Query about a Sobolev inequality (interpolation)

Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary. Let $\phi \in C_c^{\infty}(\bar{\Omega})$ (yes, in the closure of $\Omega$ so support of $\phi$ can include the boundary of $\Omega$). In that case, we all know (can be proven by simply doing Fourier transform) that there exist constants $K_1,K_2$ (independent of $\phi$) such that $$||\phi||_{r,\mathbb{R}^n} \leq K_1 ||\phi||_{r-\epsilon,\mathbb{R}^n} + K_2 ||\phi||_{-m,\mathbb{R}^n}$$ where $r, r -\epsilon, m, \epsilon >0$ and $|| . ||_{r, \mathbb{R}^n}$ is the $r$ th Sobolev norm on $\mathbb{R}^n$.

However, I am struggling to prove the same for the $|| . ||_{., \Omega}$ norm. The problem I am facing is because of the negative Sobolev norm on the domain $\Omega$ here. Do we have constants $K_1,K_2$ (independent of $\phi$) such that $$||\phi||_{r,\Omega} \leq K_1 ||\phi||_{r-\epsilon, \Omega} + K_2 ||\phi||_{-m,\Omega}$$ where $r, r -\epsilon, \epsilon, m >0$ . Any proof or reference will be highly appreciated.

Note: From the first inequality above it is easy to see the following: $$||\phi||_{b,\Omega} \leq K_1 ||\phi||_{a, \Omega} + K_2 ||\phi||_{c,\Omega}$$ where $a \geq b \geq c \geq 0$.