A generalization of Kolmogorov’s theorem to biorthogonal systems

The fundamental Kolmogorov’s theorem about divergent trigonometric Fourier series is generalized to bounded biorthonormal systems defined on a separable metric space with Borel regular outer measure. Sharp lower bounds at points and on sets of positive measure are obtained for the arithmetic means of the symmetrized Lebesgue functions of biorthonormal systems defined on an arbitrary measure space. Earlier, similar results were obtained by the author for orthogonal systems on an interval.