Global Estimates for Solutions of Partial Differential Equations∗

For a polynomial P (ξ) in ξ in R with constant complex coefficients, the operator defined by R(z)f = F((P (·)−z)f̂), where ∧ denotes the Fourier transform and F denotes its inverse, is not bounded from L to L when z is in the spectrum of P (D). What are suitable spaces B and C so that R(z) is bounded from B to C? When P (ξ) is simply characteristic, we prove that the operator R(z) is bounded from Bs to B 1−s, 0 ≤ s ≤ 1, where Bs are spaces reasonably smaller than L and B 1−s are spaces reasonably larger than L .