{"id":"12f9e133-1747-4862-b8d2-3f45475544e9","arxiv_id":"2507.21604","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For regular non-local Dirichlet forms without killing, the strong parabolic Harnack inequality is equivalent to the weak parabolic Harnack inequality plus upper jumping smoothness, proved analytically.","lead":"This paper proves, by pure analysis, that the parabolic Harnack inequality for non-local Dirichlet forms is equivalent to a weak Harnack inequality combined with an upper jumping smoothness condition. The result supplies a purely analytic route to heat kernel estimates on jump-type spaces, previously obtained with probabilistic tools.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central equivalence in Theorem 1.1 depends on unverified results from the same author's unpublished preprint [19]; a failure or hypothesis mismatch there would collapse the main result.","rationale":"I read the main proof in good faith. The internal steps I checked, including the iteration in Proposition 4.1, the (PMV+) to (PHI0) scaling in Theorem 4.5 Step (3), and Lemma 3.2, are plausible and contain no obvious circularity. The paper is honest about the source of the weak-Harnack machinery. Nevertheless, the central claim is not self-contained: its two independent routes to (PHI0) both invoke [19] for (PGL1), (FK), (Gcap), and (cap<=). Because these are arXiv preprints by the same author and are used as black boxes, the correctness of the main theorem is conditional on them. I therefore support the reader's conditional verdict. I do not see an internal contradiction that would force rejection, and I would not accept unconditionally until the [19] implications are independently verified.","tokens_in":25411,"tokens_out":20820,"duration_ms":234769,"concrete_test":"Take [19, Theorems 2.1 and 2.2] and verify each implication (LLE) iff (wPH1), (wPH1) implies (PGL1), and (wPH1)+(UJS) implies (FK)+(Gcap)+(cap<=) under exactly the standing assumptions of this paper: regular Dirichlet form without killing, (VD), (RVD), and arbitrary scale W satisfying (1.5). If any of these implications requires an assumption not present here, such as a fixed power scale W(x,r)=r^alpha, the upper estimate (UE), or a tail condition (TJ), then the corresponding use at (1.14), Proposition 4.1 Step (3), or Theorem 4.5 Step (2) is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is proved through Theorem 4.5, Proposition 2.7, Proposition 3.4, and Proposition 3.1. The first of these uses [19, Theorem 2.2] to obtain (FK), (Gcap), and (cap<=) from (wPH1)+(UJS), and Proposition 4.1 Step (3) applies (PGL1) obtained from (wPH1) via [19, Theorem 2.1] as recorded in (1.14). Proposition 3.4 additionally invokes [19, Proposition 3.9] and the implication (LLE) implies (cap<=). Thus every path from (wPH1)+(UJS) to (PHI0) and from (PHI) to (PHI+) passes through results that are not proved in this manuscript and are not independently verified. The reader is told only that [19] is the author's own preprint. If [19, Theorems 2.1 or 2.2] has an error, or if its hypotheses require a different scale W, an extra volume condition, or the a priori presence of (UJS) or (UE), the chain of equivalences in Theorem 1.1 and Corollary 1.2 loses its foundation. This is a structural, not a stylistic, dependency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies parabolic Harnack inequalities for regular Dirichlet forms without killing part on metric measure spaces satisfying (VD) and (RVD). Its main result, Theorem 1.1, states that the weak L1-parabolic Harnack inequality together with the upper jumping smoothness condition is equivalent to the basic, standard, and complete parabolic Harnack inequalities. The proof combines a Nash-type blow-up argument and a Moser-type iteration argument, and Corollary 1.2 enlarges the list of equivalent characterizations. Many of the auxiliary equivalences are taken from the author's unpublished preprint [19].","tokens_in":25579,"tokens_out":12102,"duration_ms":124848,"significance":"If correct, the result is significant: it provides a purely analytic route to the strong parabolic Harnack inequality from a weak Harnack inequality plus a jump-kernel regularity condition, extending the probabilistic theory of [5] and unifying the Nash and Moser approaches. The paper contains some genuinely analytic contributions, including the derivation of (UJS) from (PHI0), the tail estimates in Section 3.3, and the two proofs of (PHI0) in Section 4. The main caveat is the heavy reliance on the unpublished preprint [19] for load-bearing equivalences; the manuscript is not self-contained.","major_comments":[{"comment":"The central theorem is not self-contained and depends on the author's own unpublished preprint [19] for load-bearing results. Eq. (1.14) uses [19, Theorem 2.1] for (LLE)⇔(wPH1) and (wPH1)⇒(PGL1); Proposition 4.1 Step (3) applies (PGL1); Theorem 4.5 Step (2) uses [19, Theorem 2.2] to obtain (FK) and (Gcap); Proposition 3.4 uses [19, Proposition 3.9]; and Corollary 1.2 repeatedly relies on [19, Theorems 2.1, 2.2 and Proposition 5.4]. Since the statements and hypotheses of these results are not reproduced, the reader cannot verify that they apply under exactly the assumptions (VD), (RVD), regularity, and no killing part. If any of these results has an error or requires an extra condition such as a different scale W or a priori (UE), the equivalences in Theorem 1.1 and Corollary 1.2 collapse. Please include the precise statements, proofs, or an appendix with the needed results, or replace them with published references.","section":"Eq. (1.14), Prop. 4.1, Thm. 4.5, Cor. 1.2"},{"comment":"The proof of Proposition 3.4 invokes Proposition 3.1 to conclude (UJS) from (PHI), but Proposition 3.1 is stated and proved only under (PHI0). The needed observation that (PHI) implies (PHI0) for globally nonnegative caloric functions is not made. The same missing step appears in Theorem 1.1's step (PHI+)⇒(wPH1)+(UJS). This is a local gap in the written proof; it is easily repaired by noting that the tail term in (PHI) vanishes for globally nonnegative caloric functions, but it should be stated explicitly.","section":"Prop. 3.4 and Thm. 1.1"},{"comment":"The conditions (FK) and (Gcap) are used as hypotheses in Lemma 4.2 and Lemma 4.3 and are invoked in Theorem 4.5 Step (2), but they are not defined in the manuscript; the reader is referred to [19]. Since [19] is an unpublished preprint, the Moser approach in Section 4.2 is not verifiable as written. The manuscript should define (FK), (Gcap), (PI), and the other conditions appearing in Corollary 1.2, or at least state the precise results from [19] that supply them.","section":"Lemmas 4.2 and 4.3; Thm. 4.5"}],"minor_comments":[{"comment":"In the proof of Proposition 2.7, the reference to 'Proposition 2.1' should be to Lemma 2.1; there is no Proposition 2.1 in the paper.","section":"Prop. 2.7, p. 9"},{"comment":"In the proof of Lemma 3.6, the text says 'by (UJS) and (UE)', but (UE) is not assumed in the lemma; the displayed estimate appears to use only (UJS), (VD), and contractivity of the heat semigroup. Please correct the citation.","section":"Lemma 3.6"},{"comment":"The notions 'subcaloric' and 'supercaloric' are used throughout (e.g., Lemma 2.1, Lemma 2.3, (wPH1)) but are not defined in the text; please add definitions or a precise reference.","section":"Section 2, definition of caloric functions"},{"comment":"The final sentence of Proposition 2.7 asserts that (PHI0), (PHI+), and the Harnack inequalities of [5] are all equivalent, but the proof only establishes (2.5); please indicate explicitly how (2.5) implies (PHI) and how (PHI) implies (PHI0), since the latter implication is used later.","section":"Prop. 2.7"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's main theorem rests on the author's own unpublished preprint [19]. I recommend that the editor require either publication of [19] or inclusion of its needed theorems (with proofs or precise statements) in this manuscript before acceptance. The reviewer should also be asked to confirm that [19, Theorems 2.1 and 2.2] hold under the exact hypotheses (VD), (RVD), regularity, no killing part, and the given scale W."