We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account anomalous diffusion, with possible fractional components in both space and time. The operators studied comprise the case of the sum of classical and fractional Laplacians, possibly of different orders, in the space variables, and classical or fractional derivatives in the time variables.This type of approximation results shows that space-fractional and time-fractional equations exhibit a variety of solutions which is much richer and more abundant than in the case of classical diffusion.

We show that any given function can be approximated with arbitrary precision by solutions of linear, time-fractional equations of any prescribed order.This extends a recent result by Claudia Bucur, which was obtained for time-fractional derivatives of order less than one, to the case of any fractional order of differentiation.In addition, our result applies also to the ψ-Caputo-stationary case, and it will provide one of the building blocks of a forthcoming paper in which we will establish general approximation results by operators of any order involving anisotropic superpositions of classical, space-fractional and time-fractional diffusions.

Taking inspiration from a recent paper by Bergounioux et al., we study the Riemann-Liouville fractional Sobolev space W s,p RL,a+ (I), for I = (a, b) for some a, b ∈ R, a < b, s ∈ (0, 1) and p ∈ [1, ∞]; that is, the space of functions u ∈ L p (I) such that the left Riemann-Liouville (1 − s)-fractional integral I 1−s a+ [u] belongs to W 1,p (I). We prove that the space of functions of bounded variation BV (I) and the fractional Sobolev space W s,1 (I) continuously embed into W s,1 RL,a+ (I). In addition, we define the space of functions with left Riemann-Liouville s-fractional bounded variation, BV s RL,a+ (I), as the set of functions u ∈ L 1 (I) such that I 1−s a+ [u] ∈ BV (I), and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives.

We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we provide a lower bound for the $\Gamma$-liminf of the rescaled energy in terms of the horizontal perimeter.

We prove the Γ-convergence of the renormalised Gaussian fractional s-perimeter to the Gaussian perimeter as s → 1 - {s\to 1^{-}} . Our definition of fractional perimeter comes from that of the fractional powers of Ornstein–Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the Γ-limit does not depend on the dimension.