinary partial sums, which take every frequency from −N to N. It asks about the full sequence N = 0, 1, 2 and so on, not a hand-picked subsequence.

The answer depends on how wild the function may be, especially in its tall spikes. Picture a spike of height about 22,026, which is e to the power 10. Let it cover one part in 22,026 of the circle. Its total area is 1. Square the height before averaging, and the spike scores about 22,026. Multiply the height by its logarithm instead, and it scores only about 10. That second score defines the class L log L. A function belongs to it when the average of |f| times log(2 + |f|) is finite.

Carleson proved almost-everywhere convergence for square-integrable functions in 1966. In 1968, Hunt extended it to functions whose |f| raised to any fixed power above 1 has finite average. In 1923, Kolmogorov's counterexample showed that the conclusion cannot hold for every integrable function. Integrable means only that the average of |f| is finite. Between these extremes, progress came in small steps above L log L. Sjölin reached a class with an extra log log factor in 1969. Antonov reached one with an extra log log log factor in 1996. Arias de Reyna and later Carro, Mastyło and Rodríguez-Piazza built larger spaces still. A 2017 paper by Lie states the L log L conjecture through a bound on the largest partial sum.

This paper proves that the L log L condition is enoughSource 1. For every complex-valued function in L log L, the partial sums converge to the function almost everywhereSource 2. The convergence holds along the full sequence of partial sumsSource 3. The paper also derives a weak bound on the largest partial sum, measured in a standard L log L normSource 4.

The obstacle is that the partial sums must be controlled for every frequency at once. A bound that grows with the number of frequencies is useless hereSource 5. The paper reduces the problem to one spike of height at most e to the power d. Away from a small exceptional set, it shows the largest partial sum has average at most a constant times dSource 6. That cost grows like the logarithm of the height, which is exactly what L log L can pay forSource 7. To get there, the proof counts information in the sense of information theorySource 8. It splits the input into pieces and charges the cost of locating the right frequencies to that information countSource 9.

Go back to the spike. The theorem lets a function carry many such spikes, with total L log L score finite. Its partial sums still settle on the right value at almost every point. Kolmogorov's example shows that finite area alone is not enough. The paper does not claim that L log L is the largest class that works. It also gives no rate for how fast the partial sums settle.

How sure is this? A computer proof checker called Lean has a formalization of the main theorem. It covers every measurable complex-valued function on the circle whose |f| times log(2 + |f|) has a finite integral. The weak maximal bound is not in the Lean scope note. No specialist review has been published.

What it doesn't show. This does not cover every integrable function or claim that L log L is the largest class that works.