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Reverse Riesz Inequality on Manifolds with Ends

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.17107 v1 pith:4DCBWYB5 submitted 2024-11-26 math.AP

Dangyang He This is my paper · ORCID
classification math.AP MSC 42B2047F05
keywords Riesztransformreverseinequalitymanifoldswithendsharmonicannihilationlow-energyparametrixHardycriticaldimensionL^pboundedness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

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.

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 (1)
  1. [Section 6, proof of Theorem 1.3] 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.
minor comments (4)
  1. [Abstract] The phrase 'with in the framework' contains a typo; it should be 'within the framework'.
  2. [Section 3, equation (3.6)] 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.
  3. [Section 4, Section 6] The name 'Bessell' appears several times; the standard spelling is 'Bessel'.
  4. [Section 6, Lemma 6.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper proves the reverse Riesz inequality from forward-Riesz bounds, resolvent parametrix estimates, Hardy inequalities, and duality; the only flagged concern is a Section 6 correctness gap, not a self-referential reduction.

full rationale

The derivation does not feed the target inequality back into itself. For Theorem 1.2, the forward Riesz theorem [23, Theorem 1.2] is used only to obtain the dual range p > n'_*, and the remaining range p <= n'_* is proved directly via the resolvent decomposition and the harmonic-annihilation identity (3.6), which uses that Phi_i is harmonic. No parameter is fitted to the quantity being predicted; the constants are controlled by Lemma 2.1 and the resolvent estimates (2.6)-(2.7), all independent of the reverse inequality. For Theorem 1.3, the proof uses [21, Theorem 1.2] for p >= 2 and then handles 1 < p < 2 through Lemma 6.1, the Hardy inequality Lemma 6.2, and the T_+- and T_- estimates; none of these assumes the conclusion. The only self-citation, [24], appears in the introduction as a descriptive survey sentence and is not a proof input, so it is not load-bearing. The manuscript also contains an honest limitation statement near Section 3, where it says that the naive ilg(k) strategy gives an integral integral_0^{k0} ilg(k) k^{-2/p} dk that diverges for 1<p<2; the later parametrix construction is designed to avoid that. The one substantive concern found is a correctness gap, not circularity: Lemma 6.1 supplies only (Delta + k^2)u_+ = v_+ + O((ilgk)^q |z|^{-infinity}), while the integration by parts in Section 6 writes Delta u_+ = v_+ - k^2 u_+ as an exact identity and never bounds the omitted error. This may leave Theorem 1.3 unproved as written, but it is an omitted estimate in an approximate-solution construction, not an instance where the conclusion is assumed or is equivalent to an input by definition. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim is a pure theorem. The paper introduces no fitted constants and no new objects. The load is carried by prior parametrix and resolvent estimates (Hassell-Sikora and Hassell-Nix-Sikora), which are cited as black boxes, plus standard spectral and Hardy tools. The only internally questionable input is the approximate-solution error term in the critical case, which the proof silently discards.

assumptions (7)
  • standard math Δ is a positive, essentially self-adjoint operator, and the spectral formulas (1.3) and (1.4) hold.
    Used in Section 3 to write Δ^{1/2} as an integral of resolvents and to justify the duality pairing.
  • domain assumption Hassell-Sikora parametrix: Lemma 2.1 resolvent estimates and the decomposition (Δ+k^2)^{-1} = G1+G2+G3+G4 with estimates (2.4)-(2.7).
    Core of the proof of Theorem 1.2; cited from [23] and not reproven in the paper.
  • domain assumption Forward Riesz transform boundedness [23, Theorem 1.2]: (R_p) holds iff 1<p<n_*.
    Used to cover the range p > n'_* in Theorem 1.2 via duality [12, Proposition 2.1].
  • standard math Duality (R_p) implies (RR_{p'}) [12, Proposition 2.1].
    Converts known forward boundedness into a range of the reverse inequality.
  • domain assumption For the critical case, the parametrix estimates and the approximate-solution lemma [21, Lemma 2.14] are assumed, including the error term O((ilgk)^q|z|^{-∞}) being negligible after integration by parts.
    Theorem 1.3 proof relies on [21]'s construction; the error term's contribution to J_± is not computed.
  • domain assumption Hardy inequality on ends [31, Theorem 2.3] and p-hyperbolicity [16, 33] for the compact-set estimate.
    Used in Lemma 6.2 for the critical case with n_* = 2.
  • standard math Bessel function asymptotic estimates for the one-dimensional model.
    Used in Sections 4-5 for the broken line model; standard ODE asymptotics.

