Positivity of $|p|^a|q|^b+|q|^b|p|^a$

We show that $J_{a,b,n}:=\frac12(|p|^a|q|^b+|q|^b|p|^a)$ is positive, if $n\geq b+a$. (Here $q$ is the multiplication by $x$ and $p:= \mathrm{i}^{-1}\nabla$.) Furthermore we show that it generalizes the generalized Hardy inequalities for the fractional Laplacians.

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