On the H\"older regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian

E. B. dos Santos; Raimundo Leit\~ao

REVIEW 2 major objections 4 minor 26 references

On the H\"older regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian

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Pith's one-line read The paper extends nonlocal ABP, Harnack and Hölder regularity to arbitrary anisotropic exponents b_i and 0<s<4/bmax, but the written proof has a gap in the barrier construction.

desk verdict A genuine but unproved extension: the smooth and viscosity C^gamma theorems both rest on gaps, the worse being a barrier lemma that treats the anisotropic norm as rotation invariant. read the letter →

arxiv 1908.00525 v2 pith:LYBEMTKT submitted 2019-08-01 math.AP

E. B. dos Santos , Raimundo Leitão This is my paper · ORCID

classification math.AP

keywords regularitysolutionsanisotropicequationsfractionalgammaintegro-differentiallaplacian

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Many models in science use jumps instead of smooth motion, and here the jump rule can be different in different directions. The paper studies equations built from a kernel that compares a function at nearby points with weights of the form one over an anisotropic norm, where each coordinate direction has its own exponent b_i. It claims two types of interior regularity: a Hölder estimate C^\gamma for classical C^2 solutions, obtained by De Giorgi iteration, and a Harnack inequality together with C^\gamma and C^{1,\gamma} estimates for weaker viscosity solutions, obtained from an ABP estimate. If true, this would generalize the regularity theory of Caffarelli, Leitão and Urbano, which the paper says treated a particular family of exponents. The structure of the proofs follows the standard nonlocal roadmap. Two supports, however, are missing or wrong. The anisotropic form of Silvestre's inequality, which drives the smooth case, is stated without proof. More seriously, the barrier construction in Lemma 4.9 rotates the coordinate system as though the anisotropic norm were invariant under rotations. That is false when the exponents differ. Since that barrier is what produces the Harnack inequality and the viscosity regularity, the written proof does not currently support the main theorems.

Extended reading notes

Core claim

The central claim is Theorem 4.19: if u is bounded, M^-u is bounded above by C0 and M^+u is bounded below by -C0 in B1, and if 0<s0<s<4/bmax, then u belongs to C^\gamma(B_{1/2}) with the estimate |u|_{C^\gamma(B_{1/2})} <= C(sup|u| + C0). The smooth-solution version is Theorem 3.3, which gives the analogous C^\gamma estimate for solutions of the anisotropic fractional Laplacian equation in anisotropic ellipses.

Load-bearing premise

The proof of Lemma 4.9 (Section 4.2, equations (4.15)-(4.17)) assumes that the anisotropic norm ||y|| = sum |y_i|^{b_i} is invariant under coordinate rotations. This is false unless all b_i coincide. The barrier function built in Lemma 4.9 is the tool that produces the measure estimate in Lemma 4.13 and the Harnack inequality, so this incorrect change of variables is a load-bearing premise for the viscosity regularity theorems.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The paper genuinely extends the anisotropic nonlocal regularity theory from the special case in Caffarelli-Leitão-Urbano to arbitrary b_i > 0 and 0 < s < 4/bmax, and it adds a smooth-solution De Giorgi argument. But as written it does not prove its advertised theorems: the smooth theorem relies on Lemma 2.3, a Silvestre inequality that is stated without proof, and the viscosity theorem relies on Lemma 4.9, whose barrier computation treats the anisotropic norm as rotation invariant and Euclidean homogeneous. Both are load-bearing. The central claims may well be true, but the manuscript needs repair.

The extension is real. The paper says clearly that [8] covered b_i = n + sigma_i, s = 2 - c, and it pushes the method to the full range. The geometric setup with ellipses E_{r,l}, the anisotropic scaling T_{\beta,r}, and the ABP/covering structure are recognizable and mostly clean. I found no sign of circularity: it leans on [8], coauthored by one of the present authors, but for a special case, not for the conclusion.

Soft spots, in order of seriousness. Lemma 2.3 is a missing proof, not a wrong statement; a referee can ask for the argument. Lemma 4.9 is worse. To show M^-f >= 0 for 1 <= |x| <= R, the proof chooses a Euclidean rotation with x = |x|T_x e_1 and then replaces || |x|T_x y || by ||y||. That requires ||·|| to be invariant under rotations and homogeneous under Euclidean scaling, which fails unless all b_i coincide. The change of variables also seems to mix c and n in the determinant. Since this barrier drives the construction of Psi, Lemma 4.13, and the Harnack inequality, Theorem 4.19 is not established by this text. These are proof gaps rather than evidence of falsity; a barrier built with the anisotropic scaling T_{\beta,r} might repair the argument.

The paper is honest: it flags the s restriction, mentions possible extension via a metric, and cites competing work. If the gaps are fixable, the result is a useful unification within the regularity-theory subfield. As is, I would not cite it or rely on it. I would send it to a serious referee, with a request to focus on Lemma 4.9 and the missing proof of Lemma 2.3. A major-revision verdict, not a terminal reject, seems right if the authors can supply the missing pieces.

