Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set (probably we can weaken these hypothesis), and let $W^{l,p}(\Omega)$ denote the usual Sobolev space with $l \in \mathbb{N}$ being the order of derivatives and $p \in [1,\infty)$ the rate of integrability. I know that there exists a result of this type: fix $\varepsilon > 0$, then $$||f||_{W^{l-1,p}(\Omega)} \leq \varepsilon ||f||_{W^{l,p}(\Omega)} + c(\varepsilon) ||f||_{L^1(\Omega)}$$ for any function $f \in W^{l,p}(\Omega)$, where the constant $c(\varepsilon)>0$ depends on $\varepsilon$ but not on $f \in W^{l,p}(\Omega)$.
Do you have any reference on a book/paper where I can find the proof of this result?
If your problem is just about the fact that we can take a constant $\varepsilon$ in front of one of the norms, then it just follows from the Young's inequality $$ a^θ \,b^{1-\theta} ≤ θ \, a + (1-\theta) \,b $$ which also gives $$ a^θ b^{1-\theta} = (\varepsilon a)^θ \, \varepsilon^{-\theta}\, b^{1-\theta} ≤ \varepsilon \, \theta\, a + \frac{(1-\theta)}{\varepsilon^{\theta/(1-\theta)}}\, b $$ so you actually just need to find an inequality of the form $$ \|f\|_{W^{l-1,p}(\Omega)} \leq C\, \|f\|_{W^{l,p}(\Omega)}^{\theta} \,\|f\|_{L^1(\Omega)}^{1-\theta} $$ which is usually called a Gagliardo-Nirenberg-Sobolev type inequality. Some references for this latter inequality are indicated in https://en.wikipedia.org/wiki/Gagliardo%E2%80%93Nirenberg_interpolation_inequality