A fractional modular Poincaré--Wirtinger-type inequality in Sobolev spaces with variable exponents
Abstract
Let Ω⊂ℝn be a bounded Lipschitz domain and let p:Ω→[1,∞) be a measurable, essentially bounded exponent; no regularity of p -- in particular, no log-Hölder continuity -- is assumed. For a general non-negative measurable kernel 𝒦(x,y) admitting a radial bounding function κ(x,r) in the second variable, we establish the fractional modular Poincaré--Wirtinger-type inequality iintΩ×Ω𝒦(x,y) |u(x)−u(y)|p(x) d y d x leqslant C∫Ωc(x)(∫Ω (|∇ u(z)|p(x))/(|z−x|n−1+θ) d z) d x, valid for every u∈ W1,1(Ω), in particular every u∈ W1,p(·)(Ω), and every θ∈[0,1). Here c(x)=∫0Drn+p(x)−2+θκ(x,r) d r, D is the diameter of any fixed ball containing Ω̅, and the constant C depends only on n,p+,θ and the fixed domain Ω. For the fractional kernel 𝒦(x,y)=|x−y|−n−λp(x) we obtain a variable-exponent Gagliardo-modular estimate. The proof replaces the starshaped chain decompositions used there by a combination of a Morrey-type estimate on starshaped domains, a positive-kernel proof of weighted Poincaré inequalities, and one reflection extension for the weights |z−x|−(n−1+θ). The constants are uniform down to exponent $1$, and the inequalities are understood in the extended nonnegative sense when either side is infinite.
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BibTeX
@misc{fratta2026fractional,
title = {{A fractional modular Poincaré--Wirtinger-type inequality in Sobolev spaces with variable exponents}},
author = {Giovanni Di Fratta and Alberto Fiorenza and Leon Happ},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36293},
eprint = {2610.36293},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36293},
}
RIS
TY - UNPB
TI - A fractional modular Poincaré--Wirtinger-type inequality in Sobolev spaces with variable exponents
AU - Giovanni Di Fratta
AU - Alberto Fiorenza
AU - Leon Happ
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36293
UR - https://arxiv.org/abs/2610.36293
AB - Let Ω⊂ℝ^n be a bounded Lipschitz domain and let p:Ω→[1,∞) be a measurable, essentially bounded exponent; no regularity of p -- in particular, no log-Hölder continuity -- is assumed. For a general non-negative measurable kernel 𝒦(x,y) admitting a radial bounding function κ(x,r) in the second variable, we establish the fractional modular Poincaré--Wirtinger-type inequality iint_Ω×Ω𝒦(x,y) |u(x)−u(y)|^p(x) d y d x leqslant C∫_Ωc(x)(∫_Ω (|∇ u(z)|^p(x))/(|z−x|^n−1+θ) d z) d x, valid for every u∈ W^1,1(Ω), in particular every u∈ W^1,p(·)(Ω), and every θ∈[0,1). Here c(x)=∫_0^Dr^n+p(x)−2+θκ(x,r) d r, D is the diameter of any fixed ball containing Ω̅, and the constant C depends only on n,p^+,θ and the fixed domain Ω. For the fractional kernel 𝒦(x,y)=|x−y|^−n−λp(x) we obtain a variable-exponent Gagliardo-modular estimate. The proof replaces the starshaped chain decompositions used there by a combination of a Morrey-type estimate on starshaped domains, a positive-kernel proof of weighted Poincaré inequalities, and one reflection extension for the weights |z−x|^−(n−1+θ). The constants are uniform down to exponent $1$, and the inequalities are understood in the extended nonnegative sense when either side is infinite.
ER -