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

Abstract

We continue the study of the space BV α (R n ) of functions with bounded fractional variation in R n and of the distributional fractional Sobolev space S α,p (R n ), with p ∈ [1, +∞] and α ∈ (0, 1), considered in the previous works [27,28]. We first define the space BV 0 (R n ) and establish the identificationswhere H 1 (R n ) and L α,p (R n ) are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient ∇ α strongly converges to the Riesz transform as α → 0 + for H 1 ∩W α,1 and S α,p functions. We also study the convergence of the L 1 -norm of the α-rescaled fractional gradient of W α,1 functions. To achieve the strong limiting behavior of ∇ α as α → 0 + , we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.

Failure of the local chain rule for the fractional variation

“…In the case n = 1, we prove a stronger result, exhibitingThe failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the results of [9] and the distributional approach of the previous papers [5][6][7][8]. As a byproduct, we refine the fractional Hardy inequality obtained in [25,28] and we prove a fractional version of the closely related Meyers-Ziemer trace inequality.…”

“…On the one side, we refer the reader to [15, 19-22, 24, 25] and to [3,4,13] for what concerns the study of PDEs and functionals involving this fractional operator. On the other side, the properties of ∇ α led to the discovery of new (optimal) embedding inequalities [23,27,28] and the development of a distributional and asymptotic analysis in this fractional framework [5][6][7][8][9]26]. For a general panoramic on the fractional framework, the reader may consult the survey [29] and the monograph [17].…”

“…is the fractional α-divergence of the vector field ϕ ∈ Lip c (R n ; R n ). Equality (1.2) is the fundamental basis of the distributional theory in the present fractional setting developed in the previous papers [5][6][7][8][9]. In more precise terms, by imitating the classical definition of BV functions, for a given exponent p ∈ [1, +∞], we define the (total) fractional variation of a function f ∈ L p (R n ) as…”

“…The study of the space BV α (R n ) = BV α,1 (R n ) in the geometric regime p = 1 was initiated in [7], also in connection with the naturally associated notion of fractional Caccioppoli perimeter (see [7,Definition 4.1]), and then further investigated in the subsequent works [5,8]. The fractional variation of an L p function for an arbitrary exponent p ∈ [1, +∞] has been explored in [6,8,9].…”

“…The fractional variation of an L p function for an arbitrary exponent p ∈ [1, +∞] has been explored in [6,8,9]. Throughout this paper, with a slight abuse of notation (that, however, can be rigorously justified thanks to the analysis done in the previous works [5][6][7][8][9]), in the integer case α = 1 we let…”