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Estimates for oscillatory integrals with phase having $D$ type singularities

Ibrokhimbek Akramov, Isroil A. Ikromov

Abstract

In this paper, we consider estimates for the two-dimensional oscillatory integrals. The phase function of the oscillatory integrals is the linear perturbation of a function having $D$ type singularities. We consider estimates for the oscillatory integrals in terms of the Randol's type maximal functions. We obtain a sharp $L^p_{loc}$ estimates for the Randol's maximal functions. Moreover, we investigate the sharp exponent $p$ depending on whether, the phase function has linearly adapted coordinates system or not.

Estimates for oscillatory integrals with phase having $D$ type singularities

Abstract

In this paper, we consider estimates for the two-dimensional oscillatory integrals. The phase function of the oscillatory integrals is the linear perturbation of a function having type singularities. We consider estimates for the oscillatory integrals in terms of the Randol's type maximal functions. We obtain a sharp estimates for the Randol's maximal functions. Moreover, we investigate the sharp exponent depending on whether, the phase function has linearly adapted coordinates system or not.
Paper Structure (8 sections, 8 theorems, 133 equations)

This paper contains 8 sections, 8 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Newton polyhedrons and adapted coordinates systems
  3. On the normal form of the function
  4. Formulation of the main results
  5. Preliminaries
  6. Linearly adapted case
  7. The exceptional case $n=2m+1$
  8. Non-linearly adapted case

Key Result

Proposition 3.1

Assume that $\partial_1^{\alpha_1}\partial_2^{\alpha_2}\phi(0, 0)=0$ for any multi-index $\alpha:=(\alpha_1, \alpha_2)\in \mathbb{Z}_+^2$ with $|\alpha|:=\alpha_1+\alpha_2\le2$. Then the following statements hold: If $\phi_3$, which is the homogeneous part of degree $3$ of the Taylor polynomial of $ where $b, b_0$ are smooth functions, and $b(0, 0) =0,\, \partial_1 b(0, 0)\neq0, \,\partial_2 b(0,

Theorems & Definitions (16)

  • Proposition 3.1
  • Remark 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3
  • Lemma 5.1
  • proof
  • Theorem 6.1
  • proof
  • Proposition 7.1
  • ...and 6 more