Some directional microlocal classes defined using wavelet transforms

TL;DR

This paper introduces a framework for defining microlocal classes of distributions using wavelet transforms, enabling analysis of singularities and regular directions, with applications to elliptic regularity in domains with cusp-like singularities.

Contribution

It develops conditions for independence of regions in wavelet space and defines microlocal classes based on wavelet analysis, advancing the understanding of distribution behavior.

Findings

Conditions for independence of wavelet regions established

Microlocal classes characterized by wavelet behavior defined

Application to elliptic regularity in cusp domains

Abstract

In this short paper we discuss how the position - scale half-space of wavelet analysis may be cut into different regions. We discuss conditions under which they are independent in the sense that the T\"oplitz operators associated with their characteristic functions commute modulo smoothing operators. This shall be used to define microlocal classes of distributions having a well defined behavior along lines in wavelet space. This allows us the description of singular and regular directions in distributions. As an application we discuss elliptic regularity for these microlocal classes for domains with cusp-like singularities.