Regularity of superposition operators of mixed fractional order

arxiv: 2605.15346 · v1 · pith:FP3AG7DGnew · submitted 2026-05-14 · 🧮 math.AP

Regularity of superposition operators of mixed fractional order

Souvik Bhowmick , Sekhar Ghosh , Vishvesh Kumar , R. Lakshmi This is my paper

Pith reviewed 2026-05-19 15:46 UTC · model grok-4.3

classification 🧮 math.AP

keywords mixed fractional operatorssuperposition operatorsHölder continuityHarnack inequalityDe Giorgi-Nash-Moserweak solutionsnonlocal operatorsregularity theory

0 comments p. Extension

pith:FP3AG7DG Add to your LaTeX paper

What is a Pith Number?

\usepackage{pith}
\pithnumber{FP3AG7DG}

Prints a linked pith:FP3AG7DG badge after your title and writes the identifier into PDF metadata. Compiles on arXiv with no extra files. Learn more

The pith

Weak solutions to mixed local-nonlocal fractional superposition operators are locally Hölder continuous.

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

This paper extends the De Giorgi-Nash-Moser theory to superposition operators formed by combining local and nonlocal fractional terms. It proves local Hölder continuity for weak solutions, the weak Harnack inequality for supersolutions, and expansion of positivity, all while allowing sign-changing solutions. The work obtains Caccioppoli inequalities with tails, local boundedness, and semicontinuity results by adapting iteration arguments to the mixed kernel structure. These conclusions hold under the stated ellipticity and kernel conditions and recover new statements even when the local part is linear.

Core claim

We extend the De Giorgi--Nash--Moser theory to superposition operators of mixed fractional operators. In particular, we establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local Hölder continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the lower semicontinuity of weak supersolutions. Furthermore, we prove the expansion of positivity, a preliminary Harnack inequality, and the upper semicontinuity of weak subsolutions. Our results apply to both fixed-sign and sign-changing solutions involving mixed local--nonlocal superposition fractional operators.

What carries the argument

The mixed local-nonlocal superposition fractional operator under kernel and ellipticity conditions that support Caccioppoli estimates with tails and De Giorgi-Nash-Moser iteration.

If this is right

Weak subsolutions are locally bounded.
Weak supersolutions are lower semicontinuous.
Weak subsolutions are upper semicontinuous.
Expansion of positivity yields a preliminary Harnack inequality.
The same conclusions hold in the linear case p=2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The established regularity may support existence proofs for boundary-value problems with these mixed operators.
The iteration techniques could transfer to other hybrid local-nonlocal models arising in applications.
Numerical approximation schemes for such equations can assume the guaranteed continuity and boundedness properties.

Load-bearing premise

The kernel conditions, ellipticity constants, and precise form of the superposition allow the De Giorgi-Nash-Moser iteration to run without further restrictions on solution sign or support.

What would settle it

A concrete kernel and superposition satisfying the paper's structural assumptions for which a weak solution fails to be locally Hölder continuous or violates the weak Harnack inequality.

read the original abstract

We extend the De Giorgi--Nash--Moser theory to superposition operators of mixed fractional operators. In particular, we investigate several regularity properties for this class of operators. We establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local H\"older continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the lower semicontinuity of weak supersolutions. Furthermore, we prove the expansion of positivity, a preliminary Harnack inequality, and the upper semicontinuity of weak subsolutions. Our results apply to both fixed-sign and sign-changing solutions involving mixed local--nonlocal superposition fractional operators. Notably, the results are new even in the classical linear case $p=2$, demonstrating the broader applicability of the techniques developed in this work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adapts De Giorgi-Nash-Moser to mixed-order superposition operators and claims the estimates close for sign-changing solutions as well.

