Abstract
Our aim is to prove that all non{degenerate second order elliptic operators L with Dirichlet, Neumann, or other two{point boundary conditions on an interval satisfy the estimates L1=2u p du dx p + k ukp when 1 < p <1 . The study of the square root of L reduces to proving that the holomorphic functional calculus of a related rst order elliptic system is bounded. The interpolation theory which is developed in [AMcN] is a major tool in proving the L2 theory. The Lp results follow once we have derived bounds on the Green's functions of the systems. x
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