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

Abstract

We continue the study of the space $B{V}^{\alpha }({\mathbb{R}}^n)$ 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 arXiv:1809.08575 and arXiv:1910.13419. We first define the space $B{V}^0({\mathbb{R}}^n)$ and establish the identifications $B{V}^0({\mathbb{R}}^n)=H^1({\mathbb{R}}^n)$ 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 $\alpha \to 0^{+}$ 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 $\alpha \to 0^{+}$, we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.

43 pages

References in corpus (4)

• Gamma-convergence of nonlocal perimeter functionals
• Non-linear ground state representations and sharp Hardy inequalities
• On the limit as $s\to 0^{+}$ of fractional Orlicz-Sobolev spaces
• New Brezis-Van Schaftingen-Yung Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Cited by in corpus (8)

• The fractional variation and the precise representative of $B{V}^{\alpha ,p}$ functions
• Leibniz rules and Gauss-Green formulas in distributional fractional spaces
• On a Class of Nonlocal Obstacle Type Problems Related to the Distributional Riesz Fractional Derivative
• Extending linear growth functionals to functions of bounded fractional variation
• Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula
• $H$-compactness for nonlocal linear operators in fractional divergence form
• Nonlocal Lagrange multipliers and transport densities
• Recovering functions via doubly homogeneous nonlocal gradients