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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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Volume 30, Issue 3

# Fourier transforms of powers of well-behaved 2D real analytic functions

Michael Greenblatt
Published Online: 2017-09-27 | DOI: https://doi.org/10.1515/forum-2016-0256

## Abstract

This paper is a companion paper to [6], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [6] are stated in a rather general form. In this paper, we expand on the results of [6] and show that there is a class of “well-behaved” functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way.

Keywords: Fourier transform; real analytic function

MSC 2010: 42B20

## References

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V. Arnold, S. Gusein-Zade and A. Varchenko, Singularities of Differentiable Maps. Vol. II, Birkhäuser, Basel, 1988. Google Scholar

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M. Greenblatt, Newton polygons and local integrability of negative powers of smooth functions in the plane, Trans. Amer. Math. Soc. 358 (2006), 657–670.

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M. Greenblatt, Oscillatory integral decay, sublevel set growth, and the Newton polyhedron, Math. Ann. 346 (2010), no. 4, 857–895.

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M. Greenblatt, Singular integral operators with kernels associated to negative powers of real analytic functions, J. Funct. Anal. 269 (2015), no. 11, 3663–3687.

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M. Greenblatt, Uniform bounds for Fourier transforms of surface measures in ${ℝ}^{3}$ with nonsmooth density, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6601–6625. Google Scholar

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M. Greenblatt, Convolution kernels of 2D Fourier multipliers based on real analytic functions, J. Geom. Anal. (2017), 10.1007/s12220-017-9842-z. Google Scholar

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I. Ikromov, M. Kempe and D. Müller, Estimates for maximal functions associated to hypersurfaces in ${ℝ}^{3}$ and related problems of harmonic analysis, Acta Math. 204 (2010), no. 2, 151–271. Google Scholar

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D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), no. 1, 105–152.

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D. H. Phong, E. M. Stein and J. Sturm, On the growth and stability of real-analytic functions, Amer. J. Math. 121 (1999), 519–554.

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A. N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (1976), no. 3, 175–196. Google Scholar

Revised: 2017-04-09

Published Online: 2017-09-27

Published in Print: 2018-05-01

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 723–732, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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