Isroil A. Ikromov, Michael Kempe, Detlef Müller:

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(R³) 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.

Electronic Print: Front for the Mathematics ArXiv

Mathematics Subject Classification (1991): 42B10, 42B25

Bibliographical note: erscheint im Duke Math. Journal

Keywords: Maximal operator, hypersurface, Gaussian curvature, mixed homogeneous

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[Thu Feb 19 18:56:36 2009]

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