Even more generally, on $\mathbb{R}^d$, for $s>d/2$, we have a continuous embedding
$C^{k,\alpha} \rightarrow \dot{H}^s(\mathbb{R})$, where the right hand side is homogeneous Sobolev space of degree $s$, and those indices is $k=[s-d/2]$ and $\alpha$ is de 'decimal' part of $s-d/2$. And these orders are exactly compatible with the Sobolev embedding theorem, specified to the C^k function space cases.

And clearly, being in the (homogeneous) Sobolev spaces is a decay condition on the Fourier side.

For more details, see page 37, Theorem 1.50 of the textbook 'Fourier analysis and nonlinear partial differential equations' by Bahouri-Chemin-Dauchin.