The celebrated Carleson-Hunt theorem asserts that if is an function for some , then the partial Fourier series
of converge to almost everywhere. (The claim fails for , as shown by a famous counterexample of Kolmogorov.) The theorem follows easily from the inequality
where , and depends only on . Indeed, one first verifies Carleson’s theorem for a dense subclass of (e.g. the space of test functions) and then uses a standard limiting argument involving (1) (this is an example of the trick “give yourself an epsilon of room“).
The Carleson-Hunt theorem shows that converges as for almost every , but does not say much more about the nature of that convergence. One way to measure the strength of the convergence is to introduce the variational norms
for various . For this is the Carleson maximal function ; for this is the total variation of the sequence , which one can verify to be the norm of .
Our main result is to obtain the following variational strengthening of (1)
whenever and ; these conditions on are optimal. (For those readers familiar with martingales, the relationship of (2) to (1) is analogous to the relationship between Lepingle’s inequality (a variant of the more well known Doob’s inequality) and the Hardy-Littlewood maximal inequality.)
Because a sequence with finite -variation for some finite is necessarily convergent, this leads to a new proof of the Carleson-Hunt theorem without the need for a dense subclass. In particular, we obtain ergodic theory analogues of this result, in the case where no obvious dense subclass is available; more precisely, we obtain a new (and more “quantitative”) proof of a Wiener-Wintner-type theorem (first obtained by Lacey and Terwilleger), namely that given any measure-preserving group on a measure space , and a function for some , one has for almost every that for every real number , the integrals converge as for every (not merely almost every ).
The estimate (2) also provides a new proof of a result of Christ and Kiselev on the almost everywhere boundedness of eigenfunctions of Schrodinger operators with potentials with . Unfortunately, due to various endpoint issues, this barely fails to settle the endpoint case , a conjecture known as the nonlinear Carleson conjecture (discussed in this previous post).
The approach here follows the Lacey-Thiele approach to Carleson’s theorem (which is in turn based on an earlier approach of Fefferman), based on linearising the Carleson maximal function by picking the integer which attains the supremum , dividing phase space into “tiles”, and organising these tiles into “trees” and then into “forests” based on the distribution of the phase space “energy” of , together with the “mass” distribution of the graph of the function . One then needs to combine various “Bessel” type bounds on the energy, “Vitali-type” bounds on the mass, and “Calderon-Zygmund” type estimates on the trees together to obtain the result.
In our setting, the main new difficulty is that there are multiple integers associated to each point rather than one, which requires a more detailed analysis of the “multiplicity” of forests that was not present in earlier work. (Also, the Calderon-Zygmund estimates need to be replaced with Lepingle type estimates, though this is a relatively standard change, being first introduced in a paper of Bourgain.)