Let $M_q: W^{1,\tilde{p}}(\mathbb{R}^2) \rightarrow L^p(\mathbb{R}^2)$, where $\tilde{p}$ is the Sobolev conjugate of p, $\left(\frac{1}{\tilde{p}}=\frac{1}{p}-\frac{1}{2}\right)$.

What we know is that $q\in L^p(\mathbb{R}^2)$ and we want to show that this operator is compact. Further, we have the embedding $W^{1,\tilde{p}}(\mathbb{R}^2) \subset C_B(\mathbb{R}^2)$.

I have been trying to use Fréchet-Kolmogorov theorem, but I'm not able to show the uniform translation condition.

Here is what I have done up to now: https://i.stack.imgur.com/AOcTK.png


I found it. Basically take a sequence of smooth compactly supported functions $(q_n)$ which converge to $q$ in $L^p$. Show that multiplication by this operator is compact by Rellich-Kondrachov and then Operator convergence follows by Sobolev Embedding.


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