Mathematical analysis demonstrates refined Hardy and Rellich inequalities via non-standard remainder terms, indicating sharper bounds based on distance from virtual extremals.
Key Points
To refine the Hardy, critical Hardy, and Rellich inequalities by introducing scaling invariant non-standard remainder terms.
Evaluated the Hardy inequality on R^N, the critical Hardy inequality on a ball, and the Rellich inequality on R^N.
Applied the critical Hardy inequality on R^N established by Machihara, Ozawa, and Wadade to construct remainder terms.
Formulated the remainder terms using distances from families of virtual extremals.
Refined three classical Hardy-type inequalities by incorporating explicit, non-standard remainder terms.
Established that the remainder terms accurately quantify the distance of functions from optimal virtual extremal families while maintaining scaling invariance.