We continue the study of the space BV α (R n ) of functions with bounded fractional variation in R n of order α ∈ (0, 1) introduced in our previous work [10], by dealing with the asymptotic behaviour of the fractional operators involved. After some technical improvements of certain results of [10], we prove that the fractional α-variation converges to the standard De Giorgi's variation both pointwise and in the Γ-limit sense as α → 1 − . We also prove that the fractional β-variation converges to the fractional α-variation both pointwise and in the Γ-limit sense as β → α − for any given α ∈ (0, 1).

We continue the study of the space $$BV^\alpha ({\mathbb {R}}^n)$$ B V α ( R n ) of functions with bounded fractional variation in $${\mathbb {R}}^n$$ R n of order $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) 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 $$\alpha \rightarrow 1^-$$ α → 1 - . We prove that the $$\alpha $$ α -gradient of a $$W^{1,p}$$ W 1 , p -function converges in $$L^p$$ L p to the gradient for all $$p\in [1,+\infty )$$ p ∈ [ 1 , + ∞ ) as $$\alpha \rightarrow 1^-$$ α → 1 - . Moreover, we prove that the fractional $$\alpha $$ α -variation converges to the standard De Giorgi’s variation both pointwise and in the $$\Gamma $$ Γ -limit sense as $$\alpha \rightarrow 1^-$$ α → 1 - . Finally, we prove that the fractional $$\beta $$ β -variation converges to the fractional $$\alpha $$ α -variation both pointwise and in the $$\Gamma $$ Γ -limit sense as $$\beta \rightarrow \alpha ^-$$ β → α - for any given $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) .

We continue the study of the space BV 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 [27,28]. We first define the space BV 0 (R n ) and establish the identificationswhere 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 S α,p functions. We also study the convergence of the L 1 -norm of the α-rescaled fractional gradient of W α,1 functions. To achieve the strong limiting behavior of ∇ α as α → 0 + , we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.

We continue the study of the fractional variation following the distributional approach developed in the previous works Bruè et al. (2021), Comi and Stefani (2019), Comi and Stefani (2019). We provide a general analysis of the distributional space $$BV^{\alpha ,p}({\mathbb {R}}^n)$$ B V α , p ( R n ) of $$L^p$$ L p functions, with $$p\in [1,+\infty ]$$ p ∈ [ 1 , + ∞ ] , possessing finite fractional variation of order $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) . Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a $$BV^{\alpha ,p}$$ B V α , p function.

We prove two new approximation results of H-perimeter minimizing boundaries by means of intrinsic Lipschitz functions in the setting of the Heisenberg group H n with n ≥ 2. The first one is an improvement of [19] and is the natural reformulation in H n of the classical Lipschitz approximation in R n . The second one is an adaptation of the approximation via maximal function developed by De Lellis and Spadaro, [11]. 1 2 R. MONTI AND G. STEFANIHere, we prove two new intrinsic Lipschitz approximation theorems for H-perimeter minimizers in the setting of the Heisenberg group H n with n ≥ 2.The first result is an improvement of [19] and is the natural reformulation in H n of the classical Lipschitz approximation in R n , see [17, Theorem 23.7]. Let W = R × H n−1 be the hyperplane passing through the origin and orthogonal to the direction ν = −X 1 . The disk D r ⊂ W centered at the origin is defined using the natural box norm of H n and the cylinder C r (p), p ∈ H n , is defined as C r (p) = p * C r , where C r = D r * (−r, r). We denote by e(E, C r (p), ν) the excess of E in C r (p) with respect to the fixed direction ν, that is, the L 2 -averaged oscillation of ν E , the inner horizontal unit normal to E, from the direction ν in the cylinder. The 2n + 1-dimensional spherical Hausdorff measure S 2n+1 is defined by the natural distance of H n . Finally, ∇ ϕ ϕ is the intrinsic gradient of ϕ. We refer the reader to Section 2 for precise definitions.Theorem 1.1. Let n ≥ 2. There exist positive dimensional constants C 1 (n), ε 1 (n) and δ 1 (n) with the following property. If E ⊂ H n is an H-perimeter minimizer in the cylinder C 5124 with 0 ∈ ∂E and e(E, C 5124 , ν) ≤ ε 1 (n) then, letting