I have a problem which is related to the homogeneous Sobolev spaces. The problem is the following. Let $n\ge3, \frac{1}{2}<\gamma<1$. And let $\sigma,q$ satisfy $$\frac{n}{2}-\gamma<\sigma<\min\left\{n,\frac{n}{2}+2\right\}-\gamma$$ and $$\max\left\{\frac{2}{n}-\frac{1}{2},0\right\}+\frac{\gamma}{n}<\frac{1}{q}<\min\left\{\frac{1}{2},\frac{2}{n}\right\}.$$ Under these assumptions, prove that $$L^\infty(\mathbb{R}^n) \cap \dot{H}_2^\sigma(\mathbb{R}^n)\subset \dot{H}^2_q(\mathbb{R}^n),$$ where $\dot{H}_p^s(\mathbb{R}^n)$ denotes the homogeneous Sobolev space $\dot{H}_p^s(\mathbb{R}^n):=\{f\in{\cal S}';\|f\|_{\dot{H}_p^s}=\|(-\Delta)^{s/2}f\|_p<\infty\}$.
What I could show is the following. Let $f\in L^\infty(\mathbb{R}^n) \cap \dot{H}_2^\sigma(\mathbb{R}^n)$. Note that $$\frac{n}{2}-\frac{n}{q}<\frac{n}{2}-\gamma<\sigma.$$ Then, by the Sobolev embedding theorem, I have $\|f\|_{\dot{H}_q^2}=\|(-\Delta)f\|_q\le C\|(-\Delta)f\|_{H^\sigma}$, where $C$ depends only on the dimension $n$. But I could not go further.
I appreciate any advice. Thank you in advance.