Abstract In this paper, we obtain a criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also, we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the Varchenko height of the hypersurface is greater than two. We obtain the exact value of the boundedness index for degenerate smooth hypersurfaces, i.e., for hypersurfaces satisfying the assumptions of the classical Hartman–Nirenberg theorem. The obtained results justify the Stein–Iosevich–Sawyer conjecture for arbitrary convex analytic hypersurfaces as well as for smooth degenerate hypersurfaces. Also, we discuss related problems of the theory of oscillatory integrals.
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