# Multiplication operator from Sobolev space to $L^p$ space is compact?

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.