A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I.

We continue the study of the space  of functions with bounded fractional variation in  of order introduced in our previous work (Comi and Stefani in J Funct Anal 277(10):3373-3435, 2019). After some technical improvements of certain results of Comi and Stefani (2019) which may be of some separated insterest, we deal with the asymptotic behavior of the fractional operators involved as . We prove that the -gradient of a -function converges in to the gradient for all as .