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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.

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

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