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A reverse Riesz estimate combined with a spectral gap implies a Poincar\'e inequality

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Pith's one-line read A reverse Riesz estimate plus an abstract spectral gap forces a Poincaré inequality for Abel-ergodic sectorial operators on any Banach space.

desk verdict Short, correct abstract principle (reverse Riesz + spectral gap ⇒ Poincaré) that cleanly unifies a lot of classical and noncommutative inequalities. read the letter →

arxiv 2607.08322 v1 pith:SZEMOTIV submitted 2026-07-09 math.FA math.DGmath.OA

Cédric Arhancet This is my paper · ORCID
classification math.FAmath.DGmath.OA MSC 43A1547D0347B9058B34
keywords PoincaréinequalitysectorialoperatorsAbel-ergodicspectralgapreverseRieszestimatenoncommutativeLp-spacesMarkovsemigroupsBanachspaces
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

The paper proves that two mild conditions already give Poincaré control: a reverse Riesz bound that compares a fractional power of an Abel-ergodic sectorial operator A to a gradient-like operator ∂, together with the condition that zero lies in the resolvent of the part of A on the range of A. The second condition is the natural Banach-space stand-in for a spectral gap and is sharp even on Hilbert space. The resulting inequality says that the distance from any vector to the kernel of A is controlled by the size of its gradient. The same mechanism produces a dual divergence inequality. Because the argument never uses commutativity or Hilbert-space structure, it recovers classical Poincaré inequalities on manifolds and Lie groups while simultaneously covering noncommutative Lp-spaces, quantum tori, q-Ornstein–Uhlenbeck semigroups and Schur-multiplier semigroups from a single short proof.

What carries the argument

The identity A0^α (x−P(x)) = A^α x on the range of A, combined with the boundedness of the negative fractional power A0^{−α} that follows from 0∈ρ(A0). This turns the reverse Riesz hypothesis into Poincaré control in two lines.

What would settle it

Find a concrete Abel-ergodic sectorial operator A with spectral gap 0∈ρ(A0) for which a reverse Riesz bound holds, yet the Poincaré inequality fails on some vector in the domain of the gradient; or, conversely, exhibit a geometry where the reverse Riesz bound is known and verify that the predicted Poincaré constant is finite and of the expected order.

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Extended reading notes

Core claim

If A is an Abel-ergodic sectorial operator on a Banach space X with ergodic projection P onto Ker A, and if 0 belongs to the resolvent of the part A0 of A acting on the closed range of A, then any reverse Riesz estimate ||A^α x||_X ≲ ||∂x||_Y for some α in (0,1) automatically upgrades to the Poincaré inequality ||x−P(x)||_X ≲ ||∂x||_Y. The same spectral mechanism yields a companion divergence inequality that bounds ||y|| by the norm of an abstract adjoint applied to y.

Load-bearing premise

The reverse Riesz estimate that bounds the fractional power of A by the size of the gradient must hold for the given geometry; if it fails, the abstract implication produces nothing.

Editorial extensions

If this is right

  • Classical Lp-Poincaré inequalities on compact Riemannian manifolds, Cartan–Hadamard manifolds with negative curvature, and compact Lie groups follow from a single abstract theorem once the corresponding reverse Riesz estimates are known.
  • The same theorem recovers and extends the noncommutative Poincaré inequality of Jiao–Luo–Zanin–Zhou on von Neumann algebras without requiring hypercontractivity or Markovianity.
  • New Poincaré inequalities become available for quantum tori, q-Ornstein–Uhlenbeck semigroups, group von Neumann algebras and Schur-multiplier semigroups as soon as a reverse Riesz estimate is verified.
  • A dual divergence inequality holds under the same spectral-gap hypothesis, controlling the size of a vector by the norm of its abstract divergence.
  • The method applies verbatim to metric-measure spaces satisfying an RCD(K,N) condition and to spin manifolds via the Dirac operator.

