Sobolev interpolation inequality ($\Vert u{\Vert }_{H^{(1-\theta )a+\theta b}}\le \Vert u{\Vert }_{H^a}^{1-\theta }\Vert u{\Vert }_{H^b}^{\theta }$)

For $a<b$ and $0\le \theta \le 1$, the displayed estimate holds on Euclidean space and a torus with the Fourier definition of the Sobolev norm. In the squared norm, write the weighted squared Fourier coefficient as the product of its weight-$a$ value to the power $1-\theta$ and its weight-$b$ value to the power $\theta$. Holder inequality then proves the estimate. On a compact manifold, a finite coordinate cover and a partition of unity give the same estimate up to a constant. Young inequality implies the useful absorption form $\Vert u{\Vert }_{H^{s+1}}\le \epsilon \Vert u{\Vert }_{H^{s+2}}+C_{s,\epsilon }\Vert u{\Vert }_2$ for nonnegative $s$.