# Inverse theorems for Gowers norms for unbounded functions

The inverse theorem for Gowers norm over finite fields says that if a bounded function $$f: V \to C$$ where $$V$$ is a vector space over the finite field $$\mathbb{F}$$, has large Gowers uniformity norm $$\|f\|_{U^{s+1}(V)}$$, then there exists a non-classical polynomial $$P: V \to T$$ of degree at most $$s$$ such that $$f$$ has high correlation with the phase $$e^{2 \pi i P}$$.

My question is whether or not the condition that $$f$$ be bounded can be relaxed, if so to what extent?

For example, in the case of $$s = 2$$, it seems to me that it is enough to assume that $$\mathbb{E}[f^2]$$ is bounded, by some slight modifications to the proof presented in Ben Green's notes here - https://arxiv.org/abs/math/0604089. Because the proofs for $$s > 2$$ are so different, it's not clear to me if the same should hold for these cases. Additionally, I don't see much discussion about this requirement that $$f$$ be bounded in the literature, so I'm not sure if I'm missing something obvious.