I don't understand why this inequality holds (i found it in some notes): Let $ v\in L^1 \cap C_c, $ then
$$ \Vert v\Vert_{L^{q}} \leq \Vert v\Vert_{L^{1}}^{\frac{1}{q}} \cdot \Vert v\Vert_{L^{\infty}}^{1-\frac{1}{q}}. $$
I think that this could be a "consequence" or a particular case of Holder inequality, but i don't know how to prove it. Can anyone please help me?
Moreover, it is necessary that $ v\in L^1 \cap C_c $ or i can suppose that $ v\in L^p \cap H^s, $ with $ 1\leq p\leq 2$ and $ s\geq 0$?
Best Answer
You do not need any assumption on $v$ other than $v \in L^1 \cap L^{\infty}.$
Indeed:
$$\int |v|^q \leq \|v\|_{L^\infty}^{q-1} \int |v|.$$