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

We continue the study of the space BV alpha(R-n) of functions with bounded fractional variation in R-n and of the distributional fractional Sobolev space S-alpha,S-p (R-n), with p is an element of [1, +infinity] and alpha is an element of (0, 1), considered in the previousworks [27,28]. We first define the space BV0(R-n) and establish the identifications BV0(R-n) = H-1(R-n) and S-alpha,S-p (R-n) = L-alpha,L-p (R-n), where H-1(R-n) and L-alpha,L-p (R-n) are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient del(alpha) strongly converges to the Riesz transform as alpha -> 0(+) for H-1 boolean AND W-alpha,W-1 and S-alpha,S-p functions. We also study the convergence of the L-1-norm of the alpha-rescaled fractional gradient of W-alpha,W-1 functions. To achieve the strong limiting behavior of del(alpha) as alpha -> 0(+), we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.