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 $\psi\equiv 1$.