Some oscillatory integral estimates via real analysis

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 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.2010 Mathematics Subject Classification. Primary 42B20.

On Archimedean zeta functions and Newton polyhedra

“…We would like to emphasize that our approach to the study of complex zeta functions avoids the use of a toric resolution of singularities, thus the relevant information about the poles of Z φ (s, f ) relies just in the geometry of the Newton polyhedron of f , more precisely in the geometry of its normal fan (see Remark 2.2). This point of view is on the same line that the work of Gilula in [16] for real oscillatory integrals. Perhaps our methods, combined with those of [16] may give a better estimation and/or asymptotic expansions of complex oscillatory integrals like the ones studied in [33].…”

Oscillatory Loomis-Whitney and Projections of Sublevel Sets

“…The first author showed in his thesis [7] that for any λ > 2, β ∈ (−1, ∞) d , and S(x) satisfying Varchenko's condition,…”

“…The proof is nearly identical to the proof of [7, Lemma 2.1], so we won't recreate it here. Moreover, "close enough"can be quantified by analyzing the argument in [7]. For a more heuristic argument, see [8, Lemma 3.1].…”

Sharp Bounds for Oscillatory Integral Operators with Homogeneous Polynomial Phases

“…In this paper, our purpose is to establish sharp L p bounds for these operators with homogeneous polynomial phases. In the (1+1)−dimensional setting, Phong and Stein [16] proved the remarkable theorem that the sharp L 2 decay estimate of T λ is determined by the Newton polyhedron of the real -analytic phase S; for some related results on oscillatory integrals and oscillatory integral operators see [30,12,14,15,20,6,4]. This result was extended to the case of smooth phases by Greenblatt [7]; see Rychkov [19] for a partial result.…”