A function in the Dirichlet space such that its Fourier series diverges almost everywhere

Abstract:

An analytic function $F$ on the disc belongs to $B$ if $||F|{|_B} = \int _0^1 {\int _0^{2\pi } {|{F’}(r{e^{i\theta }})|d\theta dr < \infty } }$. Notice that $B \varsubsetneqq {H^1} \varsubsetneqq {L^1}$, where ${H^1}$ is the Hardy space of all analytic functions $F$ so that \[ ||F|{|_{{H^1}}} = \sup \limits _{0 < r < 1} \int _0^{2\pi } {|F(r{e^{i\theta }})| d\theta < \infty ,} \] ${L^1}$ is the Lebesgue space of integrable functions on $[0,2\pi ]$, and the inclusion ${H^1} \varsubsetneqq {L^1}$ is taken in the sense of boundary values, that is, $F \in {H^1} \Rightarrow {\lim _{r \to 1}} - \Re F(r{e^{i\theta }}) \in {L^1}$. Kolmogorov in 1923 showed that there exists an $f$ in ${L^1}$ so that its Fourier series diverges almost everywhere. In 1953 Sunouchi showed that there exists an $f$ in ${H^1}$ with an almost everywhere divergent Fourier series. The purpose of this note is to announce. Theorem 1. There exists an $f \in B$, whose Fourier series diverges a.e. This problem was mentioned to the first author by Professor Guido Weiss.