Some remarks on rearrangement for nonlocal functionals

Abstract

t r a c tWe prove that a nonlocal functional approximating the standard Dirichlet p-norm fails to decrease under two-point rearrangement. Furthermore, we get other properties related to this functional such as decay and compactness.

Some characterizations of magnetic Sobolev spaces

“…where Q 2,N is the same positive constant appearing in (3) for p = 2. Other results related to the nonlocal operator I δ can be found in [9][10][11][12][13]. The aim of this note is to survey recent results contained in [14][15][16][17][18], where the authors have extended the aforementioned results to the magnetic setting.…”

On Hardy and Caffarelli-Kohn-Nirenberg inequalities

“…The quantity I δ with p = d has its roots in estimates for the topological degree of continuous maps from a sphere into itself in [5,22]. This also appears in characterizations of Sobolev spaces [6,11,12,21,24] and related contexts [8,11,12,23,25,26,28,29]. It is known that (see [21,Theorem 2] and [12, Proposition 1]), for p ≥ 1, (1.3)…”