{"id":"e0874e42-2f2d-4280-8408-b334362ea97b","arxiv_id":"2509.05914","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Any locally bounded weak solution to ∂t(|u|^{q-1}u) + P.V. ∫ |u(x)-u(y)|^{p-2}(u(x)-u(y)) / |x-y|^{n+sp} dy = 0, with 0<s<1, p>2, 0<q<p-1, is locally Hölder continuous under a parabolic tail condition.","lead":"This paper proves that bounded, sign-changing weak solutions of a nonlocal doubly degenerate parabolic equation are locally Hölder continuous. It extends the known regularity theory for the local doubly nonlinear equation to a nonlocal, fractional setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Away-from-zero regularity hinges on omitted proofs of Lemmas 5.3-5.5.","rationale":"The reader returned CONDITIONAL, citing the parabolic tail condition as the weakest assumption while noting the omitted proofs of Lemmas 5.3-5.5. I agree with the CONDITIONAL verdict but not with the prioritization. The tail condition is explicitly part of the theorem's hypotheses and is used to define omega in (3.4). Section 4.3 Step 2 proves that after shrinking the cylinder, the tail becomes small relative to omega (inequality (4.41)); this is a standard and, from a local point of view, unavoidable step in nonlocal De Giorgi arguments. The dependence of the constants on Tail_m is transparent and does not undermine the Holder claim. By contrast, the away-from-zero part (Section 5) is formally incomplete. Lemmas 5.3-5.5 are stated without proofs, yet Proposition 5.7 and the iteration in Section 5.3 rely on them. The equation satisfied by v differs qualitatively from the near-zero case: it has a drift term involving bar v outside B_rho, and the Caccioppoli inequality (5.14) controls an energy for v with a tail involving bar v. It is not automatic that the standard De Giorgi machinery from reference [9] carries over; the proof of Lemma 5.2 is given, but Lemmas 5.3-5.5 would require analogous tail controls under the time-scaling t -> t0+(mu^-)^(q-p+1) tau. The omission is therefore a genuine gap in the written proof. Since the gap is one of missing derivations rather than a demonstrated contradiction, a conditional acceptance with a request for the full proofs is the appropriate verdict. I would not escalate to REJECT because the overall strategy is coherent and the omitted lemmas are plausibly provable by adapting Lemma 4.2 and reference [9], Lemmas 4.2-4.4. The concrete test is to supply those proofs and check the tail estimates; if they fail, the theorem's second half collapses.","tokens_in":1211,"tokens_out":2527,"duration_ms":192741,"concrete_test":"Independently write out the proof of Lemma 5.3 following the template of Lemma 4.2 but using the Caccioppoli inequality (5.14). Verify specifically that the boundary tail term in (5.14) is bounded by gamma r^(-sp) (xi tilde omega)^p |Q_r_j| (Y_j + Z_j^(1+kappa)) using assumption (5.28), after the change of variables and the estimate (bar v - k_j^(1/q))_- <= gamma xi tilde omega + (bar v - mu^-_bar v)_+. If the tail term instead requires a bound on (v - mu^-_v)_-, which is not assumed, then Lemma 5.3 fails. As a second check, re-derive Lemma 5.4's constant (5.33) and confirm that the drift term in (5.6) does not introduce an additional delta-dependence beyond that stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5 proves Holder continuity when u is away from zero by applying Proposition 5.7 to the rescaled function v=(u/mu^-)^q. Proposition 5.7 depends on Lemmas 5.3, 5.4 and 5.5, whose proofs are explicitly omitted. These lemmas are load-bearing: Lemma 5.3 provides forward-in-time propagation from an initial measure lower bound, Lemma 5.4 propagates measure-theoretic information, and Lemma 5.5 gives a measure-shrinking estimate that closes the De Giorgi iteration. They are not immediate corollaries of Lemma 5.2, because equation (5.6) for v contains a non-divergence-form drift term involving bar v outside B_rho, and the Caccioppoli inequality (5.14) has a tail term with (bar v - k^(1/q))_+/- rather than (v-k)_+/-. In particular, controlling the drift term requires the tail conditions (5.28), (5.31), (5.35) and the change of variables t -> t0+(mu^-)^(q-p+1) tau. If any of these estimates cannot be completed, the oscillation decay (5.39) and hence the Holder conclusion in the