Lp estimates for the wave equation

Abstract

Let u(x, t) be the solution of utt - Δxu = 0 with initial conditions u(x, 0) = g(x) and ut(x, 0) = f{hook};(x). Consider the linear operator T: f{hook}; → u(x, t). (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in Lp if and only if | p-1 - 2-1 | = (n - 1)-1 and ∥ Tf{hook}; ∥LαP = ∥f{hook};∥LP with α = 1 -(n - 1) | p-1 - 2-1 |. Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for | p-1 - 2-1 | (n - 1)-1 we prove the existence of f{hook}; ε{lunate} LP in such a way that Tf{hook}; ∉ LP. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers m(ξ) = ψ(ξ) ei|ξ| | ξ | -b and finally we get that the convolution against the kernel K(x) = θ{symbol}(x)(1 - | x |)-1 is bounded in H1. © 1980.