{"id":"dfdf8872-99b2-438f-817b-2d24b6c8ee90","arxiv_id":"2411.17107","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The reverse Riesz inequality ‖Δ^{1/2}f‖_p ≤ C‖|∇f|‖_p holds for all 1 < p < ∞ on connected sums of Euclidean-ended manifolds, contrasting with the Riesz transform's sharp range p < min n_i.","lead":"On manifolds built by gluing together ends of the form R^{n_i} × M_i, the reverse Riesz inequality is shown to hold in every L^p space for 1 < p < ∞, even though the ordinary Riesz transform becomes unbounded once p reaches the smallest end dimension. The proof uses a new cancellation mechanism, harmonic annihilation, to remove the leading term that obstructs the forward inequality.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 6 integration by parts drops the O((ilgk)^q |z|^-infinity) error from Lemma 6.1; with stated estimates the missing term is not controlled and may diverge for 1<p<2, so Theorem 1.3 is not proved as written.","rationale":"The reader's weakest-assumption analysis identified exactly the same Section 6 step: Lemma 6.1 provides only an approximate solution, yet the integration by parts for J± treats Delta u± = v± - k^2 u± as exact. My stress-test sharpens that concern: the omitted error term is not merely unestimated; with the estimates stated in the paper, the natural bound for the error contribution involves the divergent integral integral_0^{k0} (ilgk)^q k^(-2/p) dk for 1<p<2. Consequently, the proof of Theorem 1.3 is incomplete as written, and the gap is not cosmetic. I do not conclude the theorem is false; the error may vanish because of unstated support or orthogonality properties in the parametrix construction of [21], but those properties are not in the paper and need to be supplied. The main Theorem 1.2 for n_* >= 3 and the broken-line results in Sections 4-5 appear well supported and are not affected by this concern. Therefore the appropriate verdict remains CONDITIONAL, unchanged from the reader's assessment, pending an explicit bound or elimination of the error term in Section 6.","tokens_in":1103,"tokens_out":3136,"duration_ms":294649,"concrete_test":"Use the explicit construction in Hassell-Nix-Sikora [21, Lemma 2.14] to write E±(z,k) = (Delta + k^2)u± - v±. Check whether E±(·,k) can be chosen so that (a) it is supported outside a fixed compact set (then R± = 0 for f in C_c^infinity), or (b) it carries an extra k^beta factor with beta>0. If neither holds, compute R± on the model manifold R^2 # R^3 with f supported near the neck: determine the leading k -> 0 behavior of the expression; if it is of order k^(-2/p)/|log k|^q, the current proof fails for p<2 and Theorem 1.3 requires a new estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.3 (Section 6), after isolating the G1 and G3 terms, the paper reduces to the bilinear forms J± and applies integration by parts, using the exact identity Delta u± = v± - k^2 u± = -Delta phi± - k^2 u±. However, Lemma 6.1 only provides an approximate solution: (Delta + k^2) u± = v± + E± with E± = O((ilgk)^q |z|^-infinity). Hence the exact identity is false. Integration by parts therefore produces an additional term R±(f,g) = integral f(z) integral_0^{k0} E±(z,k) [integral_{E±} (Delta± + k^2)^-1(z0±,z') phi±(z') g(z') dz'] dk dz. No bound for R± appears in the paper. This is not a harmless omission: using (6.6)-(6.9) and Lemma 6.1, the only direct estimate available is, for f supported in a fixed compact set and 1<p<2, |R-| is bounded by ||f||_p ||g||_{p'} times integral_0^{k0} (ilgk)^q ||(Delta- + k^2)^-1(z0-,·)||_{L^p(E-)} dk. The L^p norm is at least of order k^(-2/p), so the k-integral behaves like integral_0^{k0} k^(-2/p)/|log k|^q dk, which diverges for p<2. A similar issue occurs on E+ for p close to 1. Thus, unless the error E± has extra structure not stated in Lemma 6.1 (for example, support disjoint from the support of f, or an additional factor with positive power of k), the integration-by-parts step is invalid and Theorem 1.3 is unproven as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the reverse Riesz inequality (RR_p) ||Δ^{1/2}f||_p ≤ C||∇f||_p on connected sums of Euclidean-ended manifolds. Theorem 1.2 asserts (RR_p) for all 1<p<∞ on M=(R^{n_1}×M_1)#...