Hölder interpolation inequality
= Hölder interpolation inequality
{c}
On a ball of radius $R$, lower derivative norms of $u\in C^{l,\alpha}$ are bounded by an arbitrarily small multiple of the top Hölder seminorm plus a multiple of $\|u\|_{C^0}$. One endpoint form is
$$
R^l\|D^lu\|_0\leq\varepsilon R^{l+\alpha}[D^lu]_\alpha+C_\varepsilon\|u\|_0.
$$