Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pandey-Upadhyay's wavelet transform and microlocal Sobolev singularities of functions

Akira Lee, Shinya Moritoh

TL;DR

The paper addresses how to define microlocal Sobolev singularities using Pandey-Upadhyay's wavelet transform and how this notion compares with Hörmander's microlocal framework. It introduces a double-integral criterion over a neighborhood and a conical region $\Gamma_{\psi}(\xi^0)$ to capture microlocal $H^s$ regularity and proves a two-part theorem linking PU-based conditions to PU microlocal regularity and Hörmander microlocality. The main contributions include a wavelet-based microlocal Sobolev criterion that aligns with classical microlocal theory, a clear comparison to Hörmander's approach, and discussion of properties such as locality and support, situating the work among related shearlet and microlocal analyses. This provides a bridge between wavelet-centric microlocal analysis and established microlocal Sobolev theory with potential implications for signal processing and PDE analysis.

Abstract

The aim of the paper is to define the microlocal Sobolev singularities of functions using Pandey-Upadhyay's wavelet transform and provide a comparison with Hörmander's microlocal singularities.

Pandey-Upadhyay's wavelet transform and microlocal Sobolev singularities of functions

TL;DR

The paper addresses how to define microlocal Sobolev singularities using Pandey-Upadhyay's wavelet transform and how this notion compares with Hörmander's microlocal framework. It introduces a double-integral criterion over a neighborhood and a conical region to capture microlocal regularity and proves a two-part theorem linking PU-based conditions to PU microlocal regularity and Hörmander microlocality. The main contributions include a wavelet-based microlocal Sobolev criterion that aligns with classical microlocal theory, a clear comparison to Hörmander's approach, and discussion of properties such as locality and support, situating the work among related shearlet and microlocal analyses. This provides a bridge between wavelet-centric microlocal analysis and established microlocal Sobolev theory with potential implications for signal processing and PDE analysis.

Abstract

The aim of the paper is to define the microlocal Sobolev singularities of functions using Pandey-Upadhyay's wavelet transform and provide a comparison with Hörmander's microlocal singularities.

Paper Structure

This paper contains 2 sections, 1 theorem, 16 equations.

Table of Contents

  1. Introduction
  2. Definitions, main theorem, and the proof

Key Result

Theorem 1

Let $f\in L^2({\mathbb R}^2)$, $(x_0, \xi^0)\in {\mathbb R}^2\times ({\mathbb R}_{>0})^2$, and $s\ge0$. (a) If (2) holds for a cutoff function $\phi\in C_0^{\infty}({\mathbb R}^2)$ identically equal to 1 in a neighborhood of $x_0$ and the conical neighborhood $\Gamma_{\psi}(\xi^0)$ of $\xi^0$ define

Theorems & Definitions (7)

  • Definition 1: Hörmander H
  • Definition 2
  • Theorem 1
  • Remark 1
  • proof
  • Remark 2
  • Remark 3