Hardy inequalities for weighted Dirac operatos

TL;DR

This paper establishes Hardy inequalities for weighted Dirac operators in Euclidean space, determines the exact constants, and explores cases where the constants vanish, extending known results to higher dimensions.

Contribution

It introduces new Hardy inequalities for weighted Dirac operators, finds exact constants, and generalizes minimizers, extending previous results beyond the two-dimensional case.

Findings

Exact Hardy constants $c_b(n)$ are determined.

Generalized minimizers are characterized.

Hardy inequalities are extended to cases with zero constants and bounded domains.

Abstract

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight $r^{-b}$ for functions in $\R^n$. The exact Hardy constant $c_b=c_b(n)$ is found and generalized minimizers are given. The constant $c_b$ vanishes on a countable set of $b$, which extends the known case $n=2$, $b=0$ which corresponds to the trivial Hardy inequality in $\R^2$. Analogous inequalities are proved in the case $c_b=0$ under constraints and, with error terms, for a bounded domain.