Interpolation between two Lebesgue norms

Statement

Let $1\le p_0,p_1\le\infty$ , $0\le\theta\le1$ , and

\frac1p=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}.

For $f\in L^{p_0}\cap L^{p_1}$ , the Lebesgue interpolation inequality is

\|f\|_p\le\|f\|_{p_0}^{1-\theta}\|f\|_{p_1}^{\theta}.

Direct proof

For finite distinct endpoints and $0<\theta<1$ , apply Hölder's inequality to $|f|^{p(1-\theta)}|f|^{p\theta}$ , with conjugate exponents $p_0/[p(1-\theta)]$ and $p_1/(p\theta)$ . If an endpoint is infinity, bound its factor by the essential supremum. Coincident endpoints and $\theta=0,1$ give equality. This is an estimate for one function, distinct from an operator interpolation theorem.