Lp interpolation inequality ($\Vert u{\Vert }_q\le \Vert u{\Vert }_p^{\theta }\Vert u{\Vert }_r^{1-\theta }$)

If $1\le p,r\le \infty$, $0\le \theta \le 1$ 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.