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

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