Abstract
Since Varchenko’s seminal paper, the asymptotics of oscillatory integrals and related problems have been elucidated through the Newton polyhedra associated with the phase P. The supports of those integrals are concentrated on sufficiently small neighborhoods. The aim of this paper is to investigate the estimates of sublevel sets and oscillatory integrals whose supports are global domains D. A basic model of D is \( {\mathbb {R}}^d\). For this purpose, we define the Newton polyhedra associated with (P, D) and establish analogues of Varchenko’s theorem in global domains D, under nondegeneracy conditions of P.
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Notes
If \({\mathbb {P}}_{\textrm{for}} \supset {\mathbb {R}}\textbf{1}\) or \( {\mathbb {P}}_{\textrm{bac}} \supset {\mathbb {R}}{} \textbf{1}\), then \(d({\mathbb {P}}_{\textrm{for}}){:}{=}-\infty \) or \(d({\mathbb {P}}_{\textrm{bac}}){:}{=}\infty \) respectively.
Every minimal (under \(\subset \)) face of \({\mathbb {P}}\) has dimension \(k_0=d-\text {rank}({\mathbb {P}}^{\vee })\).
If \(\delta _{\textrm{bac}}=\infty \), then \(\delta _{\textrm{bac}}\le d(\pi _{\mathfrak {q}_i,r_i}^+)=\infty \) for all i of \({\mathbb {F}}=\bigcap _{i=1}^{d_0}\pi _{\mathfrak {q}_i,r_i} \). This with Definition 5.2 implies \(\mathfrak {q}_i\cdot \textbf{1}=0\) for all \(i \le d_0\). So, \(n=0\) in (8.15)–(8.17) so that (8.16)\( =\sum _{(\alpha _{n+1},\ldots ,\alpha _{d_0}) } \lesssim \frac{1}{\lambda ^{1/\tau }}\).
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This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea NRF-2015R1A2A2A01004568.
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Kim, J. Sublevel-Set Estimates Over Global Domains. J Geom Anal 33, 379 (2023). https://doi.org/10.1007/s12220-023-01424-5
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DOI: https://doi.org/10.1007/s12220-023-01424-5