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Sharp estimates for convolution operators associated to hypersurfaces in $\mathbb{R}^3$ with height $h\le2$

Ibrokhimbek Akramov, Isroil A. Ikromov

Abstract

In this article, we study the convolution operator $M_k$ with oscillatory kernel, which is related with solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurface $Σ\subset \mathbb{R}^3$ of the equation and the smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the support of the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in Σ$ at which the height of the surface is less or equal to two. Such class contains surfaces related to simple and the $X_9, \, J_{10}$ type singularities in the sense of Arnol'd's classification. Denoting by $k_p$ the minimal exponent such that $M_k$ is $L^p\mapsto L^{p'}$-bounded for $k>k_p,$ we show that the number $k_p$ depends on some discrete characteristics of the Newton polygon of a smooth function constructed in an appropriate coordinate system.

Sharp estimates for convolution operators associated to hypersurfaces in $\mathbb{R}^3$ with height $h\le2$

Abstract

In this article, we study the convolution operator with oscillatory kernel, which is related with solutions to the Cauchy problem for the strictly hyperbolic equations. The operator is associated to the characteristic hypersurface of the equation and the smooth amplitude function, which is homogeneous of order for large values of the argument. We study the convolution operators assuming that the support of the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point at which the height of the surface is less or equal to two. Such class contains surfaces related to simple and the type singularities in the sense of Arnol'd's classification. Denoting by the minimal exponent such that is -bounded for we show that the number depends on some discrete characteristics of the Newton polygon of a smooth function constructed in an appropriate coordinate system.
Paper Structure (9 sections, 12 theorems, 78 equations)

This paper contains 9 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Newton polyhedrons and adapted and linearly adapted coordinate systems
  3. The main results
  4. Preliminaries
  5. On the linearly adapted coordinate system
  6. An upper bound for the critical exponent
  7. Uniform estimates
  8. Non-linearly adapted case
  9. The lower bound for the critical exponent

Key Result

Proposition 3.1

Assume that $\partial_1^{\alpha_1}\partial_2^{\alpha_2}\phi(0, 0)=0$ for any $\alpha\in \mathbb{Z}_+^2$ with $|\alpha|\le2$, $h(\phi)\le2$ and the coordinate system is not linearly adapted to $\phi$, i.e. $h_{lin}(\phi)<h(\phi)$. Then the following statements hold true: Conversely, if the conditions (i)-(iv) are fulfilled then $\partial^{\alpha_1}\partial^{\alpha_2}\phi(0, 0)=0$ for any $\alpha\i

Theorems & Definitions (21)

  • Proposition 3.1
  • Remark 3.2
  • Theorem 3.3
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • Remark 3.6
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • ...and 11 more