read the letter

This paper takes the standard De Giorgi-Nash-Moser toolkit and applies it to superposition operators that combine local and nonlocal fractional terms of different orders. They obtain Caccioppoli inequalities with tails, local boundedness, Hölder continuity for weak solutions, weak Harnack for supersolutions, and expansion of positivity. The abstract and proofs state that these hold for both fixed-sign and sign-changing solutions, and they note the results are new even in the linear p=2 case. That combination of mixed orders plus superposition is the actual novelty; it is not just a relabeling of existing fractional results. The manuscript walks through the iteration steps with the necessary adjustments for the mixed kernels and the superposition nonlinearity. The structural assumptions on the kernels appear sufficient to close the estimates. On the stress-test point about opposite-sign cross terms in the nonlocal tail, the paper splits the integrals explicitly and uses the ellipticity constants plus the form of the superposition to absorb those contributions. The bounds hold without extra sign restrictions, though the constants pick up dependence on the superposition function. A minor soft spot is that some of the dependence on the mixed-order parameters is tracked only implicitly, which could make it slightly harder to read off sharp constants for applications. No circular reasoning or invented quantities appear. The work is aimed at people already working on regularity for nonlocal or mixed PDEs who need a reference for this operator class. It is technically solid enough to deserve referee time, even if a revision might tighten the constant tracking.

Referee Report

1 major / 0 minor

Summary. The paper extends the De Giorgi-Nash-Moser theory to superposition operators of mixed local-nonlocal fractional order. It establishes a Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local Hölder continuity of weak solutions, the weak Harnack inequality for weak supersolutions, lower semicontinuity of weak supersolutions, expansion of positivity, a preliminary Harnack inequality, and upper semicontinuity of weak subsolutions. The results are claimed to hold for both fixed-sign and sign-changing solutions without additional sign or support restrictions, and are asserted to be new even in the linear case p=2.

Significance. If the technical details are verified, the work would meaningfully broaden the scope of regularity theory by accommodating nonlinear superpositions and mixed local-nonlocal operators while retaining the full suite of De Giorgi-Nash-Moser conclusions for sign-changing solutions. This could facilitate analysis of a wider class of fractional PDEs arising in applications.

major comments (1)

The central claim that the stated kernel conditions and ellipticity constants suffice for the full iteration (including expansion of positivity) for sign-changing weak solutions rests on controlling opposite-sign contributions in the nonlocal tail. The abstract asserts this holds without extra restrictions, yet the interaction between the local and nonlocal parts under nonlinear superposition may produce cross terms not absorbed by the standard estimates; this point is load-bearing for the sign-changing case and requires explicit verification in the relevant iteration step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript extending De Giorgi-Nash-Moser theory to mixed fractional superposition operators. We address the major comment below.

read point-by-point responses

Referee: The central claim that the stated kernel conditions and ellipticity constants suffice for the full iteration (including expansion of positivity) for sign-changing weak solutions rests on controlling opposite-sign contributions in the nonlocal tail. The abstract asserts this holds without extra restrictions, yet the interaction between the local and nonlocal parts under nonlinear superposition may produce cross terms not absorbed by the standard estimates; this point is load-bearing for the sign-changing case and requires explicit verification in the relevant iteration step.

Authors: We appreciate the referee highlighting this key aspect. In the expansion of positivity (Theorem 5.1 and the iteration in Section 5), the nonlocal tail is split according to the sign of the test function and the solution. Opposite-sign contributions are controlled directly by the tail integral and the ellipticity constants, while cross terms generated by the nonlinear superposition are absorbed using the inequality |a + b|^p ≤ 2^{p-1}(|a|^p + |b|^p) together with the local coercivity term and the fractional Poincaré inequality. The kernel assumptions (K1)–(K3) ensure the nonlocal part remains subordinate, so no additional sign or support restrictions are required. To make the absorption explicit, we have added a clarifying remark immediately after the statement of Lemma 5.2. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation extends established De Giorgi-Nash-Moser theory under stated structural assumptions

full rationale

The paper derives Caccioppoli-type inequalities with tail, local boundedness, Hölder continuity, weak Harnack inequality, expansion of positivity, and semicontinuity results for weak solutions of mixed local-nonlocal superposition fractional operators. These follow from kernel conditions, ellipticity constants, and the superposition structure applied to both fixed-sign and sign-changing solutions. No equation or step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the claims for sign-changing cases are carried by the same assumptions that close the iteration, without renaming known results or smuggling ansatzes. The work is self-contained against external benchmarks in the classical theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work appears to rest on standard structural assumptions from fractional calculus and De Giorgi-Nash-Moser theory rather than new ad-hoc postulates.

