Some trace Hardy type inequalities and trace Hardy-Sobolev-Maz'ya type inequalities

TL;DR

This paper establishes new trace Hardy and Hardy-Sobolev-Maz'ya inequalities on polyhedral cones and upper half spaces, including fractional Laplacian cases, with explicit best constants in radial scenarios.

Contribution

It generalizes recent inequalities to polyhedral cones and fractional Laplacians, providing explicit constants and new logarithmic inequalities.

Findings

Proved trace Hardy inequalities on polyhedral convex cones.

Extended Hardy-Sobolev-Maz'ya inequalities to fractional Laplacians.

Derived explicit best constants for radial cases.

Abstract

We prove a trace Hardy type inequality with the best constant on the polyhedral convex cones which generalizes recent results of Alvino et al. and of Tzirakis on the upper half space. We also prove some trace Hardy-Sobolev-Maz'ya type inequalities which generalize the recent results of Filippas et al.. In applications, we derive some Hardy type inequalities and Hardy-Sobolev-Maz'ya type inequalities for fractional Laplacian. Finally, we prove the logarithmic Sobolev trace inequalities and logarithmic Hardy trace inequalities on the upper half spaces. The best constants in these inequalities are explicitly computed in the radial case.