Reading between the lines

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

  • The brevity of the argument suggests that many existing proofs of Poincaré inequalities can be shortened to a verification of reverse Riesz plus spectral gap, potentially clarifying which geometric features are truly essential.
  • Because the exponent α is free in (0,1), the principle may produce new inequalities on fractals or other irregular spaces where the classical square-root Riesz transform is unavailable.
  • Tracking the operator norm of A0^{−α} with respect to p should give explicit p-dependence of Poincaré constants in the noncommutative examples, a quantitative refinement left open by the paper.
  • The same two-line argument may adapt to other functional inequalities (logarithmic Sobolev, concentration) once a suitable reverse estimate replaces the reverse Riesz bound.
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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

0 major / 4 minor

Summary. The paper proves that for an Abel-ergodic sectorial operator A on a Banach space X with ergodic projection P, the condition 0 ∈ ρ(A0) (A0 the part of A on Ran A) together with a reverse Riesz estimate ‖A^α x‖_X ≲ ‖∂x‖_Y for some α ∈ (0,1) and ∂ : dom ∂ o Y with dom ∂ ⊂ dom A^α implies the abstract Poincaré inequality ‖x − P(x)‖_X ≲ ‖A0^{-α}‖ · ‖∂x‖_Y (Theorem 4.1). A companion divergence inequality is obtained by exchanging roles of ∂ and an adjoint (Theorem 5.1). The argument is short once fractional-power identities and boundedness of negative powers of A0 are granted. The principle is applied across Riemannian manifolds, spin manifolds, RCD spaces, compact Lie groups, quantum tori, q-Ornstein–Uhlenbeck semigroups, group von Neumann algebras and Schur multipliers, recovering and extending the noncommutative L^p result of Jiao–Luo–Zanin–Zhou.

Significance. If correct, the result supplies a single, geometry-independent mechanism that converts a spectral-gap condition plus a reverse Riesz estimate into a Poincaré inequality on arbitrary Banach spaces, treating commutative and noncommutative settings uniformly. The derivation itself is elementary and transparent; its value lies in the identification of the two hypotheses and in the breadth of the illustrations, which recover classical inequalities and produce new ones (e.g., for spin manifolds and certain noncommutative semigroups) from the same principle. The Hilbertian necessity of the gap (Propositions 3.5–3.6) confirms sharpness at the classical point. The paper therefore organizes a large literature around one short abstract implication.

minor comments (4)
  1. In the abstract and introduction the reverse Riesz estimate is written with A^{1/2}; Theorem 4.1 correctly generalizes to arbitrary α ∈ (0,1). A single sentence noting that the square-root case is the typical one would avoid any impression of inconsistency.
  2. Section 7.2 (spin manifolds) and Section 7.1 (Cartan–Hadamard) invoke Riesz estimates from the literature; a brief pointer to the precise hypotheses under which those estimates hold would help the reader verify the domain inclusions dom ∂ ⊂ dom A^α without consulting the cited works.
  3. The constant tracking in Proposition 7.13 and Proposition 7.15 is useful; it would be clearer if the dependence on the spectral gap ω were written explicitly rather than absorbed into the ≲ symbol.
  4. A few typographical slips remain (e.g., “Abel-ergodic” hyphenation, occasional missing spaces around “≲”). They do not affect readability but should be cleaned in production.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main implication is a short, one-directional operator-theoretic argument whose hypotheses are independent of the conclusion.

