PreprintCC BY-NC-ND 4.0arXiv
Coercivity and regularity for nonlocal divergence-form operators with measurable kernels
Abstract
We consider nonlocal divergence-form operators of the type PV∫ℝn(u(x)−u(y))K(x, y) dy=0 in B2, where the kernel K is symmetric (K(x, y) = K(y, x)). We first show that there exists a kernel satisfying the conditions of \cite[Open Question 2.1]{FRRO24} for which this problem admits a bounded discontinuous weak solution, answering that question negatively. We then impose the stronger one-sided lower bound suggested in \cite{CS20,IS20}, together with an Lp bound of K over annuli, and prove a coercivity estimate and interior Hölder regularity of weak solutions. In particular, this establishes the coercivity conjecture of \cite{CS20,IS20} under the additional Lp hypothesis for p>1.
Related
- Isometries on algebras of locally measurable operators
- Explicit Measurable Counterexamples to Finite-Difference Operators
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- Hölder regularity for nonlocal equations governed by measures and non-standard growth
- Interior regularity of mixed local-nonlocal parabolic semilinear equations
BibTeX
@misc{biswas2026coercivity,
title = {{Coercivity and regularity for nonlocal divergence-form operators with measurable kernels}},
author = {Anup Biswas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06408},
eprint = {2610.06408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06408},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Coercivity and regularity for nonlocal divergence-form operators with measurable kernels
AU - Anup Biswas
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06408
UR - https://arxiv.org/abs/2610.06408
AB - We consider nonlocal divergence-form operators of the type PV∫_ℝ^n(u(x)−u(y))K(x, y) dy=0 in B_2, where the kernel K is symmetric (K(x, y) = K(y, x)). We first show that there exists a kernel satisfying the conditions of \cite[Open Question 2.1]{FRRO24} for which this problem admits a bounded discontinuous weak solution, answering that question negatively. We then impose the stronger one-sided lower bound suggested in \cite{CS20,IS20}, together with an L^p bound of K over annuli, and prove a coercivity estimate and interior Hölder regularity of weak solutions. In particular, this establishes the coercivity conjecture of \cite{CS20,IS20} under the additional L^p hypothesis for p>1.
ER -