On kato’s square root problem

Abstract

We consider abstract versions, H = − Σni, j=1 AicijAj + Σni=1(ciAi + Aici’) + c0, of second-order partial differential operators defined by sectorial forms on a Hilbert space H. The Ai are closed skew-symmetric operators with a common dense domain H1 and the cij, ci etc. are bounded operators on H with the real part of the matrix C = (cij) strictly positive-definite. We assume that D(L) ⊆ ∩i, j=1n D(AiAj) where L = − \sumi=1nAi2 is defined as a form on H1 × H1. We further assume the cij are bounded operators on one of the Sobolev spaces Hγ = D((I + L)γ2), γ ϵ 〈0,1〈, equipped with the graph norm. Then we prove that D((λ I + H)1/2) = D((λ I + H*)1/2) = H1 (1) for all large λ ϵ R . As a corollary we deduce that in any unitary representation of a Lie group all secondorder subelliptic operators in divergence form with Hölder continuous principal coefficients satisfy (1). Show more