An overview of regularity results for the Laplacian and $p$-Laplacian in metric spaces

Ivan Yuri Violo

An overview of regularity results for the Laplacian and $p$-Laplacian in metric spaces

T0 review · 0 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read On bounded RCD(0,∞) spaces, square-integrable p-Laplacian data forces the nonlinear flux |Du|^{p−2}∇u into the Sobolev class H^{1,2}_C(TX), with an explicit L2 estimate.

desk verdict A useful survey of p-Laplacian regularity in RCD spaces; the central theorem is imported from the author's companion paper, and the stress-test objection about the absorption step misses a sign. read the letter →

arxiv 2502.00825 v1 pith:OBGMUELL submitted 2025-02-02 math.AP math.MG

Ivan Yuri Violo This is my paper · ORCID

classification math.APmath.MG MSC 35B6535J9246E3658J05

keywords nonlinearpotentialtheorymetricspacesLaplacianp-LaplacianellipticPDEsregularityestimatesRiccicurvatureRCD

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 survey argues that second-order elliptic regularity survives in metric measure spaces with a synthetic lower Ricci bound. Its central claim is a recent theorem: on a bounded RCD(0,∞) space, if a function $u$ lies in the domain of the $p$-Laplacian and $\Delta_p u$ is square-integrable for $p\in(1,3)$, then the nonlinear flux $|Du|^{p-2}\nabla u$ has an $L^2$ covariant derivative and $|Du|^{p-1}$ belongs to $W^{1,2}(X)$. The estimate is quantitative, with a constant depending only on $p$. This is the $p$-analogue of the known linear fact that $\Delta u\in L^2$ forces $u\in W^{2,2}$, and it matters because a general difference-quotient method is unavailable in nonsmooth settings. The paper also surveys the surrounding landscape: H\"older and boundary regularity under doubling plus Poincar\'e inequalities, Lipschitz estimates under Ricci bounds, and a detailed outline of the proof via regularization and fixed-point arguments.

What carries the argument

The proof is carried by four devices: the $p$-Bochner identity (5.1), which expresses divergence of $|\nabla u|^{p-2}A(\nabla|\nabla u|^p)$ as a Hessian-square term plus curvature and Laplacian terms; the regularized $(\varepsilon,p)$-Laplacian $\Delta_{p,\varepsilon}u := \mathrm{div}((|Du|^2+\varepsilon)^{(p-2)/2}\nabla u)$, which is uniformly elliptic; the developed operator $D_{\varepsilon,p}u := \Delta u + (p-2)\frac{\mathrm{Hess}(u)(\nabla u,\nabla u)}{|\nabla u|^2+\varepsilon}$, which makes the freezing step algebraic; and the improved Bochner inequality (4.9) on RCD spaces, together with an absorption device that controls a remainder by $\lambda_p|\mathrm{Hess}(u)|^2$ with $\lambda_p<1$. The overall mechanism is: regularize, prove uniform a priori bounds on the flux, solve the regularized equation by a fixed point using the compact inclusion $D_0(\Delta)\subset W^{1,2}$, then pass $\varepsilon\to 0^+$.

What would settle it

Perform the calculation of Section 6.2 on a concrete bounded RCD(0,∞) space, for instance a Euclidean cone or a product manifold, and check whether the displayed remainder $R$ after (6.13) obeys $R\le \lambda_p|\mathrm{Hess}(u)|^2$ at almost every point with $\lambda_p<1$ for all $p\in(1,2)$. Finding a value of $p$ where this pointwise absorption fails would break the uniform estimate (6.6) and with it the claimed $H^{1,2}_C$ regularity.

Watch

Extended reading notes

Core claim

The paper's central asserted content is Theorem 5.1: on a bounded RCD(0,∞) space, for $p\in(1,3)$, if $u\in D(\Delta_p)$ and $\Delta_p u\in L^2(m)$, then $|Du|^{p-2}\nabla u\in H^{1,2}_C(TX)$ and $|Du|^{p-1}\in W^{1,2}(X)$, with $\int |\nabla(|Du|^{p-2}\nabla u)|^2\,dm \le C_p(\|\Delta_p u\|_{L^2}^2 + \||Du|^{p-1}\|_{L^1})$. The space $H^{1,2}_C(TX)$ is the space of $L^2$ vector fields whose covariant derivative is again $L^2$. This is the natural nonlinear analogue of the linear inclusion $u\in W^{1,2}$, $\Delta u\in L^2 \Rightarrow u\in W^{2,2}$; the flux vector field, not $u$ itself, gains the second derivative, and the paper notes that $u\in W^{2,2}$ would be false even in the Euclidean setting.

Load-bearing premise

The load-bearing premise is that the nonsmooth space satisfies the improved Bochner inequality and that, for every $p\in(1,3)$, the remainder term in the $p$-Bochner computation can be pointwise absorbed into a fraction of $|\mathrm{Hess}(u)|^2$; for $p<2$ that absorption is asserted with proof deferred to the companion paper, and if it fails the uniform bounds and the limit argument collapse.

Editorial extensions

If this is right

For $p\in(1,3)$, the $p$-Laplacian behaves like its linear brother: an $L^2$ right-hand side upgrades the natural stress vector field $|Du|^{p-2}\nabla u$ to a vector field with $L^2$ covariant derivative, and the estimate is quantitative in $p$.
$u$ itself does not gain two derivatives; the paper notes this would be false even in Euclidean space, so the theorem identifies the right object to regularize.
On bounded RCD(0,N) spaces with finite $N$, a datum in $L^q$ for $q>N$ forces solutions of the $p$-Poisson equation to be Lipschitz (Theorem 5.3).
$p$-electrostatic potentials solving $\Delta_p u=0$ away from a compact set are locally Lipschitz outside the obstacle (Theorem 5.4).
For $K<0$ or finite dimension $N$, the $p$-range can be widened, so the regularity phenomenon is not tied to $p=2$ nor to nonnegative curvature (Remark 5.5).

