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Improved H\"older regularity of fractional $(p,q)$-Poisson equation with regular data

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Pith's one-line read This paper proves that viscosity solutions of a fractional (p,q)-Poisson equation with Hölder-continuous modulating coefficient $\xi$ and Hölder data $f$ are locally $C^{0,\gamma}$ for every $\gamma<\gamma_0$, and locally Lipschitz when…

desk verdict Real new regularity result for nonlocal double-phase operators, but the proof's core comes from an unpublished companion paper—check that lemma before accepting. read the letter →

arxiv 2507.09920 v1 pith:CC7DVNSX submitted 2025-07-14 math.AP

Anup Biswas , Aniket Sen This is my paper · ORCID
classification math.AP MSC 35B6535J7035R09
keywords fractionalp-Laplaciannonlocaldoublephase(pq)-PoissonequationHölderregularityLipschitzviscositysolutionsIshii-Lionsmethodtailspaces
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

This paper establishes a quantitative Hölder-regularity theorem for viscosity solutions of a fractional double-phase equation: the fractional $p$-Laplacian plus a second $q$-Laplace-type term modulated by a coefficient $\xi(x,z)$. The main result, Theorem 1.1, says that when $\xi$ is $\alpha$-Hölder and the right-hand side $f$ is $\beta$-Hölder, every viscosity solution belongs to $C^{0,\gamma}$ on compact subsets for every $\gamma$ below a single explicit exponent $\gamma_0$ that combines the $p$-phase tail exponents $sp/(p-1)$ and $sp/(p-2)$ with the data smoothness $(\alpha\wedge\beta)/(p-1)$. When that combined exponent exceeds 1, the solution is locally Lipschitz; when the data term is the smallest bottleneck, the endpoint exponent is actually attained. If correct, this converts earlier qualitative Hölder results for nonlocal double phase problems into a sharp, quantitative statement and, for the pure fractional $p$-Laplacian, yields local Lipschitz regularity in new parameter ranges without any boundedness assumption on the solution.

What carries the argument

The proof is an Ishii–Lions doubling argument in the viscosity setting. One forms $\Phi(x,y)=u(x)-u(y)-L\varphi(|x-y|)-m_1\psi(x)$, with $\varphi$ a Hölder profile $t^\gamma$ or a log-Lipschitz profile, and shows $\Phi\leq 0$ for large $L$ by contradiction. The nonlocal operator is decomposed into a cone integral $I_1$ near the diagonal, intermediate and far-field terms $I_2, I_3, I_4$ from the $p$-phase, and matching $J_1$–$J_4$ terms from the $q$-phase; each is estimated with the data Hölder exponents. The engine is the cone lower bound $I_1 \geq C L^{p-1}|a|^{\gamma(p-1)-sp}$ (Lemma 3.1, imported verbatim from the companion paper [5]), which supplies the critical power that the bootstrap then iterates: starting from any $C^{0,\kappa}$ regularity, the argument upgrades $\kappa$ until it reaches $\gamma_0$. The paper's Remark 3.1 notes that with $\alpha=\beta=1$ the $k$-th iterate gains $sp(1/(p-1)+\cdots+1/(p-1)^k)$, whose limit is $sp/(p-2)$—this is why the $p$-phase exponent $sp/(p-2)$ appears in the final formula.

What would settle it

Verify Lemma 3.1 for the model profile $\varphi(t)=t^\gamma$ with $\gamma=sp/(p-1)$: compute the cone integral $I_1$ for the doubled test functions and check that $I_1 \geq C L^{p-1}|a|^{\gamma(p-1)-sp}$ with a positive $C$ independent of $L$ as $L\to\infty$ and $|a|\to 0$. A counterexample to this inequality, or to the underlying second-difference estimates of [5, Lemma 2.2], would invalidate the Hölder and Lipschitz conclusions.