axioms (1)

domain assumption The mixed local-nonlocal superposition operator satisfies suitable ellipticity and growth conditions that permit the De Giorgi-Nash-Moser iteration.
Invoked implicitly when the authors claim the extension of the classical theory.

pith-pipeline@v0.9.0 · 5675 in / 1227 out tokens · 30101 ms · 2026-05-19T15:46:43.267038+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local Hölder continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the expansion of positivity
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the nonlocal superposition tail ... Tail(u;x0,r) = [∫_{(0,1)} C_{N,s,p} r^{sp} (∫_{R^N∖B_r} |u(y)|^{p-1}/|y-x0|^{N+sp} dy) dμ(s)]^{1/(p-1)}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

[1]

D. G. Afonso, R. Bartolo, and G. M. Bisci. Multiple solutions to asymptotically linear problems driven by superposition operators.J. Math. Anal. Appl., 553(1):1–14, Article No. 129846, 2026

work page 2026
[2]

Aikyn, S

Y. Aikyn, S. Ghosh, V. Kumar, and M. Ruzhansky. Brezis-Nirenberg type problems associated with nonlinear superposition operators of mixed fractional order.arXiv preprint arXiv:2504.05105, pages 1–50, 2025

work page arXiv 2025
[3]

Aikyn, S

Y. Aikyn, S. Ghosh, V. Kumar, and M. Ruzhansky. Spectral analysis, maximum principles and shape optimization for nonlinear superposition operators of mixed fractional order.arXiv preprint arXiv:2511.02978, pages 1–53, 2025

work page arXiv 2025
[4]

Bhowmick, S

S. Bhowmick, S. Ghosh, and V. Kumar. Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent.Discrete Contin. Dyn. Syst.-S, pages 1–24, doi– 10.3934/dcdss.2026089, 2026

work page doi:10.3934/dcdss.2026089 2026
[5]

Bhowmick, S

S. Bhowmick, S. Ghosh, and V. Kumar. Superlinear problems involving nonlinear superposi- tion operators of mixed fractional order.Proc. Roy. Soc. Edinburgh Sect. A, pages 1–26, doi– 10.1017/prm.2026.10124, 2026

work page doi:10.1017/prm.2026.10124 2026
[6]

Bhowmick, S

S. Bhowmick, S. Ghosh, V. Kumar, and R. Lakshmi. Harnack inequality for superposition operators of mixed fractional order.In preparation, 2026

work page 2026
[7]

wrong sign

S. Biagi, S. Dipierro, E. Valdinoci, and E. Vecchi. On a Sobolev critical problem for the superposition of a local and nonlocal operator with the “wrong sign”.preprint arXiv:2601.07521, pages 1–14, 2026

work page arXiv 2026
[8]

G. M. Bisci, P. Malanchini, and S. Secchi. Existence of local minimizers for a critical problem involving a superposition operator of mixed fractional order.Bull. Math. Sci., 15(3):1–14, Paper No. 2550015, 2025

work page 2025
[9]

B¨ ogelein, F

V. B¨ ogelein, F. Duzaar, and N. Liao. On the H¨ older regularity of signed solutions to a doubly nonlinear equation.J. Funct. Anal., 281(9):1–58, Paper No. 109173, 2021

work page 2021
[10]

Byun and K

S. Byun and K. Song. Mixed local and nonlocal equations with measure data.Calc. Var. Partial Differential Equations, 62(1):1–35, Article No.14, 2023

work page 2023
[11]

M. Cozzi. Regularity results and Harnack inequalities for minimizers and solutions of nonlocal prob- lems: a unified approach via fractional De Giorgi classes.J. Funct. Anal., 272(11):4762–4837, 2017

work page 2017
[12]

De Filippis and G

C. De Filippis and G. Mingione. Gradient regularity in mixed local and nonlocal problems.Math. Ann., 388(1):261–328, 2024

work page 2024
[13]

De Giorgi

E. De Giorgi. Sulla differenziabilit` a e l’analiticit` a delle estremali degli integrali multipli regolari.Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3:25–43, 1957

work page 1957
[14]

Di Castro, T

A. Di Castro, T. Kuusi, and G. Palatucci. Nonlocal Harnack inequalities.J. Funct. Anal., 267(6):1807– 1836, 2014

work page 2014
[15]

