Calderón–Zygmund estimates for general concave nonlocal equations
Fix $n \ge 2$, $0 < s < 1$ and $0 < \lambda \le \Lambda$, and let $\{K_a\}_{a \in A}$ be any family of even measurable kernels with $$\lambda |y|^{-n-2s} \le K_a(y) \le \Lambda |y|^{-n-2s}.$$ Set $$L_a u(x) = \operatorname{PV} \int_{\mathbb{R}^n} \bigl[u(x) - u(x+y)\bigr] K_a(y)\,dy, \qquad Iu = \inf_{a \in A} (-L_a u).$$ For second-order concave equations, Caffarelli proved interior $W^{2,p}$ estimates when the right-hand side is in $L^p$ with $p \ge n$, as Fernández-Real and Ros-Oton recall. Kitano proved $W^{2s,p}$-type estimates for concave nonlocal equations whose kernels have the form $y^{\mathsf T} A_\beta(x) y\, |y|^{-n-2-2s}$, under conditions on the matrices $A_\beta$ that include a uniform modulus of continuity.
Problem. Is there a finite $p_* = p_*(n, s, \lambda, \Lambda)$ such that, for every $p > \max\{2, p_*\}$, bounded viscosity solutions of $Iu = f$ in $B_1$ with smooth $f$ satisfy $$\|u\|_{W^{2s,p}(B_{1/2})} \le C \bigl(\|u\|_{L^\infty(\mathbb{R}^n)} + \|f\|_{L^p(B_1)}\bigr),$$ where $C$ depends only on $n, s, p, \lambda, \Lambda$? Here $W^{2s,p}$ is the Slobodeckij space when $2s$ is not an integer and the ordinary Sobolev space when $2s = 1$. The estimate must be uniform over the kernel families and must not depend on stronger norms of $f$.
Source: Fernández-Real and Ros-Oton (2024), §3.6.2, Open question 3.2. Stated by AIM — Open Applied Problems as an a priori estimate uniform over the kernel family, for smooth right-hand sides.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Bellman equations, fully nonlinear integro-differential equations, Sobolev estimates
References
@book{fernandezreal2024,
author = {Fern{\'a}ndez-Real, Xavier and Ros-Oton, Xavier},
title = {Integro-Differential Elliptic Equations},
series = {Progress in Mathematics},
volume = {350},
publisher = {Birkh{\"a}user},
address = {Cham},
year = {2024},
doi = {10.1007/978-3-031-54242-8},
eprint = {2411.12455},
archivePrefix = {arXiv},
}
@article{kitano2022,
author = {Kitano, Shuhei},
title = {{$W^{\sigma,p}$} a priori estimates for fully nonlinear integro-differential equations},
journal = {Calc. Var. Partial Differential Equations},
volume = {61},
number = {4},
year = {2022},
pages = {Paper No. 153},
doi = {10.1007/s00526-022-02276-7},
eprint = {2207.06728},
archivePrefix = {arXiv},
}
@misc{aim-concave-nonlocal-calderon-zygmund,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Calderón–Zygmund estimates for general concave nonlocal equations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/463-concave-nonlocal-calderon-zygmund.md},
note = {Entry 463 at commit 24edaad, 2026-10-15},
}
Resolution claims
We consider interior full-order Sobolev estimates for concave, translation-invariant nonlocal equations with even kernels comparable to that of the fractional Laplacian. Such estimates are required to be uniform over the kernel family and to depend on the forcing only through its $L^p$ norm. We show that they fail for every strict ellipticity ratio and every finite $p>\max\{2,n/\sigma\}$, where $0<\sigma<2$ is …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00783, "Calderón–Zygmund estimates for general concave nonlocal equations", clopen.solutions.