Characterization of function spaces via low regularity mollifiers

Smoothness of a function $f:{\mathbb R}^n\to {\mathbb R}$ can be measured in terms of the rate of convergence of $f\astρ_\varepsilon$ to $f$, where $ρ$ 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 $ρ$ is adapted to a given scale of spaces. Finally, we examine in detail the case where $ρ$ is a characteristic function.

No reviews yet.

No comments yet.