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Optimal Lp Maximal Bounds for Averages over Degenerate Hypersurfaces

Establishes the optimal Lp boundedness of the maximal average operator over dilations of a smooth hypersurface in the degenerate regime where the surface measure's Fourier transform decays at rate exactly 1/2 -- the case Fourier decay alone cannot settle -- thereby resolving a conjecture of Stein. Also proves that for any non-flat hypersurface the maximal average is bounded on Lp for some finite p, generalizing the Sogge-Stein result to surfaces with vanishing Gaussian curvature.