I am currently studying numerical analysis and stumbled upon the following task:
Prove that for any $q \in [2, +\infty)$, there exists $C \gt 0$ such that
$\Vert f\Vert_{L^{q}(\mathbb{R}^2)} \le C\Vert f\Vert_{H^{1}(\mathbb{R}^2)},\ \forall\ f \in H^{1}(\mathbb{R}^2)$
$\textit{Hint: use the inequality}$
$\Vert\phi\Vert_{L^{\frac{d}{d-1}}(\mathbb{R}^d)} \le \Vert\nabla\phi\Vert_{L^{1}(\mathbb{R}^d)},\ \forall\ d \in \mathbb{N}\setminus\{1\},$
$\textit{and apply it to a well-chosen}\ \phi.$
My ideas so far: set $\phi = f^{\frac{q}{2}}$ or $\phi = f^{\frac{k}{2}}$ with $k = \lfloor q \rfloor\ \text{or}\ k = \lceil q\rceil$ and apply the hint. However, I have unfortunately no idea how to further solve this problem. I am grateful for any help.