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This set of notes discusses aspects of one of the oldest questions in Fourier analysis, namely the nature of convergence of Fourier series.

If ${f: {\bf R}/{\bf Z} \rightarrow {\bf C}}$ is an absolutely integrable function, its Fourier coefficients ${\hat f: {\bf Z} \rightarrow {\bf C}}$ are defined by the formula

$\displaystyle \hat f(n) := \int_{{\bf R}/{\bf Z}} f(x) e^{-2\pi i nx}\ dx.$

If ${f}$ is smooth, then the Fourier coefficients ${\hat f}$ are absolutely summable, and we have the Fourier inversion formula

$\displaystyle f(x) = \sum_{n \in {\bf Z}} \hat f(n) e^{2\pi i nx}$

where the series here is uniformly convergent. In particular, if we define the partial summation operators

$\displaystyle S_N f(x) := \sum_{|n| \leq N} \hat f(n) e^{2\pi i nx}$

then ${S_N f}$ converges uniformly to ${f}$ when ${f}$ is smooth.

What if ${f}$ is not smooth, but merely lies in an ${L^p({\bf R}/{\bf Z})}$ class for some ${1 \leq p \leq \infty}$? The Fourier coefficients ${\hat f}$ remain well-defined, as do the partial summation operators ${S_N}$. The question of convergence in norm is relatively easy to settle:

Exercise 1
• (i) If ${1 < p < \infty}$ and ${f \in L^p({\bf R}/{\bf Z})}$, show that ${S_N f}$ converges in ${L^p({\bf R}/{\bf Z})}$ norm to ${f}$. (Hint: first use the boundedness of the Hilbert transform to show that ${S_N}$ is bounded in ${L^p({\bf R}/{\bf Z})}$ uniformly in ${N}$.)
• (ii) If ${p=1}$ or ${p=\infty}$, show that there exists ${f \in L^p({\bf R}/{\bf Z})}$ such that the sequence ${S_N f}$ is unbounded in ${L^p({\bf R}/{\bf Z})}$ (so in particular it certainly does not converge in ${L^p({\bf R}/{\bf Z})}$ norm to ${f}$. (Hint: first show that ${S_N}$ is not bounded in ${L^p({\bf R}/{\bf Z})}$ uniformly in ${N}$, then apply the uniform boundedness principle in the contrapositive.)

The question of pointwise almost everywhere convergence turned out to be a significantly harder problem:

Theorem 2 (Pointwise almost everywhere convergence)
• (i) (Kolmogorov, 1923) There exists ${f \in L^1({\bf R}/{\bf Z})}$ such that ${S_N f(x)}$ is unbounded in ${N}$ for almost every ${x}$.
• (ii) (Carleson, 1966; conjectured by Lusin, 1913) For every ${f \in L^2({\bf R}/{\bf Z})}$, ${S_N f(x)}$ converges to ${f(x)}$ as ${N \rightarrow \infty}$ for almost every ${x}$.
• (iii) (Hunt, 1967) For every ${1 < p \leq \infty}$ and ${f \in L^p({\bf R}/{\bf Z})}$, ${S_N f(x)}$ converges to ${f(x)}$ as ${N \rightarrow \infty}$ for almost every ${x}$.

Note from Hölder’s inequality that ${L^2({\bf R}/{\bf Z})}$ contains ${L^p({\bf R}/{\bf Z})}$ for all ${p\geq 2}$, so Carleson’s theorem covers the ${p \geq 2}$ case of Hunt’s theorem. We remark that the precise threshold near ${L^1}$ between Kolmogorov-type divergence results and Carleson-Hunt pointwise convergence results, in the category of Orlicz spaces, is still an active area of research; see this paper of Lie for further discussion.

Carleson’s theorem in particular was a surprisingly difficult result, lying just out of reach of classical methods (as we shall see later, the result is much easier if we smooth either the function ${f}$ or the summation method ${S_N}$ by a tiny bit). Nowadays we realise that the reason for this is that Carleson’s theorem essentially contains a frequency modulation symmetry in addition to the more familiar translation symmetry and dilation symmetry. This basically rules out the possibility of attacking Carleson’s theorem with tools such as Calderón-Zygmund theory or Littlewood-Paley theory, which respect the latter two symmetries but not the former. Instead, tools from “time-frequency analysis” that essentially respect all three symmetries should be employed. We will illustrate this by giving a relatively short proof of Carleson’s theorem due to Lacey and Thiele. (There are other proofs of Carleson’s theorem, including Carleson’s original proof, its modification by Hunt, and a later time-frequency proof by Fefferman; see Remark 18 below.)

