Characterization of function spaces via low regularity mollifiers

TL;DR

This paper characterizes how mollifiers can be used to measure the smoothness of functions in fractional Sobolev spaces, providing conditions for mollifier suitability and analyzing specific cases like characteristic functions.

Contribution

It introduces a characterization of suitable mollifiers for fractional Sobolev spaces and offers near-necessary conditions for their adaptation to various function scales.

Findings

Identifies conditions for mollifiers to characterize fractional Sobolev spaces

Provides near-necessary criteria for mollifier adaptation

Analyzes the case of characteristic function mollifiers

Abstract

Smoothness of a function $f:{\mathbb R}^n\to {\mathbb R}$ can be measured in terms of the rate of convergence of $f\ast\rho_\varepsilon$ to $f$, where $\rho$ is an appropriate mollifier. In the framework of fractional Sobolev spaces, we characterize the "appropriate" mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that $\rho$ is adapted to a given scale of spaces. Finally, we examine in detail the case where $\rho$ is a characteristic function.