Watch

Extended reading notes

Core claim

The central claim is that the regularity of solutions is controlled by the single exponent $\gamma_0 = \min\{1, (sp+\alpha\wedge\beta)/(p-1), sp/(p-2)\}$ for $p>2$ and $\gamma_0 = \min\{1, (sp+\alpha\wedge\beta)/(p-1)\}$ for $p\in(1,2]$, under the parameter conditions $1<p\leq q$, $t,s\in(0,1)$, $tq\leq sp$, and $\xi\geq 0$ symmetric in its second variable. Theorem 1.1 proves that any viscosity solution $u$ in the appropriate tail space belongs to $C^{0,\gamma}(\overline{B}_1)$ for every $\gamma<\gamma_0$, and, moreover, belongs to $C^{0,\gamma_0}$ when $\gamma_0 = (sp+\alpha\wedge\beta)/(p-1) < \min\{1, sp/(p-2)\}$ for $p>2$, or $<1$ for $p\leq 2$. It also proves local Lipschitz regularity whenever $\gamma_0>1$, i.e. when $\min\{(sp+\alpha\wedge\beta)/(p-1), sp/(p-2)\}>1$ for $p>2$, or $(sp+\alpha\wedge\beta)/(p-1)>1$ for $p\leq 2$. Theorem 1.2 transfers the result to weak solutions of the fractional $p$-Laplacian via the known equivalence of weak and viscosity solutions, giving $C^{0,\gamma}$ bounds and, in particular, local Lipschitz regularity for $p$-harmonic functions when $p\in(1,2]$ or $sp>p-2$ with $p>2$.

Load-bearing premise

The cone lower bound of Lemma 3.1—the inequality that makes the $p$-phase contribution dominate near the maximum point—is imported without proof from a companion preprint, and the entire bootstrap rests on it.

Editorial extensions

If this is right

  • For the pure fractional $p$-Laplacian with $\beta$-Hölder data, local weak solutions are $C^{0,\gamma}$ for every $\gamma<\min\{1,(sp+\beta)/(p-1), sp/(p-2)\}$ when $p>2$, and for every $\gamma<\min\{1,(sp+\beta)/(p-1)\}$ when $p\in(1,2]$.
  • In the homogeneous case $f=0$, local weak solutions are locally Lipschitz whenever $p\in(1,2]$ or, for $p>2$, $sp>p-2$.
  • When the data term is the bottleneck, the solution reaches the endpoint regularity $C^{0,\gamma_0}$ with $\gamma_0=(sp+\alpha\wedge\beta)/(p-1)$, so the quantitative bound is sharp in that regime.
  • Boundedness of the solution is not assumed: the estimates depend only on the tail-functionals (weighted $L^{p-1}$ and $L^{q-1}$ norms) and on the Hölder norms of $\xi$ and $f$.
  • The theorem answers the question raised in [20, p. 551] about higher regularity of nonlocal double phase problems, by supplying a quantitative exponent under condition (1.1).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The bootstrap's limiting exponent for smooth data is the fixed point of $\kappa \mapsto (sp+\kappa)/(p-1)$, namely $sp/(p-2)$; this suggests that attaining $sp/(p-2)$ itself, rather than every $\gamma$ below it, would require a mechanism beyond the present iteration.
  • Because the estimates are not stable as $s\to 1$ or $t\to 1$ (Remark 1.2), the result does not pass to the classical local double-phase limit; a uniform-in-fractional-order version would be needed to recover local regularity theory from the same argument.
  • The formula's dependence on $\alpha\wedge\beta$ suggests that smoothing either the coefficient $\xi$ or the right-hand side $f$ yields the same gain; one can test this by comparing solutions of the pure $p$-Laplacian with an $\alpha$-Hölder $\xi$ against those with $\beta$-Hölder $f$, using the endpoint $\gamma_0$ as the prediction.
  • If Theorem 1.2 is correct, the condition $sp>p-2$ marks a plausible exact boundary where fractional $p$-harmonic functions become Lipschitz for $p>2$; constructing solutions near this boundary could test sharpness.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proves a quantitative Hölder regularity estimate for viscosity solutions of a fractional double-phase equation in which a fractional p-Laplacian is perturbed by a q-growth nonlocal term with a nonnegative Hölder-continuous modulating coefficient ξ and a Hölder-continuous forcing f. The main theorem gives, for p>2, C^{0,γ} regularity for every γ<γ0 = min{1,(sp+α∧β)/(p-1), sp/(p-2)}, and for 1<p≤2, γ0 = min{1,(sp+α∧β)/(p-1)}, with an endpoint statement when the data term (sp+α∧β)/(p-1) is the bottleneck, and a Lipschitz conclusion when γ0>1. The proof uses the Ishii-Lions doubling method with Hölder and logarithmic-Lipschitz profiles, organized into four bootstrap steps in each of the superquadratic and subquadratic regimes. A corollary for weak solutions of the fractional p-Laplacian is also stated and proved by invoking the viscosity/weak equivalence, yielding local Lipschitz regularity for p-harmonic functions when p∈(1,2] or sp>p-2 with p>2.

