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

Abstract

We show that $$J_{a,b,n}:=\frac{1}{2}(|p{|}^a|q{|}^b+|q{|}^b|p{|}^a)$$ 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.

6 pages