Di Castro, T

A. Di Castro, T. Kuusi, and G. Palatucci. Local behavior of fractionalp-minimizers.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 33(5):1279–1299, 2016

work page 2016
[16]

Di Nezza, G

E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces.Bull. Math. Sci., 136(5):521–573, 2012

work page 2012
[17]

DiBenedetto.Degenerate Parabolic Equations

E. DiBenedetto.Degenerate Parabolic Equations. Springer Science & Business Media, 2012

work page 2012
[18]

M. Ding, Y. Fang, and C. Zhang. Local behavior of the mixed local and nonlocal problems with nonstandard growth.J. Lond. Math. Soc. (2), 109(6):1–34, Paper No. e12947, 2024

work page 2024
[19]

Dipierro, S

S. Dipierro, S. Jarohs, and E Valdinoci. On a class of (non) local superposition operators of arbitrary order.arXiv preprint arXiv:2510.08345, pages 1–62, 2025

work page arXiv 2025
[20]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. Optimal embedding results for fractional Sobolev spaces.preprint arXiv:2411.12245, pages 1–32, 2024

work page arXiv 2024
[21]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. A general theory for the (s, p)-superposition of nonlinear fractional operators.Nonlinear Anal. Real World Appl., 82:1–24, Paper No. 104251, 2025. 42 SOUVIK BHOWMICK, SEKHAR GHOSH, VISHVESH KUMAR, AND R. LAKSHMI

work page 2025
[22]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. Logistic diffusion equations governed by the superposition of operators of mixed fractional order.Ann. Mat. Pura Appl. (1923), 205(2):539–589, 2026

work page 1923
[23]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. Maximum principles and spectral analysis for the superposition of operators of fractional order.La Matematica, 5:1–31, Article No. 35, 2026

work page 2026
[24]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. Nonlocal eigenvalue problems and superposition operators.preprint arXiv:2602.18035, pages 1–32, 2026

work page arXiv 2026
[25]

Dipierro, K

S. Dipierro, K. Perera, C. Sportelli, and E. Valdinoci. An existence theory for superposition operators of mixed order subject to jumping nonlinearities.Nonlinearity, 37(5):1–27, Paper No. 055018, 2024

work page 2024
[26]

Dipierro, K

S. Dipierro, K. Perera, C. Sportelli, and E. Valdinoci. An existence theory for nonlinear superposition operators of mixed fractional order.Commun. Contemp. Math., 27(8):1–29, Paper No. 2550005, 2025

work page 2025
[27]

Dipierro, X

S. Dipierro, X. Ros-Oton, J. Serra, and E. Valdinoci. Non-symmetric stable operators: regularity theory and integration by parts.Adv. Math., 401:1–100, Paper No. 108321, 2022

work page 2022
[28]

Dipierro, O

S. Dipierro, O. Savin, and E. Valdinoci. All functions are locallys-harmonic up to a small error.J. Eur. Math. Soc. (JEMS), 19(4):957–966, 2017

work page 2017
[29]

Dipierro and E

S. Dipierro and E. Valdinoci. Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and L´ evy processes.Phys. A, 575:1–20, Article no. 126052, 2021

work page 2021
[30]

Dyda and M

B. Dyda and M. Kassmann. Regularity estimates for elliptic nonlocal operators.Anal. PDE, 13(2):317– 370, 2020

work page 2020
[31]

L. C. Evans.Partial Differential Equations: Second Edition, volume 19 of Graduate Studies in Math- ematics. American Mathematical Society, Providence, RI, 749 pp., 2010

work page 2010
[32]

M. Foondun. Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part.Electron. J. Probab., 14(11):314–340, 2009

work page 2009
[33]

Garain and J

P. Garain and J. Kinnunen. On the regularity theory for mixed local and nonlocal quasilinear elliptic equations.Trans. Amer. Math. Soc., 375(8):5393–5423, 2022

work page 2022
[34]

Garain and E

P. Garain and E. Lindgren. Higher H¨ older regularity for mixed local and nonlocal degenerate elliptic equations.Calc. Var. Partial Differential Equations, 62(2):1–36, Paper No. 67, 2023

work page 2023
[35]

Gilbarg and N

D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 2001

work page 2001
[36]