full rationale

Theorem 4.1 (and its special case Theorem 1.1) states that an Abel-ergodic sectorial operator A with 0∈ρ(A0) together with a reverse Riesz estimate ‖A^α x‖_X ≲ ‖∂x‖_Y yields the Poincaré inequality ‖x-P(x)‖_X ≲ ‖A0^{-α}‖ · ‖∂x‖_Y. The proof (Section 4) is the elementary identity A0^α(x-P(x))=A^α x followed by boundedness of the negative power; both hypotheses are taken as given and are independently checkable. In the Hilbertian case Propositions 3.5–3.6 prove the gap condition is necessary, confirming independence. The companion divergence inequality (Theorem 5.1) and the Riesz-to-reverse implication (Proposition 6.1) are likewise direct. Section 7 merely verifies the two hypotheses case-by-case via classical spectral facts (compact resolvent, hypercontractivity, curvature bounds) and known Riesz equivalences (some of which appear in the author’s earlier papers). Those self-citations function as black-box estimates; they do not redefine the spectral gap or the reverse Riesz estimate in terms of the Poincaré constant, nor do they force the conclusion by construction. There are no fitted parameters, no uniqueness theorems imported to exclude alternatives, and no renaming of a known empirical pattern. The derivation is therefore self-contained against its stated inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is pure functional analysis. It rests on standard sectorial-operator calculus, the definition of Abel-ergodicity, and the classical notion of spectral gap for the reduced operator A0. No free parameters are fitted; the reverse Riesz estimate and the gap condition are hypotheses verified externally. No new physical or mathematical entities are postulated.

assumptions (4)
  • domain assumption A is an Abel-ergodic sectorial operator on a Banach space X, so X=Ker A⊕Ran A topologically and the ergodic projection P exists.
    Definition 2.2 and the standing hypothesis of Theorem 4.1; automatic on reflexive spaces.
  • domain assumption 0 lies in the resolvent set of the part A0 of A on Ran A (the abstract spectral-gap condition).
    Hypothesis of Theorem 4.1; shown equivalent to uniform exponential stability of the reduced semigroup when A generates a bounded holomorphic semigroup.
  • standard math Fractional powers of sectorial operators satisfy Ker A^α=Ker A, Ran A^α=Ran A and the usual functional-calculus identities.
    Invoked throughout §2 and the proof of Theorem 4.1; standard references Haase, Hytönen–van Neerven–Veraar–Weis.
  • domain assumption A reverse Riesz estimate ||A^α x||_X ≲ ||∂x||_Y holds for some α∈(0,1) on dom ∂⊂dom A^α.
    Hypothesis (4.2) of the main theorem; verified case-by-case via known Riesz equivalences or duality (Proposition 6.1).

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Pith. "Pith review of A reverse Riesz estimate combined with a spectral gap implies a Poincar\'e inequality." pith.science (2026). https://pith.science/paper/SZEMOTIV

@misc{pith2026260708322,
  author       = {Pith},
  title        = {Pith review of: A reverse Riesz estimate combined with a spectral gap implies a Poincar\'e inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZEMOTIV}},
  note         = {Machine review of arXiv:2607.08322}
}
abstract

Working at the level of an Abel-ergodic sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\partial$ defined on a subspace $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\|A^\alpha x\|_X \lesssim \|\partial x\|_Y$ for some $0 < \alpha < 1$, combined with the condition $0 \in \rho(A_0)$, where $A_0$ is the part of $A$ on the closure of the range of $A$, implies the Poincar\'e inequality $\|x - P(x)\|_X \lesssim \|\partial x\|_Y$, where $P$ is the Abel-ergodic projection onto the kernel of $A$. The condition $0 \in \rho(A_0)$ is the natural abstract substitute for a spectral gap, and is sharp already in the Hilbertian case. We also obtain a companion divergence inequality. The arguments are remarkably short, yet the principle is genuinely unifying: it covers commutative and noncommutative situations on the same footing and can be used with arbitrary Banach spaces. As a consequence, we recover, and considerably extend, a recent theorem of Jiao, Luo, Zanin and Zhou [CMP2024] on (possibly noncommutative) $\mathrm{L}^p$-spaces. We then illustrate the flexibility of the method across a wide spectrum of geometries, ranging from Riemannian manifolds, Lie groups, metric measure spaces, spin manifolds to genuinely noncommutative settings such as quantum groups, semigroups of Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups and quantum tori, where we sometimes establish new inequalities and otherwise recover classical ones from a single principle.

