Divergent Fourier Series with Respect to Biorthonormal Systems in Function Spaces Near $L^1$

Abstract

In this paper, we generalize Bochkarev's theorem, which states that for any uniformly bounded biorthonormal system $\Phi$, there exists a Lebesgue integrable function whose Fourier series with respect to the system $\Phi$ diverges on a set of positive measure. We find the class of variable exponent Lebesgue spaces $L^{p(\cdot)}([0,1]^n)$, where $1 < p(x) < \infty$ almost everywhere on $[0,1]^n$, such that the aforementioned Bochkarev's theorem holds.