Reading between the lines

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

If the absorption inequality used for $p<2$ can be proved directly from the Bochner inequality rather than by a deferred argument, the method would likely give local, not only global, $H^{1,2}_C$ regularity for the $p$-Laplacian; this is a testable extension the survey leaves open.
The frozen-coefficient fixed-point scheme used to solve the regularized equation looks transplantable to other divergence-form operators whose coefficients depend on $|\nabla u|$, such as weighted or anisotropic $p$-Laplacians; on RCD spaces the same two-step contraction argument might yield second-order estimates for those operators.
Because Theorem 5.1 controls $|\nabla(|Du|^{p-2}\nabla u)|$ in $L^2$, it may feed into higher integrability and compactness arguments for $p$-harmonic approximations, and ultimately into boundary regularity or nodal-domain estimates for $p$-eigenfunctions in singular spaces, though the paper does not pursue these.

Share X Bluesky LinkedIn Reddit HN

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

The bottom line: this is a well-organized survey of Laplacian and p-Laplacian regularity in metric measure spaces, with the centerpiece being a proof sketch of a second-order regularity theorem for the p-Laplacian in bounded RCD(0,∞) spaces. That theorem is not new here—it comes from the author's companion paper with Benatti [13]—but the survey frames it usefully against the first-order theory, the Laplacian case, and open questions.

What it does well: the paper gives a clean map of the area. Sections 3 and 4 summarize existence, boundary regularity, Harnack and Hölder estimates, Lipschitz regularity, eigenfunction bounds, and unique continuation, with the right pointers to the literature. The proof sketch in Section 6 is a genuine attempt to convey how the ε-regularization argument works, including the fixed-point construction in Section 6.3. The survey statements look consistent with the cited results, and the heavy citation to [13] is legitimate because the main theorem is from there.

Soft spots: the paper is not self-contained, which is normal for a survey but more pronounced here because the proof of Theorem 5.1 repeatedly defers to [13], especially the p<2 absorption step and the approximation in Theorem 6.9. There is also a blank citation after "lower semicontinuity (see e.g. )" in Section 6.2. These are minor, not load-bearing. The "sufficiently close to two" phrasing in the introduction undersells the actual p∈(1,3) range; that is a harmless imprecision.

I checked the stress-test objection about the absorption inequality after (6.13). It doesn't hold up. For p<2 the proof sets M=∞, so the indicator inside the first remainder term gives 1−2χ=−1, not +1. On the 1D example the remainder is then bounded by −(p−2)²+2(2−p)=p(2−p), which is strictly less than 1 for p∈(1,2). So the claimed λ_p∈(0,1) is consistent with that example. The concern appears to be a sign error, not a real gap in the survey's argument.

Who this is for: anyone wanting an entry point into p-Laplacian regularity on RCD spaces, or a preview of the proof of the main theorem before reading [13]. It deserves a serious referee; a published survey would be useful. I'd recommend sending it to review, with the blank reference and the p-range wording fixed.

Referee Report

0 major / 5 minor

Summary. The manuscript is a survey of regularity theory for the Laplacian and p-Laplacian on metric measure spaces, with emphasis on RCD spaces. It first reviews the variational formulation, boundary value problems, and Hölder regularity under doubling/Poincaré assumptions. It then recalls second-order and Lipschitz regularity for the Laplacian under Ricci bounds, eigenfunction estimates, and unique continuation. The final two sections present recent results by Benatti and Violo: Theorem 5.1 (|Du|^{p−2}∇u ∈ H^{1,2}_C and |Du|^{p−1} ∈ W^{1,2} when Δ_p u ∈ L², p ∈ (1,3)) and related Lipschitz statements, together with a proof overview via ε-regularization. The survey is clearly written and the statements are consistent with the cited literature.

Significance. If correct, Theorem 5.1 is a substantial contribution: it extends to nonsmooth RCD spaces the Euclidean second-order regularity f ∈ L² ⇒ |∇u|^{p−2}∇u ∈ W^{1,2} for the p-Laplacian, and the range p ∈ (1,3) is plausible from the structure of the proof. The paper also provides a useful state-of-the-art account and lists open questions. The proof overview is not fully self-contained—several key claims (Prop. 6.3, Lemma 6.6, the p<2 absorption after (6.13), and the p<2 case of Theorem 6.9) are imported from the authors' preprint [13]—but this is typical for a survey. I checked the specific stress-test objection: the one-dimensional computation with R = (2−p)(4−p)|Hess|² does not apply to Eq. (6.13), because for p<2 the factor (1−2χ_{|Du|≤M}) is −1 when M = +∞, making the first contribution negative. Thus I find no mathematical contradiction in the manuscript, though the p<2 bound needs to be stated explicitly for clarity.

minor comments (5)