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Key takeaway: this is the first purely analytic proof of the strong parabolic Harnack inequality for regular non-local Dirichlet forms without killing, with both Nash and Moser routes. That is real. The paper also enlarges the equivalence dictionary and proves (UJS) from (PHI0) analytically rather than by stochastic reasoning.\n\nWhat is new: the Nash proof in Prop 4.1, the Moser route in Prop 4.4/4.5, and the equivalences (wPH1)+(UJS) ⇔ (PHI0) ⇔ (PHI) ⇔ (PHI+), plus the large Corollary 1.2 list. The tail absorption Lemma 2.3 and the chaining argument in Prop 2.7 are well crafted. The paper is honest that the strong PHI was hinted probabilistically in [19, Cor 2.6]; the contribution is the analytic proof, not the bare statement.\n\nThe math I can check is internally coherent. I did not find circular reasoning: Theorem 1.1's chain is acyclic, and [19] concerns the weak Harnack inequality, not the target strong PHI.\n\nSoft spots, in order of importance.\n\n1. Load-bearing dependence on [19]. Theorem 4.5 Step (2) uses [19, Thm 2.2] to get (FK), (Gcap), (cap≤) from (UJS); Prop 4.1 Step (3) uses (PGL1) obtained from (wPH1) via [19, Thm 2.1]; Prop 3.4 uses [19, Prop 3.9] plus (LLE)⇒(cap≤). Corollary 1.2 leans on the same chain. If [19] has an error, or if its hypotheses require a different scale or an extra volume condition, the main equivalence collapses. This is structural, not stylistic. The stress-test note is right about that.\n\n2. Some delegated steps are opaque. 'Repeat the proof of [10]' in Lemma 4.3 and 'by [19]' in several places make independent verification slow. That is normal for this subfield, but with an unpublished source it becomes a real cost.\n\n3. Minor: Lemma 3.6 says it reduces (UE), yet the proof line says 'by (UJS) and (UE)'. I suspect (UE) is not actually used or is a leftover from [3]; either way it should be cleaned up.\n\nWho should read it: people working on analytic equivalence theory for nonlocal Dirichlet forms, heat kernel estimates, and Harnack inequalities. It deserves a serious referee. My recommendation: send to peer review, but ask the author to either post [19] in final form or fold the needed theorems into an appendix, and to state explicitly which hypotheses of [19] are needed. The analytic contribution is substantial enough to warrant that effort.","headline":"Strong analytic proof of PHI for non-local Dirichlet forms, but the central equivalence leans on the author's own unpublished preprint [19]—worth refereeing, not yet fully verifiable in isolation.","tokens_in":26215,"tokens_out":2803,"would_cite":true,"duration_ms":32771,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34K30","31C25","35K08","47D07","60J46"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for non-local Dirichlet forms without killing, the strong parabolic Harnack inequality is equivalent to the weak parabolic Harnack inequality together with a pointwise smoothness condition on the jump kernel, and…","keywords":["parabolic Harnack inequality","non-local Dirichlet forms","upper jumping smoothness","weak parabolic Harnack inequality","local heat kernel","jump kernel","Moser iteration","Nash inequality"],"falsifier":"Read the unpublished preprint [19] and check its Theorems 2.1 and 2.2 under exactly the hypotheses used here, namely a regular Dirichlet form without killing part on a (VD)$+$(RVD) space with an arbitrary scaling function; a single counterexample to (LLE)$\\Leftrightarrow$(wPH1), to (wPH1)$\\Rightarrow$(PGL1), or to (wPH1)$\\Rightarrow$(FK)$+$(Gcap)$+$(cap$\\le$) would break Theorem 1.1. Independently, one could construct a jump kernel satisfying