Let $d\in\mathbb{N}^+$, $s\geq 0$, and consider the uniformly local Sobolev space

$$H^s_{u,loc}(\mathbb{R}^d):=\{f\in H^s_{loc}(\mathbb{R}^d)\,s.t.\,\|f\|_{H^s_{u,loc}}:=\sup_{x\in \mathbb{R}^d} \|f\|_{H^s(B_x)} < \infty\},$$

where $B_x$ is the ball of radius one centered at x. I expect that, given $0\leq s_0<s_1$, $\theta\in(0,1)$, and setting $s_{\theta}:=\theta s_1+(1-\theta)s_0$ the following interpolation inequality holds true:

$$(*)\quad\|f\|_{H^{s_{\theta}}}\leq \|f\|_{H^{s_{1}}_{u,loc}}^{\theta}\|f\|^{1-\theta}_{H^{s_{0}}} $$

Observe that, replacing $H^{s_{1}}_{u,loc}$ with $H^{s_1}$, we reduce to the classical interpolation inequality for Sobolev space.

**Is estimate (*) actually true? In case, is there an explicit reference?**

Thank you for any suggestions.