# A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

created by stefani on 08 Nov 2020
modified by comi on 26 Jan 2021

[BibTeX]

Preprint

Inserted: 8 nov 2020
Last Updated: 26 jan 2021

Year: 2020

ArXiv: 2011.03928 PDF

Abstract:

We continue the study of the space $BV^\alpha(\mathbb R^n)$ of functions with bounded fractional variation in $\mathbb R^n$ and of the distributional fractional Sobolev space $S^{\alpha,p}(\mathbb R^n)$, with $p\in [1,+\infty]$ and $\alpha\in(0,1)$, considered in the previous works A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up and A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I. We first define the space $BV^0(\mathbb R^n)$ and establish the identifications $BV^0(\mathbb R^n)=H^1(\mathbb R^n)$ and $S^{\alpha,p}(\mathbb R^n)=L^{\alpha,p}(\mathbb R^n)$, where $H^1(\mathbb R^n)$ and $L^{\alpha,p}(\mathbb R^n)$ are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient $\nabla^\alpha$ strongly converges to the Riesz transform as $\alpha\to0^+$ for $H^1\cap W^{\alpha,1}$ and $S^{\alpha,p}$ functions. We also study the convergence of the $L^1$-norm of the $\alpha$-rescaled fractional gradient of $W^{\alpha,1}$ functions. To achieve the strong limiting behavior of $\nabla^\alpha$ as $\alpha\to0^+$, we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.