Besov Wavefront Set

TL;DR

This paper introduces a new Besov wavefront set concept to analyze distribution singularities in Fourier space, providing equivalent characterizations, a multiplication criterion, and applications to hyperbolic operator singularity propagation.

Contribution

It develops a novel Besov wavefront set framework with equivalent characterizations and a generalized multiplication criterion, extending classical results and enabling propagation of singularities analysis.

Findings

Defined a Besov wavefront set in Fourier space.

Provided an equivalent pseudo-differential operator characterization.

Proved a propagation of singularities theorem for hyperbolic operators.

Abstract

We develop a notion of wavefront set aimed at characterizing in Fourier space the directions along which a distribution behaves or not as an element of a specific Besov space. Subsequently we prove an alternative, albeit equivalent characterization of such wavefront set using the language of pseudo-differential operators. Both formulations are used to prove the main underlying structural properties. Among these we highlight the individuation of a sufficient criterion to multiply distributions with a prescribed Besov wavefront set which encompasses and generalizes the classical Young's theorem. At last, as an application of this new framework we prove a theorem of propagation of singularities for a large class of hyperbolic operators.