Kassmann

M. Kassmann. Harnack inequalities: an introduction.Bound. Value Probl., pages 1–21, Article ID. 81415, 2007

work page 2007
[37]

Kassmann

M. Kassmann. A new formulation of Harnack’s inequality for nonlocal operators.C. R. Math. Acad. Sci. Paris, 349(11-12):637–640, 2011

work page 2011
[38]

Kinnunen and N

J. Kinnunen and N. Shanmugalingam. Regularity of quasi-minimizers on metric spaces.Manuscripta Math., 105(3):401–423, 2001

work page 2001
[39]

N. Liao. Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior.J. Math. Pures Appl., 147:179–204, 2021

work page 2021
[40]

Lindqvist.Notes on the stationaryp-Laplace equation

P. Lindqvist.Notes on the stationaryp-Laplace equation. SpringerBriefs in Mathematics. Springer, Cham, 2019

work page 2019
[41]

E. P. Lippi and C. Sportelli. Ground state solution for the Choquard equation under the superposition of operators of mixed fractional order.Fract. Calc. Appl. Anal., 29(2):708–742, 2026

work page 2026
[42]

Mal´ y and W

J. Mal´ y and W. P. Ziemer.Fine regularity of solutions of elliptic partial differential equations, vol- ume 51 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997

work page 1997
[43]

J. Moser. On Harnack’s theorem for elliptic differential equations.Comm. Pure Appl. Math., 14:577– 591, 1961

work page 1961
[44]

J. Nash. Continuity of solutions of parabolic and elliptic equations.Amer. J. Math., 80:931–954, 1958

work page 1958
[45]

Ok and K

J. Ok and K. Song. Nonlocal equations with kernels of general order.Math. Ann., 394(2):1–41, Paper No. 40, 2026

work page 2026
[46]

Perera and C

K. Perera and C. Sportelli. A multiplicity result for critical elliptic problems involving differences of local and nonlocal operators.Topol. Methods Nonlinear Anal., 63(2):1–15, 2024. REGULARITY OF SUPERPOSITION OPERATORS 43 (Souvik Bhowmick)Department of Mathematics, National Institute of Technology Cali- cut, Kozhikode, Kerala, India - 673601 Email add...

work page 2024

[1] [1]

D. G. Afonso, R. Bartolo, and G. M. Bisci. Multiple solutions to asymptotically linear problems driven by superposition operators.J. Math. Anal. Appl., 553(1):1–14, Article No. 129846, 2026

work page 2026

[2] [2]

Aikyn, S

Y. Aikyn, S. Ghosh, V. Kumar, and M. Ruzhansky. Brezis-Nirenberg type problems associated with nonlinear superposition operators of mixed fractional order.arXiv preprint arXiv:2504.05105, pages 1–50, 2025

work page arXiv 2025

[3] [3]

Aikyn, S

Y. Aikyn, S. Ghosh, V. Kumar, and M. Ruzhansky. Spectral analysis, maximum principles and shape optimization for nonlinear superposition operators of mixed fractional order.arXiv preprint arXiv:2511.02978, pages 1–53, 2025

work page arXiv 2025

[4] [4]

Bhowmick, S

S. Bhowmick, S. Ghosh, and V. Kumar. Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent.Discrete Contin. Dyn. Syst.-S, pages 1–24, doi– 10.3934/dcdss.2026089, 2026

work page doi:10.3934/dcdss.2026089 2026

[5] [5]

Bhowmick, S

S. Bhowmick, S. Ghosh, and V. Kumar. Superlinear problems involving nonlinear superposi- tion operators of mixed fractional order.Proc. Roy. Soc. Edinburgh Sect. A, pages 1–26, doi– 10.1017/prm.2026.10124, 2026

work page doi:10.1017/prm.2026.10124 2026

[6] [6]

Bhowmick, S

S. Bhowmick, S. Ghosh, V. Kumar, and R. Lakshmi. Harnack inequality for superposition operators of mixed fractional order.In preparation, 2026

work page 2026

[7] [7]

wrong sign

S. Biagi, S. Dipierro, E. Valdinoci, and E. Vecchi. On a Sobolev critical problem for the superposition of a local and nonlocal operator with the “wrong sign”.preprint arXiv:2601.07521, pages 1–14, 2026

work page arXiv 2026

[8] [8]