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Pith. "Pith review of Reverse Riesz Inequality on Manifolds with Ends." pith.science (2026). https://pith.science/paper/4DCBWYB5

@misc{pith2026241117107,
  author       = {Pith},
  title        = {Pith review of: Reverse Riesz Inequality on Manifolds with Ends},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DCBWYB5}},
  note         = {Machine review of arXiv:2411.17107}
}
abstract

In our investigation, we focus on the reverse Riesz transform within the framework of manifolds with ends. Such manifolds can be described as the connected sum of finite number of Cartesian products $\mathbb{R}^{n_i} \times \mathcal{M}_i$, where $\mathcal{M}_i$ are compact manifolds. We rigorously establish the boundedness of this transform across all $L^p$ spaces for $1<p<\infty$. Notably, existing knowledge indicates that the Riesz transform in such a context demonstrates boundedness solely within a specific range of $L^p$ spaces, typically observed for $1<p<n_*$, where $n_*$ signifies the smallest dimension of the manifold's ends on a large scale. This observation serves as a significant counterexample to the presumed equivalence between the Riesz and reverse Riesz transforms. Our study illuminates the nuanced behaviour of these transforms within the setting of manifolds with ends, providing valuable insights into their distinct properties. Although the lack of equivalence has been previously noted in relevant literature, our investigation contributes to a deeper understanding of the intricate interplay between the Riesz and reverse Riesz transforms.

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Works this paper leans on

35 extracted references · 34 canonical work pages

  1. [24]

    D. He. Endpoint estimates for riesz transform on manifolds with ends. Annali di Matematica Pura ed Applicata (1923-), pages 1–15, 2024. 3, 4

    work page 1923

  2. [1]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and math- ematical tables , volume No. 55 of National Bureau of Standards Applied Mathematics Series . U. S. Gov- ernment Printing Office, Washington, DC, 1964. For sale by the Supe rintendent of Documents. 15

    work page 1964

  3. [2]

    P. Auscher. On Lp estimates for square roots of second order elliptic operators on Rn. Publ. Mat. , 48(1):159–186, 2004. 3

    work page 2004

  4. [3]

    Auscher and T

    P. Auscher and T. Coulhon. Riesz transform on manifolds and Poin car´ e inequalities.Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 4(3):531–555, 2005. 2, 3, 4, 7, 12, 13

    work page 2005

  5. [4]

    Auscher, T

    P. Auscher, T. Coulhon, X. T. Duong, and S. Hofmann. Riesz tra nsform on manifolds and heat kernel regularity. Ann. Sci. ´Ecole Norm. Sup. (4) , 37(6):911–957, 2004. 2, 3, 4

    work page 2004

  6. [5]

    Bailey and A

    J. Bailey and A. Sikora. Vertical and horizontal square function s on a class of non-doubling manifolds. J. Differential Equations , 358:41–102, 2023. 3, 12

    work page 2023

  7. [6]

    D. Bakry. Transformation de riesz pour les semi-groupes sym´ e triques. premi` ere partie: ´ etude de la dimen- sion 1. S´ eminaire de probabilit´ es de Strasbourg, 19:130–144, 1985. 2

    work page 1985

  8. [7]

    D. Bakry. ´Etude des transformations de Riesz dans les vari´ et´ es riemanniennes ` a courbure de Ricci minor´ ee. In S´ eminaire de Probabilit´ es, XXI, volume 1247 of Lecture Notes in Math., pages 137–172. Springer, Berlin,

    work page

Show all 35 references
  1. [8]

    T. A. Bui, X. T. Duong, J. Li, and B. D. Wick. Functional calculus of operators with heat kernel bounds on non-doubling manifolds with ends. Indiana Univ. Math. J. , 69(3):713–747, 2020. 3

    2020

  2. [9]

    A. P. Calderon and A. Zygmund. On the existence of certain singu lar integrals. Acta Math. , 88:85–139,

  3. [10]

    Carron, T

    G. Carron, T. Coulhon, and A. Hassell. Riesz transform and Lp-cohomology for manifolds with Euclidean ends. Duke Math. J. , 133(1):59–93, 2006. 2, 3, 4, 5, 6, 7, 17

    2006

  4. [11]

    Coulhon and X

    T. Coulhon and X. T. Duong. Riesz transforms for 1 ≤ p ≤ 2. Trans. Amer. Math. Soc., 351(3):1151–1169,

  5. [12]

    Coulhon and X

    T. Coulhon and X. T. Duong. Riesz transform and related inequa lities on noncompact Riemannian man- ifolds. Comm. Pure Appl. Math. , 56(12):1728–1751, 2003. 2, 3, 4, 7