the (UJS) averaging bound only at scales below a fixed threshold while (wPH1) still holds; the theorem predicts the strong Harnack inequality must hold, so a violation of (PHI) at that threshold would refute it.","tokens_in":25102,"feed_emoji":"♨️","tokens_out":10062,"duration_ms":100718,"temperature":0.7,"pith_summary":"This paper claims that, for symmetric non-local Dirichlet forms with no killing part, the strong parabolic Harnack inequality is an analytic consequence of two ingredients: the $L^1$ weak parabolic Harnack inequality and a pointwise smoothness condition on the jump kernel called upper jumping smoothness. The author proves the implication both ways, so the weak inequality plus the jump-kernel bound becomes equivalent to the strong inequality, and he gives two independent analytic proofs of the hard direction, one following Nash's blow-up idea and one following Moser's iteration. The point of doing this purely analytically is that previously the strong Harnack inequality for non-local forms had only a stochastic proof; the analytic route also yields tail estimates, capacity bounds, and heat-kernel consequences, collected in a long list of equivalent characterizations. The equivalence chain rests on the author's own unpublished weak-Harnack results, so the unconditional content is the reduction of the strong inequality to the weak one plus upper jumping smoothness.","feed_headline":"Weak Harnack plus jump-kernel bound forces strong Harnack","feed_subtitle":"Nash's and Moser's iterations show weak inequality plus one jump condition equals the strong inequality for non-local forms.","key_machinery":"The load-bearing condition is upper jumping smoothness (UJS): for the jump kernel $J$, one has $J(x,y)\\le C\\,V(x,r)^{-1}\\int_{B(x,r)}J(z,y)\\,d\\mu(z)$ whenever $r\\le \\tfrac12 d(x,y)$, meaning no jump probability exceeds a constant multiple of its local average at the relevant separation scale. The arguments also rest on three further mechanisms: the tail functional $T^{(\\lambda)}_Q(u)$, which measures the mass of $u$ outside a ball that can leak in through jumps; the semi-homogeneous caloric extension, which builds a caloric function inside a ball from data supported outside it, turning the tail into a source term; and the two iteration schemes, Nash's chained-ball blow-up that duplicates a large value of a caloric function outward until boundedness is contradicted, and Moser's cylinder-by-cylinder iteration driven by the parabolic mean-value inequalities (PMV$-$) and (PMV$+$). A recurring device (Lemma 2.3) adds a tail term to a function multiplied by a cutoff so that the result is again subcaloric.","core_discovery":"The central theorem (Theorem 1.1) asserts that on a locally compact metric measure space satisfying volume doubling and reverse volume doubling, for any regular symmetric Dirichlet form without killing part that admits a jump kernel, the following are equivalent: the $L^1$ weak parabolic Harnack inequality together with upper jumping smoothness, written (wPH1)$+$(UJS), and the three strong forms (PHI0), (PHI), and (PHI$+$), where (PHI$+$) is the tail-enhanced version that also controls the positive tail of a caloric function. The proof reduces to two analytic implications: (wPH1)$+$(UJS) implies (PHI0), established by a Moser-type iteration through parabolic mean-value inequalities (Lemma 4.3, Proposition 4.4, Theorem 4.5), with a Nash-type route (LLE)$+$(UJS)$\\Rightarrow$(PHI0) also given in Proposition 4.1; and conversely (PHI0) implies (UJS) in Proposition 3.1, so the jump-kernel smoothness is necessary as well as sufficient. Corollary 1.2 then expands the equivalence to roughly twenty characterizations in terms of heat-kernel lower and upper estimates, capacity and Poincar\\'e-type inequalities, mean-value inequalities, and elliptic Harnack-type conditions.","pith_inferences":["If the unpublished weak-Harnack equivalences hold under exactly these hypotheses, the paper effectively