{"id":"9c35f9a8-1bdc-467b-9ad2-19de4210eb1a","arxiv_id":"2501.06866","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For symmetric Dirichlet forms on doubling spaces, diagonal heat kernel upper bounds follow from Faber-Krahn, cutoff Sobolev, tail, and a new integrated jump condition, with a nearly sharp range of exponents.","lead":"This mathematics paper proves a stable set of conditions that force on-diagonal heat kernel upper bounds for symmetric Dirichlet forms on doubling metric measure spaces, including cases without a jump density. It adds one new integral condition on the jump kernel, shows the parameter range is nearly optimal, and settles an open question in the negative.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The forward implication depends on the unproved Lemma 5.4, and the optimality theorem is delegated to the unpublished preprint [30]; these two gaps make the current conditional verdict appropriate.","rationale":"The reader's stated weakest assumption was the no-killing-part hypothesis, which I view as a scope condition rather than the most load-bearing risk: the paper explicitly assumes it and the central implications are stated within that framework. However, the reader's rationale also flagged Lemma 5.4 and the delegation of Theorem 1.2 to [30], and those are the concerns I find most load-bearing. I therefore partially agree: the same two proof-completeness gaps motivate the conditional verdict, while the no-killing assumption is not the decisive issue for the central claim. The omitted proof of Lemma 5.4 matters because it is an internal step in the only forward direction WFK+CS+IJ⇒DUE, and the use of WFK in place of FK is non-trivial. The omitted proof of Theorem 1.2 matters because the claimed optimality of (2.13), and the negative answer to Question 1, depend on the unpublished change-of-metric result [30]. Neither gap has been shown to be fatal, and the rest of the argument contains substantial supporting structure, including explicit constructions and estimates, so CONDITIONAL remains the correct verdict rather than ACCEPT or REJECT.","tokens_in":59771,"tokens_out":21726,"duration_ms":204388,"concrete_test":"Write out the full proof of Lemma 5.4 following [29, Cor. 10.3 and Lemma 11.2], replacing FK_ν by WFK_ν at every occurrence and tracking the additive constant from (5.1). In particular, verify that the constants δ,η can be chosen independent of a>0 and of the radius r in the class r<φ^{-1}(x0,δ2T0), using only VD, TJ, WFK and CS. If the induction step forces the un-truncated FK_ν, then Theorem 2.9(ii) needs an additional RVD hypothesis; if it succeeds, the remaining check is to supply the omitted metric-change argument for Theorem 1.2 and verify that the counterexample's DUE failure is preserved under the transformation from Theorem 2.11.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.9(ii), the core implication WFK(φ)+CS(φ)+IJ_{2,γ}(φ) ⇒ DUE(φ), is obtained through Proposition 5.9, whose proof of LRE_κ(φ) uses Lemma 5.4. Lemma 5.4 is stated after the sentence 'The details are omitted' and requires adapting [29, Cor. 10.3 and Lemma 11.2] from FK_ν to the weaker WFK_ν. This is precisely the place where the weaker hypothesis could fail: Lemma 5.2 shows how the additive WFK constant is absorbed into the estimate (5.1), but the missing Lemma 5.4 must yield uniform constants δ,η independent of the ball and of the level a, and the text does not demonstrate that the [29] argument survives the truncation of the Faber-Krahn term. If Lemma 5.4 requires the un-truncated FK_ν rather than WFK_ν, then Proposition 5.9, and hence Theorem 2.9(ii), would need additional hypotheses such as RVD(α0) with α0>0. Separately, the sharpness claim Theorem 1.2 is not self-contained: its proof states 'By applying the change of metric from Proposition 7.3... The details are omitted,' and Proposition 7.3 is cited from the unpublished preprint [30]. Thus the negative answer to Question 1 and the optimality of the exponent range (2.13) rest on an external, unverified input. These are proof-completeness concerns rather than demonstrations of falsity, but they are load-bearing because they sit exactly on the two central claims of the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a stable characterization of on-diagonal upper bounds for heat kernels of regular symmetric Dirichlet forms without killing part on metric measure spaces satisfying volume