{"id":"8e549022-26b2-4997-94e5-85b3c5fb0060","arxiv_id":"2607.05181","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":4,"one_line_summary":"The parabolic Riesz transform for non-autonomous divergence-form operators with bounded measurable coefficients is bounded on L^p for the maximal open range 1<p≤2, with sharpness in dimension n≥2.","lead":"This paper proves L^p boundedness of Riesz transforms for parabolic operators with measurable coefficients in all variables, for 1<p≤2. It identifies the optimal exponent range and proves sharpness in dimension n≥2.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The p-sensitivity of the off-diagonal decay in Proposition 4.4 is the linchpin: the exponent ε = 1 + 1/(1+p') must remain > 1 uniformly as p ↓ p_-(H), but the proof's Jensen step is only valid for p > 1, creating a potential gap at the endpoint of the bootstrap.","rationale":"The reader identified the commutator locality in Proposition 4.4 as the weakest assumption. This is structurally correct — the locality of [H*, η] = -(∂_t η) is indeed what makes the entire off-diagonal framework work, and it is verified directly from the variational definition. However, the more load-bearing concern is whether the p-sensitive bootstrap in Theorem 6.1 actually closes: the off-diagonal exponent ε = 1 + 1/(1+p') depends on p and degrades as p → 1, and the iteration must work within the range where ε > 1 while simultaneously satisfying the convergence conditions of Propositions 5.1 and 5.10. Having traced the argument carefully, the bootstrap is constructed correctly: it starts at s_0 = 2 and produces a decreasing sequence converging to any fixed p > p_-(H), with each step using the off-diagonal estimates at the current exponent s_n (which is bounded away from 1 when p_-(H) ≥ 2★ - ε_0). The real-coefficient case (p_-(H) = 1) is handled separately via Gaussian bounds in Section 8, avoiding the bootstrap endpoint issue entirely. The sharpness argument in Section 9 using Mooney's construction is clean and correctly establishes that p_-(H) can be arbitrarily close to 2★. The necessity proof (Proposition 7.12) uses a backward induction on Sobolev conjugates that is standard and correct. The functional calculus extension in Lemma 7.1 from real λ to a complex sector via Neumann series is valid under the stated L^p boundedness assumption. The Blunck–Kunstmann criterion (Theorem 5.1) is correctly adapted to the two-geometry setting, and the proof carefully handles the non-metric nature of the annuli by introducing j-dependent maximal functions M_j with uniform doubling constants. The iterated maximal function M_t M_x in Proposition 5.10 is L^{r/q} bounded for r > q, which is standard. The convergence of the three integrals in (7.10) and the two series in (7.11) is achieved by choosing α and N large, and the conditions are correctly verified. I do not find a load-bearing concern that undermines the central claim. The paper is a substantial and technically careful contribution.","tokens_in":43863,"tokens_out":1426,"duration_ms":345827,"concrete_test":"Verify the convergence of the bootstrap sequence in Theorem 6.1 Step 2 explicitly: fix p_-(H) = 2★ - ε_0 (the worst case) and trace one iteration step to confirm that the output exponent r satisfies r < s (the input) and r > p_-(H), and that the sequence s_n converges to p_-(H) in finitely many or countably many steps with the off-diagonal exponent ε = 1 + 1/(1+s_n') remaining sufficient for summability in Theorem 5.1 at each step. Specifically, check that when s_n is close to 2★, the condition θ > 1/q in Proposition 5.10 is compatible with ε(s_n) > 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that R_H is L^p bounded for all p ∈ (p_-(H), 2], with p_-(H) = q_-(H) ∈ [1, 2★). The proof of Theorem 6.1 (Step 2) bootstraps q_-(H) ≤ p_-(H) by iterating Proposition 5.10, which requires the off-diagonal decay exponent ε = 1 + 1/(1+p') > 1 from Theorem 4.10(i)/Proposition 4.4(i). This ε is p-dependent and approaches 1 as p → 1. The bootstrap in Theorem 6.1 constructs a decreasing sequence s_n → p (where p > p_-(H) is fixed but arbitrary). At each step, the off-diagonal estimates from Theorem 4.10 are applied with exponent p replaced by the current s_n. The convergence condition in Proposition 5.10 requires θ > 1/q where q = [p, s]_θ, and the summability condition S = Σ ε_j (2^n N^2)^{j/q'} < ∞ in Theorem 5.1 requires ε > 2/q' (roughly). As s_n → p, if p is close to 1, then ε = 1 + 1/(1+s_n') approaches 1 + 1/2 = 3/2, which is fine. But the real concern is more subtle: in Proposition 4.4, the dual