[Section 6.2, after Eq. (6.13)] The stress-test counterexample with R = (2−p)(4−p)|Hess|² is not applicable because the first term in R contains the factor (1−2χ_{|Du|≤M}), which is −1 for p<2 (M=+∞), so the first contribution is negative. Nevertheless, the bound R ≤ λ_p|Hess(u)|² for p<2 is only asserted with a reference to [13, Prop. 4.3]; since the full range p∈(1,3) of Theorem 5.1 relies on this absorption, the survey should state explicitly what that proposition gives (or at least its hypotheses) so that the reader can verify the argument.
[Section 6.2, below Eq. (6.6)] The sentence 'Therefore by lower semicontinuity (see e.g. )' contains an empty citation; please provide a precise reference (e.g., a proposition in [13] or in [52]).
[Section 6.3, proof of Proposition 6.8] The notation T_{f,w}(w) and the subsequent expression T_{f,w}(U_1−U_2) appear to be a typo; the contraction argument should involve T_{f,w}(U_1)−T_{f,w}(U_2) (or the definition of the map should be clarified), otherwise the displayed estimate is confusing.
[Section 3.2, Proposition 3.5] The right-hand side of inequality (D) is missing the integration domain; it should read ∫_Ω |Du|^p dm (or the domain should be understood from context, but it is better to write it explicitly).
[Throughout] Please fix the typos: 'brtieﬂy' (Section 1), 'Corolalry' (Section 3.3), 'Haj/suppress lasz' in reference [58], and 'a version of the classical Weyl’s lemma' → 'a version of the classical Weyl lemma' (Section 4.2).

Circularity Check

2 steps flagged · score 4.0 of 10

Central p<2 absorption and regularization steps are deferred to the authors' own preprint [13], but the result is not definitionally circular.

self citation load bearing [Section 6.2, after Eq. (6.13)]
"In the case p < 2 it still can be proved (see [13, Prop. 4.3]) that there exists a constant λp ∈ (0, 1) such that R ≤ λp|Hess(u)|2 m-a.e., which allows to absorb the reminder term R into the left-hand side obtaining again (6.9)."

This absorption is the decisive step that makes the uniform estimate (6.6) hold for p < 2. Without it, the remainder R in (6.13) cannot be controlled and the limit argument in Section 6.4 does not deliver Theorem 5.1 for p ∈ (1, 2). The survey does not prove the absorption; it imports [13, Prop. 4.3], a preprint co-authored by the present author. Since the same preprint is also the source of the theorem being reviewed, the presented derivation is not self-contained for p < 2 and leans on the authors' own prior work. This is load-bearing self-citation, not a definitional circle: the underlying result still has independent mathematical content in [13].
self citation load bearing [Section 6.3, end of proof of Theorem 6.9]
"In the case p < 2 the core of the argument is the same only that the function h(ε, w) = ( |∇w|2 + ε)^{(2−p)/2} is not necessarily in L2(m). Hence we need first to perform a cut-off and consider the function h(ε, w) := ((|∇w| ∧ M )2 + ε)^{(2−p)/2} and then send M → +∞ (see [13, Theorem 5.6] for the details)."

Existence of W^2,2 regularized solutions for p < 2, which is required to run Steps 3 and 4 and to apply Theorem 6.4, is delegated to the same authors' preprint [13, Theorem 5.6]. The survey gives only an outline; the actual argument supporting p ∈ (1, 2) in Theorem 5.1 is contained in [13]. This is another load-bearing self-citation. It does not make the statement definitionally circular, but it means the derivation shown here cannot stand alone for the full claimed range p ∈ (1, 3).

full rationale

The paper is a survey, and its expository derivation of Theorem 5.1 is not a claim of a new self-contained proof. The main theorem is not defined into existence: the p-Laplacian is defined by integration by parts, and the conclusion |Du|^{p-2}∇u ∈ H^{1,2}_C(TX) is a genuine second-order regularity statement, not an immediate rewriting of the definition. The proof of Theorem 6.4 uses external black boxes, principally the improved Bochner inequality (4.9) and Corollary 4.5 from [44]; these are citations to other authors and are not circular. The circularity concern is concentrated in the p < 2 range. After (6.13), the remainder absorption R ≤ λp |Hess(u)|² with λp < 1 is asserted with citation to the authors' own preprint [13, Prop. 4.3], and the W^2,2 regularization for p < 2 is deferred to [13, Theorem 5.6]. These are load-bearing self-citations: if [13, Prop. 4.3] were false or carried extra hypotheses, the derivation of Theorem 5.1 as presented would fail for p ∈ (1, 2). The critic's one-dimensional Euclidean computation bears on whether [13, Prop. 4.3] is true as stated; that is a correctness concern, not a circularity concern, and I do not count it toward the circularity score. Because the core result has independent content and the self-citations point to a separate proof rather than to the paper's own conclusion, the appropriate score is 4, not 0 and not 8.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on established RCD calculus, plus a chain of auxiliary lemmas from the author's own companion paper [13]. No new entities are postulated and no data-dependent parameters are fitted; the constants C_P, C_D, and s are structural, and ε is a regularization parameter sent to zero.

