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15 February 2005 Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces
Isroil A. Ikromov, Michael Kempe, Detlef Müller
Duke Math. J. 126(3): 471-490 (15 February 2005). DOI: 10.1215/S0012-7094-04-12632-6

Abstract

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 that 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 Lp($\mathbb{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 corresponding theorems on mixed homogeneous functions by A. Iosevich and E. Sawyer that allowed only for critical points of f at the origin. If c ≠ 0, our result is sharp.

Citation

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Isroil A. Ikromov. Michael Kempe. Detlef Müller. "Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces." Duke Math. J. 126 (3) 471 - 490, 15 February 2005. https://doi.org/10.1215/S0012-7094-04-12632-6

Information

Published: 15 February 2005
First available in Project Euclid: 11 February 2005

zbMATH: 1152.42304
MathSciNet: MR2120115
Digital Object Identifier: 10.1215/S0012-7094-04-12632-6

Subjects:
Primary: 42B10 42B25

Rights: Copyright © 2005 Duke University Press

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Vol.126 • No. 3 • 15 February 2005
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