#(R^{n_l}×M_l) with l≥2 and n_*=min_i n_i≥3. Theorem 1.3 asserts the same conclusion in the critical case M=(R^2×M_-)#(R^{n_+}×M_+) with n_+≥3. Sections 4–5 develop a one-dimensional radial model and prove an intermediate range of (RR_p) there. The proof of Theorem 1.2 uses the parametrix decomposition of Hassell–Sikora, identifies a harmonic term that is annihilated after integration by parts, and reduces the remaining operator to explicit pointwise estimates. The proof of Theorem 1.3 follows the same strategy but relies on an approximate-solution lemma from Hassell–Nix–Sikora.","tokens_in":28154,"tokens_out":21763,"duration_ms":204180,"significance":"If Theorem 1.2 is correct, it is a clean counterexample to the expected equivalence of Riesz and reverse Riesz boundedness: on these manifolds the forward Riesz transform is known to be bounded only for p<n_*, while the reverse inequality is claimed for all p. The 'harmonic annihilation' mechanism in Section 3 is an interesting and potentially reusable idea, and the one-dimensional model in Sections 4–5 gives corroborating evidence. The proof of Theorem 1.2 is detailed and, as far as I could verify, internally consistent. However, the proof of the critical-case Theorem 1.3 contains a concrete gap: an approximate solution is treated as an exact solution in an integration-by-parts step, and the error term is never estimated. The critical-case claim is therefore not established as written.","major_comments":[{"comment":"After the definition of J_±(f,g), the proof says 'Apply integration by parts and use the fact that Δu_± = v_± − k^2u_±' and then writes the resulting bilinear form as an exact identity. This is not justified, because Lemma 6.1 provides only an approximate solution: (Δ+k^2)u_± = v_± + E_± with E_± = O((ilgk)^q |z|^{-∞}). The integration by parts therefore produces an additional term R_±(f,g) = ∫_M f(z) ∫_0^{k_0} E_±(z,k) [∫_{E_±} (Δ_±+k^2)^{-1}(z^0_±,z') φ_±(z') g(z') dz'] dk dz, and no estimate for this term appears anywhere. Using the stated bounds (6.6)–(6.9) and Lemma 6.1, the natural estimate for R_- is bounded by ||f||_p ||g||_{p'} times ∫_0^{k_0} (ilgk)^q ||(Δ_-+k^2)^{-1}(z^0_-,·)||_{L^p(E_-)} dk, and the resolvent norm is at least of order k^{-2/p}. For 1<p<2 this integral behaves like ∫_0^{k_0} k^{-2/p}/|log k|^q dk, which diverges. A similar divergence occurs for R_+ when p is close to 1. Theorem 1.3 is therefore not proved as written; a stronger property of E_±, or a different argument that estimates this error term, is needed.","section":"Section 6, proof of Theorem 1.3"}],"minor_comments":[{"comment":"The phrase 'with in the framework' contains a typo; it should be 'within the framework'.","section":"Abstract"},{"comment":"The notation C(g) is used before it is defined; please define it explicitly and note that it depends on g through the resolvent integral.","section":"Section 3, equation (3.6)"},{"comment":"The name 'Bessell' appears several times; the standard spelling is 'Bessel'.","section":"Section 4, Section 6"},{"comment":"The lemma is stated for manifolds 'defined by (1.2)' but (1.2) requires n_i≥3, whereas the critical case has n_-=2; please clarify the hypotheses of the lemma, since the proof explicitly extends the statement to n_*=2.","section":"Section 6, Lemma 6.2"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the unestimated error term in Section 6. If the author can supply a valid bound for the additional term R_±, the paper is likely publishable. The proof of Theorem 1.2 appears sound, and the critical case seems to be the only place where the argument breaks. The heavy reliance on [21] is acceptable if that reference is publicly available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Key takeaway: Theorem 1.2 is a real, new result, and the harmonic-annihilation method is the right mechanism. Theorem 1.3, the critical case, has a gap in Section 6 and is not proved as written.