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Some oscillatory integral estimates via real analysis

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Abstract

We study oscillatory integrals in several variables with analytic, smooth, or \(C^k\) phases satisfying a nondegeneracy condition attributed to Varchenko. With only real analytic methods, Varchenko’s sharp estimates are rediscovered and generalized. The same methods are pushed further to obtain full asymptotic expansions of such integrals with analytic and smooth phases, and finite expansions with error assuming the phase is only \(C^k\). The Newton polyhedron appears naturally in the estimates; in particular, we show precisely how the exponents appearing in the asymptotic expansions depend only on the geometry of the Newton polyhedron of the phase. All estimates proven hold for oscillatory parameter real and nonzero, not just asymptotically.

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Notes

  1. This smoothness is not sharp as a byproduct of the proofs. (Compare to van der Corput in \(d=1\): here we demand \(\phi \) to be \(C^{k+4}\) instead of the known sharp smoothness, \(C^k\).)

  2. One can reduce “\(n\in \mathbb {N}\)”to “\(n\in \{0,1,\dots , c_d\}\),”where \(c_d\) depends only on the dimension, by standard results borrowed from the theory of finitely generated cones.

  3. This follows from Theorem 1.2 of Ziegler’s textbook [18]. To see that the Newton polyhedron is a polyhedron in the sense of Ziegler, one can read Proposition 1 in [4].

  4. Instead of breaking the integral into orthants, we can consider a multi-dyadic partition of unity similar to Gressman’s bi-dyadic partition in [7]. Since Varchenko’s bounds are sharp for \(t>1,\) and we recover this sharp bound in just orthants, it is an interesting question whether the bounds proven here are sharpest possible over orthants for all t, or at least in the dyadic pieces.

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Acknowledgements

I want to deeply thank the many people who read and discussed early drafts of this paper. Most importantly I would like to thank my thesis adviser, Professor Philip T. Gressman, for everything he has taught me. Without him surely this would have been impossible for me. Moreover, I am very grateful to him for funding a year of my studies with his NSF grant.

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  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Maxim Gilula

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Gilula, M. Some oscillatory integral estimates via real analysis . Math. Z. 289, 377–403 (2018). https://doi.org/10.1007/s00209-017-1956-2

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