Significance. If the main theorem is correct, it is a meaningful quantitative advance over the qualitative Hölder theory for nonlocal double-phase problems of De Filippis and Palatucci and later papers, and it improves the known regularity statements for fractional p-harmonic functions by identifying the precise role of the data smoothness α∧β. The exponents are explicit and falsifiable, and the bootstrap structure is natural and well matched to the operator. The paper is not machine-checked and does not contain reproducible code; its main weakness is that several load-bearing estimates are imported without proof from the companion preprint [5], which is not available to the reader in refereed form. My assessment of significance is therefore conditional: the claimed endpoint and Lipschitz results are plausible and well motivated, but the present manuscript does not, on its own, contain enough of the proof of its own engine to be fully verified.

major comments (4)
  1. [Section 3, Lemma 3.1; proof of Theorem 3.7, inequalities (3.6) and (3.8)] The main positive lower bound I1 ≥ C L^{p-1}|a|^{γ(p-1)-sp} on the cone C is the engine of the contradiction argument, but its proof is not contained in this manuscript. The proof cites [5, Lemma 3.1] for p≥2, [5, Lemma 4.1] for p∈(1,2), and [5, Lemma 2.2] for the second-difference bounds, and the log-profile analogue is likewise imported from [5]. Since [5] is an unpublished arXiv preprint, the referee cannot verify from the present text that the relevant regimes are covered: in particular the regime γ(p-1)<sp used in Step 1, the cone-aperture choice δ0=η0∈(0,1/2), and the log-profile scaling δ0=δ1(log2|a|)^{-1}. All subsequent terms in (3.6) and (3.8) are either negative corrections that are made small by choosing ε1, or lower-order terms, so if Lemma 3.1 fails or requires extra hypotheses, the Hölder and Lipschitz conclusions of Theorems 3.7 and 4.3 collapse. A complete proof of Lemma 3.1, or a precise statement of which assertions of [5] are being assumed and why they apply in each of the needed parameter regimes, is necessary.
  2. [Lemmas 3.2 and 3.5 and their log-profile versions] The estimates for I2 and J1 are also quoted from [5, Lemmas 3.2 and 4.2] without proof. These estimates are not cosmetic: they are exactly the negative corrections that are absorbed into I1 by choosing ε1 small uniformly in L, and the logarithmic-profile versions are needed for the Lipschitz endpoint in Steps 2 and 4 of Theorems 3.7 and 4.3. As with Lemma 3.1, the manuscript should either provide the proofs or state these estimates as explicit hypotheses whose validity is part of the theorem statement.
  3. [Theorem 1.2, proof] The proof of Theorem 1.2 says that the result 'follows directly from Theorem 1.1' after invoking the equivalence of weak and viscosity solutions. To apply Theorem 1.1 to the pure fractional p-Laplacian one must specify that ξ≡0 and choose the Hölder exponent of ξ; the natural choice is α=β (or any fixed α∈(0,1]), which gives the stated exponent with β. This is a small but real gap in the application of the theorem, and should be written out explicitly.
  4. [Lemma 3.6 and the q>2 analysis in Theorem 4.3] In Lemma 3.6 the displayed last term of the lower bound for J2 is written with the p,s exponents, namely |a|^{κ+θ(κ(p-2)-sp)}, whereas the direct analogue of Lemma 3.3 would give the q,t exponent |a|^{κ+θ(κ(q-2)-tq)}. Since q≥p and tq≤sp, the q,t exponent is larger, so the displayed p,s version is a weaker bound and is therefore usable; however the proof's sentence that one 'only uses' the inequality between the two radial integrals is terse and should be expanded. In the q>2 case in Step 3 of Theorem 4.3 the same comparison is load-bearing for the final contradiction, so a short justification of the direction of the inequality would remove this ambiguity.
minor comments (5)
  1. [Equation (2.8)] In the definition of J3 the denominator is written as |z|^{z+tq}; it should be |z|^{n+tq}.
  2. [Lemma 3.4, display after (3.4)] The quantity being estimated is J4, but the display reads |J5|; the symbol J5 is not defined elsewhere and should be corrected to J4.
  3. [Theorem 3.7, Step 1, display (3.6)] The display says I1+I2+J2 ≥ ..., but the lemma being invoked in that step is Lemma 3.5, which estimates J1, not J2. This is a notation slip; the same inequality appears correctly with J1 in Step 3.
  4. [Lemma 3.4, proof] The proof uses 'Since sup_{B2}|u|≤1' in estimating J2, but the paper has not normalized A=sup_{B2}|u|+tails to one. The estimate should read '≤A' and absorb A into the constant, as the lemma statement already declares dependence on A.
  5. [Lemma 3.1 and Section 2] Section 2 fixes δ0=η0∈(0,1/2), while Lemma 3.1 states δ0=η0∈(0,1). These ranges should be reconciled, since δ0 controls both the cone radius and the aperture parameter and is used later in the proof.