estimate (4.7) uses Jensen's inequality to bound ||II||_{p'} by pulling out an L^{p'}(F) factor with exponent p'-1. This step requires p' < ∞, i.e., p > 1. When p_-(H) = 1 (real coefficients), the bootstrap must approach p = 1, and at p = 1 the dual exponent p' = ∞ makes the Jensen step in (4.7) break down — the factor |F|^{1-1/p'} = |F|^0 = 1 loses all decay. The paper handles real coefficients separately in Section 8 via Gaussian bounds, bypassing the bootstrap entirely. So the concern is whether the complex-coefficient bootstrap in Theorem 6.1 can actually reach p_-(H) ∈ [1, 2★) when p_-(H) is close to 1. The answer is that Theorem 6.1 only claims p_-(H) ∈ [1, 2★), and the bootstrap starts from p = 2 and works downward. The key question is whether the iteration converges to a value strictly below 2★. The proof argues it converges to any p > p_-(H), but p_-(H) itself is defined as the infimum of L^p boundedness of (E_λ). The bootstrap shows q_-(H) ≤ p for any p > p_-(H), hence q_-(H) ≤ p_-(H). This is logically sound: the iteration need not reach p_−","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper establishes L^p bounds for parabolic Riesz transforms R_H = D H^{-1/2} associated with non-autonomous second-order parabolic operators H = ∂_t − div_x(A∇_x) with bounded measurable coefficients on R^{n+1}. The main result (Theorem 1.3) identifies the maximal open range (p_-(H), 2] for L^p boundedness, where p_-(H) = q_-(H) ∈ [1, 2★) with 2★ = 2(n+2)/(n+4). For real coefficients, p_-(H) = 1 and the spatial component ∇_x H^{-1/2} is weak type (1,1). The range is shown to be sharp for n ≥ 2. The proof combines novel space-time off-diagonal estimates (Section 4) exploiting two complementary geometries—parabolic cubes on small scales and time-stretched annuli on large scales—with a Blunck–Kunstmann-type extrapolation framework (Section 5) and a bootstrap argument (Section 6) identifying p_-(H) = q_-(H).","tokens_in":44712,"tokens_out":1209,"duration_ms":360336,"significance":"This is a substantial contribution to the limited-range extrapolation theory for Riesz transforms with rough coefficients. The key technical innovation is the space-time off-diagonal framework of Section 4, where the commutator locality [H*, η] = −(∂_t η) (Proposition 4.4) is leveraged to convert spatial exponential decay into temporal algebraic decay with a p-sensitive exponent ε = 1 + 1/(1+p') > 1. The dual geometry (parabolic cubes vs. time-stretched annuli) is a genuine new idea that falls outside standard Calderón–Zygmund theory on spaces of homogeneous type. The sharpness result (Section 9) via Mooney's irregular solutions provides a falsifiable endpoint. The identification p_-(H) = q_-(H) via the bootstrap in Theorem 6.1, together with the iterative extrapolation in Theorem 7.3, constitutes the central achievement. The paper builds on the L^2 theory of [1, 9] and resolvent bounds of [1] as external inputs.","major_comments":[{"comment":"The stress-test concern regarding the Jensen step in (4.7) at the endpoint p = 1 does not, on careful reading, create a gap in the complex-coefficient case. Proposition 4.4 is stated for p ∈ (1, 2], and the exponent ε = 1 + 1/(1+p') is strictly greater than 1 for all such p. The bootstrap in Theorem 6.1, Step 2 constructs a decreasing sequence s_n → p where p > p_-(H) is fixed but arbitrary. Since p_-(H) ≥ 1 for complex coefficients and the bootstrap only needs to approach p > p_-(H) ≥ 1, the sequence s_n stays bounded away from 1 whenever p > 1. The case p_-(H) = 1 is handled separately in Section 8 via Gaussian bounds, bypassing the bootstrap entirely. The concern is well-raised but ultimately does not land: the paper's architecture correctly separates the complex-coefficient bootstrap (which never needs p = 1) from the real-coefficient case (which uses a different mechanism). I have a","section":null}],"minor_comments":[{"comment":"Section 5.1 title: 'Exrapolation' should read 'Extrapolation'.","section":null},{"comment":"Proof of Theorem 8.5: 'we imnose' should be 'we impose'.","section":null},{"comment":"In the proof of Proposition 4.4, the transition from the dual estimate to the primal estimate via duality is stated briefly. A one-line clarification that the L^{p'} bound for (λ E*_λ D^{1/2}_t) transfers to the L^p bound for (λ D^{1/2}_t E_λ) by duality would improve readability.","section":null},{"comment":"The notation 2★ is used before its definition in the statement of Theorem 1.3. Moving the definition 2★ = 2(n+2)/(n+4) to appear before part (1) of the theorem would help the reader.","section":null},{"comment":"In the proof of Theorem 7.3, the condition 3α ≥ β + 1 is imposed mid-proof. A brief remark explaining the role of this condition (ensuring sufficient decay in the functional calculus) would aid the reader.","section":null},{"comment":"Figure 1: the caption mentions 'typically N ≥ 2^n' but the role of N in controlling the kernel decay via (1.2) could be stated more explicitly. A brief sentence connecting N to the requirement ε > 1 on the full annulus would avoid confusion.","section":null},{"comment":"In Lemma 7.5, the exponents ϱ_1, ϱ_2 are introduced but their relationship to p_-(H) could be stated more explicitly. Clarifying that ϱ_1, ϱ_2 ∈ (p_-(H), p) would improve clarity.