Bounding the decay of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function

We call a function "constructible" if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. Our main theorem gives uniform bounds on the decay of parameterized families of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function, assuming that the amplitude function is integrable and that the phase function satisfies a certain natural condition called the hyperplane condition. As a simple application of this theorem, we also show that any continuous, integrable, constructible function of a single variable has an integrable Fourier transform.