doubling. The main forward result, Theorem 2.9(ii), states that under VD, CS(ϕ) and TJ(ϕ), the weak Faber–Krahn inequality WFKν(ϕ) together with the new integrated jump condition IJ2,γ(ϕ) implies DUE(ϕ)+SE(ϕ), provided (1−ν)γ<1+ν. Theorem 2.11 gives an explicit counterexample showing the parameter range cannot be improved in general, and Theorems 1.2 and 2.14 record the sharpness statement and an extension to jump densities satisfying TJq(ϕ) for some q<2. The applications in Section 9 concern variable-order nonlocal operators and singular jump kernels.","tokens_in":60092,"tokens_out":4290,"duration_ms":46739,"significance":"If the proofs are completed as indicated, this would be a substantial contribution: it removes the density assumption from the existing stability theory, introduces a flexible integrated jump condition IJ2,γ, and supplies an explicit Cantor-type counterexample demonstrating optimality. The technical route through truncated Dirichlet forms, lower resolvent estimates, and a change of metric is well structured and contains many new comparison and self-improvement arguments. The explicit counterexample in Theorem 2.11 is a genuine strength of the paper. However, two load-bearing passages are currently left as omitted details or delegated to an unpublished preprint, so the central claims are not yet verifiable from the manuscript alone.","major_comments":[{"comment":"Lemma 5.4 is introduced with 'The details are omitted' and is stated to follow from adapting [29, Cor. 10.3 and Lemma 11.2] from FKν to WFKν. This lemma is essential: it is used in Proposition 5.9 to obtain LREκ(ϕ), which then yields SE(ϕ), and through Theorem 7.2 it feeds into Theorem 2.9(ii). The text does not prove that the additive constant C′ in WFKν can be absorbed uniformly over all balls and all levels a, nor that the truncation of the Faber–Krahn term preserves the uniform constants δ,η required by Lemma 5.4. Because this is the exact point where the weaker hypothesis WFKν is used, the forward implication is currently incomplete. Please provide the full proof or a precise statement of the modified argument.","section":"Section 5, Lemma 5.4"},{"comment":"The sharpness claim, including the negative answer to Question 1 and the optimality of the range (2.13), is obtained by 'applying the change of metric from Proposition 7.3' with 'The details are omitted.' Proposition 7.3 itself, as well as parts of Proposition 7.6(i)–(ii), is cited from the unpublished preprint [30]. Since Theorem 1.2 is one of the headline results and the main optimality statement, this is a load-bearing gap rather than a presentation issue. Please include a self-contained proof of the metric transformation (or of the needed consequence of [30]) and give the full verification that the counterexample from Theorem 2.11 satisfies the required conditions after the transformation with the stated constants and with (1−ν)γ<1+ν+ε.","section":"Section 8.2, proof of Theorem 1.2"},{"comment":"The proof of Theorem 2.9(ii) is compressed into the chain (8.1): WFK+CS ⇒ GFK+LRE, GFK ⇔ Nash, LRE ⇒ SE, followed by an application of Theorem 7.2. The conclusion is therefore valid only if Proposition 5.9, Lemma 5.4, and Theorem 7.2 are fully established. Given that Lemma 5.4 and parts of Section 7 depend on omitted details or on [30], the main implication should be regarded as conditional until those gaps are filled. I am not asking for more detail in the chain (8.1) itself, but for the missing inputs to be made available within the manuscript.","section":"Section 8.1, proof of Theorem 2.9"}],"minor_comments":[{"comment":"The condition '0<R< R+ r < R′' is written with 'R+ r' rather than 'R + r'; the spacing should be fixed to avoid misreading.","section":"Section 2, Definition 2.6"},{"comment":"The blanket assumption that (E,F) has no killing part is stated only in Section 2. Since it is essential for extending E to F′ and for several conservativeness and truncation arguments, it should be advertised in the abstract or introduction as a standing restriction.","section":"Section 2, framework"},{"comment":"There are several typographical errors, e.g. 