A multi-dimensional resolution of singularities with applications to analysis

Collins, Tristan C.; Greenleaf, Allan; Pramanik, Malabika

doi:10.1353/ajm.2013.0042

Cited by 44 publications

(41 citation statements)

References 55 publications

(113 reference statements)

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“…More detailed investigations into meromorphic continuation of Z f (ϕ) in various situations are in [8], [9], [7], [24], [2], etc. An interesting work [6] treating the equality c 0 (f ) = 1/d(f ) is from another approach. We remark that these results treat the general dimensional case.…”

Section: Known Results and Description Of The Problemsmentioning

confidence: 99%

Meromorphy of local zeta functions in smooth model cases

Kamimoto

Nose

2020

Journal of Functional Analysis

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It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general (C ∞ ) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as u(x, y)x a y b + flat function, where u(0, 0) = 0 and a, b are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each cases. Our results show new interesting phenomena in one of these cases. Actually, when a < b, local zeta functions can be meromorphically extended to the half-plane Re(s) > −1/a and their poles on the half-plane are contained in the set {−k/b : k ∈ N with k < b/a}.2000 Mathematics Subject Classification. 11S40 (26E10).

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Section: Known Results and Description Of The Problemsmentioning

confidence: 99%

Meromorphy of local zeta functions in smooth model cases

Kamimoto

Nose

2020

Journal of Functional Analysis

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“…Note that the vertices of the Newton polyhedron Γ + (f ) are contained in Q n + . (Refer to [6] for the details.) (v) f ∈ C ∞ 1/p (U + ) is said to be nondegenerate over R with respect to Γ + (f ) if the γ-part f γ satisfies ∇f γ = (0, .…”

Section: Generalization Of Varchenko's Results To the Puiseux Series mentioning

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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“…If we restrict ourselves in this definition to the affine tangent hyperplane H = x 0 + T x 0 S at the point x 0 , we shall call the corresponding index the contact index γ (x 0 , S) of the hypersurface S at the point x 0 ∈ S. Here, we shall always assume that ρ(x 0 ) = 0. Note that if we change coordinates so that x 0 = 0 and S is the graph of φ near the origin, where φ satisfies (1.2), then the contact index is just the supremum over all γ such that there is some neighborhood U of the origin so that |φ| −γ ∈ L 1 (U ), so that the contact index agrees with the critical integrability index of φ as defined for instance in [15].…”

Section: Remark 43 Notice That Condition (41) Is Easily Seen To Bementioning

confidence: 98%

“…(b) Greenblatt [28], [30], and independently Collins, Greenleaf, and Paramanik [15], have devised (quite different) algorithms of resolution of singularities which in principle allow us to compute the contact index also in higher dimensions. (c) If p ≤ 2, then examples (see, e.g., [46]) show that neither the notion of height nor that of contact index will determine the range of exponents p for which the maximal operator M is L p -bounded.…”

Section: For Linear Perturbations Of a Given Function φ Our Theorem mentioning

confidence: 99%

“…The reason for this is that strong information on the resolution of singularities is available for analytic functions of two variables, for instance, by means of Puiseux series expansions of roots, whereas the situation for multivariate functions of more than two variables is substantially more complex. Nevertheless, there has been a lot of progress on the question as to how to construct more elementary and "concrete" resolutions of singularities for real analytic multivariate functions, giving more detailed information than what Hironaka's celebrated theorem [35] on the resolution of singularities would yield, for instance, in work by Bierstone and P. D. Milman [7], [8], Sussmann [64], Parusiński [53], [54], Greenblatt [28], [30], Collins, Greenleaf, and Pramanik [15], among others. These techniques have already led to a very good understanding of, for instance, the sublevel estimation problem for slices of the surfaces in direction to the Gaussian normal, and the related determination of critical integrability indices, in independent work by Greenblatt [28], [30], and also Collins, Greenleaf, and Pramanik [15], by rather different methods.…”

Section: Introductionmentioning

confidence: 99%

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