It is well known that if $\varphi$ is a Schwartz function on $\mathbb{R}$ (i.e. smooth and decaying at infinity faster than polynomials), then its Fourier transform decays faster than polynomials. More precisely, for any $M>0$ there exists a constant $C_M>0$ such that \begin{equation}\tag{1}\label{1} |\widehat{\varphi}(\lambda)|=\left|\int_{\mathbb{R}} \varphi(x)e^{-2\pi i x \lambda}\,dx\right|\le C_M \lambda^{-M}, \end{equation} for any $\lambda>0$. The proof of \eqref{1} uses integration by parts.

My questions regard potential relaxing of the smoothness assumption imposed on $\varphi$:

- Is there a function which is non-differentiable at some point, but for which \eqref{1} still holds?
- Take $\varphi(x)=|x|e^{-x^2}$. Does \eqref{1} hold for this function, and if not, what is the optimal rate of decay of $|\widehat{\varphi}(\lambda)|$?

I think the answer to the second question should be $\lambda^{-1}$ (same as in van der Corput estimate with a non-smooth cutoff function), but I am not able to work out the details. All hints will be appreciated.