My question is about the van der Corput lemma for
$$
\int\_a^b e^{i t \phi(x)} \psi(x) dx
$$
The version you find everywhere, e.g. on


<http://www.tricki.org/article/The_van_der_Corput_lemma_for_oscillatory_integrals>


requires $\psi$ to be absolutely continuous. However, what are weaker conditions
on $\psi$ such that the lemma still holds?


I was able to prove it when $\psi$ is either of bounded variation or in the Wiener class (summable Fourier coefficients).


Now I found a paper


Boldrighini, Carlo and Pellegrinotti, Alessandro and Triolo, Livio,
Convergence to stationary states for infinite harmonic systems,
J. Stat. Phys., 30 (1983), 123-155


where the authors claim in Prop A.3 that continuity of $\psi$ is
sufficient for the case k=2. The proof refers to a book by T. Kawata "Fourier Analysis in Probabity Theory" (Academic Press, New York, 1972) but in this book I could only find the case $\psi\equiv 1$.



van der Corput lemma for oscillatory integrals
Andreas Seeger pointed out that the fact that an estimate of the form $\frac{C\_{k,\psi}}{t^{1/k}}$ implies an estimate of the form $\frac{C\_k\|\psi\|\_\infty}{t^{1/k}}$ via Banach-Steinhaus. Since the latter is wrong, so is the first and thus
the claim from the paper.