identifies a minimal analytic package, volume doubling, reverse doubling, no killing, and a scale-local averaging bound on the jump kernel, that completely determines the parabolic Harnack inequality; a testable corollary would be stability of the inequality under kernel perturbations that prese","The necessity of (UJS) suggests a concrete way to certify failure of the strong Harnack inequality: a kernel with a sharp gap in its local averages, such as a truncated stable kernel whose cutoff scale is comparable to the ball radius, should violate (PHI) even if the weak inequality holds, a family one could test numerically or analytically.","The no-killing hypothesis $\\kappa\\equiv 0$ is used throughout, and extensions to forms with a killing part are left open; since killing acts like a zeroth-order term, one would expect analogous equivalences with a suitable tail term for the killing measure, but this is not established in the paper.","The two routes to (PHI0) plausibly give different quantitative constants, Nash's chaining growing with the number of intermediate balls and Moser's tied to the mean-value inequality; extracting explicit Harnack constants from the constants of (wPH1) and (UJS) alone would test how sharp the arguments are."],"forward_implications":["The strong parabolic Harnack inequality for non-local forms needs no stochastic input: the weak inequality plus the single pointwise bound (UJS) on the jump kernel is both necessary and sufficient.","Every condition in Corollary 1.2, including local heat-kernel lower and upper estimates, capacity upper bounds, Poincar\\'e-type inequalities, and elliptic Harnack-type conditions, becomes an equivalent face of one analytic object, matching the stability theory long available for local operators.","Because (PHI0) implies (UJS), any jump kernel failing the local-averaging bound cannot support the parabolic Harnack inequality at all; (UJS) is a genuine obstruction, not a technical convenience.","The equivalence (PHI)$\\Leftrightarrow$(PHI$+$) shows that the tail terms appearing in the two formulations carry identical information, making the tail-enhanced inequality the canonical strong form for non-local operators.","The two proofs are complementary: the Nash route derives (PHI0) directly from the local lower estimate of the heat kernel, while the Moser route works from the weak inequality through mean-value inequalities and yields subcaloric and supercaloric control along the way."],"supporting_citations":[{"why":"The author's own unpublished preprint; its Theorems 2.1 and 2.2 supply the links (LLE)$\\Leftrightarrow$(wPH1), (wPH1)$\\Rightarrow$(PGL1), and (wPH1)$\\Rightarrow$(FK)$+$(Gcap)$+$(cap$\\le$), invoked at (1.14), in Proposition 4.1 Step (3), and throughout Corollary 1.2.","marker":"[19]"},{"why":"The stochastic stability theory for non-local parabolic Harnack inequalities that this paper reproduces analytically; also the source of the idea behind Proposition 3.1, (PHI0)$\\Rightarrow$(UJS).","marker":"[5]"},{"why":"Supplies the parabolic mean-value inequality (PMV$-$) whose proof Lemma 4.3 adapts, plus the definitional framework for caloric functions and the comparison used in Lemma 2.5.","marker":"[10]"},{"why":"Lemma 3.6's tail comparison is stated to be essentially contained in its Lemma 5.3, which the paper adapts while dropping the upper-estimate condition.","marker":"[3]"},{"why":"Nash's original blow-up strategy, which Proposition 4.1 follows for the route (LLE)$+$(UJS)$\\Rightarrow$(PHI0).","marker":"[21]"},{"why":"Moser's iteration strategy, which underlies Proposition 4.4 and the construction of (PMV$+$) in the second proof of the strong inequality.","marker":"[20]"},{"why":"Provides capacity and mean-value equivalences used in Corollary 1.2 (the step (e)$'\\Leftrightarrow$(k)) and the on-diagonal heat-kernel upper bound argument in Lemma 4.6.","marker":"[12]"}],"fun_headline_variants":["Jump smoothness upgrades weak Harnack to