assumptions (6)

domain assumption Standing assumptions i) and ii): strong p-independency of weak upper gradients and infinitesimal Hilbertianity (Section 2).
The p-Laplacian and Sobolev calculus in Sections 3 and 6 rely on these assumptions; for RCD spaces they are known from [48] and [41], but not proved here.
domain assumption Local doubling and local Poincaré inequality (Definitions 3.13 and 3.4).
Used for the Sobolev inequality, Harnack inequalities, Hölder regularity, and Morrey embedding in Section 3.3; standard background assumptions.
standard math RCD(K,∞) Bochner machinery including Theorem 4.4 and Corollary 4.5 from [44].
The W^{2,2} and Hessian estimates for the Laplacian are imported as established tools, and the p-Laplacian proof in Section 6 relies on them directly.
ad hoc to paper Convergence of ε-regularized p-Laplacian solutions and identity Δ_{p,ε}u = (|Du|²+ε)^{(p−2)/2}D_{ε,p}u (Prop 6.3, Lemma 6.6).
Stated with proofs deferred to the companion preprint [13, Prop. 3.3 and Lemma 4.2]; these are load-bearing for the regularization scheme.
ad hoc to paper Absorption estimate R ≤ λp|Hess(u)|² for p < 2 (Equation (6.13), [13, Prop. 4.3]).
This is the key technical step allowing the remainder term to be absorbed into the Hessian term; no proof appears in this paper.
standard math Schauder fixed point theorem and compact embedding D0(Δ) ↪ W^{1,2} (Prop 6.1).
Used in Step 3 to produce regularized solutions u_ε in W^{2,2} for the regularized equation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of An overview of regularity results for the Laplacian and $p$-Laplacian in metric spaces." pith.science (2026). https://pith.science/paper/OBGMUELL

@misc{pith2026250200825,
  author       = {Pith},
  title        = {Pith review of: An overview of regularity results for the Laplacian and $p$-Laplacian in metric spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBGMUELL}},
  note         = {Machine review of arXiv:2502.00825}
}

abstract

We review some regularity results for the Laplacian and $p$-Laplacian in metric measure spaces. The focus is mainly on interior H\"older, Lipschitz and second-regularity estimates and on spaces supporting a Poincar\'e inequality or having Ricci curvature bounded below.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

100 extracted references · 76 canonical work pages

[13]

Second-order estimates for the $p$-Laplacian in RCD spaces

L. Benatti and I. Y. Violo , Second-order estimates for the p-Laplacian in RCD spaces , arXiv preprint arXiv:2401.09982, (2024)

work page Pith review arXiv 2024
[1]

Agmon, Unicit´ e et convexit´ e dans les probl` emes diﬀ´ erentiels, vol

S. Agmon, Unicit´ e et convexit´ e dans les probl` emes diﬀ´ erentiels, vol. No. 13 ( ´Et´ e, 1965) of S´ eminaire de Math´ ematiques Sup´ erieures [Seminar on Higher Mathematics], Les Presses de l’Universit´ e de Montr´ eal, Montreal, QC, 1966

1965
[2]

F. J. Almgren, Jr., Dirichlet’s problem for multiple valued functions and the r egularity of mass minimizing integral currents , in Minimal submanifolds and geodesics (Proc. Japan- United States Sem., Tokyo, 1977), North-Holland, Amsterda m-New York, 1979, pp. 1–6

1977
[3]

Ambrosio , Calculus, heat ﬂow and curvature-dimension bounds in metri c measure spaces, in Proceedings of the International Congress of Mathemati cians—Rio de Janeiro

L. Ambrosio , Calculus, heat ﬂow and curvature-dimension bounds in metri c measure spaces, in Proceedings of the International Congress of Mathemati cians—Rio de Janeiro
[4]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar ´e, Bakry- ´Emery curvature-dimension condition and Riemannian Ricci curvature bounds , The Annals of Probability, 43 (2015), pp. 339– 404

2015
[5]

Ambrosio, S

L. Ambrosio, S. Honda, J. W. Portegies, and D. Tewodrose , Embedding of RCD∗(K, N) spaces in L2 via eigenfunctions , J. Funct. Anal., 280 (2021), pp. Paper No. 108968, 72

2021
[6]

Ambrosio, S

L. Ambrosio, S. Honda, and D. Tewodrose, Short-time behavior of the heat kernel and Weyl’s law on RCD∗(K, N) spaces, Ann. Global Anal. Geom., 53 (2018), pp. 97–119

2018
[7]

Ambrosio, A

L. Ambrosio, A. Mondino, and G. Savar´e, On the Bakry- ´Emery condition, the gradient estimates and the local-to-global property of RCD∗(K,N) metric measure spaces , J. Geom. Anal., 26 (2016), pp. 24–56

2016

Show all 100 references

[8]

C. A. Antonini, A. Cianchi, G. Ciraolo, A. Farina, and V. Maz’ ya, Global second- order estimates in anisotropic elliptic problems , arXiv preprint arXiv:2307.03052, (2023)

2023 arXiv
[9]

C. A. Antonini, G. Ciraolo, and F. Pagliarin , Global second order regular- ity for degenerate p-Laplace type equations with log-conca ve weights , arXiv preprint arXiv:2501.02106, (2025)

2025 arXiv
[10]

Aronszajn, A

N. Aronszajn, A. Krzywicki, and J. Szarski , A unique continuation theorem for exte- rior diﬀerential forms on Riemannian manifolds , Arkiv f¨ or Matematik, 4 (1962), pp. 417– 453

1962
[11]

Bakry, Transformations de Riesz pour les semi-groupes sym´ etriques

D. Bakry, Transformations de Riesz pour les semi-groupes sym´ etriques. II. ´Etude sous la condition Γ 2 ≥ 0, in S´ eminaire de probabilit´ es, XIX, 1983/84, vol. 1123 o f Lecture Notes in Math., Springer, Berlin, 1985, pp. 145–174

1983
[12]

Benatti et al., Monotonicity formulas in nonlinear potential theory and th eir geomet- ric applications , Ph.D

L. Benatti et al., Monotonicity formulas in nonlinear potential theory and th eir geomet- ric applications , Ph.D. thesis, (2022)

