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If $ x^{\alpha} g \in L^1 ( \Bbb R^n)$ for $| \alpha | \leqslant k$, then how can I prove that its Fourier transform $$ \mathscr{F} g \in C^k ( \Bbb R^n) ?$$ Here $\alpha$ is a multi-index.

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Hint: Integration by parts. However, I think you'll only find that weak derivatives exist, not strong derivatives. – Christopher A. Wong Oct 15 '12 at 21:09
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You know that if $g\in L^{1}(\mathbb{R}^{n})$ then $\mathscr{F}g\in C(\mathbb{R}^{n})$

Now try to prove this formula: $$x^{\alpha}D^{\beta}_{x}\mathscr{F}f(x)=(-1)^{|\beta|}\mathscr{F}(D^{\alpha}x^{\beta}f)(x)$$

Then conclude

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