Referee Report

2 major / 4 minor

Summary. This paper studies Hölder regularity for integro-differential equations with anisotropic kernels K(y) comparable to ‖y‖^{-(c+s)}, where ‖y‖^2 = Σ_{i=1}^n |y_i|^{b_i}, under the restriction 0 < s < 4/b_max. The authors claim a C^γ estimate for smooth solutions of the anisotropic fractional Laplacian (Theorem 3.3) and for viscosity solutions of fully nonlinear integro-differential equations (Theorem 4.19). The smooth case is approached through Silvestre's De Giorgi technique, relying on an anisotropic Silvestre inequality (Lemma 2.3). The viscosity case follows Caffarelli–Leitão–Urbano: a nonlocal ABP estimate (Theorem 4.7), a barrier construction (Lemma 4.9), a measure estimate (Lemma 4.13), and a Harnack inequality (Theorem 4.18). The paper also states C^{1,γ} estimates under additional kernel assumptions.

Significance. If the proofs were correct, the results would provide a meaningful extension of the regularity theory for anisotropic nonlocal operators to kernels with different directional homogeneities, with explicit dependence on the lower bound s0. The geometric framework—anisotropic ellipses, the Caffarelli–Calderón covering lemma, and the nonlocal ABP structure—is well chosen and the theorem statements are precise. However, the central claims are not currently supported: Lemma 2.3 is unproved, and the proof of Lemma 4.9 contains an invalid change of variables that is load-bearing for the viscosity regularity theorem. These are not cosmetic issues.

major comments (2)

[§4.2, Lemma 4.9] The proof of Lemma 4.9 is not sound as written. In equations (4.15)–(4.17), the change of variables treats the anisotropic norm ‖y‖ (defined by ‖y‖^2 = Σ_{i=1}^n |y_i|^{b_i}) as if it were invariant under Euclidean rotations and homogeneous under scalar multiplication. Specifically, the denominator (Σ_{i=1}^n ||x|(T_x y)_i|^{b_i})^{(c+s)/2} is converted into |x|^{-n}‖y‖^{c+s} with no additional factor, which is valid only when all b_i coincide. Moreover, the replacement of ⟨T_x^{-1} y, |x|^{-1} e_1⟩^2 by ⟨y, x⟩^2 omits a factor |x|^{-4}, and the integration domain is changed from B_{1/4} to another ball without justification. Consequently, the lower bound for the term I_1 and the conclusion M^- f(x) ≥ 0 for 1 ≤ |x| ≤ R do not follow. Since this barrier is used in Corollaries 4.10–4.11 and Lemma 4.12 to construct Ψ, which is the input to Lemma 4.13, the measure estimate, the Harnack inequality (Theorem 4.18), and the central Theorem 4.19 all rest on this lemma. The viscosity regularity theorem is therefore not established by the manuscript.
[§2.1, Lemma 2.3] Lemma 2.3 (the anisotropic Silvestre inequality) is stated without proof. It is a key ingredient in the proof of Lemma 3.1 (Growth lemma), which in turn is used in Lemma 3.2 and Theorem 3.3. The introduction indicates that the inequality should follow from the anisotropic scaling T_{β,r} and radial barriers, but no derivation appears in Section 2.1. As a consequence, the smooth C^γ regularity theorem (Theorem 3.3) also lacks a complete proof.

minor comments (4)

[Introduction and Section 2] The notation for the norm is confusing: the paper writes ‖y‖^2 = Σ_{i=1}^n |y_i|^{b_i} but then uses ‖y‖^{c+s} in the kernel. This should be clarified by explicitly defining ‖y‖ = (Σ|y_i|^{b_i})^{1/2} and consistently using that throughout.
[§3, Lemma 3.1] In the proof of Lemma 3.1, the last displayed inequality does not match the form of (2.7): the term (3/4+|y|)^τ appears instead of a scaled variable such as |8y|^τ, suggesting a missing rescaling of the integration variable.
[§4.2, Lemma 4.9] The statement 'without loss of generality, we can assume that x ∈ {y : x_i ≥ 0} and x_1 ≥ 1/n' after equation (4.17) is not justified, since a Euclidean rotation does not preserve the anisotropic kernel and an arbitrary vector cannot be rotated into the positive orthant.
[General] There are numerous typographical errors, including 'Leito' in the abstract, inconsistent use of 'qmax,s' versus 'q_{max,s}', and undefined centers in the balls in Lemma 4.2. These should be corrected in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the C^gamma estimates are derived from external De Giorgi and ABP tools rather than assumed as inputs.

