# How to show this Sobolev inequality?

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.

• What exactly do you mean by $H^1$? Is that a $W^{1,2}$, a Hilbert case of Sobolev space? – Wham Bang Shang-a-Lang May 13 at 20:21
• Yes exactly. $H^1=W^{1,2}$. – fidel castro May 14 at 11:50