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Camil Muscalu, Christoph Thiele and I have just uploaded to the arXiv our joint paper, “Multi-linear multipliers associated to simplexes of arbitrary length“, submitted to Analysis & PDE. This paper grew out of our project from many years ago to attempt to prove the nonlinear (or “scattering”) version of Carleson’s theorem on the almost everywhere convergence of Fourier series. This version is still open; our original approach was to handle the nonlinear Carleson operator by multilinear expansions in terms of the potential function V, but while the first three terms of this expansion were well behaved, the fourth term was unfortunately divergent, due to the unhelpful location of a certain minus sign. [This survey by Michael Lacey, as well as this paper of ourselves, covers some of these topics.]

However, what we did find out in this paper was that if we modified the nonlinear Carleson operator slightly, by replacing the underlying Schrödinger equation by a more general AKNS system, then for “generic” choices of this system, the problem of the ill-placed minus sign goes away, and each term in the multilinear series is, in fact, convergent (though we did not yet verify that the series actually converged, though in view of the earlier work of Christ and Kiselev on this topic, this seems likely). The verification of this convergence (at least with regard to the scattering data, rather than the more difficult analysis of the eigenfunctions) is the main result of our current paper. It builds upon our earlier estimates of the bilinear term in the expansion (which we dubbed the “biest”, as a multilingual pun). The main new idea in our earlier paper was to decompose the relevant region of frequency space $\{ (\xi_1,\xi_2,\xi_3) \in {\Bbb R}^3: \xi_1 < \xi_2 < \xi_3 \}$ into more tractable regions, a typical one being the region in which $\xi_2$ was much closer to $\xi_1$ than to $\xi_3$. The contribution of each region can then be “parafactored” into a “paracomposition” of simpler operators, such as the bilinear Hilbert transform, which can be treated by standard time-frequency analysis methods. (Much as a paraproduct is a frequency-restricted version of a product, the paracompositions that arise here are frequency-restricted versions of composition.)

A similar analysis happens to work for the multilinear operators associated to the frequency region $S := \{ (\xi_1,\ldots,\xi_n): \xi_1 < \ldots < \xi_n \}$, but the combinatorics are more complicated; each of the component frequency regions has to be indexed by a tree (in a manner reminiscent of the well-separated pairs decomposition), and a certain key “weak Bessel inequality” becomes considerably more delicate. Our ultimate conclusion is that the multilinear operator

$T(V_1,\ldots,V_n) := \int_{(\xi_1,\ldots,\xi_n) \in S} \hat V_1(\xi_1) \ldots \hat V_n(\xi_n) e^{2i (\xi_1+\ldots+\xi_n) x}\ d\xi_1 \ldots d\xi_n$ (1)

(which generalises the bilinear Hilbert transform and the biest) obeys Hölder-type $L^p$ estimates (note that Hölder’s inequality related to the situation in which the (projective) simplex S is replaced by the entire frequency space ${\Bbb R}^n$).

For the remainder of this post, I thought I would describe the “nonlinear Carleson theorem” conjecture, which is still one of my favourite open problems, being an excellent benchmark for measuring progress in the (still nascent) field of “nonlinear Fourier analysis“, while also being of interest in its own right in scattering and spectral theory.