= Sobolev interpolation inequality
{c}
{title2=$\|u\|_{H^{(1-\theta)a+\theta b}}\leq\|u\|_{H^a}^{1-\theta}\|u\|_{H^b}^{\theta}$}
= Sobolev interpolation
{c}
{synonym}
For $a<b$ and $0\leq\theta\leq1$, 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 $\|u\|_{H^{s+1}}\leq\varepsilon\|u\|_{H^{s+2}}+C_{s,\varepsilon}\|u\|_2$ for nonnegative $s$.
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