2022
[14]

Bj ¨orn and J

A. Bj ¨orn and J. Bj ¨orn, Nonlinear potential theory on metric spaces , vol. 17 of EMS Tracts in Mathematics, European Mathematical Society (EMS ), Z¨ urich, 2011

2011
[15]

Bj ¨orn and N

A. Bj ¨orn and N. Marola , Moser iteration for (quasi)minimizers on metric spaces , Manuscripta Math., 121 (2006), pp. 339–366

2006
[16]

Bj¨orn, Fine continuity on metric spaces , Manuscripta Math., 125 (2008), pp

J. Bj¨orn, Fine continuity on metric spaces , Manuscripta Math., 125 (2008), pp. 369–381. [17] , Sharp exponents and a Wiener type condition for boundary reg ularity of quasimini- mizers, Adv. Math., 301 (2016), pp. 804–819

2008
[18]

Bj¨orn and N

J. Bj¨orn and N. Shanmugalingam , Poincar´ e inequalities, uniform domains and exten- sion properties for Newton–Sobolev functions in metric spa ces, Journal of Mathematical Analysis and Applications, 332 (2007), pp. 190–208

2007
[19]

Cavalletti and S

F. Cavalletti and S. Farinelli, Indeterminacy estimates and the size of nodal sets in singular spaces, Adv. Math., 389 (2021), pp. Paper No. 107919, 38

2021
[20]

Cavalletti and E

F. Cavalletti and E. Milman , The globalization theorem for the curvature-dimension condition, Invent. Math., 226 (2021), pp. 1–137

2021
[21]

Cavalletti and A

F. Cavalletti and A. Mondino , Sharp and rigid isoperimetric inequalities in metric- measure spaces with lower ricci curvature bounds , Invent. Math., 208 (2017), pp. 803–849. OVER VIEW OF REGULARITY FOR THE p-LAPLACIAN IN METRIC SPACES 29

2017
[22]

Topol., 21 (2017), pp

, Sharp geometric and functional inequalities in metric meas ure spaces with lower Ricci curvature bounds , Geom. Topol., 21 (2017), pp. 603–645

2017
[23]

Cheeger , Diﬀerentiability of Lipschitz functions on metric measure s paces, Geom

J. Cheeger , Diﬀerentiability of Lipschitz functions on metric measure s paces, Geom. Funct. Anal., 9 (1999), pp. 428–517

1999
[24]

Cheeger and T

J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I , J. Diﬀerential Geom., 46 (1997), pp. 406–480

1997
[25]

, On the structure of spaces with Ricci curvature bounded belo w. II , J. Diﬀerential Geom., 54 (2000), pp. 13–35

2000
[26]

S. Y. Cheng and S. T. Yau , Diﬀerential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28 (1975), pp. 333–354

1975
[27]

Cianchi and V

A. Cianchi and V. G. Maz’ya , Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal., 229 (2018), pp. 569–599

2018
[28]

T. H. Colding , New monotonicity formulas for Ricci curvature and applicat ions. I , Acta Math., 209 (2012), pp. 229–263

2012
[29]

T. H. Colding and W. P. Minicozzi II , Large scale behavior of kernels of schr¨ odinger operators, American Journal of Mathematics, 119 (1997), pp. 1355–139 8

1997
[30]

R. Courant, Ein allgemeiner satzt zur theorie der eigenfunktionen selb sadjungierter dif- ferentialausdr¨ ucke, Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse, 1923 (1923), pp. 81–8 4

1923
[31]

X. Dai, S. Honda, J. Pan, and G. Wei, Singular Weyl’s law with Ricci curvature bounded below, Trans. Amer. Math. Soc. Ser. B, 10 (2023), pp. 1212–1253

2023
[32]

De Philippis and J

G. De Philippis and J. N´u˜nez-Zimbr´on, The behavior of harmonic functions at singular points of rcd spaces , manuscripta mathematica, 171 (2023), pp. 155–168

2023
[33]

De Ponti, S

N. De Ponti, S. Farinelli, and I. Y. Violo , Pleijel nodal domain theorem in non- smooth setting , Transactions of the American Mathematical Society, Serie s B, 11 (2024), pp. 1138–1182

2024
[34]

Deng and X

Q. Deng and X. Zhao , Failure of strong unique continuation for harmonic functio ns on RCD Spaces, Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2023 (2023), pp. 221–241

2023
[35]

DiBenedetto , C 1+α local Regularity of Weak Solutions of Degenerate Elliptic E qua- tions., Nonlinear Analysis: Theory, Methods & Applications, 7 (19 83), pp

E. DiBenedetto , C 1+α local Regularity of Weak Solutions of Degenerate Elliptic E qua- tions., Nonlinear Analysis: Theory, Methods & Applications, 7 (19 83), pp. 827–850
[36]

Erbar, K

M. Erbar, K. Kuwada, and K.-T. Sturm, On the equivalence of the entropic curvature- dimension condition and Bochner’s inequality on metric mea sure spaces, Invent. Math., 201 (2015), pp. 993–1071

2015
[37]

L. C. Evans, A new proof of local C 1,α regularity for solutions of certain degenerate elliptic p.d.e, J. Diﬀerential Equations, 45 (1982), pp. 356–373

1982
[38]

19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998

, Partial diﬀerential equations , vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998

1998
[39]

Garofalo and F.-H

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, Ap weights and unique continuation , Indiana University Mathematics Journal, 35 (1986), pp. 24 5–268

1986
[40]

