Some remarks on Kato's inequality. Issue 1 (2001) - Horiuchi, Toshio
Abstract : LetN ≥ 1 andN > 1 . LetΩ be a domain ofℝ N . In this article we shall establish Kato's inequalities forp -harmonic operatorsL p . HereL p is defined asL p u = div ( | ∇ u | p − 2 ∇ u ) foru ∈ K p ( Ω ), whereK p ( Ω ) is an admissible class. Ifp = 2 for example, then we haveK p ( Ω ) = { u ∈ L loc 1 ( Ω ) : ∂ j u, ∂ j, k 2 u ∈ L loc 1 ( Ω )   for   j, k = 1, 2, …, N } . Then we shall prove thatL p | u | ≥ ( sgn u ) L p u andL p u + ≥ ( sgn + u ) p − 1 L p u in𝒟 ′ ( Ω ) withu ∈ K p ( Ω ) . These inequalities are called Kato's inequalities provided thatp = 2 .