Let $\hat{f}(\xi)$ be a smooth function on $\mathbb{R}^n$ that decays like $|D^\alpha_\xi \hat{f}(\xi)| \lesssim (1 + |\xi|^2)^{-\frac{1}{4}(1 + |\alpha|)}$, where $\alpha$ is a multi-index such that $D^\alpha_\xi = D^{\alpha_1}_{\xi_1}...D^{\alpha_n}_{\xi_n}$, and $|\alpha| = \alpha_1 + ... + \alpha_n$. The question is, since $\hat{f}$ has sufficient decay properties, would the inverse Fourier transform $f(x)$ of $\hat{f}(\xi)$ be smooth? Thanks in advance for any help!
Edit: In view of user225318's answer, now I am curious whether $f$ can be said to be smooth away from the origin.
In view of this comment clarifying the question, we can give a counter example. Consider
$$ f = \exp( - |x|) $$
which is in any $L^p$. Its Fourier transform is well known to be (up to a constant depending on the dimension)
$$ (1 + |\xi|^2)^{-(n+1)/2} $$
This function is smooth and decays much faster (with all derivatives) than what you supposed. (In fact it is $L^1$ and so the inverse Fourier transform converges everywhere to $f$.)
Clearly $f$ is not differentiable at the origin.