Let $f\in \dot{H}^s$ with $s\in (0,1)$ and $g\in \dot{H}^{-s}$, where $\dot{H}^s$ denote the fractional homogeneous Sobolev space $\dot{W}^{p,s}$with $p=2$. Does one have the following Cauchy-Schwarz type inequality? $$ (f,g)_{L^2}\leq \| f \|_{\dot{H}^s} \| g \|_{\dot{H}^{-s}}, $$ where $(.,.)_{L^2}$ is the $L^2$-inner product and on the right are the seminorms corresponding to the homogeneous spaces (given by the Gagliardo seminorms).
I have seen an inequality of this type for fractional Sobolev spaces but without any proof. Does one have a reference (also for the case above)?
Thanks for your help!