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

We continue the study of the space [Formula: see text] of functions with bounded fractional variation in [Formula: see text] of order [Formula: see text] 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 [Formula: see text] . We prove that the [Formula: see text] -gradient of a [Formula: see text] -function converges in [Formula: see text] to the gradient for all [Formula: see text] as [Formula: see text] . Moreover, we prove that the fractional [Formula: see text] -variation converges to the standard De Giorgi’s variation both pointwise and in the [Formula: see text] -limit sense as [Formula: see text] . Finally, we prove that the fractional [Formula: see text] -variation converges to the fractional [Formula: see text] -variation both pointwise and in the [Formula: see text] -limit sense as [Formula: see text] for any given [Formula: see text] .