'mainfolds' in the introduction and 'Alfhors' for 'Ahlfors'; these should be corrected in the final version.","section":"Introduction and Section 9"}],"recommendation":"major_revision","confidential_remarks":"The two 'details are omitted' passages sit precisely on the two central claims of the paper: the forward implication via Lemma 5.4 and the sharpness claim via the change-of-metric argument relying on [30]. An editor should insist that the author supply these proofs in the manuscript or in a publicly verifiable supplement before acceptance. The paper is otherwise well motivated and the explicit counterexample in Theorem 2.11 is a strong point in its favor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a genuinely new stable condition (IJ2,γ) for on-diagonal heat kernel upper bounds without assuming a jump density, and it proves a sharp range of exponents with an explicit Cantor-set counterexample. The positive theorem (WFK+CS+IJ2,γ ⇒ DUE) is elaborate but the proof structure is coherent. The counterexample construction is explicit and the computations I checked are consistent. This is a step beyond [18,28], and the applications to variable-order non-local operators and singular kernels look real.\n\nNow the soft spots. Two load-bearing steps are not actually proved in the text. First, Lemma 5.4—the growth lemma for superharmonic functions—is stated with 'The details are omitted,' and it is exactly where the proof must adapt the Faber-Krahn argument from FK to the weaker WFK. The text shows in Lemma 5.2 how the additive WFK constant is absorbed, but the uniform constants in Lemma 5.4 need to be independent of the ball and the level. If that adaptation silently requires the untruncated FK, then Proposition 5.9 and hence the forward implication would need extra hypotheses like RVD(α0) with α0>0. I can't see the missing argument, and it matters.\n\nSecond, the negative answer to Question 1 (Theorem 1.2) is delegated to the unpublished preprint [30] via Proposition 7.3, with one sentence 'The details are omitted.' Since this is a headline claim, the author must supply the change-of-metric proof or a verifiable reference. The counterexample in Theorem 2.11 is for a variable-order scale function; converting it to fixed β is exactly where the composition of the Dirichlet form with the metric change needs checking. I trust the main idea but not the missing verification.\n\nThe no-killing-part assumption is a scope condition; I would not call it a flaw, just a stated restriction.\n\nBottom line: this is a serious paper that deserves a proper referee. The editor should send it to review, not desk-reject. The referee should ask the author to fill in Lemma 5.4 and either prove or provide a checked proof of the change-of-metric step. If those details come out clean, the paper would be a solid contribution to the subfield.","headline":"Genuinely new condition and an explicit counterexample, but two load-bearing proofs are omitted—worth refereeing, not desk-rejecting.","tokens_in":60630,"tokens_out":5329,"would_cite":true,"duration_ms":48483,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K08","31C25","60J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"On doubling metric measure spaces, the paper proves that a Faber–Krahn inequality, a cutoff Sobolev inequality, a tail bound, and a new integrated jump condition with $(1-\\nu)\\gamma<1+\\nu$ characterize the on-diagonal heat kernel upper…","keywords":["heat kernel","Dirichlet form","doubling space","Faber–Krahn inequality","cutoff Sobolev inequality","jump kernel tail estimate","integrated jump condition","variable-order stable-like process"],"falsifier":"Run the paper's explicit counterexample: on the n-fold product of a middle-ξ Cantor set with a variable-order jump kernel whose order jumps from 1 to β2 near two points, all stated hypotheses hold with $(1-\\nu)\\gamma<1+\\nu+\\varepsilon$, yet the heat kernel lower bound $t^{-((n-1)\\alpha_\\xi+\\beta_2)}$ outgrows the on-diagonal upper bound $t^{-(1+1/\\beta_2)n\\alpha_\\xi/2}$ as $t\\to 0$; verifying this computation (or finding the error) settles whether the sharpness claim stands.","tokens_in":59547,"feed_emoji":"🎯","tokens_out":7707,"duration_ms":69268,"temperature":0.7,"pith_summary":"The paper seeks a stable, two-way characterization of on-diagonal heat kernel upper bounds for symmetric Dirichlet forms, covering local and non-local parts, on metric measure spaces with volume doubling. It proves that, once a cutoff Sobolev inequality and a jump-tail bound are in place, the on-diagonal bound $\\mathrm{DUE}(\\beta)$ follows from a Faber–Krahn inequality together with a newly introduced integrated jump condition $IJ_{2,\\gamma}(\\beta)$, and that the permissible