Decay of fourier transform of a function with two variables

Question

In the paragraph about 'use in harmonic analysis', the modulus of fourier transform of a (one variable) function $f(\xi)$ is shown to be bounded by a quantity involving powers of $\xi$. http://en.wikipedia.org/wiki/Integration_by_parts Is there a similar result for a function $f(\xi_1,\xi_2)$ of two variables? How can it be obtained? (I am not very familiar with fourier transforms)

Answer 1

Yes, there is a similar result. In fact it is very standard: Let $f\in L^{1}(\mathbb R^{n}), n\in \mathbb N.$

Result: If $f\in C^{k}, \partial^{\alpha}f\in L^{1}$ for $|\alpha| \leq k,$ and $f\in C_{0}$ for $|\alpha|\leq k-1$, then $\hat{(\partial^{\alpha}f)}(\xi)= (2\pi i \xi)^{\alpha} \hat {f}(\xi).$ (For the proof, you may see: a book Real analysis :Modern Techniques and Their applications by Folland, theorem 8.22 (e), p. 249)