away-from-zero case is unsupported. The tail-condition issue raised in the reader's weakest_assumption is a stated hypothesis and is handled in Section 4.3 Step 2; the omitted proofs are a more direct formal gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves local Hölder continuity for locally bounded weak solutions of a nonlocal doubly degenerate parabolic equation ∂t(|u|^{q-1}u) + L_K u = 0, where L_K is a fractional p-Laplacian-type operator with kernel comparable to |x-y|^{-n-sp}, under the assumptions p>2, 0<q<p-1, and a parabolic tail condition on the solution. The proof follows the intrinsic-scaling De Giorgi method. When the solution is near zero, the author derives oscillation decay via Caccioppoli inequalities, De Giorgi-type lemmas, and measure propagation arguments. When the solution is away from zero, the equation is transformed by v = (u/μ^-)^q, an approach that introduces a nonlocal drift term; Hölder continuity is then obtained from an oscillation-decay proposition for v. The paper states the main theorem (Theorem 2.2) and provides detailed arguments for the near-zero case, but several central lemmas in the away-from-zero case are stated without proofs.","tokens_in":47702,"tokens_out":2609,"duration_ms":25235,"significance":"If the result is correct, it extends the known Hölder regularity theory for local doubly degenerate parabolic equations to the nonlocal setting, a direction with only few prior results. The paper is carefully structured, and the proof in the near-zero case follows a demanding intrinsic-scaling machinery with explicit Caccioppoli inequalities and tail estimates; the author also credits the relevant literature. The main caveat is that the away-from-zero case, which is essential for sign-changing solutions, relies on Lemmas 5.3–5.5 whose proofs are entirely omitted. Because these lemmas are used in Proposition 5.7 and hence in the final Hölder estimate, the central claim is not yet fully verified in the manuscript as written. The tail condition is a stated hypothesis rather than an unstated assumption; it is a genuinely nonlocal condition and makes the theorem depend on global decay, which is consistent with the nonlocal nature of the operator but should be emphasized as a limitation.","major_comments":[{"comment":"The proofs of Lemmas 5.3, 5.4, and 5.5 are explicitly omitted (the text states \"The proofs of Lemma 5.3-5.5 are omitted\"). These lemmas are load-bearing: Lemma 5.3 propagates a pointwise lower bound forward in time, Lemma 5.4 propagates measure-theoretic information, and Lemma 5.5 provides the measure-shrinking estimate that closes the De Giorgi iteration in the away-from-zero case. Proposition 5.7 invokes all three, and the final Hölder continuity in §5.3–§5.4 depends on Proposition 5.7. The lemmas are not immediate corollaries of Lemma 5.2 because the equation (5.6) for v contains a non-divergence-form drift term involving ¯v outside B_ρ, and the Caccioppoli inequality (5.14) contains a tail with (¯v − k^{1/q})_± instead of (v − k)_±. The manuscript must either supply complete proofs of these lemmas or give a precise reduction showing that they follow from Lemma 5.2 and the stated tail conditions (5.28), (5.31), (5.35). As written, the away-from-zero oscillation decay (5.39) is unsupported.","section":"§5.2 (Lemmas 5.3–5.5)"},{"comment":"Lemma 3.1, the Caccioppoli inequality, is the foundation of all subsequent De Giorgi arguments, but its proof is only sketched. After the time-mollification and limiting steps for the parabolic term, the text says \"The rest of the proof is similar to that of [9, Lemma 2.5] and [23, Proposition 2.1], and so is omitted.\" In particular, the handling of the nonlocal tail term on R^n \\ B_R and the convergence of the nonlinear integral terms under the mollification are not shown. Since this inequality is applied iteratively in Lemmas 4.1, 4.2, 4.4, 4.6, and 4.7, and in Lemma 5.1, a complete verification of the terms unique to the doubly degenerate structure—especially the g_± terms and the tail term with (u−k)_±^{p-1}—should be included or, at minimum, the proof should be given in an appendix.","section":"§3 (Lemma 3.1)"},{"comment":"Lemma 5.1 states the Caccioppoli inequality for the transformed function v, but its proof also ends with \"the desired estimate (5.14) follows from the proof of [1, Lemma 7.6] and we omit the details.