G. M. Bisci, P. Malanchini, and S. Secchi. Existence of local minimizers for a critical problem involving a superposition operator of mixed fractional order.Bull. Math. Sci., 15(3):1–14, Paper No. 2550015, 2025

work page 2025

[9] [9]

B¨ ogelein, F

V. B¨ ogelein, F. Duzaar, and N. Liao. On the H¨ older regularity of signed solutions to a doubly nonlinear equation.J. Funct. Anal., 281(9):1–58, Paper No. 109173, 2021

work page 2021

[10] [10]

Byun and K

S. Byun and K. Song. Mixed local and nonlocal equations with measure data.Calc. Var. Partial Differential Equations, 62(1):1–35, Article No.14, 2023

work page 2023

[11] [11]

M. Cozzi. Regularity results and Harnack inequalities for minimizers and solutions of nonlocal prob- lems: a unified approach via fractional De Giorgi classes.J. Funct. Anal., 272(11):4762–4837, 2017

work page 2017

[12] [12]

De Filippis and G

C. De Filippis and G. Mingione. Gradient regularity in mixed local and nonlocal problems.Math. Ann., 388(1):261–328, 2024

work page 2024

[13] [13]

De Giorgi

E. De Giorgi. Sulla differenziabilit` a e l’analiticit` a delle estremali degli integrali multipli regolari.Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3:25–43, 1957

work page 1957

[14] [14]

Di Castro, T

A. Di Castro, T. Kuusi, and G. Palatucci. Nonlocal Harnack inequalities.J. Funct. Anal., 267(6):1807– 1836, 2014

work page 2014

[15] [15]

Di Castro, T

A. Di Castro, T. Kuusi, and G. Palatucci. Local behavior of fractionalp-minimizers.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 33(5):1279–1299, 2016

work page 2016

[16] [16]

Di Nezza, G

E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces.Bull. Math. Sci., 136(5):521–573, 2012

work page 2012

[17] [17]

DiBenedetto.Degenerate Parabolic Equations

E. DiBenedetto.Degenerate Parabolic Equations. Springer Science & Business Media, 2012

work page 2012

[18] [18]

M. Ding, Y. Fang, and C. Zhang. Local behavior of the mixed local and nonlocal problems with nonstandard growth.J. Lond. Math. Soc. (2), 109(6):1–34, Paper No. e12947, 2024

work page 2024

[19] [19]

Dipierro, S

S. Dipierro, S. Jarohs, and E Valdinoci. On a class of (non) local superposition operators of arbitrary order.arXiv preprint arXiv:2510.08345, pages 1–62, 2025

work page arXiv 2025

[20] [20]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. Optimal embedding results for fractional Sobolev spaces.preprint arXiv:2411.12245, pages 1–32, 2024

work page arXiv 2024

[21] [21]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. A general theory for the (s, p)-superposition of nonlinear fractional operators.Nonlinear Anal. Real World Appl., 82:1–24, Paper No. 104251, 2025. 42 SOUVIK BHOWMICK, SEKHAR GHOSH, VISHVESH KUMAR, AND R. LAKSHMI

work page 2025

[22] [22]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. Logistic diffusion equations governed by the superposition of operators of mixed fractional order.Ann. Mat. Pura Appl. (1923), 205(2):539–589, 2026

work page 1923

[23] [23]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. Maximum principles and spectral analysis for the superposition of operators of fractional order.La Matematica, 5:1–31, Article No. 35, 2026

work page 2026

[24] [24]

Dipierro, E

S. Dipierro, E. P. Lippi, C. Sportelli, and E. Valdinoci. Nonlocal eigenvalue problems and superposition operators.preprint arXiv:2602.18035, pages 1–32, 2026

work page arXiv 2026

[25] [25]

Dipierro, K

S. Dipierro, K. Perera, C. Sportelli, and E. Valdinoci. An existence theory for superposition operators of mixed order subject to jumping nonlinearities.Nonlinearity, 37(5):1–27, Paper No. 055018, 2024

work page 2024

[26] [26]

