Fonctionne avec Indico
v3.2.9
Departing from Lusin's conjecture on 1913 about the pointwise convergence of the Fourier series of L^2 periodic functions (proved positively by Carleson in 1966, and extended to L^p, p>1, by Hunt in 1967), as well as some paradigmatic results like a divergence one of Kolmogorov (there exists a periodic L^1 function whose partial Fourier sum S_Nf(x) is unbounded in N for a.e. x), we will explore this so-called Carleson problem, from its aforementioned original formulation to its version in PDE theory (the latter reading as what is the minimal Sobolev regularity on the initial data in order to guarantee a.e. pointwise convergence of the solution to the initial datum), touching some relevant techniques to deal with it. Once we have walked this path, I will briefly talk of a recent joint work with R. Lucà about such a Carleson problem for the cubic nonlinear Klein-Gordon equation, both from a deterministic and from a probabilistic point of view. We will see how s=1/2 and s=0 are, respectively, the barriers to get positive answers.