\n\nWhat is new and what works: on M = (R^{n_1}×M_1)#...#(R^{n_l}×M_l) with n_* = min n_i ≥ 3, the paper proves the reverse Riesz inequality for all 1<p<∞. Prior forward Riesz results only gave the reverse range by duality, so the full range is genuinely new. The proof is detailed and internally consistent: the parametrix decomposition into G1–G4, the replacement of z by z0_i, the annihilation of the harmonic part Φ_i in (3.6), and the estimates for T_i in Lemma 3.2 all check out. The decay cases S1–S3 are handled carefully. This settles a concrete question: reverse Riesz boundedness is not tied to the forward Riesz range p<n_*, and the paper explains why—the leading harmonic term, which blocks the forward inequality, disappears under the reverse pairing.\n\nThe soft spot is exactly where the stress-test points. In Section 6, after reducing to J±, the proof integrates by parts using the exact identity Δu± = -Δφ± - k²u±. But Lemma 6.1 only provides an approximate solution: (Δ+k²)u± = v± + O((ilgk)^q |z|^{-∞}). The extra error term is dropped without comment. This is not a harmless omission. With the stated estimates, the residual on E− contributes something like ∫_0^{k0} (ilgk)^q k^{-2/p} dk, which diverges for 1<p<2. A similar issue appears on E+ for p close to 1. So Theorem 1.3 is unproven as it stands. It may be fixable—for example, by constructing an exact solution via a correction term, or by giving the error extra structure—but the fix is not in the manuscript.\n\nThe toy model in Sections 4–5 is a reasonable warm-up and seems okay, but it is not needed for the main result. The citation pattern is fine; the self-citation [24] appears only in the introduction and does not feed the proof. The writing has thesis-note roughness (typos, \"Bessell\", the odd \"ilg\" notation), but nothing that affects substance.\n\nRecommendation: send this to a serious referee. The main theorem deserves referee time, but the paper should not be accepted as is—Section 6 needs a real repair before Theorem 1.3 can stand.","headline":"New all-p reverse Riesz result for n*≥3 is solid and worth citing; the critical 2D case in Section 6 is not proved because an approximate solution is treated as exact.","tokens_in":28696,"tokens_out":4135,"would_cite":true,"duration_ms":42206,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B20","47F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the reverse Riesz inequality on every connected sum of Euclidean-ended manifolds: it holds for all $1<p<\\infty$, including the critical two-dimensional-end case, despite the forward Riesz transform being bounded only in a…","keywords":["Riesz transform","reverse Riesz inequality","manifolds with ends","harmonic annihilation","low-energy parametrix","Hardy inequality","critical dimension","L^p boundedness"],"falsifier":"On $M=(\\mathbb{R}^2\\times M_-)\\#(\\mathbb{R}^{n_+}\\times M_+)$ with $1<p<2$, compute the contribution of the error term $(\\Delta+k^2)u_\\pm-v_\\pm$ in the integration-by-parts step of Section 6; if that contribution is not bounded by $C\\|\\nabla f\\|_p\\|g\\|_{p'}$ uniformly as $k\\to 0$, the claimed all-$p$ reverse inequality fails.","tokens_in":27516,"feed_emoji":"📐","tokens_out":16511,"duration_ms":134125,"temperature":0.7,"pith_summary":"On a manifold formed by gluing together Euclidean-ended pieces, the forward Riesz transform $\\nabla\\Delta^{-1/2}$ is known to be $L^p$-bounded only for a limited range of exponents, controlled by the smallest end dimension. The paper sets out to prove that the reverse Riesz inequality, $\\|\\Delta^{1/2}f\\|_p \\le C\\|\\nabla f\\|_p$, holds for every $1<p<\\infty$ on such manifolds, and also in the critical case where one end has dimension two. That conclusion separates the two transforms: the dimensional restriction that stops the forward transform does not