I am reading the book "Partial Differenatial Equations III: Nonlinear Equations" by M. Taylor (Google Books) and I am stuck on a claim he makes at page 379. I will reformulate it here.
Suppose we have a sequence $(u_m)$ such that for all $m$ $$u_m\in C^0(I,H^{l}(\mathbb{R}^n))\cap C^1(I,H^{l-2}(\mathbb{R}^n))$$ and that $(u_m)$ is uniformly bounded in both $C^0(I,H^{l}(\mathbb{R}^n))$ and $C^1(I,H^{l-2}(\mathbb{R}^n))$, where $l>\frac{n}{2}+2$ and $I$ is a bounded real interval.
Then it is claimed that for all $\sigma\in (0,1)$, $(u_m)$ is uniformly bounded in $C^{0,\sigma}(I,H^{l-2\sigma}(\mathbb{R}^n))$ by interpolation.
I assume one might use the Sobolev interpolation inequality: $$ \|u\|_{H^r}\leq C_s \|u\|_{L^2}^{1-r/s}\|u\|_{H^s}^{r/s}\quad\textrm{for }0<r<s$$ and the Hölder interpolation inequalities: \begin{align} \|u\|_{C^{0,\beta}}& \leq \|u\|_{C^{0,\alpha}}^{\frac{1-\beta}{1-\alpha}}\|u\|_{C^{0,1}}^{\frac{\beta-\alpha}{1-\alpha}}\\ \|u\|_{C^{0,\beta}}& \leq \|u\|_{C^0}^{1-\beta/\gamma}\|u\|_{C^{0,\gamma}}^{\beta/\gamma} \end{align} valid for $0<\alpha<\beta<\gamma\leq 1$. However, I can't seem to be able to get the powers right to prove the claim. I was wondering if there were any stronger interpolation inequalities, which might be useful to prove the claim.
Take $s,t\in I$, $s\ne t$. Then $$ |s-t|^{-\sigma}\|u(t)-u(s)\|_{H^{l-2\sigma}(\Omega)} \le |s-t|^{-\sigma}\|u(t)-u(s)\|_{H^{l-2}(\Omega)}^\sigma\|u(t)-u(s)\|_{H^{l}(\Omega)}^{1-\sigma}\\ \le \left( |s-t|^{-1}\|u(t)-u(s)\|_{H^{l-2}(\Omega)}\right)^\sigma\|u(t)-u(s)\|_{H^{l}(\Omega)}^{1-\sigma}\\ \le \|u\|_{C^1(I,H^{l-2}(\Omega))}^\sigma \|u\|_{C^0(I,H^l(\Omega))}^{1-\sigma}. $$ Now take the supremum over $s \ne t$, $s,t\in I$. This shows $$ \|u\|_{C^{0,\sigma}(I,H^{l-2\sigma}(\Omega))}\le \|u\|_{C^1(I,H^{l-2}(\Omega))}^\sigma \|u\|_{C^0(I,H^l(\Omega))}^{1-\sigma} + \|u\|_{C^0(I,H^l(\Omega))}. $$