Sharp Hardy inequalities in the half space with trace remainder term

TL;DR

This paper introduces a new class of inequalities in the half space that interpolate between Hardy's and Kato's inequalities, featuring an optimal trace remainder term that varies with a parameter.

Contribution

It establishes sharp Hardy inequalities with trace remainder terms in the half space, generalizing classical inequalities with a parameter-dependent optimal constant.

Findings

The trace remainder term's constant is optimal.

The trace term constant tends to zero as the parameter approaches the space dimension.

The inequality reduces to Kato's inequality when the parameter equals 2.

Abstract

In this paper we deal with a class of inequalities which interpolate the Kato's inequality and the Hardy's inequality in the half space. Starting from the classical Hardy's inequality in the half space $\rnpiu =\R^{n-1}\times(0,\infty)$, we show that, if we replace the optimal constant $\frac{(n-2)^2}{4}$ with a smaller one $\frac{(\beta-2)^2}{4}$, $2\le \beta <n$, then we can add an extra trace-term equals to that one that appears in the Kato's inequality. The constant in the trace remainder term is optimal and it tends to zero when $\beta$ goes to $n$, while it is equal to the optimal constant in the Kato's inequality when $\beta=2$.