\" This is a substantial local estimate whose proof involves a nonlocal drift term, a new tail term with (¯v − k^{1/q})_±, and a convolution argument. The dependence on [1] should be made precise: either state the exact steps from [1, Lemma 7.6] that are being reused, or provide a self-contained verification. As written, the proof of Lemma 5.1 is incomplete, and Lemma 5.2 and the subsequent away-from-zero arguments rely on it directly.","section":"§5.1 (Lemma 5.1)"}],"minor_comments":[{"comment":"The section heading contains a typo: \"aw ay from zero\" should be \"away from zero\".","section":"§5 (heading)"},{"comment":"Before Lemma 5.2, the sentence \"The proofs of Lemma 5.3-5.5 are omitted\" is a fragment in context; it would be clearer to state explicitly that these proofs will be provided in a longer version or to give a short outline of the method, since the reader is otherwise left without guidance.","section":"§5.2"},{"comment":"The parabolic tail condition appears in Definition 2.1 through u ∈ L^m_loc(0,T; L^{p-1}_{sp}(R^n)), but it is not discussed in the introduction. Because the main theorem's constants depend on Tail_m(|u|; Q_R), the authors should explicitly mention, already in the introduction, that the result is not purely local and requires a global tail condition; this is a noteworthy difference from the local theory.","section":"§2 (Definition 2.1)"},{"comment":"In the decomposition of the tail term, the notation \"gTail_m((u−μ_-)_-; Q_0)\" is introduced but the reader may confuse Q_0 with the cylinder Q_0 defined in §3; it would help to use a different symbol or to explicitly restate the definition of Q_0 in this context.","section":"§4.3, Step 2"},{"comment":"In the proof of (5.53), the notation \"T, T', T''\" is introduced without explicitly writing the corresponding integrals for T and T''; this makes the estimates harder to follow. Adding explicit definitions would improve readability.","section":"§5.3, Step 2"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely and challenging problem, and the near-zero case is worked out in substantial detail. However, the away-from-zero case—which is indispensable for sign-changing solutions—is presented with three key lemmas whose proofs are omitted and with a Caccioppoli inequality in §5.1 that is delegated to another reference. This is not a mere presentation issue: Proposition 5.7 and the final Hölder estimate depend on these results. I recommend major revision with the expectation that the author supply complete proofs or a rigorous reduction to the cited lemmas. The citation to [1] (an arXiv preprint) should also be checked for availability and stability."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the expected Hölder regularity result for nonlocal doubly degenerate parabolic equations in the range p>2, 0<q<p-1, proved with the standard intrinsic-scaling De Giorgi machinery. The statement is plausible and the paper is honestly written, but the formal status is conditional because the away-from-zero part of the proof is not actually in the paper.\n\nWhat is new and good: the local doubly degenerate result is extended to the nonlocal fractional setting, sign-changing solutions are allowed, and the paper follows the known template from Adimurthi, Byun-Kim, and the local theory. The Caccioppoli inequality in Lemma 3.1 is stated in full with a credible sketch, and Section 4 is detailed: the iteration and the tail decomposition in Step 2 of Section 4.3 are substantial and look coherent. The citation pattern is appropriate; the relevant local and nonlocal literature is covered.\n\nWhere the soft spots are: the stress-test note is right. Section 5 proves the away-from-zero case through Proposition 5.7, which depends on Lemmas 5.3, 5.4, and 5.5. The text explicitly says their proofs are omitted. These are not routine restatements. Equation (5.6) contains a non-divergence drift term involving \\bar v outside B_rho, and the Caccioppoli inequality (5.14) has a tail term with (\\bar v - k^{1/q})_\\pm rather than (v-k)_\\pm. The tail conditions (5.28), (5.31), (5.35) and the time rescaling have to do real work. Without seeing those arguments, the oscillation decay (5.39) is unsupported. The same is true, to a lesser extent, for Lemma 3.1, where the proof is sketched and then deferred to references. That is more defensible because the statement is explicit and the references are standard, but it is still load-bearing for Section 4.