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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Alberto Ruiz
Journal: Trans. Amer. Math. Soc. 287 (1985), 167-188
MSC: Primary 35L15
MathSciNet review: 766212
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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]).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0766212-9
PII: S 0002-9947(1985)0766212-9
Article copyright: © Copyright 1985 American Mathematical Society