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Works this paper leans on

129 extracted references · 129 canonical work pages

  1. [1]

    Adamczak and P

    R. Adamczak and P. Wolff. Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. Probab. Theory Related Fields 162 (2015), 531--586

    work page 2015

  2. [2]

    Alexopoulos

    G. Alexopoulos. An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Canad. J. Math. 44 (1992), no. 4, 691--727

    work page 1992

  3. [3]

    G. Allaire. \`A la recherche de l'in\'egalit\'e perdue. Matapli 98 (2012), 52--64

    work page 2012

  4. [4]

    Ammann and N

    B. Ammann and N. Gro e. L^p -spectrum of the Dirac operator on products with hyperbolic spaces. Calc. Var. Partial Differential Equations 55 (2016), no. 5, art. 127

    work page 2016

  5. [5]

    Ambrosio, N

    L. Ambrosio, N. Gigli and G. Savar\'e. Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163 (2014), no. 7, 1405--1490

    work page 2014

  6. [6]

    Ambrosio, S

    L. Ambrosio, S. Honda and D. Tewodrose. Short-time behavior of the heat kernel and Weyl's law on RCD^*(K,N) spaces. Ann. Global Anal. Geom. 53 (2018), no. 1, 97--119

    work page 2018

  7. [7]

    Ambrosio, A

    L. Ambrosio, A. Mondino and G. Savar\'e. Nonlinear diffusion equations and curvature conditions in metric measure spaces. Mem. Amer. Math. Soc. 262 (2019), no. 1270

    work page 2019

  8. [8]

    Ambrosio, S

    L. Ambrosio, S. Honda, J. W. Portegies and D. Tewodrose. Embedding of ^* (K,N) spaces in L^2 via eigenfunctions. J. Funct. Anal. 280 (2021), no. 10, Paper No. 108968, 72 pp

    work page 2021

Show all 129 references
  1. [9]

    Applebaum

    D. Applebaum. Some L^2 properties of semigroups of measures on Lie groups. Semigroup Forum 79 (2009), no. 2, 217--228

    2009

  2. [10]

    Applebaum

    D. Applebaum. Probability on Compact Lie Groups. Probability Theory and Stochastic Modelling, vol. 70. Springer, Cham, 2014

    2014

  3. [11]

    N. Arcozzi. Riesz transforms on compact Lie groups, spheres and Gauss space. Ark. Mat. 36 (1998), no. 2, 201--231

    1998

  4. [12]

    W. Arendt. Semigroups and evolution equations: functional calculus, regularity and kernel estimates. Evolutionary equations. Vol. I, 1--85, Handb. Differ. Equ., North-Holland, Amsterdam, 2004

    2004

  5. [13]

    W. Arendt. Heat kernels. Lecture notes, internet seminar, 2005

    2005

  6. [14]

    Arendt, C

    W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander. Vector-valued Laplace transforms and Cauchy problems. Second edition. Monographs in Mathematics, 96. Birkh\"auser/Springer Basel AG, Basel, 2011

    2011

  7. [15]

    Arhancet

    C. Arhancet. On Matsaev's conjecture for contractions on noncommutative L^p -spaces. J. Operator Theory 69 (2013), no. 2, 387--421

    2013

  8. [16]

    Arhancet

    C. Arhancet. Sobolev algebras on Lie groups and noncommutative geometry. J. Noncommut. Geom. 18 (2024), no. 2, 451--500

    2024

  9. [17]

    Arhancet

    C. Arhancet. Spectral triples, Coulhon--Varopoulos dimension and heat kernel estimates. Adv. Math. 451 (2024), Paper No. 109794

    2024

  10. [18]

    Arhancet

    C. Arhancet. Curvature, Dolbeault–-Dirac operators, and an L^p -index theorem on compact K\"ahler manifolds. Preprint, arXiv:2401.04203

    arXiv

  11. [19]