range $(1-\\nu)\\gamma<1+\\nu$ is optimal. The result requires no density for the jump kernel, which is the usual bottleneck in earlier stability results. A companion sharpness construction answers an open question in the negative: the tail, Faber–Krahn and cutoff Sobolev conditions alone do not force the heat kernel bound. A density version broadens the range of the $L^q$-tail condition to some $q<2$.","feed_headline":"Heat kernel bounds characterized by an integrated jump condition","feed_subtitle":"Works without jump densities and proves the exponent window is optimal, settling an open question.","key_machinery":"The central new object is the integrated jump condition $IJ_{2,\\gamma}(\\beta)$, which averages the jump kernel over annuli against inverse square roots of ball volumes; it carries the inhomogeneity of the space in a way a pointwise tail bound cannot. The proof runs through truncated Dirichlet forms (removing jumps longer than a scale $\\rho$), an $L^2$-mean value inequality for subharmonic functions, lower resolvent estimates of the semigroup, and survival estimates for truncated processes, using comparison inequalities between truncated and untruncated semigroups. A change-of-metric argument reduces a general scale function to a power-law scaling, and an explicit counterexample is built from products of Cantor-type spaces with variable-order jump kernels to prove sharpness.","core_discovery":"On a metric measure space satisfying volume doubling, a regular symmetric Dirichlet form without killing part, whose jump kernel satisfies the tail bound $\\mathrm{TJ}(\\beta)$, admits an on-diagonal heat kernel upper bound $\\mathrm{DUE}(\\beta)$ if its Faber–Krahn inequality $\\mathrm{FK}_\\nu(\\beta)$, cutoff Sobolev inequality $\\mathrm{CS}(\\beta)$, and the new integrated jump condition $IJ_{2,\\gamma}(\\beta)$ hold with constants obeying $(1-\\nu)\\gamma<1+\\nu$. The integrated condition measures how the jump kernel distributes mass across annuli, weighted by inverse square roots of ball volumes, and is what lets the argument run without assuming the jump kernel has a density. The same statement is false if the exponent inequality is relaxed by any positive amount: the paper builds metric measure spaces satisfying all hypotheses with $(1-\\nu)\\gamma<1+\\nu+\\varepsilon$ for any $\\varepsilon>0$ where $\\mathrm{DUE}(\\beta)$ fails, so the range is optimal and the previously open Question 1 about whether tail and Faber–Krahn and cutoff Sobolev conditions alone suffice has a negative answer. When a jump density exists, the result extends the known $L^q$-tail implication to some exponents $q<2$.","pith_inferences":["Because the arguments never touch the density, the method should transfer directly to cylindrical fractional Laplacians and other singular kernels for which no density exists; constructing such an example at the boundary exponent would test the sharpness further.","The product-Cantor counterexample suggests that variable-order jump kernels whose order varies in space produce heat kernels with exponents that mix the local orders; this could be used to build models of anisotropic walk dimension on fractals.","A concrete next test is whether $IJ_{2,\\gamma}$ is preserved under rough isometries or quasisymmetric maps; if it is, the characterization would be stable under the same equivalence classes as the classical heat kernel estimates.","For applied analysis, the integrated condition can be read as a quantitative measure of metric inhomogeneity, and one could check numerically whether it is the sharp diagnostic for failure of diagonal heat kernel bounds in random media."],"forward_implications":["Under volume doubling, the on-diagonal bound for a wide class of non-local and mixed Dirichlet forms can be certified by the four local-to-global conditions, without any density for the jump kernel.","The negative answer to Question 1 means that tail-type jump bounds alone, even with a Faber–Krahn inequality and cutoff Sobolev inequality, are insufficient; the integrated condition is essentially necessary.","For strongly local forms, and for jump forms whose volume growth is not too anisotropic, the full equivalence $\\mathrm{WFK}_\\nu(\\phi)+\\mathrm{CS}(\\phi)\\iff \\mathrm{DUE}(\\phi)+\\mathrm{SE}(\\phi)$ holds (Corollary 2.10).","When the jump kernel has a density, the $L^q$-tail range is widened: the implication holds for some $q<2$, improving the previous $q\\ge 2$ threshold.","The applications include stable-like variable-order operators on $\\mathbb{R}^d$ with no jump density and product spaces whose factor forms have singular jump kernels, both cases not covered by earlier stability results."],"supporting_citations":[{"why":"Establishes the stability framework for heat kernel estimates of non-local Dirichlet forms under the pointwise jump kernel bound that the present paper relaxes.","marker":"[18]"},{"why":"Introduces the Lq-tail condition TJq(β) and proves the on-diagonal implication for q≥2 that Theorem 2.14 extends to some q<2.","marker":"[28]"},{"why":"Supplies the L2-mean value inequality and generalized capacity technique adapted here through WFKν and lower resolvent estimates.","marker":"[29]"},{"why":"Introduces the cutoff Sobolev inequality CS(β) on which the paper builds.","marker":"[1]"},{"why":"Shows the equivalence between on-diagonal upper bounds and Faber–Krahn inequalities on doubling spaces, providing the backward direction of the characterization.","marker":"[23]"},{"why":"Gives the truncation and comparison estimates for non-local Dirichlet forms without a killing part that the proof exploits.","marker":"[33]"},{"why":"Provides the change-of-metric reduction used to pass from power-law scaling to general scale functions.","marker":"[30]"}],"fun_headline_variants":["Integrated jump condition yields optimal heat bounds","Heat kernel bounds from integrated jump without densities","Optimal exponent range for heat kernel upper bounds","Answering open question: integrated condition optimal for heat"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The framework assumes the Dirichlet form has no killing part—mass is never destroyed in place, only moved elsewhere—and the truncation, energy-measure, and conservativeness arguments all depend on that; introduce killing and the characterization may fail.","fun_headline_variants_meta":{"raw":{"variants":["Integrated jump condition yields optimal heat bounds","Heat kernel bounds from integrated jump without densities","Optimal exponent range for heat kernel upper bounds","Answering open question: integrated condition optimal for heat"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000928,"raw_usage":{"total_tokens":3939,"prompt_tokens":872,"completion_tokens":3067,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":3010}},"tokens_in":488,"tokens_out":3067,"duration_ms":22562,"temperature":1.0,"reasoning_tokens":3010,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:49:52.086726+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's explicit counterexample: on the n-fold product of a middle-ξ Cantor set with a variable-order jump kernel whose order jumps from 1 to β2 near two points, all stated hypotheses hold with $(1-\\nu)\\gamma<1+\\nu+\\varepsilon$, yet the heat kernel lower bound $t^{-((n-1)\\alpha_\\xi+\\beta_2)}$ outgrows the on-diagonal upper bound $t^{-(1+1/\\beta_2)n\\alpha_\\xi/2}$ as $t\\to 0$; verifying this computation (or finding the error) settles whether the sharpness claim stands.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the stability framework for heat kernel estimates of non-local Dirichlet forms under the pointwise jump kernel bound that the present paper relaxes."},{"cited_title":"Grigor’yan, E","cited_arxiv_id":null,"evidence_quote":"Introduces the Lq-tail condition TJq(β) and proves the on-diagonal implication for q≥2 that Theorem 2.14 extends to some q<2."},{"cited_title":"Grigor’yan, E","cited_arxiv_id":null,"evidence_quote":"Supplies the L2-mean value inequality and generalized capacity technique adapted here through WFKν and lower resolvent estimates."},{"cited_title":"Andres, M","cited_arxiv_id":null,"evidence_quote":"Introduces the cutoff Sobolev inequality CS(β) on which the paper builds."},{"cited_title":"Grigor’yan, J","cited_arxiv_id":null,"evidence_quote":"Shows the equivalence between on-diagonal upper bounds and Faber–Krahn inequalities on doubling spaces, providing the backward direction of the characterization."},{"cited_title":"Grigor’yan, J","cited_arxiv_id":null,"evidence_quote":"Gives the truncation and comparison estimates for non-local Dirichlet forms without a killing part that the proof exploits."},{"cited_title":"Grigor’yan, E","cited_arxiv_id":null,"evidence_quote":"Provides the change-of-metric reduction used to pass from power-law scaling to general scale functions."}],"review_version":1}