Pointwise convergence of Fourier series and a Calderon-Zygmund decomposition for modulation invariant singular integrals

The Calderon-Zygmund decomposition is a fundamental tool in
the analysis of singular integral operators, prime examples of which are
the Hilbert and Riesz transforms occurring, for instance, in elliptic
PDEs. Its prime purpose is the extension of L^2 boundedness results for
operators of this type to (all) other L^p spaces. The exploited
mechanism is the extra cancellation occurring when the singular integral
operator is applied to functions with mean zero.

On the other hand, pointwise convergence of Fourier series of L^p
functions is governed by L^p bounds for maximally modulated versions of
singular integrals, for which the single zero frequency of f plays no
particular role. In this talk, we describe a novel Calderon-Zygmund
decomposition adapted to the maximally modulated setting and its
applications to both pointwise convergence of Fourier series of
functions near L^1 and sharp bounds for the bilinear Hilbert transform
near the critical exponent L^{2/3}. The presentation will be
non-technical and suitable for an advanced undergraduate audience.

Partly joint work with Ciprian Demeter (IU), Christoph Thiele (Bonn),
Andrei Lerner (Bar-Ilan U, Israel) and Yumeng Ou (Brown U).