On maximal functions associated to hypersurfaces and the Cauchy problem for strictly hyperbolic operators

Sogge, Christopher D.

doi:10.1090/S0002-9947-1987-0911093-5

On maximal functions associated to hypersurfaces and the Cauchy problem for strictly hyperbolic operators
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by Christopher D. Sogge

Trans. Amer. Math. Soc. 304 (1987), 733-749

DOI: https://doi.org/10.1090/S0002-9947-1987-0911093-5

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Abstract:

In this paper we prove a maximal Fourier integral theorem for the types of operators which arise in the study of maximal functions associated to averaging over hypersurfaces and also the Cauchy problem for hyperbolic operators. We apply the Fourier integral theorem to generalize Stein’s spherical maximal theorem (see [8]) and also to prove a sharp theorem for the almost everywhere convergence to ${L^p}$ initial data of solutions to the Cauchy problem for second order strictly hyperbolic operators. Our results improve those of Greenleaf [3] and Ruiz [6]. We also can prove almost everywhere convergence to ${L^2}$ initial data for operators of order $m \geqslant 3$.

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