  We study Sobolev $H^s(\mathbb{R}^n)$, $s \in \mathbb{R}$, stability of the
Fourier phase problem to recover $f$ from the knowledge of $|\hat{f}|$ with an
additional Bessel potential $H^{t,p}(\mathbb{R}^n)$ a priori estimate when $t
\in \mathbb{R}$ and $p \in [1,2]$. These estimates are related to the ones
studied recently by Steinerberger in "On the stability of Fourier phase
retrieval" J. Fourier Anal. Appl., 28(2):29, 2022. While our estimates in
general are different, they share some comparable special cases and the main
improvement given here is that we can remove an additional imaginary term and
obtain sharper constants. We also consider these estimates for the quotient
distances related to the non-uniqueness of the Fourier phase problem. Our
arguments closely follow the Fourier analysis proof of the Sobolev embeddings
for Bessel potential spaces with minor modifications.
