Positivity of $\mathrm{\mid }p{\mathrm{\mid }}^a\mathrm{\mid }q{\mathrm{\mid }}^b+\mathrm{\mid }q{\mathrm{\mid }}^b\mathrm{\mid }p{\mathrm{\mid }}^a$

We show that $$J_{a,b,n}:=\frac{1}{2}(\mathrm{\mid }p{\mathrm{\mid }}^a\mathrm{\mid }q{\mathrm{\mid }}^b+\mathrm{\mid }q{\mathrm{\mid }}^b\mathrm{\mid }p{\mathrm{\mid }}^a)$$ is positive, if $n\ge b+a$. (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.