Fourier proof of Hölder regularity from a Sobolev norm

If $s>n/2+\alpha$ with $0<\alpha \le 1$, then Fourier inversion, $|e^{it}-1|\le C_{\alpha }|t{|}^{\alpha }$, and Cauchy-Schwarz giveThe weighted frequency integral converges precisely because $s-\alpha >n/2$.