Interpolation inequality in Holder spaces
= Interpolation inequality in Holder spaces
{c}
For $u\in C^{l,\alpha}(B_R)$, $0\leq j<l$, and every $\varepsilon>0$,
$$
R^j[D^ju]_{0;B_R}
\leq \varepsilon R^{l+\alpha}[D^lu]_{\alpha;B_R}
+C\varepsilon^{-j/(l+\alpha-j)}[u]_{0;B_R},
$$
with analogous estimates for lower Hölder seminorms. Scaling reduces the result to the unit ball.