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

We show thatJa,b,n:=12(|p|a|q|b+|q|b|p|a) is positive under suitable conditions on the exponents a and b and the underlying dimension n  . (Here qq is the multiplication by x   and p:=i−1∇p:=i−1∇.) Furthermore we show a generalization of the generalized Hardy inequalities for the fractional Laplacians.