For an application to analytic number theory, I'm considering the Fourier transform of test functions $h(x)$ on $\mathbb R$ with compact support, so the transforms $\hat h(y)$ are smooth. But I want to add a mild condition on $h$ that will force $\hat h(y) \ll 1/y^2$ as $y\to\infty$. Based on the examples I know, it looks like "continuous and piecewise differentiable" works, but I'd like the largest space of test functions possible. I'm familiar with the definition of $L^2$-based Sobolev Spaces $H^k(\mathbb R)$, but $k=1$ seems too weak, and $k=2$ too strong.

What is the right space of test functions $h(x)$ so that $\hat h(y)\ll 1/y^2$