A distributional approach to fractional Sobolev spaces and fractional variation

In this thesis, we present the distributional approach to fractional Sobolev spaces and fractional variation developed in [20, 22, 23]. The new space BVᵅ(ℝⁿ) of functions with bounded fractional variation in ℝⁿ of order α ∈ (0, 1) is distributionally defined by exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical BV theory, we give a new notion of set E of (locally) finite fractional Caccioppoli α-perimeter and we define its fractional reduced boundary FᵅE. We are able to show that Wᵅ,¹(ℝⁿ) ⊂ BV ᵅ(ℝⁿ) continuously and, similarly, that sets with (locally) finite standard fractional α-perimeter have (locally) finite fractional Caccioppoli α-perimeter, so that our theory provides a natural extension of the known fractional framework. We first extend De Giorgi’s Blow-up Theorem to sets of locally finite fractional Caccioppoli α-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets. We then 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- and, similarly, 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). Finally, by exploiting some new interpolation inequalities on the fractional operators involved, we prove that the fractional α-gradient converges to the Riesz transform as α → 0⁺ in Lp for p ∈ (1,+∞) and in the Hardy space and that the α-rescaled fractional α-variation converges to the integral mean of the function as α → 0⁺.