    Arhancet

    C. Arhancet. The L^p -index of the Hodge--Dirac operator on compact Riemannian manifolds. Preprint, arXiv:2512.22517

    arXiv

  12. [20]

    Arhancet and C

    C. Arhancet and C. Kriegler. Riesz transforms, Hodge--Dirac operators and functional calculus for multipliers. Lecture Notes in Mathematics, 2304. Springer, Cham, 2022

    2022

  13. [21]

    D. Bakry. \'Etude des transformations de Riesz dans les vari\'et\'es riemanniennes \`a courbure de Ricci minor\'ee. (French) [A study of Riesz transforms in Riemannian manifolds with minorized Ricci curvature]. S\'eminaire de Probabilit\'es, XXI, 137--172, Lecture Notes in Mat...

    1987

  14. [22]

    Bakry, I

    D. Bakry, I. Gentil and M. Ledoux. Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften 348. Springer, 2014

    2014

  15. [23]

    Ba\ nuelos, F

    R. Ba\ nuelos, F. Baudoin and L. Chen. Gundy--Varopoulos martingale transforms and their projection operators on manifolds and vector bundles. Math. Ann. 378 (2020), 359--388

    2020

  16. [24]

    C. B\"ar. Spectral bounds for Dirac operators on open manifolds. Ann. Global Anal. Geom. 36 (2009), no. 1, 67--79

    2009

  17. [25]

    Bergh and J

    J. Bergh and J. L\"ofstr\"om. Interpolation spaces. An Introduction. Springer-Verlag, Berlin, Heidelberg, New York, 1976

    1976

  18. [26]

    Bebendorf

    M. Bebendorf. A note on the Poincar\'e inequality for convex domains. Z. Anal. Anwend. 22 (2003), no. 4, 751--756

    2003

  19. [27]

    P. Biane. Free hypercontractivity. Comm. Math. Phys. 184 (1997), no. 2, 457--474

    1997

  20. [28]

    M. Bo\. z ejko, B. K\"ummerer and R. Speicher. q -Gaussian processes: non-commutative and classical aspects. Comm. Math. Phys. 185 (1997), no. 1, 129--154

    1997

  21. [29]

    M. Bo\. z ejko and R. Speicher. Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. Math. Ann. 300 (1994), no. 1, 97--120

    1994

  22. [30]

    M. Bo\. z ejko and R. Speicher. An example of a generalized Brownian motion. Comm. Math. Phys. 137 (1991), no. 3, 519--531

    1991

  23. [31]

    Bourbaki

    N. Bourbaki. Th\'eories spectrales: Chapitres 3 \`a 5. Springer, Cham, 2023

    2023

  24. [32]

    Brannan, L

    M. Brannan, L. Gao and M. Junge. Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II. J. Topol. Anal. 15 (2023), no. 3, 741--794

    2023

  25. [33]

    Cavalletti and E

    F. Cavalletti and E. Milman. The globalization theorem for the curvature-dimension condition. Invent. Math. 226 (2021), no. 1, 1--137

    2021

  26. [34]

    Carbonaro, L

    A. Carbonaro, L. Tamanini, and D. Trevisan. Boundedness of Riesz transforms on RCD(K, ) spaces. Preprint, arXiv:2308.16294

    arXiv

  27. [35]

    Charalambous and N

    N. Charalambous and N. Gro e. On the L^p spectrum of the Dirac operator. J. Geom. Anal. 33 (2023), no. 2, Art. 44

    2023

  28. [36]

    I. Chavel. Riemannian Geometry: A Modern Introduction, second edition. Cambridge Studies in Advanced Mathematics, vol. 98. Cambridge University Press, Cambridge, 2006

    2006

  29. [37]

    Z. Chen, Q. Xu and Z. Yin. Harmonic analysis on quantum tori. Comm. Math. Phys. 322 (2013), no. 3, 755--805

    2013

  30. [38]

