$\DeclareMathOperator\rad{rad}$
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ be compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
In $H_{\rad}^1(\mathbb{R}^3)$, by Struass estimate $|f(x)| \lesssim |x|^{-1} \| f \|_{H^1(\mathbb{R}^3)}$ and with Rellich compactness theorem, we can get that $H_{\rad}^1(\mathbb{R}^3) $ is compactly embedded in $L^q(\mathbb{R}^3)$, where $2 < q < 2^*=6$. And the similar fact that $|f(x)| \lesssim |x|^{-(d-1)/2} \| f \|_{H^1(\mathbb{R}^d)}$ for $d\ge2$ can also imply the same conclusion. However, if we restrict our attention in $\mathbb{R}^3$, then can we get the similar decay as $|x|$ tends to infinity so that we can deduce that $H^s(\mathbb{R^3}) \subset \subset L^q(\mathbb{R}^3)$ for any $\frac{1}{2}\le s \le 1$, where $2 < q < \frac{6}{3-2s}$.
I'm particularly interested in the particular case: can $H^{1/2}(\mathbb{R}^3)$ be compactly embedded in $L^q(\mathbb{R}^3)$ for $2 <q <3$?
