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Uniform estimates for oscillatory integrals with parameter-dependent phases

Zihua Guo

Abstract

We consider the oscillatory integrals with parameter-dependent phases. We decompose the integrals into a leading term and a remainder term. Instead of the pointwise estimate, we use some $L^p$-estimate for the remainder term and get various uniform estimates when the phase functions satisfy certain conditions. This enables us to reduce the requirement of the smoothness on the phase functions, and hence improve the results in \cite[Theorem 7.7.5]{Hormander} and also obtain a refined version of the well-known Van der Corput Lemma. Some applications on the uniform expansion of the Bessel functions and dispersive estimates are also given.

Uniform estimates for oscillatory integrals with parameter-dependent phases

Abstract

We consider the oscillatory integrals with parameter-dependent phases. We decompose the integrals into a leading term and a remainder term. Instead of the pointwise estimate, we use some -estimate for the remainder term and get various uniform estimates when the phase functions satisfy certain conditions. This enables us to reduce the requirement of the smoothness on the phase functions, and hence improve the results in \cite[Theorem 7.7.5]{Hormander} and also obtain a refined version of the well-known Van der Corput Lemma. Some applications on the uniform expansion of the Bessel functions and dispersive estimates are also given.
Paper Structure (7 sections, 8 theorems, 110 equations)

This paper contains 7 sections, 8 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Uniform estimates for oscillatory integrals
  3. One-dimensional estimates
  4. High-dimensional estimates
  5. Applications
  6. Uniform estimates on Bessel function
  7. Dispersive estimates

Key Result

Lemma 1

Assume $\phi$ is real-valued and smooth in $(a,b)$, and $|\phi^{(k)}(x)|\geq 1$ for all $x\in (a,b)$. Then holds when (1) $k\geq 2$, or (2) $k=1$ and $\phi'(x)$ is monotonic. Here $c_k$ is a constant depending only on $k$.

Theorems & Definitions (18)

  • Lemma : Van der Corput
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3: Implicit function
  • proof
  • Corollary 2.4
  • proof
  • ...and 8 more