    Coulhon, E

    T. Coulhon, E. Russ and V. Tardivel-Nachef. Sobolev algebras on Lie groups and Riemannian manifolds. Amer. J. Math. 123 (2001), no. 2, 283--342

    2001

  31. [39]

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

    1989

  32. [40]

    E. B. Davies, L. Gross, and B. Simon. Hypercontractivity: a bibliographic review. In: Ideas and Methods in Quantum and Statistical Physics, Vol. 2 (S. Albeverio, J. E. Fenstad, H. Holden, and T. Lindstr m, eds.), pp. 370--389. Cambridge University Press, Cambridge, 1992

    1992

  33. [41]

    C. R. de Oliveira. Intermediate Spectral Theory and Quantum Dynamics. Progress in Mathematical Physics, vol. 54. Birkh\"auser, Basel, 2009

    2009

  34. [42]

    Dieudonn\'e

    J. Dieudonn\'e. Treatise on analysis. Vol. III. Translated from the French by I. G. MacDonald. Pure Appl. Math., Vol. 10-III. Academic Press, New York-London, 1972

    1972

  35. [43]

    Dieudonn\'e

    J. Dieudonn\'e. Treatise on analysis. Vol. VII. Translated from the French by Laura Fainsilber. Pure Appl. Math., 10-VII. Academic Press, Inc., Boston, MA, 1988

    1988

  36. [44]

    Dungey, A

    N. Dungey, A. F. M. ter Elst and D. Robinson. Analysis on Lie groups with polynomial growth. Progress in Mathematics, 214. Birkh\"auser Boston, Inc., Boston, MA, 2003

    2003

  37. [45]

    L. B. Efraim and F. Lust-Piquard. Poincar\'e type inequalities on the discrete cube and in the CAR algebra. Probab. Theory Related Fields 141 (2008), no. 3-4, 569--602

    2008

  38. [46]

    E. Y. Emel'yanov. Non-spectral asymptotic analysis of one-parameter operator semigroups. Operator Theory: Advances and Applications, 173. Birkh\"auser Verlag, Basel, 2007

    2007

  39. [47]

    Engel and R

    K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000

    2000

  40. [48]

    Erbar, K

    M. Erbar, K. Kuwada and K.-T. Sturm. On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces. Invent. Math. 201 (2015), no. 3, 993--1071

    2015

  41. [49]

    J. Feneuil. Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion. Preprint, arXiv:2606.05475

    arXiv

  42. [50]

    R. L. Frank, A. Laptev, and T. Weidl. Schr\"odinger Operators: Eigenvalues and Lieb--Thirring Inequalities. Cambridge Studies in Advanced Mathematics, vol. 200. Cambridge University Press, Cambridge, 2022

    2022

  43. [51]

    Friedrich

    T. Friedrich. Dirac Operators in Riemannian Geometry. Graduate Studies in Mathematics, vol. 25. American Mathematical Society, Providence, RI, 2000

    2000

  44. [52]

    N. Ginoux. The Dirac Spectrum. Lecture Notes in Mathematics, vol. 1976. Springer, Berlin, 2009

    1976

  45. [53]

    J. A. Goldstein. Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs. Oxford University Press, New York, 1985

    1985

  46. [54]

    auser Advanced Texts: Basler Lehrb\

    J. M. Gracia-Bondia, J. C. Varilly and H. Figueroa. Elements of noncommutative geometry. Birkh\"auser Advanced Texts: Basler Lehrb\"ucher. Birkh\"auser Boston, Inc., Boston, MA, 2001

    2001

  47. [55]

    Grigor'yan

    A. Grigor'yan. Heat kernel and analysis on manifolds. AMS/IP Studies in Advanced Mathematics, 47. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009

    2009

  48. [56]

    uneysu. Covariant Schr\

    B. G\"uneysu. Covariant Schr\"odinger semigroups on Riemannian manifolds. Oper. Theory Adv. Appl., 264. Birkh\"auser/Springer, Cham, 2017