stop the reverse one. The engine is a mechanism the paper calls harmonic annihilation, in which the would-be obstruction is a harmonic leading term that disappears when paired with a gradient under integration by parts.","feed_headline":"Reverse Riesz inequality holds for every p on ended manifolds","feed_subtitle":"The forward Riesz transform fails for large p, but the reverse bound survives, breaking the two-sided equivalence.","key_machinery":"The load-bearing mechanism is harmonic annihilation. In the low-energy resolvent expansion, the kernel pieces $G_1$ and $G_3$ produce terms whose leading behaviour is governed by $\\nabla\\Phi_i$, where $\\Phi_i=\\varphi_i+u_i(\\cdot,0)$ is the unique harmonic function tending to $1$ at the infinity of the $i$-th end and to $0$ at the others. Because $\\Phi_i$ is harmonic, $\\langle\\nabla f,\\nabla\\Phi_i\\rangle=\\langle f,\\Delta\\Phi_i\\rangle=0$ for $f\\in C_c^\\infty$, so the leading singular term vanishes identically. What remains is controlled with the resolvent estimates (2.6)--(2.7), the uniform low-energy difference estimate (3.9), and, in the critical case, the logarithmic-factor approximate solutions from Lemma 6.1 together with the Hardy inequality of Lemma 6.2. An implicit version of the same cancellation, using integration by parts and Hardy's inequality rather than an explicit harmonic function, is what carries the broken-line model and the critical two-dimensional end.","core_discovery":"The central claim is Theorem 1.2: on $M=(\\mathbb{R}^{n_1}\\times M_1)\\#\\cdots\\#(\\mathbb{R}^{n_l}\\times M_l)$ with $l\\ge 2$ and $n_*=\\min_i n_i\\ge 3$, the reverse Riesz inequality $\\|\\Delta^{1/2}f\\|_p\\le C\\|\\nabla f\\|_p$ holds for all $f\\in C_c^\\infty(M)$ and all $1<p<\\infty$. Theorem 1.3 extends the same all-$p$ conclusion to the critical manifold $M=(\\mathbb{R}^2\\times M_-)\\#(\\mathbb{R}^{n_+}\\times M_+)$ with $n_+\\ge 3$. On the same manifolds the forward Riesz inequality $\\|\\nabla\\Delta^{-1/2}f\\|_p\\le C\\|f\\|_p$ is known to hold only for $1<p<n_*$, or only for $1<p\\le 2$ in the critical case, so the paper's result is a direct counterexample to the expected equivalence between the Riesz and reverse Riesz transforms.","pith_inferences":["Because the cancellation step (3.6) uses the fact that $f$ vanishes at infinity, the paper's proof does not automatically extend from $C_c^\\infty$ to the full homogeneous Sobolev space; extending the inequality there would need a new argument.","The same low-energy cancellation recipe suggests a general principle: whenever a resolvent has a harmonic leading term and a Hardy inequality is available, the reverse Riesz inequality should hold on a wider $p$-range than the forward one; this could be tested on other non-doubling geometries such as exterior domains with mixed boundary conditions.","The method indicates that reverse Riesz boundedness can be obtained without the doubling and Poincar\\'e conditions used by previous approaches, since the needed control comes from parametrix decay and Hardy inequalities instead."],"forward_implications":["On every manifold covered by Theorem 1.2, the full two-sided inequality $(E_p)$ holds exactly for $1<p<n_*$; outside that range only the reverse side survives (Corollary 3.5).","On the critical manifold $(\\mathbb{R}^2\\times M_-)\\#(\\mathbb{R}^{n_+}\\times M_+)$, the reverse inequality holds for all $p$ while the forward Riesz transform is bounded only for $1<p\\le 2$.","The Hodge projector $d_M\\Delta^{-1}d_M^*$ is bounded on $L^p$ exactly for $n'_*<p<n_*$ (Corollary 3.6).","On the broken line with measure $|r|^{d-1}dr$, the reverse Riesz inequality holds for all $1<p<\\infty$ when $d\\ge 2$, and for $p\\in(1,d)\\cup(d,\\infty)$ when $1<d<2$ (Theorem 5.1).","On $M=\\mathbb{R}^n\\#\\mathbb{R}^n$, the Sobolev inequality $\\|f\\|_q\\lesssim\\|\\nabla f\\|_p$ with $1/p-1/q=1/n$ holds for $1<p<n$ (Corollary 3.7)."],"supporting_citations":[{"why":"Supplies the low-energy parametrix Lemma 2.1, the