THE SQUARE-ROOT PROBLEM OF KATO IN ONE-DIMENSION, AND FIRST-ORDER ELLIPTIC-SYSTEMS

Our aim is to prove that all non-degenerate second order elliptic oper ators L with Dirichlet, Neumann, or other two-point boundary condition s on an interval Omega satisfy the estimates parallel to L(1/2)u paral lel to p approximate to parallel to du/dx parallel to(p)+ parallel to u parallel to(p) when 1 < p < infinity. The study of the square root o f L reduces to proving that the holomorphic functional calculus of a r elated first order elliptic system is bounded. The interpolation theor y which is developed in [AM(c)N] is a major tool in proving the L-2 th eory. The L-p results follow once we have derived bounds on the Green' s functions of the systems.