    2017

  49. [57]

    M. Haase. The functional calculus for sectorial operators. Operator Theory: Advances and Applications, 169. Birkh\"auser Verlag (2006)

    2006

  50. [58]

    M. Haase. Lectures on Functional Calculus. 21st International Internet Seminar, March 19, 2018. https://www.math.uni-kiel.de/isem21/en/course/phase1 https://www.math.uni-kiel.de/isem21/en/course/phase1

    2018

  51. [59]

    M. J. D. Hamilton. Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics. Universitext. Springer, Cham, 2017

    2017

  52. [60]

    E. Hebey. Sobolev Spaces on Riemannian Manifolds. Lecture Notes in Mathematics, vol. 1635. Springer-Verlag, Berlin, 1996

    1996

  53. [61]

    E. Hebey. Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes in Mathematics, vol. 5. Courant Institute of Mathematical Sciences, New York University, New York, 2000

    2000

  54. [62]

    Helgason

    S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics, vol. 34. American Mathematical Society, Providence, RI, 2001

    2001

  55. [63]

    Hellmich

    M. Hellmich. Decoherence in Infinite Quantum Systems. PhD thesis, University of Bielefeld, Bielefeld, Germany, 2009

    2009

  56. [64]

    Huang and J

    D. Huang and J. A. Tropp. From Poincar\'e inequalities to nonlinear matrix concentration. Bernoulli 27 (2021), no. 3, 1724--1744

    2021

  57. [65]

    Hyt\"onen, J

    T. Hyt\"onen, J. van Neerven, M. Veraar and L. Weis. Analysis in Banach spaces, Volume I: Martingales and Littlewood-Paley theory. Springer, 2016

    2016

  58. [66]

    Hyt\"onen, J

    T. Hyt\"onen, J. van Neerven, M. Veraar and L. Weis. Analysis in Banach spaces, Volume II: Probabilistic Methods and Operator Theory. Springer, 2018

    2018

  59. [67]

    Hyt\"onen, J

    T. Hyt\"onen, J. van Neerven, M. Veraar and L. Weis. Analysis in Banach spaces, Volume III: Harmonic Analysis and Spectral Theory. Springer, 2023

    2023

  60. [68]

    Jiang, H

    R. Jiang, H. Li and H. Zhang. Heat kernel bounds on metric measure spaces and some applications. Potential Anal. 44 (2016), no. 3, 601--627

    2016

  61. [69]

    Y. Jiao, S. Luo, D. Zanin, and D. Zhou. Noncommutative logarithmic Sobolev inequalities. Comm. Math. Phys. 405 (2024), art. 265

    2024

  62. [70]

    L. Ji, P. Kunstmann, and A. Weber. Riesz transform on locally symmetric spaces and Riemannian manifolds with a spectral gap. Bull. Sci. Math. 134 (2010), 37--43

    2010

  63. [71]

    J. Jost. Riemannian Geometry and Geometric Analysis, seventh edition. Universitext. Springer, Cham, 2017

    2017

  64. [72]

    Junge, C

    M. Junge, C. Le Merdy and Q. Xu. H^ functional calculus and square functions on noncommutative L^p -spaces. Ast\'erisque No. 305 (2006)

    2006

  65. [73]

    Junge, T

    M. Junge, T. Mei and J. Parcet. Noncommutative Riesz transforms--dimension free bounds and Fourier multipliers. J. Eur. Math. Soc. 20 (2018), no. 3, 529--595

    2018

  66. [74]

    Junge and Q

    M. Junge and Q. Zeng. Subgaussian 1-cocycles on discrete groups. J. Lond. Math. Soc. (2) 92 (2015), no. 2, 242--264

    2015

  67. [75]

    Junge and Q

    M. Junge and Q. Zeng. Noncommutative martingale deviation and Poincar\'e type inequalities with applications. Probab. Theory Related Fields 161 (2015), no. 3-4, 449--507

