Fourier proof of Hölder regularity from a Sobolev norm
= Fourier proof of Hölder regularity from a Sobolev norm
{c}
If $s>n/2+\alpha$ with $0<\alpha\leq1$, then Fourier inversion, $|e^{it}-1|\leq C_\alpha|t|^\alpha$, and Cauchy-Schwarz give
$$
\lVert u\rVert_{C^{0,\alpha}}
\leq C\lVert u\rVert_{H^s}.
$$
The weighted frequency integral converges precisely because $s-\alpha>n/2$.