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

TL;DR

This paper proves the positivity of a specific operator involving fractional derivatives and multiplication, extending Hardy inequalities for fractional Laplacians, with implications in analysis and PDEs.

Contribution

It establishes the positivity of the operator $$J_{a,b,n}$$ for certain dimensions and connects it to generalized Hardy inequalities for fractional Laplacians.

Findings

Proves positivity of $$J_{a,b,n}$$ when $$n \\geq b+a$$.

Shows the operator generalizes Hardy inequalities.

Provides a new perspective on fractional Laplacian inequalities.

Abstract

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.