Hardy inequalities for weighted Dirac operator

Abstract

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r-b for functions in ℝn. The exact Hardy constant cb = cb(n) is found and generalized minimizers are given. The constant cb vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in ℝ2. Analogous inequalities are proved in the case cb = 0 under constraints and, with error terms, for a bounded domain. © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2009.