Circularity Check

0 steps flagged · score 2.0 of 10

No by-construction circularity; the central claim is a genuine bootstrap, with the only imported ingredient being the cone estimate Lemma 3.1 from the authors' companion preprint [5], which is load-bearing but does not encode the target exponent.

full rationale

I walked the proof of Theorems 3.7 and 4.3. The contradiction argument needs a positive lower bound for I1 that grows like L^{p-1}|a|^{κ1(p-1)-sp}; this is supplied by Lemma 3.1, whose proof is not reproduced: 'First let p≥2. Using [5, Lemma 3.1] we can find L0 such that...' and similarly [5, Lemma 2.2] for the second-difference bounds. That makes Lemma 3.1 load-bearing: inequalities (3.6), (3.8) and the log-profile versions in Steps 2 and 4 all rest on it, and a failure of that lemma would collapse the bootstrap. However, this is not circular in the sense used here. [5] is an earlier, independent preprint by the first author and Topp; the lemma's assumptions (p∈(1,∞), cone aperture δ0=η0, φ(t)=t^γ or the Lipschitz log-profile) do not include the target exponent γ0 = min{1,(sp+α∧β)/(p-1),sp/(p-2)} nor the Hölder data α,β, and no fitted parameter is renamed as a prediction. The gain sp/(p-2) arises from iterating κ1 = min{γ, ...} with the parabola ℓ(y), not from a definitional identity. The reliance on an unpublished companion paper is a verification/reproducibility concern, not a circularity; per the hard rules, a parameter-free external lemma with stated assumptions that do not contain the target result counts as independent support. The only other self-citation, [5, Proposition 1.5] for weak-viscosity equivalence, is backed by the independent reference [32]. I therefore find no significant circularity and assign a low score reflecting the non-self-contained import.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The only inputs are the exponents s,t,p,q and the Hölder exponents α,β, all fixed hypotheses of the theorem, not fitted parameters. The proof introduces no new entities. The main external dependencies are the viscosity-solution framework of [32] and the cone estimate of [5], which are imported as ready-made tools.

assumptions (7)
  • domain assumption Viscosity solution framework of Korvenpää-Kuusi-Lindgren [32]: the operator L is evaluated on restricted test functions C^2_η and the comparison principle holds for the class of functions in C(B_2) ∩ L^{p-1}_{sp} ∩ L^{q-1}_{tq}.
    Invoked in Definition 2.1 and Lemma 2.2; needed to justify testing (1.3) at the maximizing points x̄ and ȳ in Section 2.1.
  • domain assumption Imported estimate [5, Lemma 3.1]: for the cone C, I1 ≥ C L^{p-1}|a|^{γ(p-1)-sp} for φ=φ_γ, and the log-profile version I1 ≥ C L^{p-1}|a|^{p-1-sp}|log|a||^{-θ}.
    Quoted in Lemma 3.1 without proof; this lower bound is the engine of the contradiction argument, used in (3.6) and (3.8).
  • standard math Algebraic estimate Lemma 2.3 (convexity bounds on ∫_0^1 |a+tb|^{p-2} dt) from [32].
    Used throughout the estimates for I2, I3, and J2 in Sections 3 and 4.
  • domain assumption Tail-space integrability: u ∈ L^{p-1}_{sp}(R^n) ∩ L^{q-1}_{tq}(R^n) and A = sup_{B2}|u| + Tail_{s,p}(u;0,2) + Tail_{t,q}(u;0,2) < ∞.
    Part of the hypotheses of Theorem 1.1; needed to control the far-field terms I4 and J4 in Lemma 3.4 and the tail integrals.
  • domain assumption Hölder regularity of the modulating coefficient: ξ is α-Hölder continuous in both arguments and symmetric ξ(x,z)=ξ(x,-z), with 0 ≤ ξ ≤ M.
    Assumed in Theorem 1.1; controls J3, the ξ difference term, in Lemmas 3.6 and 4.2.
  • domain assumption Regularity of the forcing term: f ∈ C^{0,β}(B̄2).
    Bounds the right-hand side difference f(x̄)-f(ȳ) ≤ C|a|^β in the master inequality (2.8).
  • standard math Equivalence of weak and viscosity solutions for the fractional p-Laplacian [32], used to pass from Theorem 1.1 with ξ≡0 to Theorem 1.2.
    Appears in the proof of Theorem 1.2 at the end of Section 4.

