Truncations and compositions in function spaces

TL;DR

This paper examines recent assertions regarding the behavior of truncations and compositions within certain function spaces, specifically the spaces $A^s_{p,q}(R^n)$, providing insights into their properties.

Contribution

It analyzes and clarifies recent claims about how truncations and compositions operate in Besov and Triebel-Lizorkin spaces, offering new theoretical understanding.

Findings

Clarifies properties of truncations in $A^s_{p,q}$ spaces

Provides insights into composition operators in these spaces

Addresses recent assertions and their validity

Abstract

These notes deal with some recent assertions about truncations $f \mapsto |f|$ and compositions $f \mapsto g\circ f$ in the spaces $A^s_{p,q}(\mathbb{R}^n)$, $A \in \{B,F \}$.