## Duke Mathematical Journal

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

#### 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 (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

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.

#### References

• J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69-85.
• J. Bruna, A. Nagel, and S. Wainger, Convex hypersurfaces and Fourier transforms, Ann. of Math. (2) 127 (1988), 333-365.
• M. Cowling and G. Mauceri, Inequalities for some maximal functions, II, Trans. Amer. Math. Soc. 296 (1986), 341-365.
• --------, Oscillatory integrals and Fourier transforms of surface carried measures, Trans. Amer. Math. Soc. 304 (1987), 53-68.
• A. Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981), 519-537.
• I. A. Ikromov, An estimate for the Fourier transform of the indicator of nonconvex sets (in Russian), Dokl. Akad. Nauk 331, no. 3 (1993), 272-274; English translation in Russian Acad. Sci. Dokl. Math. 48, no. 1 (1994), 71-74.
• A. Iosevich, Maximal operators associated to families of flat curves in the plane, Duke Math. J. 76 (1994), 633-644.
• A. Iosevich and E. Sawyer, Oscillatory integrals and maximal averages over homogeneous surfaces, Duke Math. J. 82 (1996), 103-141.
• --------, Maximal averages over surfaces, Adv. Math. 132 (1997), 46-119.
• A. Iosevich, E. Sawyer, and A. Seeger, On averaging operators associated with convex hypersurfaces of finite type, J. Anal. Math. 79 (1999), 159-187.
• A. Nagel, A. Seeger, and S. Wainger, Averages over convex hypersurfaces, Amer. J. Math. 115 (1993), 903-927.
• B. Randol, On the asymptotic behavior of the Fourier transform of the indicator function of a convex set, Trans. Amer. Math. Soc. 139 (1969), 279-285.
• H. Schulz, Convex hypersurfaces of finite type and the asymptotics of their Fourier transforms, Indiana Univ. Math. J. 40 (1991), 1267-1275.
• C. D. Sogge, Maximal operators associated to hypersurfaces with one nonvanishing principal curvature'' in Fourier Analysis and Partial Differential Equations (Miraflores de la Sierra, Spain, 1992), Stud. Adv. Math., CRC, Boca Raton, Fla., 1995, 317-323.
• C. D. Sogge and E. M. Stein, Averages of functions over hypersurfaces in $\bf R\sp n$, Invent. Math. 82 (1985), 543-556.
• E. M. Stein, Maximal functions, I: Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2174-2175.
• --------, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
• E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971.
• B. A. Vasil'ev, The asymptotic behavior of exponential integrals, the Newton diagram and the classification of minima (in Russian), Funkcional. Anal. i Priložen. 11, no. 3 (1977), 1-11, 96; English translation in Funct. Anal. Appl. 11 (1977), 163-172.