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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 thegeneralized Hardy inequalities for the fractional Laplacians.

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