Asymptotic decay of Fourier, Laplace and other integral transforms

We study the asymptotic decay as t→+∞ of integral transforms ∫0+∞g(x)Φ(tx)dx. Examples are the cosine and sine Fourier transforms, the Hankel transforms, and the Laplace transforms. Under appropriate assumptions on the kernels and on the functions involved, we prove that the integral transforms can be controlled by the support, or by the first oscillation of the kernels. We also prove that if f(x) and g(x) are asymptotic at the origin, then the associated integral transforms are asymptotic at infinity. Finally, we give an asymptotic estimate for the integral transform of h(x)⋅g(x) when h(x) is suitably slowly varying.