On the Divergence of Fourier Series in the Spaces $\varphi(L)$ Containing $L$

The paper deals with the question of the divergence of Fourier series in function spaces wider than $L=L[-\pi,\pi]$, but narrower than $L^p=L^p[-\pi,\pi]$ for all $p\in(0,1)$. It is proved that the recent results of Filippov on the generalization to the space $\varphi(L)$ of Kolmogorov's theorem on the convergence of Fourier series in $L^p$, $p\in(0,1)$, cannot be improved.