Almost Everywhere Convergent Fourier Series

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA . As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA .