Sharp Truncated Difference Estimates for Nonlocal Nonconvex Functionals Defined by Superlevel Sets
Abstract
Let N∈ℕ and p∈[1,∞). In this article, we consider a family {Φλ,pγ}λ∈(0,∞) of nonlocal nonconvex functionals with a real parameter γ and establish two truncated difference estimates bounded by these functionals with the sharp range of γ; that is, for any γ∈(−∞,−1], any λ∈(0,∞), any cube Q⊂ℝN, and any measurable function g:Q→ℝ, &∫Q∫Q[(|g(x)−g(y)|−λ|x−y|1+γ/p)/(ℓ(Q)1+γ/p)]+p dx dy & ≲ λpmathop∫Q∫Q(|g(x)−g(y)|)/(|x−y|1+γ/p)>λ[(|x−y|)/ℓ(Q)]γ−N dx dy=:Φλ,pγ(g;Q)/(ℓ(Q)γ−N), where the implicit positive constant depends only on N, p, and γ. For γ∈(0,∞), the above estimate holds with the left-most integrand replaced by (min{|g(x)−g(y)|p,λp|x−y|p+γ})/(ℓ(Q)p+γ). As applications, these estimates yield related nonlocal Poincaré inequalities with the sharp range of γ, and also recover the known Γ-liminf lower bounds and characterizations of Sobolev and BV spaces in terms of the liminf of the functionals {Φγλ,p}λ∈(0,∞) for all γ∈ℝ∖(−1,0]. We investigate further applications to compactness and variational problems as well as a new VMO criterion. Inspired by a recent work of R. L. Frank, we develop an elementary approach based on dyadic iteration of truncated differences along line segments, which allows us to treat the aforementioned problems in a unified way.
Related
- Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets
- Weak type (1,1) estimates for noncommutative maximal truncated rough singular integrals
- Sharp L6 Fourier moments and (3,δ)-energy estimates for fractal measures on the truncated cone in R3
- Truncated Dirichlet estimates for Bloch functions
- Harnack-type estimates for nonlocal double phase functionals
BibTeX
@misc{dai2026sharp,
title = {{Sharp Truncated Difference Estimates for Nonlocal Nonconvex Functionals Defined by Superlevel Sets}},
author = {Feng Dai and Dachun Yang and Wen Yuan and Yirui Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04259},
eprint = {2610.04259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04259},
}
RIS
TY - UNPB
TI - Sharp Truncated Difference Estimates for Nonlocal Nonconvex Functionals Defined by Superlevel Sets
AU - Feng Dai
AU - Dachun Yang
AU - Wen Yuan
AU - Yirui Zhao
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.04259
UR - https://arxiv.org/abs/2610.04259
AB - Let N∈ℕ and p∈[1,∞). In this article, we consider a family {Φ_λ,p^γ}_λ∈(0,∞) of nonlocal nonconvex functionals with a real parameter γ and establish two truncated difference estimates bounded by these functionals with the sharp range of γ; that is, for any γ∈(−∞,−1], any λ∈(0,∞), any cube Q⊂ℝ^N, and any measurable function g:Q→ℝ, &∫_Q∫_Q[(|g(x)−g(y)|−λ|x−y|^1+γ/p)/(ℓ(Q)^1+γ/p)]_+^p dx dy & ≲ λ^pmathop∫_Q∫_Q_(|g(x)−g(y)|)/(|x−y|^1+γ/p)>λ[(|x−y|)/ℓ(Q)]^γ−N dx dy=:Φ_λ,p^γ(g;Q)/(ℓ(Q)^γ−N), where the implicit positive constant depends only on N, p, and γ. For γ∈(0,∞), the above estimate holds with the left-most integrand replaced by (min{|g(x)−g(y)|^p,λ^p|x−y|^p+γ})/(ℓ(Q)^p+γ). As applications, these estimates yield related nonlocal Poincaré inequalities with the sharp range of γ, and also recover the known Γ-liminf lower bounds and characterizations of Sobolev and BV spaces in terms of the liminf of the functionals {Φ^γ_λ,p}_λ∈(0,∞) for all γ∈ℝ∖(−1,0]. We investigate further applications to compactness and variational problems as well as a new VMO criterion. Inspired by a recent work of R. L. Frank, we develop an elementary approach based on dyadic iteration of truncated differences along line segments, which allows us to treat the aforementioned problems in a unified way.
ER -