Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces

We study the boundedness problem for maximal operators in 3-dimensional Euclidean space associated to hypersurfaces given as the graph of c + f , where f is a mixed homogeneous function which is smooth away from the origin and c is a constant. Assuming that the Gaussian curvature of this surface nowhere vanishes of infinite order, we prove that the associated maximal operator is bounded on L p (R 3 ) whenever p > h 2. Here h denotes a "height" of the function f defined in terms of its maximum order of vanishing and the weights of homogeneity. This result generalizes a corresponding theorem on mixed homogeneous polynomial functions by A. Iosevich and E. Sawyer. In particular, it shows that a certain "ellipticity" conditon used by these authors is not necessary. If c = 0, our result is sharp.

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