On someLp-estimates for hyperbolic equations
where ak(~)ES -k and 0~suppak. (S -k is the class of smooth functions which satisfy the estimates ID~a(OI<-C,(1 + [~l)-k-I~l for all derivations D'.) The operators MR, n(D)=F-1mk, dF (F denotes the Fourier transform, and F -1 its inverse) are used to represent the fundamental solutions of Cauchy problems for Schr6dinger equations (d= 1) and wave equations (d=2). It was not so easy to obtain L pestimates for them because we cannot apply the famous Marcinkiewicz theorem: "I f [D~m(~)[~C~l~[ -I~1 for I~[<=[n/2]+l, the operator M ( D ) = F l m F is L pbounded ( l<p<oo) . " (See, for example, Stein[15].) But many authors such as Hirshmann [5], Wainger [17], Fefferman--Stein [3], Sj6strand [14], Miyachi [10], [11], and Peral [12] contributed to it and gave