I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ is $C^{\infty}$ and compactly supported on $[0,1]^2$ , then if we use the notation $\langle u\rangle = \sqrt{1 + |u|^{2}}$ and $N_{1} = \pi* N *(1,0)$, then the Fourier Transform of $f$ is such that, for all $M>0$ there exists $C_M$ such that $\widehat{f}(\omega) \leq C_{M}. \big( \langle\omega - N_{1} \rangle^{-M} + \langle\omega + N_{1} \rangle^{-M} \big)$.

Knowing that the Fourier transform is defined as $\widehat{f}(\omega ) = \int e^{-i\langle x,w\rangle}f(x)dx$, I was wondering which mathematical result could justify this estimation. Is it the Payley-Wiener theorem?

Thank you for your help.