PreprintCC BY-NC-SA 4.0arXiv
Failure of interior Lp estimates for nonlocal elliptic equations with continuous kernels
Abstract
Let 0<σ<2 and 1≤ p<∞. We construct a fixed linear nonlocal elliptic operator L and functions um∈ C∞(ℝn)∩ L∞(ℝn) such that |um|L∞(ℝn)+|Lum|Lp(B2)⟶0, liminfm→∞|(−Δ)σ/2um|Lp(B1)>0. The kernel is symmetric in the increment variable, uniformly elliptic, and satisfies the standard bounds on its first two derivatives in that variable. Its dependence on x has a uniform modulus of continuity of order (log(1/r))−1/2 as r↓0. Moreover, |Dum|Lp(B4)→0 for σ>1, and these norms remain bounded for σ=1. Thus the interior Lp estimates for translation-invariant operators do not extend to this class under mere continuity in x.
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BibTeX
@misc{liu2026failure,
title = {{Failure of interior $L^p$ estimates for nonlocal elliptic equations with continuous kernels}},
author = {Chang Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33689},
eprint = {2610.33689},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33689},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Failure of interior L^p estimates for nonlocal elliptic equations with continuous kernels
AU - Chang Liu
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33689
UR - https://arxiv.org/abs/2610.33689
AB - Let 0<σ<2 and 1≤ p<∞. We construct a fixed linear nonlocal elliptic operator L and functions u_m∈ C^∞(ℝ^n)∩ L^∞(ℝ^n) such that |u_m|_L^∞(ℝ^n)+|Lu_m|_L^p(B_2)⟶0, liminf_m→∞|(−Δ)^σ/2u_m|_L^p(B_1)>0. The kernel is symmetric in the increment variable, uniformly elliptic, and satisfies the standard bounds on its first two derivatives in that variable. Its dependence on x has a uniform modulus of continuity of order (log(1/r))^−1/2 as r↓0. Moreover, |Du_m|_L^p(B_4)→0 for σ>1, and these norms remain bounded for σ=1. Thus the interior L^p estimates for translation-invariant operators do not extend to this class under mere continuity in x.
ER -