Some oscillatory integral estimates via real analysis

@article{Gilula2016SomeOI,
  title={Some oscillatory integral estimates via real analysis},
  author={Maxim Gilula},
  journal={Mathematische Zeitschrift},
  year={2016},
  volume={289},
  pages={377-403}
}
  • Maxim Gilula
  • Published 1 November 2016
  • Mathematics
  • Mathematische Zeitschrift
We study oscillatory integrals in several variables with analytic, smooth, or $$C^k$$Ck 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 $$C^k$$Ck. The Newton polyhedron appears naturally in the… Expand

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References

SHOWING 1-10 OF 24 REFERENCES
A real analytic approach to estimating oscillatory integrals
A REAL ANALYTIC APPROACH TO ESTIMATING OSCILLATORY INTEGRALS Maxim Gilula Philip T. Gressman We develop an asymptotic expansion for oscillatory integrals with real analytic phases. We assume theExpand
Oscillatory integral operators with homogeneous polynomial phases in several variables
Abstract We obtain L 2 decay estimates in λ for oscillatory integral operators T λ whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decayExpand
Damping oscillatory integrals by the Hessian determinant via Schr\"odinger
We consider the question of when it is possible to force a degenerate scalar oscillatory integral to decay as fast as a nondegenerate one by restricting the support to the region where the HessianExpand
Newton polyhedra and weighted oscillatory integrals with smooth phases
In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term byExpand
Toric resolution of singularities in a certain class of $C^{\infty}$ functions and asymptotic analysis of oscillatory integrals
In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatory integrals with real analytic phase is precisely investigated by using the theory of toric varieties based on the geometryExpand
Maximal averages over hypersurfaces and the Newton polyhedron
Using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques, L^p boundedness theorems for p > 2 are obtained forExpand
Sharp $L^2$ bounds for oscillatory integral operators with $C^\infty$ phases
Let T be an oscillatory integral operator on L^2(R) with a smooth real phase function S(x,y). We prove that, in all cases but the one described below, after localization to a small neighborhood ofExpand
Oscillatory integral decay, sublevel set growth, and the Newton polyhedron
Extending some resolution of singularities methods (Greenblatt in J Funct Anal 255(8):1957–1994, 2008) of the author, a generalization of a well-known theorem of Varchenko (Funct Anal ApplExpand
A multi-dimensional resolution of singularities with applications to analysis
We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions $\geq 3$. Rather than using the celebrated result of Hironaka, the algorithmExpand
Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude
The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that theExpand
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