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a strong submission appropriate for a serious harmonic analysis journal. The main technical contributions—the space-time off-diagonal estimates and the dual-geometry extrapolation framework—are novel and well-executed. The stress-test concern about the Jensen step at p = 1 is a natural worry but is handled correctly by the paper's architecture: the complex-coefficient bootstrap never reaches p = 1, and the real-coefficient case uses Gaussian bounds instead. I recommend minor revision with attention to the presentation issues listed."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and the positive assessment. The referee raises a single major comment concerning a potential stress-test of the Jensen step in (4.7) at the endpoint p=1 in the complex-coefficient case. As we explain below, we agree with the referee's analysis that no gap exists: the architecture of the paper correctly separates the complex-coefficient bootstrap (which operates strictly for p>1) from the real-coefficient case (which uses Gaussian bounds and a different mechanism).","responses":[{"response":"We thank the referee for the careful and precise analysis of this point. We agree entirely with the assessment. To confirm the referee's reasoning explicitly: Proposition 4.4 is stated for p in (1,2], and the exponent epsilon = 1 + 1/(1+p') satisfies epsilon > 1 for all such p, tending to 1 only in the limit as p tends to 1. In the bootstrap of Theorem 6.1, Step 2, one fixes an arbitrary p in (p_-(H), 2] and constructs a decreasing sequence (s_n) converging to p. Since p > p_-(H) >= 1, the sequence s_n remains bounded away from 1, and thus epsilon evaluated at s_n remains strictly greater than 1 throughout the iteration. The Jensen step in (4.7) is therefore always applied with p' finite. The endpoint p_-(H) = 1 for real coefficients is treated in Section 8 via Gaussian kernel bounds (Lemma 8.2), which bypasses the bootstrap and the p-sensitive off-diagonal estimates of Section 4 entirely. No revision to the mathematical content is needed. We will add a brief clarifying remark in the text near the statement of Proposition 4.4 or in the proof of Theorem 6.1, Step 2, explicitly noting that the bootstrap sequence stays bounded away from p=1 whenever p > p_-(H) >= 1, so that epsilon > 1 is maintained throughout. This should preempt any reader having the same well-motivated concern.","revision_made":"partial","referee_comment":"The stress-test concern regarding the Jensen step in (4.7) at the endpoint p = 1 does not, on careful reading, create a gap in the complex-coefficient case. Proposition 4.4 is stated for p in (1,2], and the exponent epsilon = 1 + 1/(1+p') is strictly greater than 1 for all such p. The bootstrap in Theorem 6.1, Step 2 constructs a decreasing sequence s_n to p where p > p_-(H) is fixed but arbitrary. Since p_-(H) >= 1 for complex coefficients and the bootstrap only needs to approach p > p_-(H) >= 1, the sequence s_n stays bounded away from 1 whenever p > 1. The case p_-(H) = 1 is handled separately in Section 8 via Gaussian bounds, bypassing the bootstrap entirely. The concern is well-raised but ultimately does not land: the paper's architecture correctly separates the complex-coefficient bootstrap (which never needs p = 1) from the real-coefficient case (which uses a different mechanism)."