I have a Lemma from a text that states that if $g\in W^{1,2}(\mathbb{R})$ ($W^{k,p}$ a Sobolev space) and the weak derivative $Dg\in L^2(\mathbb{R})$ then the Fourier transform $\mathcal{F}g\in L^1(\mathbb{R})$. Assuming there a higher-dimensional analogue to this, should I check the condition for each of the partial weak derivatives?
Cheers...
The higher dimensional analog is somewhat different. The idea of the result you quoted is that the transform of $Dg$ is a multiple of $\xi \hat g(\xi)$, and once we know that $\int_{\mathbb R} (1+|\xi|^2) |\hat g(\xi)|^2 <\infty$, it follows from Cauchy-Schwarz that $$\left(\int_{\mathbb R} |\hat g(\xi)| \right)^2 \le \int_{\mathbb R} (1+|\xi|^2)^{-1} \int_{\mathbb R} (1+|\xi|^2) |\hat g(\xi)|^2 <\infty$$ Trying to do the same in dimensions $n\ge 2$, we encounter a problem: $$\int_{\mathbb R^n} (1+|\xi|^2)^{-1} =\infty$$
Solution: take more derivatives. If the weak derivatives of order $k$ are square integrable, then following the above we get $$\left(\int_{\mathbb R^n} |\hat g(\xi)| \right)^2 \le \int_{\mathbb R^n} (1+|\xi|^{2k})^{-1} \int_{\mathbb R} (1+|\xi|^{2k}) |\hat g(\xi)|^2 <\infty$$ provided that $2k>n$. Thus, the $n$-dimensional analog involves derivatives of order $\lceil (n+1)/2 \rceil$. And yes, you should check each of the weak partial derivatives.
