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

We continue the study of the space BV α(Rn) of functions with bounded fractional variation in Rn and of the distributional fractional Sobolev space Sα,p (Rn), with p 2 [1,+1] and α 2 (0, 1), considered in the previousworks [27,28].We first define the space BV 0(Rn) and establish the identifications BV 0(Rn) H1(Rn) and Sα,p (Rn) Lα,p (Rn), where H1(Rn) and Lα,p (Rn) are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient rα strongly converges to the Riesz transform as α→0+ for H1\Wα,1 and Sα,p functions.We also study the convergence of the L1-normof the α-rescaled fractional gradient ofWα,1 functions. To achieve the strong limiting behavior of rα as α→0+,we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.