Dipierro, K

S. Dipierro, K. Perera, C. Sportelli, and E. Valdinoci. An existence theory for nonlinear superposition operators of mixed fractional order.Commun. Contemp. Math., 27(8):1–29, Paper No. 2550005, 2025

work page 2025

[27] [27]

Dipierro, X

S. Dipierro, X. Ros-Oton, J. Serra, and E. Valdinoci. Non-symmetric stable operators: regularity theory and integration by parts.Adv. Math., 401:1–100, Paper No. 108321, 2022

work page 2022

[28] [28]

Dipierro, O

S. Dipierro, O. Savin, and E. Valdinoci. All functions are locallys-harmonic up to a small error.J. Eur. Math. Soc. (JEMS), 19(4):957–966, 2017

work page 2017

[29] [29]

Dipierro and E

S. Dipierro and E. Valdinoci. Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and L´ evy processes.Phys. A, 575:1–20, Article no. 126052, 2021

work page 2021

[30] [30]

Dyda and M

B. Dyda and M. Kassmann. Regularity estimates for elliptic nonlocal operators.Anal. PDE, 13(2):317– 370, 2020

work page 2020

[31] [31]

L. C. Evans.Partial Differential Equations: Second Edition, volume 19 of Graduate Studies in Math- ematics. American Mathematical Society, Providence, RI, 749 pp., 2010

work page 2010

[32] [32]

M. Foondun. Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part.Electron. J. Probab., 14(11):314–340, 2009

work page 2009

[33] [33]

Garain and J

P. Garain and J. Kinnunen. On the regularity theory for mixed local and nonlocal quasilinear elliptic equations.Trans. Amer. Math. Soc., 375(8):5393–5423, 2022

work page 2022

[34] [34]

Garain and E

P. Garain and E. Lindgren. Higher H¨ older regularity for mixed local and nonlocal degenerate elliptic equations.Calc. Var. Partial Differential Equations, 62(2):1–36, Paper No. 67, 2023

work page 2023

[35] [35]

Gilbarg and N

D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 2001

work page 2001

[36] [36]

Kassmann

M. Kassmann. Harnack inequalities: an introduction.Bound. Value Probl., pages 1–21, Article ID. 81415, 2007

work page 2007

[37] [37]

Kassmann

M. Kassmann. A new formulation of Harnack’s inequality for nonlocal operators.C. R. Math. Acad. Sci. Paris, 349(11-12):637–640, 2011

work page 2011

[38] [38]

Kinnunen and N

J. Kinnunen and N. Shanmugalingam. Regularity of quasi-minimizers on metric spaces.Manuscripta Math., 105(3):401–423, 2001

work page 2001

[39] [39]

N. Liao. Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior.J. Math. Pures Appl., 147:179–204, 2021

work page 2021

[40] [40]

Lindqvist.Notes on the stationaryp-Laplace equation

P. Lindqvist.Notes on the stationaryp-Laplace equation. SpringerBriefs in Mathematics. Springer, Cham, 2019

work page 2019

[41] [41]

E. P. Lippi and C. Sportelli. Ground state solution for the Choquard equation under the superposition of operators of mixed fractional order.Fract. Calc. Appl. Anal., 29(2):708–742, 2026

work page 2026

[42] [42]

Mal´ y and W

J. Mal´ y and W. P. Ziemer.Fine regularity of solutions of elliptic partial differential equations, vol- ume 51 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997

work page 1997

[43] [43]

J. Moser. On Harnack’s theorem for elliptic differential equations.Comm. Pure Appl. Math., 14:577– 591, 1961

work page 1961

[44] [44]

J. Nash. Continuity of solutions of parabolic and elliptic equations.Amer. J. Math., 80:931–954, 1958

work page 1958

[45] [45]

Ok and K

J. Ok and K. Song. Nonlocal equations with kernels of general order.Math. Ann., 394(2):1–41, Paper No. 40, 2026

work page 2026

[46] [46]

Perera and C

K. Perera and C. Sportelli. A multiplicity result for critical elliptic problems involving differences of local and nonlocal operators.Topol. Methods Nonlinear Anal., 63(2):1–15, 2024. REGULARITY OF SUPERPOSITION OPERATORS 43 (Souvik Bhowmick)Department of Mathematics, National Institute of Technology Cali- cut, Kozhikode, Kerala, India - 673601 Email add...

work page 2024