\n\nThe parabolic tail condition is also a real assumption, as the reader notes: solutions live in L^m_t L^{p-1}_{sp}, and the tail magnitude enters the constants and the intrinsic cylinders. That is stated honestly and is not a hidden flaw, but it means the result is not a purely local one.\n\nMy recommendation: this deserves a serious referee, not a desk reject. A referee should ask for complete proofs of Lemmas 5.3-5.5 and an expanded Lemma 3.1. If those are standard extensions, the paper will be solid. As it stands, I would not rely on the away-from-zero conclusion without seeing the missing steps.","headline":"The right result, probably true, but Section 5's away-from-zero proof leans on three lemmas whose proofs are omitted, so the paper is not fully verified as written.","tokens_in":48327,"tokens_out":2391,"would_cite":false,"duration_ms":23988,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K10","35K59","35K65","35K92"],"pacs":[],"model":"deepseek-v4-flash","headline":"For nonlocal doubly degenerate parabolic equations, locally bounded weak solutions are locally Hölder continuous under a parabolic tail condition.","keywords":["doubly degenerate parabolic equations","nonlocal parabolic equations","Hölder continuity","De Giorgi technique","intrinsic scaling","parabolic tail condition","fractional p-Laplacian","weak solutions"],"falsifier":"Take the equation with $p>2$, $0<q<p-1$ on $\\mathbb{R}^n$ and construct a family of locally bounded solutions with $\\|u\\|_{L^\\infty(Q_R)}$ fixed but with $\\operatorname{Tail}_m(|u|;Q_R)$ growing without bound (e.g., by adding a slowly decaying, high-frequency far-field term supported outside $B_R$). If the oscillation over $Q_{r,cr^{sp}}$ stops decaying in $r$ once the tail exceeds a threshold depending on $\\omega$, the smallness conditions such as (4.20) are essential and not just technical. If instead the Hölder estimate persists with $\\gamma_0$ uniformly bounded independent of the tail, the tail assumption is removable.","tokens_in":47166,"feed_emoji":"📉","tokens_out":9680,"duration_ms":81245,"temperature":0.7,"pith_summary":"This paper studies local regularity for nonlocal doubly degenerate parabolic equations, whose model is $$\\partial_t(|u|^{q-1}u)+\\mathrm{P.V.}\\int_{\\mathbb{R}^n}\\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}}\\,dy=0$$ with $p>2$ and $0<q<p-1$. The main result, Theorem 2.2, asserts that every locally bounded, sign-changing weak solution satisfying a parabolic tail condition is locally Hölder continuous in space and time. This extends the known Hölder theory for the local doubly nonlinear equation to the nonlocal setting, a step that requires controlling the nonlocal tail of the solution and a new Caccioppoli inequality. The proof combines a nonlocal version of the De Giorgi iteration with intrinsic scaling, and it splits the argument into a case where the solution is near zero and a case where it stays away from zero.","feed_headline":"Nonlocal doubly degenerate equations: solutions are Hölder continuous","feed_subtitle":"Sign-changing weak solutions decay like (r/R)^α on small cylinders, extending local regularity to nonlocal equations.","key_machinery":"The load-bearing objects are the nonlocal parabolic tail $\\mathrm{Tail}_m(f;Q)$ together with a Caccioppoli-type inequality (Lemma 3.1 and its variant Lemma 5.1). The tail measures the $L^{p-1}$ contribution of $f$ outside a cylinder, normalized by $|y-x_0|^{n+sp}$; every De Giorgi step requires a smallness condition of the form $\\varrho^{\\frac{n\\kappa}{p-1}}\\theta^{1/m}\\mathrm{gTail}_m(\\dots)\\le\\nu_*\\omega$, where $\\kappa=\\frac{sp}{n}\\frac{m-(p-1)}{m}$ encodes the time–space scaling of the fractional Sobolev embedding. The intrinsic-scaling parameter $\\theta=(4\\omega)^{q+1-p}$ stretches the time scale according to the oscillation $\\omega$, and the iteration of nested cylinders $Q_j=Q_{\\varrho_j}^{(A\\theta_j)}$ converts a measure-density estimate into geometric decay of the oscillation.","core_discovery":"The central claim is that for $p>2$ and $0<q<p-1$, any locally bounded weak solution $u$ of (2.1)--(2.3) in the sense of Definition 2.1 is locally Hölder continuous in $\\Omega_T$, with no sign restriction on $u$. Concretely, if $Q_R(z_0)\\subset\\Omega_T$, then for any smaller cylinder $Q_{r,cr^{sp}}(z_0)$ meeting $Q_{R/4}(z_0)$ one has $\\mathrm{ess\\,osc}_{Q_{r,cr^{sp}}(z_0)}u\\le \\gamma_0(r/R)^\\alpha$, where the Hölder exponent $\\alpha\\in(0,1)$ depends only on the data and the constants $\\delta,c,\\gamma_0$ depend also on $R$, $\\|u\\|_\\infty$ and the parabolic tail $\\mathrm{Tail}_m(|u|;Q_R(z_0))$. The proof treats the near-zero regime by De Giorgi iteration on intrinsic cylinders of height $\\theta\\,r^{sp}$ and the away-from-zero regime by transforming $u$ through $v=\\bar v^q$ to reduce the equation to a $p$-Laplace-type problem; no expansion of positivity is used.","pith_inferences":["The smallness thresholds such as (4.20), (4.29), and (4.39) suggest a testable trade-off: fixing the local oscillation $\\omega$ while increasing the far-field tail should shrink the admissible starting radius $\\varrho_0$ but leave the exponent $\\alpha$ unchanged; a finite-volume discretization with manufactured far-field data could check whether the pre-factor $\\gamma_0$ grows like a power of the ","The power renormalization $v=\\bar v^q$ in Section 5 is a general device: once $u$ is bounded away from zero by a controlled fraction of its oscillation, the doubly degenerate nonlinearity is converted into the standard $p$-Laplace-type structure. The same device should apply to the doubly singular range $1<p<2$ and to the borderline case, modulo a different choice of exponent, which is a natural n","Since the oscillation decay for the normalized solution $v$ is obtained at every step, the proof likely upgrades to a modulus of continuity estimate at each point rather than only on nested cylinders; extracting an explicit modulus would give a quantitative route toward a Harnack inequality for nonnegative doubly degenerate nonlocal solutions."],"forward_implications":["The estimate (2.5) is quantitative: the Hölder exponent $\\alpha$ is determined solely by $\\{n,m,p,q,s,\\Lambda\\}$, while the radius ratio $\\delta$ and the multiplicative constant $\\gamma_0$ absorb the initial data, the radius $R$, and the tail.","Because the proof treats sign-changing solutions directly, the result covers solutions that cross zero, which is exactly the case where the degeneracy of $\\partial_t(|u|^{q-1}u)$ is most delicate.","The kernel only needs to be symmetric and comparable to $|x-y|^{-(n+sp)}$, so the conclusion is uniform over that whole class of nonlocal operators, not tied to one particular kernel.","In the away-from-zero regime the rescaled function $v$ satisfies a parabolic equation with a power-reduced nonlinearity and bounded coefficient; the oscillation decay obtained there transfers back to $u$ through the Hölder-continuous power map $v\\mapsto v^{1/q}$, so the pointwise modulus persists at the vertex."],"supporting_citations":[{"why":"Supplies the two algebraic comparison lemmas for $b_\\alpha$ and $g_\\pm$, and the template for the Caccioppoli estimate.","marker":"[5]"},{"why":"The local doubly nonlinear Hölder result that the present theorem extends; its Lemma 5.1 is used in the measure-density ratio estimate.","marker":"[6]"},{"why":"Provides the parabolic fractional Sobolev embedding, the truncation comparison, and the optimal-tail De Giorgi iteration for nonlocal parabolic $p$-Laplace equations.","marker":"[9]"},{"why":"Establishes the regularity framework for general nonlocal doubly nonlinear equations, including the weak formulation and Caccioppoli-style estimates.","marker":"[4]"},{"why":"Gives the nonlocal Trudinger regularity whose away-from-zero argument and Lemma 7.6 are adapted for the $p$-Laplace-type reduction.","marker":"[1]"},{"why":"Supplies the Hölder regularity for parabolic fractional $p$-Laplacian, including the mollification and iterative estimates used in the Caccioppoli proof.","marker":"[23]"},{"why":"Provides local boundedness and Hölder continuity for the parabolic fractional $p$-Laplace equation, including the convolution properties used in Lemma 5.1.","marker":"[15]"},{"why":"The source of the intrinsic scaling method and the fast geometric convergence