}],"tokens_in":43521,"tokens_out":696,"duration_ms":47985,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This is the first paper to establish L^p boundedness of parabolic Riesz transforms for non-autonomous operators with merely measurable coefficients depending on all variables. The only prior L^p result (Ouhabaz) required autonomous real coefficients. The main theorem identifies the optimal exponent range (p_-(H), 2], proves sharpness for n >= 2 via Mooney's irregular solutions, and treats real coefficients with the full range plus weak (1,1) for the spatial component. This is a genuine advance and deserves serious attention from anyone working in limited-range extrapolation or parabolic PDEs with rough coefficients. The central technical innovation is the dual-geometry off-diagonal framework: parabolic cubes on small scales, time-stretched annuli on large scales. The key structural fact making this work is that the commutator [H*, eta] = -(partial_t eta) is a local multiplication operator, which is what lets the temporal off-diagonal estimate in Proposition 4.4 split into a local and non-local term. The p-sensitivity of the decay exponent epsilon = 1 + 1/(1+p') is then exploited iteratively in the bootstrap of Theorem 6.1. The stress-test concern about the Jensen step in (4.7) breaking down at p = 1 does not actually land as a gap. The complex-coefficient bootstrap in Theorem 6.1 only needs to approach p > p_-(H), and p_-(H) is bounded below by 1 in the complex case. The real-coefficient case with p_-(H) = 1 is handled separately in Section 8 via Gaussian bounds, bypassing the bootstrap entirely. So the logical structure is sound. The soft spots are minor. The paper is dense and the logical dependencies across Sections 4-7 are intricate enough that a referee will need to work through them carefully, particularly the parameter bookkeeping in the proof of Theorem 7.3 (the interplay of alpha, beta, N, and the convergence conditions on the series in (7.11)). The j-independence of the maximal function constants in the Blunck-Kunstmann adaptation (Theorem 5.1) is asserted but not fully justified, though the doubling constant argument is standard and I believe it checks out. No circularity issues: the L^2 theory comes from the independently verified parabolic Kato square root estimate, and the resolvent bounds are external. This paper is for harmonic analysts and PDE researchers working on Riesz transforms, limited-range extrapolation, or operators with rough coefficients. It deserves a serious referee who is willing to verify the off-diagonal estimates and the bootstrap convergence in detail.","headline":"First L^p bounds for parabolic Riesz transforms with non-autonomous measurable coefficients; the dual-geometry off-diagonal framework is the real innovation.","tokens_in":45268,"tokens_out":626,"would_cite":true,"duration_ms":62148,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Parabolic Riesz transforms bounded on L^p with rough coefficients","keywords":[],"falsifier":"Construct a bounded measurable elliptic coefficient A in dimension n ≥ 2 for which p_-(H) is strictly below 2★, or show that in dimension n = 1, p_-(H) = 1 fails for some complex coefficient.","tokens_in":43854,"feed_emoji":"","tokens_out":1184,"duration_ms":57557,"temperature":0.7,"pith_summary":"This paper establishes the first L^p boundedness results for parabolic Riesz transforms associated with non-autonomous second-order parabolic operators in divergence form with bounded measurable coefficients depending on all variables. The central object is the parabolic Riesz transform R_H = (∇_x H^{-1/2}, D_t^{1/2} H^{-1/2}), which combines a spatial gradient and a half-order time derivative applied to the inverse square root of the heat-type operator H. The authors prove that R_H is bounded on L^p for every p in the range (p_-(H), 2], where p_-(H) is a critical exponent governed by L^p resolvent bounds. For complex coefficients, this exponent always lies below the lower parabolic Sobolev conjugate 2★ = 2(n+2)/(n+4), and the result is sharp in dimension n ≥ 2. For real coefficients, p_-(H) = 1, giving the full range (1, 2], and the spatial gradient component is additionally of weak type (1,1). The key technical innovation is a set of space-time off-diagonal estimates that combine two different geometric regimes: parabolic cubes on small scales and time-stretched annuli modeled on the level sets of the half-order time derivative of a parabolic Bessel potential on large scales. This change of geometry allows the exponential spatial decay of the resolvent kernel to compensate for the insufficient temporal decay (which is 3/2, below the homogeneous dimension n+2), enabling extrapolation below the threshold 2★ that no single-metric approach could reach.","feed_headline":"","feed_subtitle":"","key_machinery":"Space-time off-diagonal estimates combining parabolic cubes (small scale) with time-stretched annuli modeled on Bessel kernel level sets (large scale); Blunck-Kunstmann extrapolation adapted to two incompatible geometries; iterative bootstrap along parabolic Sobolev conjugates; commutator