Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operator L in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 < p < ∞ and μ > 0. Then L has a bounded H_∞ functional calculus in L^p(Ω), in the sense that ||f (L + cI)u||_p ≤ C sup_|argλ|<;μ | f (λ)| ||u||_P for some constants c and C, and all bounded holomorphic functions f on the sector | arg λ| < μ. that contains the spectrum of L + cI. We prove this by showing that the operators f (L + cI) are Calderón-Zygmund singular integral operators.