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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.)

Jordan Ellenberg, Richard Oberlin, and I have just uploaded to the arXiv the paper “The Kakeya set and maximal conjectures for algebraic varieties over finite fields“, submitted to Mathematika.  This paper builds upon some work of Dvir and later authors on the Kakeya problem in finite fields, which I have discussed in this earlier blog post.  Dvir established the following:

Kakeya set conjecture for finite fields. Let F be a finite field, and let E be a subset of $F^n$ that contains a line in every direction.  Then E has cardinality at least $c_n |F|^n$ for some $c_n > 0$.

The initial argument of Dvir gave $c_n = 1/n!$.  This was improved to $c_n = c^n$ for some explicit $0 < c < 1$ by Saraf and Sudan, and recently to $c_n =1/2^n$ by Dvir, Kopparty, Saraf, and Sudan, which is within a factor 2 of the optimal result.

In our work we investigate a somewhat different set of improvements to Dvir’s result.  The first concerns the Kakeya maximal function $f^*: {\Bbb P}^{n-1}(F) \to {\Bbb R}$ of a function $f: F^n \to {\Bbb R}$, defined for all directions $\xi \in {\Bbb P}^{n-1}(F)$ in the projective hyperplane at infinity by the formula

$f^*(\xi) = \sup_{\ell // \xi} \sum_{x \in \ell} |f(x)|$

where the supremum ranges over all lines $\ell$ in $F^n$ oriented in the direction $\xi$.  Our first result is the endpoint $L^p$ estimate for this operator, namely

Kakeya maximal function conjecture in finite fields. We have $\| f^* \|_{\ell^n({\Bbb P}^{n-1}(F))} \leq C_n |F|^{(n-1)/n} \|f\|_{\ell^n(F^n)}$ for some constant $C_n > 0$.

This result implies Dvir’s result, since if f is the indicator function of the set E in Dvir’s result, then $f^*(\xi) = |F|$ for every $\xi \in {\Bbb P}^{n-1}(F)$.  However, it also gives information on more general sets E which do not necessarily contain a line in every direction, but instead contain a certain fraction of a line in a subset of directions.  The exponents here are best possible in the sense that all other $\ell^p \to \ell^q$ mapping properties of the operator can be deduced (with bounds that are optimal up to constants) by interpolating the above estimate with more trivial estimates.  This result is the finite field analogue of a long-standing (and still open) conjecture for the Kakeya maximal function in Euclidean spaces; we rely on the polynomial method of Dvir, which thus far has not extended to the Euclidean setting (but note the very interesting variant of this method by Guth that has established the endpoint multilinear Kakeya maximal function estimate in this setting, see this blog post for further discussion).

It turns out that a direct application of the polynomial method is not sufficient to recover the full strength of the maximal function estimate; but by combining the polynomial method with the Nikishin-Maurey-Pisier-Stein “method of random rotations” (as interpreted nowadays by Stein and later by Bourgain, and originally inspired by the factorisation theorems of Nikishin, Maurey, and Pisier), one can already recover a “restricted weak type” version of the above estimate.  If one then enhances the polynomial method with the “method of multiplicities” (as introduced by Saraf and Sudan) we can then recover the full “strong type” estimate; a few more details below the fold.

It turns out that one can generalise the above results to more general affine or projective algebraic varieties over finite fields.  In particular, we showed

Kakeya maximal function conjecture in algebraic varieties. Suppose that $W \subset {\Bbb P}^N$ is an (n-1)-dimensional algebraic variety.  Let $d \geq 1$ be an integer. Then we have

$\| \sup_{\gamma \ni x; \gamma \not \subset W} \sum_{y \in \gamma} f(y) \|_{\ell^n_x(W(F))} \leq C_{n,d,N,W} |F|^{(n-1)/n} \|f\|_{\ell^n({\Bbb P}^N(F))}$

for some constant $C_{n,d,N,W} > 0$, where the supremum is over all irreducible algebraic curves $\gamma$ of degree at most d that pass through x but do not lie in W, and W(F) denotes the F-points of W.

The ordinary Kakeya maximal function conjecture corresponds to the case when N=n, W is the hyperplane at infinity, and the degree d is equal to 1.  One corollary of this estimate is a Dvir-type result: a subset of ${\Bbb P}^N(F)$ which contains, for each x in W, an irreducible algebraic curve of degree d passing through x but not lying in W, has cardinality $\gg |F|^n$ if $|W| \gg |F|^{n-1}$.  (In particular this implies a lower bound for Nikodym sets worked out by Li.)  The dependence of the implied constant on W is only via the degree of W.

The techniques used in the flat case can easily handle curves $\gamma$ of higher degree (provided that we allow the implied constants to depend on d), but the method of random rotations does not seem to work directly on the algebraic variety W as there are usually no symmetries of this variety to exploit.  Fortunately, we can get around this by using a “random projection trick” to “flatten” W into a hyperplane (after first expressing W as the zero locus of some polynomials, and then composing with the graphing map for such polynomials), reducing the non-flat case to the flat case.

Below the fold, I wish to sketch two of the key ingredients in our arguments, the random rotations method and the random projections trick.  (We of course also use some algebraic geometry, but mostly low-tech stuff, on the level of Bezout’s theorem, though we do need one non-trivial result of Kleiman (from SGA6), that asserts that bounded degree varieties can be cut out by a bounded number of polynomials of bounded degree.)

[Update, March 14: See also Jordan's own blog post on our paper.]