Several Weighted Hardy, Hardy-Sobolev-Maz'ya and Related Inequalities

TL;DR

This paper establishes new weighted Hardy, Hardy-Sobolev, and related inequalities on the positive orthant and upper half space, including inequalities with remainder and interpolation terms, advancing the theoretical understanding of these inequalities.

Contribution

It introduces several new weighted Hardy and Hardy-Sobolev inequalities on the positive orthant and upper half space, with applications to inequalities with remainder and interpolation.

Findings

New weighted Hardy and Hardy-Sobolev inequalities established.

Inequalities with remainder terms derived.

Interpolation and Maz'ya type inequalities obtained.

Abstract

In this paper we establish several Hardy and Hardy-Sobolev type inequalities with homogeneous weights on the first orthant $\displaystyle \mathbb{R}_{*}^n:=\{(x_1, \ldots, x_n):x_1>0, \ldots, x_n>0 \}$. We then use some of them to produce Hardy type inequalities with remainder terms. Furthermore, we obtain some interpolation inequalities and Maz'ya type inequalities with remainder terms with the help of Maz'ya inequality and Sobolev inequality of Cabr\'e and Ros-Orton on the upper half space $\mathbb{R}_{+}^n:=\{x=(x_1, \ldots, x_n)|\, x_n>0 \}$.