identity [H*, η] = -(∂_t η) enabling local splitting of temporal off-diagonal estimates","core_discovery":"The critical exponent p_-(H) governing L^p boundedness of the parabolic Riesz transform equals q_-(H) and lies in [1, 2★), and this range is sharp. The identification rests on a two-geometry off-diagonal framework: parabolic cubes for small scales and time-stretched annuli for large scales, which together convert spatial exponential decay into sufficient temporal decay. A bootstrap argument along parabolic Sobolev conjugates then iteratively extends the boundedness range from L^2 down to (p_-(H), 2], with each extrapolation step's interval independent of the starting exponent. For real coefficients, Gaussian heat kernel bounds force p_-(H) = 1, and the spatial gradient component achieves the","pith_inferences":["The two-geometry strategy may apply more broadly to operators where a non-local component (here D_t^{1/2}) couples variables in a way that destroys single-metric structure, suggesting a general principle: when kernel decay is insufficient in one variable, a change of geometry can trade decay from another variable to compensate.","The sharpness result in n ≥ 2 but not n = 1 hints that the obstruction to p_-(H) = 1 is fundamentally tied to spatial dimension, possibly reflecting the interplay between the parabolic scaling dimension n+2 and the singularity structure of irregular solutions.","The commutator identity [H*, η] = -(∂_t η) being a local multiplication operator is the structural linchpin; if this locality were lost under different operator realizations or lower regularity, the entire off-diagonal and bootstrap framework would fail, suggesting that the result's reach depends on preserving this commutator structure."],"forward_implications":["The two-geometry off-diagonal framework may extend to other non-local operators where a single metric cannot capture both local and large-scale behavior, such as generalized Stokes operators.","The p-sensitivity of the off-diagonal decay rate (ε = 1 + 1/(1+p')) suggests that coefficient-dependent lower bounds on p_-(H) could be computed for specific operator classes.","The sharpness construction via Mooney's irregular solutions in dimension n ≥ 2 raises the question of whether p_-(H) = 1 always holds in dimension n = 1, which remains open.","The weak type (1,1) bound for the spatial gradient component with real coefficients leaves open whether the full Riesz transform including D_t^{1/2} H^{-1/2} also satisfies this endpoint estimate.","The extrapolation range for p > 2 is conjectured to be governed by a dual critical exponent q_+(H), analogous to the elliptic setting, but duality cannot be directly applied due to the limited-range phenomenon."],"fun_headline_variants":["Sharp L^p bounds for rough parabolic Riesz transforms","Critical exponents for parabolic Riesz transforms with rough coefficients","Parabolic Riesz transforms: sharp L^p bounds for rough coefficients","L^p bounds for non-autonomous parabolic Riesz transforms","Two-geometry approach to L^p bounds for parabolic Riesz transforms"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The off-diagonal estimates for temporal supports rely on the commutator [H*, η] = -(∂_t η) being a local multiplication operator, which allows the dual estimate to split into local and non-local terms and ultimately controls the decay rate. If this commutator structure were lost — for instance, under a different operator realization or with lower regularity — the off-diagonal framework and the bootstrap argument would fail.","fun_headline_variants_meta":{"raw":{"variants":["Sharp L^p bounds for rough parabolic Riesz transforms","Critical exponents for parabolic Riesz transforms with rough coefficients","Parabolic Riesz transforms: sharp L^p bounds for rough coefficients","L^p bounds for non-autonomous parabolic Riesz transforms","Two-geometry approach to L^p bounds for parabolic Riesz transforms","Sharp L^p bounds for parabolic Riesz transforms in 1 < p <= 2"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1048,"prompt_tokens":468,"completion_tokens":580,"prompt_tokens_details":null},"tokens_in":468,"tokens_out":580,"duration_ms":10600,"temperature":1.0,"reasoning_tokens":476,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T00:49:40.381071+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Construct a bounded measurable elliptic coefficient A in dimension n ≥ 2 for which p_-(H) is strictly below 2★, or show that in dimension n = 1, p_-(H) = 1 fails for some complex coefficient.","supporting_citations":[],"review_version":1}