Duke Mathematical Journal

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

Isroil A. Ikromov, Michael Kempe, and Detlef Müller

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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.

Article information

Source
Duke Math. J. Volume 126, Number 3 (2005), 471-490.

Dates
First available in Project Euclid: 11 February 2005

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1108155759

Digital Object Identifier
doi:10.1215/S0012-7094-04-12632-6

Mathematical Reviews number (MathSciNet)
MR2120115

Zentralblatt MATH identifier
1152.42304

Subjects
Primary: 42B10 42B25

Citation

Ikromov, Isroil A.; Kempe, Michael; Müller, Detlef. Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces. Duke Math. J. 126 (2005), no. 3, 471--490. doi:10.1215/S0012-7094-04-12632-6. http://projecteuclid.org/euclid.dmj/1108155759.


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