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 $Φ$, there exists a Lebesgue integrable function whose Fourier series with respect to the system $Φ$ 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.

💡 Research Summary

The paper extends Bochkarev’s theorem—originally stating that for any uniformly bounded bi‑orthonormal system Φ there exists an (L^{1}) function whose Fourier series with respect to Φ diverges on a set of positive Lebesgue measure—to the setting of variable‑exponent Lebesgue spaces (L^{p(\cdot)}(