Gidas and J

B. Gidas and J. Spruck , Global and local behavior of positive solutions of nonlinea r elliptic equations , Communications on Pure and Applied Mathematics, 34 (1981) , pp. 525– 598

1981
[41]

Gigli , On the weak closure of convex sets of probability measures , Rend

N. Gigli , On the weak closure of convex sets of probability measures , Rend. Mat. Appl. (7), 29 (2009), pp. 133–141

2009
[42]

169–2 13

, An overview of the proof of the splitting theorem in spaces wi th non-negative Ricci curvature, Analysis and Geometry in Metric Spaces, 2 (2014), pp. 169–2 13

2014
[43]

, On the diﬀerential structure of metric measure spaces and app lications, Mem. Amer. Math. Soc., 236 (2015), pp. vi+91

2015
[44]

, Nonsmooth diﬀerential geometry—an approach tailored for sp aces with Ricci curva- ture bounded from below , Mem. Amer. Math. Soc., 251 (2018), pp. v+161

2018
[45]

, De Giorgi and Gromov working together , arXiv preprint arXiv:2306.14604, (2023). 30 I. Y. VIOLO

2023 arXiv
[46]

Anal., (2023), pp

, On the regularity of harmonic maps from RCD(K, N) to CAT(0) spaces and related results, Ars Inven. Anal., (2023), pp. Paper No. 5, 55

2023
[47]

Gigli and G

N. Gigli and G. De Philippis , Non-collapsed spaces with Ricci curvature bounded from below, J. ´Ec. polytech. Math., 5 (2018), pp. 613–650

2018
[48]

Gigli and B.-X

N. Gigli and B.-X. Han , Independence on p of weak upper gradients on RCD spaces, J. Funct. Anal., 271 (2016), pp. 1–11

2016
[49]

Gigli and F

N. Gigli and F. Marconi , A general splitting principle on rcd spaces and application s to spaces with positive spectrum , arXiv preprint arXiv:2312.06252, (2023)

2023 arXiv
[50]

Gigli, A

N. Gigli, A. Mondino, and G. Savar ´e, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and h eat ﬂows , Proc. Lond. Math. Soc. (3), 111 (2015), pp. 1071–1129

2015
[51]

Gigli and F

N. Gigli and F. Nobili , A ﬁrst-order condition for the independence on p of weak gradi- ents, J. Funct. Anal., 283 (2022), pp. Paper No. 109686, 52

2022
[52]

Gigli and E

N. Gigli and E. Pasqualetto , Lectures on Nonsmooth Diﬀerential Geometry , SISSA Springer Series, Springer, 2020

2020
[53]

Gigli and C

N. Gigli and C. Rigoni , A note about the strong maximum principle on RCD spaces , Canad. Math. Bull., 62 (2019), pp. 259–266

2019
[54]

Gigli and I

N. Gigli and I. Y. Violo , Monotonicity formulas for harmonic functions in RCD(0, N) spaces, J. Geom. Anal., 33 (2023), pp. Paper No. 100, 89

2023
[55]

Gilbarg and N

D. Gilbarg and N. S. Trudinger , Elliptic partial diﬀerential equations of second order , Classics in Mathematics, Springer-Verlag, Berlin, 2001. R eprint of the 1998 edition

2001
[56]

J. Gong, J. J. Manfredi, and M. Parviainen , Nonhomogeneous variational problems and quasi-minimizers on metric spaces , Manuscripta Math., 137 (2012), pp. 247–271

2012
[57]

Guarnotta and S

U. Guarnotta and S. Mosconi, A general notion of uniform ellipticity and the regularity of the stress ﬁeld for elliptic equations in divergence form , Analysis & PDE, 16 (2023), pp. 1955–1988

2023
[58]

Haj/suppress lasz and P

P. Haj/suppress lasz and P. Koskela, Sobolev met Poincar´ e, Mem. Amer. Math. Soc., 145 (2000), pp. x+101

2000
[59]

Han and F

Q. Han and F. Lin, Elliptic partial diﬀerential equations , vol. 1 of Courant Lecture Notes in Mathematics, Courant Institute of Mathematical Science s, New York; American Math- ematical Society, Providence, RI, second ed., 2011

2011
[60]

Heinonen, P

J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson , Sobolev spaces on metric measure spaces , vol. 27 of New Mathematical Monographs, Cambridge Univers ity Press, Cambridge, 2015. An approach based on upper gradient s

2015
[61]

Jiang, Lipschitz continuity of solutions of Poisson equations in m etric measure spaces, Potential analysis, 37 (2012), pp

R. Jiang, Lipschitz continuity of solutions of Poisson equations in m etric measure spaces, Potential analysis, 37 (2012), pp. 281–301

2012
[62]

, Cheeger-harmonic functions in metric measure spaces revis ited, J. Funct. Anal., 266 (2014), pp. 1373–1394

2014
[63]

Kell , A Note on Lipschitz Continuity of Solutions of Poisson Equat ions in Metric Measure Spaces

M. Kell , A Note on Lipschitz Continuity of Solutions of Poisson Equat ions in Metric Measure Spaces. arXiv:1307.2224, 2013

2013 arXiv
[64]

, On interpolation and curvature via Wasserstein geodesics , Adv. Calc. Var., 10 (2017), pp. 125–167

2017
[65]

Ketterer , Cones over metric measure spaces and the maximal diameter th eorem, J

C. Ketterer , Cones over metric measure spaces and the maximal diameter th eorem, J. Math. Pures Appl. (9), 103 (2015), pp. 1228–1275

2015
[66]

