# Interpolation between (or: simultaneous Whitney extension for) $C^\alpha$ and $C^{1,\gamma}$ on a Lipschitz domain

I would like to know whether for a bounded Lipschitz domain $$\Omega \subset \mathbb{R}^n$$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), there holds $$\bigl(C^\alpha(\overline\Omega),C^{1,\gamma}(\overline\Omega)\bigr)_{\theta,\infty} \hookrightarrow C^{0,1}(\overline\Omega) \quad \text{for some}~\theta < 1,$$

where $$0 < \gamma \leq \alpha < 1$$ for the case I am interested in.

• $$C^\alpha(\overline\Omega)$$ is the usual space of $$\alpha$$-Hölder continuous functions on $$\Omega$$,
• $$C^{1,\gamma}(\overline\Omega)$$ consists of continuously differentiable functions on $$\Omega$$ whose derivatives are bounded and $$\gamma$$-Hölder continuous.

For both spaces we identify the respective functions with their unique extension to the closure of $$\Omega$$.

It would be sufficient to have a continuous linear extension operator from $$C^\alpha(\overline\Omega)$$ to $$\mathcal{C}^\alpha(\mathbb{R}^n)$$ whose restriction to $$C^{1,\gamma}(\overline\Omega)$$ is continuous to $$\mathcal{C}^{1+\gamma}(\mathbb{R}^n)$$, because on Euclidean space the assertion would be true.

In fact, exactly such a property is claimed for the Whitney extension operator in Hölder Classes with Boundary Conditions as Interpolation Spaces by Acquistapace and Terreni (here), Proposition 1.4, however, without a precise proof, referring to the Singular Integrals book of Elias Stein, Chapter IV§2. I fail to understand how the restriction of the extension operator for the $$C^\alpha$$ scale to $$C^{1,\gamma}$$, resulting in a linear continuous operator, would work.

(The higher order Whitney operator is also defined for the $$(1+\gamma)$$-jet space which includes $$C^{1+\gamma}(\overline\Omega)$$ since a Lipschitz domain is quasiconvex, if I understand it right from this related question and the references there. A precise proof has eluded me so far, though.)

Any help or comments would be greatly appreciated.