Positivity of $|p|^a|q|^b+|q|^b|p|^a$
Li Chen, Heinz Siedentop

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 is positive, if . (Here is the multiplication by and .) Furthermore we show that it generalizes the generalized Hardy inequalities for the fractional Laplacians.
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Nonlinear Partial Differential Equations · Mathematical Approximation and Integration