Kinnunen and N

J. Kinnunen and N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces , Manuscripta Math., 105 (2001), pp. 401–423

2001
[67]

Koskela, K

P. Koskela, K. Rajala, and N. Shanmugalingam , Lipschitz continuity of Cheeger- harmonic functions in metric measure spaces , J. Funct. Anal., 202 (2003), pp. 147–173

2003
[68]

Latvala, N

V. Latvala, N. Marola, and M. Pere, Harnack’s inequality for a nonlinear eigenvalue problem on metric spaces , J. Math. Anal. Appl., 321 (2006), pp. 793–810

2006
[69]

Li and S

P. Li and S. T. Yau , On the parabolic kernel of the Schr¨ odinger operator , Acta Mathe- matica, 156 (1986), pp. 153 – 201. OVER VIEW OF REGULARITY FOR THE p-LAPLACIAN IN METRIC SPACES 31

1986
[70]

J. Liu, F. Peng, and Y. Zhou , HW 2, 2 loc -regularity for p-harmonic functions in Heisenberg groups, Adv. Calc. Var., 16 (2023), pp. 379–390

2023
[71]

Lott and C

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal tran sport, Ann. of Math. (2), 169 (2009), pp. 903–991

2009
[72]

Lou , On singular sets of local solutions to p-Laplace equations , Chinese Ann

H. Lou , On singular sets of local solutions to p-Laplace equations , Chinese Ann. Math. Ser. B, 29 (2008), pp. 521–530

2008
[73]

J. J. Manfredi and A. Weitsman , On the Fatou theorem for p-harmonic functions , Comm. Partial Diﬀerential Equations, 13 (1988), pp. 651–66 8

1988
[74]

Mangoubi, The eﬀect of curvature on convexity properties of harmonic fu nctions and eigenfunctions, Journal of the London Mathematical Society, 87 (2013), pp

D. Mangoubi, The eﬀect of curvature on convexity properties of harmonic fu nctions and eigenfunctions, Journal of the London Mathematical Society, 87 (2013), pp. 645–662

2013
[75]

Q. Miao, F. Peng, and Y. Zhou, A global second-order Sobolev regularity for p-Laplacian type equations with variable coeﬃcients in bounded domains , Calculus of Variations and Partial Diﬀerential Equations, 62 (2023), p. 191

2023
[76]

K. Miller, Nonunique continuation for uniformly parabolic and ellipt ic equations in self- adjoint divergence form with H¨ older continuous coeﬃcients, Archive for Rational Mechanics and Analysis, 54 (1974), pp. 105–117

1974
[77]

Mondino and A

A. Mondino and A. Naber, Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds . Accepted at J. Eur. Math. Soc., arXiv:1405.2222, 2014

2014 arXiv
[78]

Mondino and D

A. Mondino and D. Semola, Polya-Szego inequality and Dirichlet p-spectral gap for non- smooth spaces with Ricci curvature bounded below , J. Math. Pures Appl. (9), 137 (2020), pp. 238–274

2020
[79]

, Lipschitz continuity and Bochner-Eells-Sampson inequali ty for harmonic maps from RCD (K, N) spaces to CAT (0) spaces , Preprint, (2022)

2022
[80]

Peng, H.-C

Y. Peng, H.-C. Zhang, and X.-P. Zhu , Weyl’s lemma on RCD (K, N) metric measure spaces, arXiv preprint arXiv:2212.09022, (2022)

2022 arXiv
[81]

A. k. Pleijel , Remarks on Courant’s nodal line theorem , Comm. Pure Appl. Math., 9 (1956), pp. 543–550

1956
[82]

Plis , On non-uniqueness in Cauchy problem for an elliptic second o rder diﬀerential equation, Bull

A. Plis , On non-uniqueness in Cauchy problem for an elliptic second o rder diﬀerential equation, Bull. Acad. Polon. Sci. S´ er. Sci. Math. Astronom. Phys, 11 (1963), pp. 95–100

1963
[83]

Rajala , Interpolated measures with bounded density in metric space s satisfying the curvature-dimension conditions of Sturm , Journal of Functional Analysis, 263 (2012), pp

T. Rajala , Interpolated measures with bounded density in metric space s satisfying the curvature-dimension conditions of Sturm , Journal of Functional Analysis, 263 (2012), pp. 896–924

2012
[84]

, Local Poincar´ e inequalities from stable curvature conditions on metric spaces , Calc. Var. Partial Diﬀerential Equations, 44 (2012), pp. 477–494

2012
[85]

, Approximation by uniform domains in doubling quasiconvex m etric spaces, Complex Analysis and its Synergies, 7 (2021), p. 4

2021
[86]

Saloff-Coste, A note on Poincar´ e, Sobolev, and Harnack inequalities, Internat

L. Saloff-Coste, A note on Poincar´ e, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices, (1992), pp. 27–38

1992
[87]

Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds , J

L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds , J. Diﬀerential Geom., 36 (1992), pp. 417–450

1992
[88]

Sarsa , Note on an elementary inequality and its application to the r egularity of p- harmonic functions , Ann

S. Sarsa , Note on an elementary inequality and its application to the r egularity of p- harmonic functions , Ann. Fenn. Math., 47 (2022), pp. 139–153

2022
[89]

Savar´e, Self-improvement of the Bakry- ´Emery condition and Wasserstein contraction of the heat ﬂow in RCD(K,∞ ) metric measure spaces , Discrete Contin