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Pith. "Pith review of Improved H\"older regularity of fractional $(p,q)$-Poisson equation with regular data." pith.science (2026). https://pith.science/paper/CC7DVNSX

@misc{pith2026250709920,
  author       = {Pith},
  title        = {Pith review of: Improved H\"older regularity of fractional $(p,q)$-Poisson equation with regular data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CC7DVNSX}},
  note         = {Machine review of arXiv:2507.09920}
}
abstract

We prove a quantitative H\"{o}lder continuity result for viscosity solutions to the equation $$ (-\Delta_p)^{s}u(x) + {\rm PV} \int_{\mathbb{R}^n} |u(x)-u(x+z)|^{q-2}(u(x)-u(x+z))\frac{\xi(x,z)}{|z|^{n+ tq}} dz=f \quad \text{in}\; B_2, $$ where $t, s\in (0, 1), 1<p\leq q, tq\leq sp$ and $\xi\geq 0$. Specifically, we show that if $\xi$ is $\alpha$-H\"{o}lder continuous and $f$ is $\beta$-H\"{o}lder continuous then any viscosity solution is locally $\gamma$-H\"{o}lder continuous for any $\gamma<\gamma_\circ $, where \[ \gamma_\circ=\left\{\begin{array}{lll} \min\{1, \frac{sp+\alpha\wedge\beta}{p-1}, \frac{sp}{p-2}\} & \text{for}\; p>2, \\ \min\{1, \frac{sp+\alpha\wedge\beta}{p-1}\} & \text{for}\; p\in (1, 2]. \end{array} \right. \] Moreover, if $\min\{\frac{sp+\alpha\wedge\beta}{p-1}, \frac{sp}{p-2}\}>1$ when $p>2$, or $\frac{sp+\alpha\wedge\beta}{p-1}>1$ when $p\in (1, 2]$, the solution is locally Lipschitz. This extends the result of [20] to the case of H\"{o}lder continuous modulating coefficients. Additionally, due to the equivalence between viscosity and weak solutions, our result provides a local Lipschitz estimate for weak solutions of $(-\Delta_p)^{s}u(x)=0$ provided either $p\in (1, 2]$ or $sp>p-2$ when $p>2$, thereby improving recent works [9, 10, 24].

Figures

Figures reproduced from arXiv: 2507.09920 by the authors.

Figure 1
Figure 1. Also, let D˜ 2 “ Bϱ˜zBδ˜. Using (2.1), (2.2) and (2.7) we arrive at LprCsw1px¯q ´ LprCsw2py¯q looooooooooooooomooooooooooooooon “I1 ` LprD1sw1px¯q ´ LprD1sw2py¯q looooooooooooooooomooooooooooooooooon “I2 ` LprD2sw1px¯q ´ LprD2sw2py¯q looooooooooooooooomooooooooooooooooon “I3 ` LprB c ϱ˜ sw1px¯q ´ LprB c ϱ˜ sw2py¯q looooooooooooooooomooooooooooooooooon “I4 ` LqrBδ˜sw˜1px¯q ´ LqrBδ˜sw˜2py¯q looooooooooooooooomoooooooo… view at source ↗
Figure 1
Figure 1. Relevant sets in the nonlocal viscosity evaluation. ż D˜2 rJqpw˜2py¯q ´ w˜2py¯ ` zqqs ξpx, z ¯ q ´ ξpy, z ¯ q |z| z`tq dz looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon “J3 ` LqrB c ϱ˜ sw˜1px¯q ´ LqrB c ϱ˜ sw˜2py¯q looooooooooooooooomooooooooooooooooon “J4 ď C|a¯| β . (2.8) Our main goal is to estimate the terms Ii , Ji , i “ 1, 2, 3, 4 suitable so that (2.8) leads to contradiction for all L large… view at source ↗

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Forward citations

Cited by 2 Pith papers

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  1. Gradient regularity for nonlocal double phase equations

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    Viscosity solutions to the given nonlocal double phase equation have Hölder continuous gradients when a is Lipschitz and |tq - sp| is sufficiently small.

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Pith tools

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