= Lp interpolation inequality
{c}
{title2=$\|u\|_q\leq\|u\|_p^\theta\|u\|_r^{1-\theta}$}
= Interpolation of Lp norms
{synonym}
If $1\leq p,r\leq\infty$, $0\leq\theta\leq1$ and $1/q=\theta/p+(1-\theta)/r$, then the displayed bound follows by applying the <Holder inequality> to $|u|^{\theta q}|u|^{(1-\theta)q}$. Interpret the endpoint cases using the essential supremum. In particular convergence in <Lp space> and boundedness in a larger-exponent <Lp space> imply convergence at intermediate exponents.
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