G. Savar´e, Self-improvement of the Bakry- ´Emery condition and Wasserstein contraction of the heat ﬂow in RCD(K,∞ ) metric measure spaces , Discrete Contin. Dyn. Syst., 34 (2014), pp. 1641–1661

2014
[90]

Sturm , On the geometry of metric measure spaces

K.-T. Sturm , On the geometry of metric measure spaces. I , Acta Math., 196 (2006), pp. 65–131

2006
[91]

II , Acta Math., 196 (2006), pp

, On the geometry of metric measure spaces. II , Acta Math., 196 (2006), pp. 133–177

2006
[92]

, Metric measure spaces and synthetic Ricci bounds: fundamen tal concepts and recent developments, in European Congress of Mathematics, EMS Press, Berlin, [2 023] ©2023, pp. 125–159

2023
[93]

Uhlenbeck , Regularity for a class of non-linear elliptic systems , Acta Math., 138 (1977), pp

K. Uhlenbeck , Regularity for a class of non-linear elliptic systems , Acta Math., 138 (1977), pp. 219–240. 32 I. Y. VIOLO

1977
[94]

N. N. Ural ′ ceva, Degenerate quasilinear elliptic systems , Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), pp. 184–222

1968
[95]

Valtorta, Sharp estimate on the ﬁrst eigenvalue of the p-Laplacian , Nonlinear Anal- ysis: Theory, Methods & Applications, 75 (2012), pp

D. Valtorta, Sharp estimate on the ﬁrst eigenvalue of the p-Laplacian , Nonlinear Anal- ysis: Theory, Methods & Applications, 75 (2012), pp. 4974–4 994

2012
[96]

Villani, Synthetic theory of Ricci curvature bounds , Jpn

C. Villani, Synthetic theory of Ricci curvature bounds , Jpn. J. Math., 11 (2016), pp. 219– 263

2016
[97]

Wang and L

X. Wang and L. Zhang, Local gradient estimate for p-harmonic functions on Riemannian manifolds, Comm. Anal. Geom., 19 (2011), pp. 759–771

2011
[98]

Yau, Harmonic functions on complete Riemannian manifolds , Communications on Pure and Applied Mathematics, 28 (1975), pp

S.-T. Yau, Harmonic functions on complete Riemannian manifolds , Communications on Pure and Applied Mathematics, 28 (1975), pp. 201–228

1975
[99]

Zhang and X.-P

H.-C. Zhang and X.-P. Zhu, Weyl’s law on RCD ∗(K, N) metric measure spaces, Comm. Anal. Geom., 27 (2019), pp. 1869–1914

2019
[100]

, Boundary regularity of harmonic maps from RCD (K, N)-space to CAT (0)-space, arXiv preprint arXiv:2402.19179, (2024). I. Y. Violo, Centro di Ricerca Matematica Ennio De Giorgi, Sc uola Normale Superiore, Piazza dei Cavalieri 3, 56126 Pisa (PI), Italy Email address : ivan.vio...

2024 arXiv
[2018]

Vol. I. Plenary lectures, World Sci. Publ., Hackensac k, NJ, 2018, pp. 301–340

2018

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [13]

Second-order estimates for the $p$-Laplacian in RCD spaces

L. Benatti and I. Y. Violo , Second-order estimates for the p-Laplacian in RCD spaces , arXiv preprint arXiv:2401.09982, (2024)

work page Pith review arXiv 2024

[2] [1]

Agmon, Unicit´ e et convexit´ e dans les probl` emes diﬀ´ erentiels, vol

S. Agmon, Unicit´ e et convexit´ e dans les probl` emes diﬀ´ erentiels, vol. No. 13 ( ´Et´ e, 1965) of S´ eminaire de Math´ ematiques Sup´ erieures [Seminar on Higher Mathematics], Les Presses de l’Universit´ e de Montr´ eal, Montreal, QC, 1966

1965

[3] [2]

F. J. Almgren, Jr., Dirichlet’s problem for multiple valued functions and the r egularity of mass minimizing integral currents , in Minimal submanifolds and geodesics (Proc. Japan- United States Sem., Tokyo, 1977), North-Holland, Amsterda m-New York, 1979, pp. 1–6

1977

[4] [3]

Ambrosio , Calculus, heat ﬂow and curvature-dimension bounds in metri c measure spaces, in Proceedings of the International Congress of Mathemati cians—Rio de Janeiro

L. Ambrosio , Calculus, heat ﬂow and curvature-dimension bounds in metri c measure spaces, in Proceedings of the International Congress of Mathemati cians—Rio de Janeiro

[5] [4]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar ´e, Bakry- ´Emery curvature-dimension condition and Riemannian Ricci curvature bounds , The Annals of Probability, 43 (2015), pp. 339– 404

2015

[6] [5]

Ambrosio, S

L. Ambrosio, S. Honda, J. W. Portegies, and D. Tewodrose , Embedding of RCD∗(K, N) spaces in L2 via eigenfunctions , J. Funct. Anal., 280 (2021), pp. Paper No. 108968, 72

2021

[7] [6]

Ambrosio, S

L. Ambrosio, S. Honda, and D. Tewodrose, Short-time behavior of the heat kernel and Weyl’s law on RCD∗(K, N) spaces, Ann. Global Anal. Geom., 53 (2018), pp. 97–119

2018

[8] [7]

Ambrosio, A

L. Ambrosio, A. Mondino, and G. Savar´e, On the Bakry- ´Emery condition, the gradient estimates and the local-to-global property of RCD∗(K,N) metric measure spaces , J. Geom. Anal., 26 (2016), pp. 24–56

2016