resolvent estimates (2.6)--(2.7), and the forward Riesz range $1<p<n_*$ used in the duality step.","marker":"[23]"},{"why":"Introduces the resolvent decomposition into $G_1,\\dots,G_4$ and the harmonic-coefficient expansion that motivates harmonic annihilation.","marker":"[10]"},{"why":"Provides the approximate-solution Lemma 6.1 and the parametrix for the critical two-dimensional-end manifold whose forward Riesz transform is bounded only for $1<p\\le 2$.","marker":"[21]"},{"why":"Gives the duality proposition that the forward Riesz inequality on $L^p$ implies the reverse inequality on $L^{p'}$, the bridge covering $p>n'_*$.","marker":"[12]"},{"why":"Supplies the Hardy inequality behind Lemma 6.2 and the reverse-Riesz comparison used for the critical case.","marker":"[31]"},{"why":"Provides the one-dimensional broken-line resolvent kernel formulas and Riesz-transform range used as a toy model in Section 4.","marker":"[22]"},{"why":"Supplies the spectral resolution formula for $\\Delta^{1/2}$ and background on the reverse Riesz inequality.","marker":"[3]"}],"fun_headline_variants":["All-p reverse Riesz inequality on manifolds with ends","Reverse Riesz bound for every p on ended manifolds","Forward Riesz limited, reverse Riesz holds for all p","Manifolds with ends: reverse Riesz beats forward for large p","Reverse Riesz inequality all-p, forward only small p"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The critical-dimension proof treats the approximate solutions $u_\\pm$ as exact solutions when integrating by parts, and the error left behind by that approximation is never estimated.","fun_headline_variants_meta":{"raw":{"variants":["All-p reverse Riesz inequality on manifolds with ends","Reverse Riesz bound for every p on ended manifolds","Forward Riesz limited, reverse Riesz holds for all p","Manifolds with ends: reverse Riesz beats forward for large p","Reverse Riesz inequality all-p, forward only small p"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000204,"raw_usage":{"total_tokens":1413,"prompt_tokens":989,"completion_tokens":424,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":339}},"tokens_in":605,"tokens_out":424,"duration_ms":4158,"temperature":1.0,"reasoning_tokens":339,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:34:30.312874+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On $M=(\\mathbb{R}^2\\times M_-)\\#(\\mathbb{R}^{n_+}\\times M_+)$ with $1<p<2$, compute the contribution of the error term $(\\Delta+k^2)u_\\pm-v_\\pm$ in the integration-by-parts step of Section 6; if that contribution is not bounded by $C\\|\\nabla f\\|_p\\|g\\|_{p'}$ uniformly as $k\\to 0$, the claimed all-$p$ reverse inequality fails.","supporting_citations":[{"cited_title":"Hassell and A","cited_arxiv_id":null,"evidence_quote":"Supplies the low-energy parametrix Lemma 2.1, the resolvent estimates (2.6)--(2.7), and the forward Riesz range $1<p<n_*$ used in the duality step."},{"cited_title":"Carron, T","cited_arxiv_id":null,"evidence_quote":"Introduces the resolvent decomposition into $G_1,\\dots,G_4$ and the harmonic-coefficient expansion that motivates harmonic annihilation."},{"cited_title":"Hassell, D","cited_arxiv_id":null,"evidence_quote":"Provides the approximate-solution Lemma 6.1 and the parametrix for the critical two-dimensional-end manifold whose forward Riesz transform is bounded only for $1<p\\le 2$."},{"cited_title":"Coulhon and X","cited_arxiv_id":null,"evidence_quote":"Gives the duality proposition that the forward Riesz inequality on $L^p$ implies the reverse inequality on $L^{p'}$, the bridge covering $p>n'_*$."},{"cited_title":"Hassell and A","cited_arxiv_id":null,"evidence_quote":"Provides the one-dimensional broken-line resolvent kernel formulas and Riesz-transform range used as a toy model in Section 4."},{"cited_title":"Auscher and T","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral resolution formula for $\\Delta^{1/2}$ and background on the reverse Riesz inequality."}],"review_version":1}