full rationale

Walking the derivation chain: Theorem 4.19 (C^gamma for viscosity solutions) follows from the Harnack inequality (Theorem 4.18), which follows from the measure decay estimates (Lemma 4.14, Theorem 4.15), which use Lemma 4.13, which combines the nonlocal anisotropic ABP estimate (Theorem 4.7) with the barrier function of Lemma 4.9. The ABP estimate is built from Lemmas 4.1-4.3, whose proofs are explicitly borrowed from the external results [9] and [8]. No step in this chain assumes the conclusion of Theorem 4.19 or Theorem 3.3. The cited work [8] is coauthored by one of the present authors, but it is an independently published paper covering the special case b_i = n + sigma_i, s = 2 - c; the present theorem covers the full range 0 < s < 4/b_max and is not a restatement of [8]. Remark 4.8 only notes that the special case is recovered, which is not circular. There are no fitted parameters renamed as predictions and no uniqueness theorem imported from the authors' own work. The proof does contain serious gaps: Lemma 2.3 (anisotropic Silvestre inequality) is stated without proof, and the rotation change of variables in Lemma 4.9, equations (4.15)-(4.17), is not valid for an anisotropic norm unless all b_i coincide. These are correctness concerns, not circularity: a false or missing step is different from a conclusion that is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The main theorems rest on a standard nonlocal regularity scaffold: kernel bounds, viscosity solution formalism, the Caffarelli-Calderón covering lemma, and a barrier function. The paper contributes anisotropic adaptations of De Giorgi and ABP, but two pieces are not independently established: Lemma 2.3 is unproved, and Lemma 4.9's proof assumes rotation invariance of the anisotropic norm. There are no fitted data parameters and no invented entities.

assumptions (5)

ad hoc to paper Anisotropic Silvestre inequality (Lemma 2.3) is true and available.
The lemma is stated without proof in Section 2.1 and is the engine of the Growth Lemma 3.1 and the smooth C^gamma theorem. The paper does not supply the anisotropic adaptation of Silvestre's argument.
ad hoc to paper The anisotropic norm ||y|| = sum |y_i|^{b_i} is invariant under coordinate rotations in the change of variables of Lemma 4.9.
Equations (4.15)-(4.17) in Section 4.2 rotate coordinates through a matrix T_x and pull out a scalar factor as if the anisotropic norm were Euclidean. This is only valid when all b_i are equal.
standard math Caffarelli-Calderón covering lemma and Lebesgue differentiation theorem for anisotropic rectangles.
Invoked via Lemma 4.5 and the Calderón-Zygmund iteration in Lemma 4.14; these are background results from the literature.
domain assumption Viscosity solution machinery for nonlocal operators from [9] (Lemmas 2.9-2.11).
The paper uses published results on C^{1,1} test functions, extremal operators, and comparison properties for fully nonlinear integro-differential equations.
domain assumption Kernel bounds (1.4) with 0 < lambda <= Lambda and 0 < s < 4/bmax define the admissible operator class.
This is the standing assumption of every theorem in the paper; the restriction s < 4/bmax is not derived from deeper principles.

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Pith. "Pith review of On the H\"older regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian." pith.science (2026). https://pith.science/paper/LYBEMTKT

@misc{pith2026190800525,
  author       = {Pith},
  title        = {Pith review of: On the H\"older regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LYBEMTKT}},
  note         = {Machine review of arXiv:1908.00525}
}

abstract

In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in [Silvestre, Indiana University Mathematics Journal 55, 2006], we adapt the De Giorgi technique to achieve the $C^{\gamma}$-regularity for solutions of class $C^{2}$ and use the geometry found in [Caffarelli, Leit\~ao, and Urbano, Math. Ann. 360, 2014] to get an ABP estimate, a Harnack inequality and the interior $C^{1, \gamma}$ regularity for viscosity solutions.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [8]

and Urbano, J

Caﬀarelli, L.; Leit˜ ao, R. and Urbano, J. M. Regularity for anisotr opic fully nonlinear integro- diﬀerential equations. Math. Ann. 360 (2014), no. 3-4, 681–714

2014

[2] [1]

and Kassmann

Bass, R.F. and Kassmann. M. Harnack inequalities for non-local o perators of variable order, Trans. Amer. Math. Soc. 357 (2005), 837-850

work page 2005

[3] [2]

and Bass, R.F

Bass, R.F. and Bass, R.F. H¨ older continuity of harmonic function s with respect to operators of variable order, Comm. Partial Diﬀerential Equations 30 (2005), 1249-1259

work page 2005

[4] [3]

and Levin D.A

Bass, R.F. and Levin D.A. Harnack inequalities for jump processes , Potential Anal. 17 (2002), 375-388. 31

work page 2002

[5] [4]

and Cabr´ e, X

Caﬀarelli, L.A. and Cabr´ e, X. Fully nonlinear elliptic equations . American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, R.I., 1995

work page 1995

[6] [5]

and Calder´ on, C.P

Caﬀarelli, L.A. and Calder´ on, C.P. Weak type estimates for the Ha rdy-Littlewood maximal functions, Studia Mathematica 49 (1974), 217-223

work page 1974

[7] [6]

and Calder´ on, C.P., On Abel summability of multiple Jac obi series, Colloq

Caﬀarelli, L.A. and Calder´ on, C.P., On Abel summability of multiple Jac obi series, Colloq. Math. 30 (1974), 277-288

work page 1974

[8] [7]

and Kassmann, M

Chaker, J. and Kassmann, M. Nonlocal operators with singular a nisotropic kernels. Comm. Partial Diﬀerential Equations. 45 (2020), no. 1, 1–31

work page 2020