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

We continue the study of the space B V 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 [28, 27]. We first define the space B V 0 ( ℝ n ) and establish the identifications B V 0 ( ℝ n ) = H 1 ( ℝ n ) and S α , p ( ℝ n ) = L α , p ( ℝ n ) , where 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 <mml:math xmlns:mml="http://www.w