strong","Weak Harnack plus jump kernel forces strong Harnack","One jump condition yields strong Harnack from weak","Harnack equivalence: weak + jump smoothness = strong"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire equivalence chain leans on the author's own unpublished preprint [19] for three links, the equivalence (LLE)$\\Leftrightarrow$(wPH1), the implication (wPH1)$\\Rightarrow$(PGL1), and the implication (wPH1)$\\Rightarrow$(FK)$+$(Gcap)$+$(cap$\\le$), used at equation (1.14), in Proposition 4.1 Step (3), and repeatedly in Corollary 1.2; if any of those results is wrong or needs hypotheses not present here, such as a different scaling function or an extra volume condition, the central theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Jump smoothness upgrades weak Harnack to strong","Weak Harnack plus jump kernel forces strong Harnack","One jump condition yields strong Harnack from weak","Harnack equivalence: weak + jump smoothness = strong"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000555,"raw_usage":{"total_tokens":2615,"prompt_tokens":888,"completion_tokens":1727,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":1660}},"tokens_in":504,"tokens_out":1727,"duration_ms":14581,"temperature":1.0,"reasoning_tokens":1660,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:35:18.272199+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Read the unpublished preprint [19] and check its Theorems 2.1 and 2.2 under exactly the hypotheses used here, namely a regular Dirichlet form without killing part on a (VD)$+$(RVD) space with an arbitrary scaling function; a single counterexample to (LLE)$\\Leftrightarrow$(wPH1), to (wPH1)$\\Rightarrow$(PGL1), or to (wPH1)$\\Rightarrow$(FK)$+$(Gcap)$+$(cap$\\le$) would break Theorem 1.1. Independently, one could construct a jump kernel satisfying the (UJS) averaging bound only at scales below a fixed threshold while (wPH1) still holds; the theorem predicts the strong Harnack inequality must hold, so a violation of (PHI) at that threshold would refute it.","supporting_citations":[{"cited_title":"Weak parabolic Harnack inequality and H\\\"older regularity for non-local Dirichlet forms","cited_arxiv_id":"2410.23732","evidence_quote":"The author's own unpublished preprint; its Theorems 2.1 and 2.2 supply the links (LLE)$\\Leftrightarrow$(wPH1), (wPH1)$\\Rightarrow$(PGL1), and (wPH1)$\\Rightarrow$(FK)$+$(Gcap)$+$(cap$\\le$), invoked at (1.14), in Proposition 4.1 Step (3), and throughout Corollary 1.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The stochastic stability theory for non-local parabolic Harnack inequalities that this paper reproduces analytically; also the source of the idea behind Proposition 3.1, (PHI0)$\\Rightarrow$(UJS)."},{"cited_title":"Grigor’yan, E","cited_arxiv_id":null,"evidence_quote":"Supplies the parabolic mean-value inequality (PMV$-$) whose proof Lemma 4.3 adapts, plus the definitional framework for caloric functions and the comparison used in Lemma 2.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lemma 3.6's tail comparison is stated to be essentially contained in its Lemma 5.3, which the paper adapts while dropping the upper-estimate condition."},{"cited_title":"Nash, Continuity of solutions of parabolic and elliptic equations, Amer","cited_arxiv_id":null,"evidence_quote":"Nash's original blow-up strategy, which Proposition 4.1 follows for the route (LLE)$+$(UJS)$\\Rightarrow$(PHI0)."},{"cited_title":"Moser, A Harnack inequality for parabolic di fferential equations, Comm","cited_arxiv_id":null,"evidence_quote":"Moser's iteration strategy, which underlies Proposition 4.4 and the construction of (PMV$+$) in the second proof of the strong inequality."},{"cited_title":"Grigor’yan, E","cited_arxiv_id":null,"evidence_quote":"Provides capacity and mean-value equivalences used in Corollary 1.2 (the step (e)$'\\Leftrightarrow$(k)) and the on-diagonal heat-kernel upper bound argument in Lemma 4.6."}],"review_version":1}