JOURNAL OF FUNCTIONAL ANALYSIS 36, 114-145 (1980) LP Estimates for the Wave Equation JUAN C. PEFLU Princeton University, Princeton, New Jersey 08540 Communicated by the Editors Received May 26, 1978; revised April 23, 1979 Let u(x, t) be the solution of utt - A,u = 0 with initial conditions u(x, 0) = g(x) and ut(x, 0) = f(x). Consider the linear operator T :f 4 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 1 p-r - 2-r 1 = (n - 1)-r and (1Tfjj~,~ = llf//rP with (Y = 1 - (rz - 1) I p-r - 2-r I. 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 I p-r - 2-r I < (n - 1)-r. (b) If n = 2k - 1, the result is valid for 1 p-r - 2-r 1 Q (n - I). This result are sharp in the sense that for p such that 1 p-r - 2-i 1 > (n - 1)-r we prove the existence off E LP in such a way that Tf 6 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( [) = I/(~) e*ieI I 6 I -* and finally we get that the convolution against the kernel K(x) = &x)(1 - I x I)-’ is bounded in HI. 1. The aim of this work is to prove some results about boundedness in&, norm of the solution u(x; t,) (t, fixed) of the following Cauchy problem for the wave equation: utt -- A,u = 0 u(x, 0) = g;x,, %(X, 0) = f(X), where x E R”, t E R, and A,u means CT=“=, L&(x, t)/axis. We will treat also the variable coefficients case. Equivalently if we consider the linear operator 7’: f(x) - u(x, k,) (assume g = 0 here), to what extent is the following true The case p = 2 is related to the well-known “conservation of energy law.” 114 0022-1236/80/040114-32$02.00/O Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.