    2003

  6. [13]

    Cowling and A

    M. Cowling and A. Sikora. A spectral multiplier theorem for a subla placian on SU(2). Math. Z. , 238(1):1– 36, 2001. 8

    2001

  7. [14]

    E. B. Davies. Heat kernels and spectral theory , volume 92 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1989. 2

    1989

  8. [15]

    E. B. Davies. Non-gaussian aspects of heat kernel behaviour . Journal of the London Mathematical Society , 55(1):105–125, 1997. 3

    1997

  9. [16]

    B. Devyver. A perturbation result for the Riesz transform. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 14(3):937–964, 2015. 24

    2015

  10. [17]

    X. T. Duong, J. Li, and A. Sikora. Boundedness of maximal func tions on non-doubling manifolds with ends. In AMSI International Conference on Harmonic Analysis and App lications, volume 45 of Proc. Centre Math. Appl. Austral. Nat. Univ. , pages 37–47. Austral. Nat. Univ., Canb...

    2013

  11. [18]

    Grafakos et al

    L. Grafakos et al. Classical fourier analysis , volume 2. Springer, 2008. 20

    2008

  12. [19]

    Grigor’yan and L

    A. Grigor’yan and L. Saloff-Coste. Heat kernel on manifolds with ends. Ann. Inst. Fourier (Grenoble) , 59(5):1917–1997, 2009. 3, 13

    1917

  13. [20]

    G. H. Hardy, J. E. Littlewood, and G. P´ olya. Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1934. 2

    1934

  14. [21]

    Hassell, D

    A. Hassell, D. Nix, and A. Sikora. Riesz transforms on a class of n on-doubling manifolds ii, 2019. 3, 4, 12, 13, 22, 23, 24, 25

    2019

  15. [22]

    Hassell and A

    A. Hassell and A. Sikora. Riesz transforms in one dimension. Indiana Univ. Math. J. , 58(2):823–852, 2009. 2, 14, 15, 16, 17

    2009

  16. [23]

    Hassell and A

    A. Hassell and A. Sikora. Riesz transforms on a class of non-do ubling manifolds. Comm. Partial Differ- ential Equations , 44(11):1072–1099, 2019. 2, 3, 4, 5, 6, 7, 8, 12, 13, 24

    2019

  17. [25]

    Jiang and F

    R. Jiang and F. Lin. Riesz transform on exterior Lipschitz domain s and applications. Adv. Math., 453:Pa- per No. 109852, 48, 2024. 2, 14

    2024

  18. [26]

    Killip, M

    R. Killip, M. Visan, and X. Zhang. Riesz transforms outside a conv ex obstacle. Int. Math. Res. Not. IMRN, pages 5875–5921, 2016. 2, 3, 14

    2016

  19. [27]

    H.-Q. Li. La transformation de riesz sur les vari´ et´ es coniques. Comptes Rendus de l’Acad´ emie des Sciences- Series I-Mathematics , 326(10):1167–1170, 1998. 2, 3

    1998

  20. [28]

    D. Nix. The resolvent and riesz transform on connected sums o f manifolds with different asymptotic dimensions. Bulletin of the Australian Mathematical Society , 104(2):344–345, 2021. 3, 14, 15, 17

    2021

  21. [29]

    M. Riesz. Sur les fonctions conjugu´ ees. Mathematische Zeitschrift , 27(1):218–244, 1928. 1

    1928

  22. [30]

    E. Russ. Riesz transforms on graphs for 1 ≤ p ≤ 2. Math. Scand. , 87(1):133–160, 2000. 13

    2000

  23. [31]

    Russ and B

    E. Russ and B. Devyver. Reverse inequality for the riesz trans forms on riemannian manifolds. arXiv preprint arXiv:2209.05083, 2022. 2, 24

    2022 arXiv

  24. [32]

    R. S. Strichartz. Analysis of the Laplacian on the complete Riema nnian manifold. J. Functional Analysis , 52(1):48–79, 1983. 2

    1983

  25. [33]

    Troyanov

    M. Troyanov. Parabolicity of manifolds. Siberian Adv. Math. , 9(4):125–150, 1999. 24

    1999

  26. [34]

    N. T. Varopoulos. Hardy-Littlewood theory for semigroups. J. Funct. Anal. , 63(2):240–260, 1985. 13 Department of Mathematics and Statistics, Macquarie Unive rsity Email address : dangyang.he@mq.edu.au

    1985

  27. [1999]

    2, 3, 4 28 DANGYANG HE

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