The Newton polyhedron and oscillatory integral operators

Phong, Duong H.; Stein, Elias M.

doi:10.1007/bf02392721

Cited by 123 publications

(161 citation statements)

References 13 publications

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“…For a fixed real-analytic u with critical point at (0, 0), satisfying ∂ j+k u/∂x j ∂y k = 0 on Q, Phong and Stein [PS2] have established the sharp bound T λ 2−2 = O(|λ| − 1 2(j∨k) ) (among many others). But of course implicit in their result are assumptions on upper bounds of higher derivatives of u.…”

Section: It Is Known That At Least When D Is Even and U(x Y)mentioning

confidence: 99%

Multidimensional van der Corput and sublevel set estimates

Carbery

Christ

Wright

1999

J. Amer. Math. Soc.

119

158

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Section: It Is Known That At Least When D Is Even and U(x Y)mentioning

confidence: 99%

Multidimensional van der Corput and sublevel set estimates

Carbery

Christ

Wright

1999

J. Amer. Math. Soc.

119

158

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“…For T λ , we are interested in the decay rate of the L p operator norm as λ → ∞. For R we ([PSt2], [PSt3], [R], [S2]). When S is real-analytic, Phong and Stein proved sharp results on the decay rate of ||T λ || L 2 →L 2 depending on the Newton polygon of S xy [PSt2].…”

Section: T λ F (X) = E Iλs(xy) F (Y)χ(x Y)dy and Radon Transforms Omentioning

confidence: 99%

“…For R we ([PSt2], [PSt3], [R], [S2]). When S is real-analytic, Phong and Stein proved sharp results on the decay rate of ||T λ || L 2 →L 2 depending on the Newton polygon of S xy [PSt2]. In [S1] Seeger obtained nearly optimal results when S is a C ∞ real function.…”

Section: T λ F (X) = E Iλs(xy) F (Y)χ(x Y)dy and Radon Transforms Omentioning

confidence: 99%

$L^p$ improving estimates for some classes of Radon transforms

Yang

2005

Trans. Amer. Math. Soc.

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Abstract. In this paper, we give L p − L q estimates and the L p regularizing estimate of Radon transforms associated to real analytic functions, and we also give estimates of the decay rate of the L p operator norm of corresponding oscillatory integral operators. For L p −L q estimates and estimates of the decay rate of the L p operator norm we obtain sharp results except for extreme points; however, for L p regularity we allow some restrictions on the phase function.

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“…Pramanik and Yang [26] obtained a similar result relating to the above equation "β(f, g) = −1/d(f, g)" in the case when the dimension is two and the weight has the form g(x) = |h(x)| ǫ where h is real analytic and ǫ is positive. Their approach is based on the Puiseux series expansions of the roots of f and h, which is inspired by the work of Phong and Stein in [25]. Their definition of Newton distance, which is different from ours, is given through the process of a good choice of coordinate system.…”

Section: Theorem 33 ([5])mentioning

confidence: 99%

“…By the way, Pramanik and Yang [26] also use simultaneous resolution of singularities in their analysis. Indeed, they extend the method of Phong and Stein [25] by considering the expressions of both the phase and the weight in terms of the Puiseux serieses of their roots.…”

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confidence: 99%

Newton polyhedra and weighted oscillatory integrals with smooth phases

Kamimoto

Nose

2015

Trans. Amer. Math. Soc.

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Abstract. 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 by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals, whose phases and weights are contained in a certain class of smooth functions including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.

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The Newton polyhedron and oscillatory integral operators

Cited by 123 publications

References 13 publications

Multidimensional van der Corput and sublevel set estimates

Multidimensional van der Corput and sublevel set estimates

$L^p$ improving estimates for some classes of Radon transforms

Newton polyhedra and weighted oscillatory integrals with smooth phases

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