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Newton polygons and local integrability of negative powers of smooth functions in the plane

Author(s): Michael Greenblatt
Journal: Trans. Amer. Math. Soc. 358 (2006), 657-670.
MSC (2000): Primary 42B20; Secondary 35H20
Posted: February 4, 2005
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Abstract | References | Similar articles | Additional information

Abstract: Let $f(x,y)$ be any smooth real-valued function with $f(0,0)=0$. For a sufficiently small neighborhood $U$ of the origin, we study the number

\begin{displaymath}\sup\left\{\epsilon:\int_U \vert f(x,y)\vert^{-\epsilon}<\infty\right\}. \end{displaymath}

It is known that sometimes this number can be expressed in a natural way using the Newton polygon of $f$. We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal $\epsilon$ is also analyzed.

References:

[CCW]
A. Carbery, M. Christ, J. Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12 (1999), 981-1015. MR 2000h:42010

[C]
M. Christ, Hilbert transforms along curves. I. Nilpotent groups, Annals of Mathematics (2) 122 (1985), no.3, 575-596. MR 87f:42039a

[PSSt]
D. H. Phong, E. M. Stein, J. Sturm, On the growth and stability of real-analytic functions, American J. Math. 121 (1999), 519-554. MR 2002a:58025

[P]
M. Pramanik, Convergence of two-dimensional weighted integrals, Trans. Amer. Math. Soc. 354 (2002), 1651-1665. MR 2003a:41033

[V]
A. N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Functional Anal. Appl. 18 (1976), no. 3, 175-196.

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Additional Information:

Michael Greenblatt
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

DOI: 10.1090/S0002-9947-05-03664-0
PII: S 0002-9947(05)03664-0
Keywords: Resolution of singularities, Newton polygon
Received by editor(s): November 11, 2002
Received by editor(s) in revised form: February 6, 2004
Posted: February 4, 2005
Additional Notes: This research was supported in part by NSF grant DMS-9988798
Copyright of article: Copyright 2005, American Mathematical Society

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