Richard Oberlin, Andreas Seeger, Christoph Thiele, Jim Wright, and I have just uploaded to the arXiv our paper “A variation norm Carleson theorem“, submitted to J. Europ. Math. Soc..

The celebrated Carleson-Hunt theorem asserts that if $f: {\Bbb R}/{\Bbb Z} \to {\Bbb C}$ is an $L^p$ function for some $1 < p \leq \infty$, then the partial Fourier series

$S_n f(x) := \sum_{k=-n}^n \hat f(k) e^{2\pi ikx}$

of $f$ converge to $f$ almost everywhere.  (The claim fails for $p=1$, as shown by a famous counterexample of Kolmogorov.)  The theorem follows easily from the inequality

$\| \sup_n |S_n f(x)| \|_{L^p({\Bbb R}/{\Bbb Z})} \leq C_p \| f \|_{L^p({\Bbb R}/{\Bbb Z})},$ (1)

where $1 < p < \infty$, and $C_p$ depends only on $p$.  Indeed, one first verifies Carleson’s theorem for a dense subclass of $L^p$ (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 $S_n f(x)$ converges as $n \to \infty$ for almost every $x$, 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

${\mathcal V}^r (S_n f(x))_{n = 0}^\infty := \sup_{n_1 \leq \ldots \leq n_k} (\sum_{j=1}^{k-1} |S_{n_{j+1}} f(x) - S_{n_j} f(x)|^r)^{1/r}$

for various $1 \leq r \leq \infty$.  For $r = \infty$ this is the Carleson maximal function $\sup_n |S_n f(x)|$; for $r=1$ this is the total variation of the sequence $S_n f(x)$, which one can verify to be the $\ell^1$ norm of $\hat f$.

Our main result is to obtain the following variational strengthening of (1)

$\| {\mathcal V}^r S_n f(x) \|_{L^p({\Bbb R}/{\Bbb Z})} \leq C_{p,r} \| f \|_{L^p({\Bbb R}/{\Bbb Z})}$ (2)

whenever $r > 2$ and $r/(r-1) < p < \infty$; these conditions on $p,r$ 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 $r$-variation for some finite $r$ 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 $(T_t)_{t \in {\Bbb R}}$ on a measure space $X$, and a function $f \in L^p(X)$ for some $p>1$, one has for almost every $x \in X$ that for every real number $\theta$, the integrals $\int_{\varepsilon \leq |t| \leq N} T^t f(x) e^{i\theta t}/t\ dt$ converge as $\varepsilon \to 0, N \to \infty$ for every $\theta$ (not merely almost every $\theta$).

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 $L^p$ potentials with $p<2$.  Unfortunately, due to various endpoint issues, this barely fails to settle the endpoint case $p=2$, 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 $n = n(x)$ which attains the supremum $\sup_n |S_n f(x)|$, 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 $f$, together with the “mass” distribution of the graph of the function $x \mapsto n(x)$.  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 $n_1(x),\ldots,n_{k(x)}(x)$ 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.)

Ciprian Demeter, Michael Lacey, Christoph Thiele and I have just uploaded our joint paper, “The Walsh model for $M_2^*$ Carleson” to the arXiv. This paper (which was recently accepted for publication in Revista Iberoamericana) establishes a simplified model for the key estimate (the “$M_2^*$ Carleson estimate”) in another (much longer) paper of ours on the return times theorem of Bourgain, in which the Fourier transform is replaced by its dyadic analogue, the Walsh-Fourier transform. This model estimate is established by the now-standard techniques of time-frequency analysis: one decomposes the expression to be estimated into a sum over tiles, and then uses combinatorial stopping time arguments into group the tiles into trees, and the trees into forests. One then uses (phase-space localised, and frequency-modulated) versions of classical Calderòn-Zygmund theory (or in this particular case, a certain maximal Fourier inequality of Bourgain) to control individual trees and forests, and sums up over the trees and forests using orthogonality methods (excluding an exceptional set if necessary).

Rather than discuss time-frequency analysis in detail here, I thought I would dwell instead on the return times theorem, and sketch how it is connected to the $M_2^*$ Carleson estimate; this is a more complicated version of the “$M_2$ Carleson estimate”, which is an estimate which is logically equivalent to Carleson’s famous theorem (and its extension by Hunt) on the almost everywhere convergence of Fourier series.