On a.e. convergence of solutions of hyperbolic equations to $L^ p$-initial data

Abstract:

We consider the Cauchy data problem $u(x,0) = 0$, $\partial u(x,0)/\partial t = f(x)$, for a strongly hyperbolic second order equation in $n$th spatial dimension, $n \geq 3$, with ${C^\infty }$ coefficients. Almost everywhere convergence of the solution of this problem to initial data, in the appropriate sense is proved for $f$ in ${L^p}$, $2n/(n + 1) < p < 2(n - 2)/(n - 3)$. The basic techniques are ${L^p}$-estimates for some maximal operators associated to the problem (see [4]), and the asymptotic expansion of the Riemann function given by D. Ludwig (see [9]).