lemma used to close the De Giorgi iteration.","marker":"[13]"},{"why":"Introduces the parabolic Harnack setting for nonlocal equations from which the parabolic tail condition is taken.","marker":"[19]"}],"fun_headline_variants":["Hölder regularity despite nonlocal double degeneracy","Nonlocal doubly degenerate equations: solutions are Hölder","Sign-changing weak solutions get local Hölder continuity","Nonlocal doubly degenerate flows: sign-changing solutions Hölder"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that the solution satisfies a parabolic tail condition—its far-away values are integrable enough that the nonlocal tail $\\operatorname{Tail}_m(|u|;Q_R)$ is finite—and that this tail is small relative to the local oscillation in every De Giorgi step; if the tail is large or the integrability exponent $m$ is too close to $p-1$, the iteration does not close and the Hölder estimate is not obtained.","fun_headline_variants_meta":{"raw":{"variants":["Hölder regularity despite nonlocal double degeneracy","Nonlocal doubly degenerate equations: solutions are Hölder","Sign-changing weak solutions get local Hölder continuity","Nonlocal doubly degenerate flows: sign-changing solutions Hölder"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001251,"raw_usage":{"total_tokens":5131,"prompt_tokens":949,"completion_tokens":4182,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":4119}},"tokens_in":565,"tokens_out":4182,"duration_ms":27897,"temperature":1.0,"reasoning_tokens":4119,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:18:46.447878+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the equation with $p>2$, $0<q<p-1$ on $\\mathbb{R}^n$ and construct a family of locally bounded solutions with $\\|u\\|_{L^\\infty(Q_R)}$ fixed but with $\\operatorname{Tail}_m(|u|;Q_R)$ growing without bound (e.g., by adding a slowly decaying, high-frequency far-field term supported outside $B_R$). If the oscillation over $Q_{r,cr^{sp}}$ stops decaying in $r$ once the tail exceeds a threshold depending on $\\omega$, the smallness conditions such as (4.20) are essential and not just technical. If instead the Hölder estimate persists with $\\gamma_0$ uniformly bounded independent of the tail, the tail assumption is removable.","supporting_citations":[{"cited_title":"Bögelein, F","cited_arxiv_id":null,"evidence_quote":"Supplies the two algebraic comparison lemmas for $b_\\alpha$ and $g_\\pm$, and the template for the Caccioppoli estimate."},{"cited_title":"Bögelein, F","cited_arxiv_id":null,"evidence_quote":"The local doubly nonlinear Hölder result that the present theorem extends; its Lemma 5.1 is used in the measure-density ratio estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the parabolic fractional Sobolev embedding, the truncation comparison, and the optimal-tail De Giorgi iteration for nonlocal parabolic $p$-Laplace equations."},{"cited_title":"Banerjee, P","cited_arxiv_id":null,"evidence_quote":"Establishes the regularity framework for general nonlocal doubly nonlinear equations, including the weak formulation and Caccioppoli-style estimates."},{"cited_title":"Local H\\\"older regularity for bounded, signed solutions to nonlocal Trudinger equations","cited_arxiv_id":"2503.07184","evidence_quote":"Gives the nonlocal Trudinger regularity whose away-from-zero argument and Lemma 7.6 are adapted for the $p$-Laplace-type reduction."},{"cited_title":"Liao: Hölder regularity for parabolic fractionalp-Laplacian, Calc","cited_arxiv_id":null,"evidence_quote":"Supplies the Hölder regularity for parabolic fractional $p$-Laplacian, including the mollification and iterative estimates used in the Caccioppoli proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides local boundedness and Hölder continuity for the parabolic fractional $p$-Laplace equation, including the convolution properties used in Lemma 5.1."},{"cited_title":"DiBenedetto: Degenerate Parabolic Equations","cited_arxiv_id":null,"evidence_quote":"The source of the intrinsic scaling method and the fast geometric convergence lemma used to close the De Giorgi iteration."},{"cited_title":"Kassmann and M","cited_arxiv_id":null,"evidence_quote":"Introduces the parabolic Harnack setting for nonlocal equations from which the parabolic tail condition is taken."}],"review_version":2}