    2015

  68. [76]

    Junge and J

    M. Junge and J. Wang. Generalized Poincar\'e inequality for quantum Markov semigroups. Preprint, arXiv:2601.06005

  69. [77]

    R. V. Kadison and J. R. Ringrose. Fundamentals of the theory of operator algebras. Vol. I. Elementary theory. Reprint of the 1983 original. Graduate Studies in Mathematics, 15. American Mathematical Society, Providence, RI, 1997

    1983

  70. [78]

    T. Kato. Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1976

    1976

  71. [79]

    K\"ummerer and R

    B. K\"ummerer and R. Nagel. Mean ergodic semigroups on W^* -algebras. Acta Sci. Math. (Szeged) 41 (1979), no. 1-2, 151--159

    1979

  72. [80]

    Kuwada and K

    K. Kuwada and K. Kuwae. Radial processes on ^*(K,N) spaces. J. Math. Pures Appl. (9) 126 (2019), 72--108

    2019

  73. [81]

    Kuwada and X.-D

    K. Kuwada and X.-D. Li. Monotonicity and rigidity of the W-entropy on RCD(0,N) spaces. Manuscripta Math. 164 (2021), no. 1-2, 119--149

    2021

  74. [82]

    Labl\'ee

    O. Labl\'ee. Spectral Theory in Riemannian Geometry. EMS Textbooks in Mathematics. EMS Press, Z\"urich, 2015

    2015

  75. [83]

    Le Donne

    E. Le Donne. Metric Lie Groups: Carnot-Carath\'eodory Spaces from the Homogeneous Viewpoint. Graduate Texts in Mathematics, vol. 306. Springer, Cham, 2025

    2025

  76. [84]

    J. M. Lee. Manifolds and differential geometry. Grad. Stud. Math., 107, American Mathematical Society, Providence, RI, 2009

    2009

  77. [85]

    J. M. Lee. Introduction to Riemannian Manifolds, second edition. Graduate Texts in Mathematics, vol. 176. Springer, Cham, 2018

    2018

  78. [86]

    G. Leoni. A First Course in Sobolev Spaces. Second edition. Graduate Studies in Mathematics, vol. 181. American Mathematical Society, Providence, RI, 2017

    2017

  79. [87]

    Leuzinger

    E. Leuzinger. Critical exponents of discrete groups and L^2 -spectrum. Proc. Amer. Math. Soc. 132 (2004), no. 3, 919--927

    2004

  80. [88]

    X.-D. Li. Martingale transforms and L^p -norm estimates of Riesz transforms on complete Riemannian manifolds. Probab. Theory Related Fields 141 (2008), no. 1-2, 247--281

    2008

  81. [89]

    X.-D. Li. On the strong L^p -Hodge decomposition over complete Riemannian manifolds. J. Funct. Anal. 257 (2009), no. 11, 3617--3646

    2009

  82. [90]

    Lott and C

    J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 (2009), no. 3, 903--991

    2009

  83. [91]

    Lohou\'e

    N. Lohou\'e. Comparaison des champs de vecteurs et des puissances du laplacien sur une vari\'et\'e riemannienne \`a courbure non positive. J. Funct. Anal. 61 (1985), 164--201

    1985

  84. [92]

    A. Lunardi. Interpolation theory. Third edition. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 16. Edizioni della Normale, Pisa, 2018

    2018

  85. [93]

    Lust-Piquard

    F. Lust-Piquard. Riesz transforms associated with the number operator on the Walsh system and the fermions. J. Funct. Anal. 155 (1998), no. 1, 263--285

    1998

  86. [94]

    Lust-Piquard

    F. Lust-Piquard. Riesz transforms on deformed Fock spaces. Comm. Math. Phys. 205 (1999), no. 3, 519--549

    1999

  87. [95]

    J. Mawhin. Henri Poincar\'e and partial differential equations. Nieuw Arch. Wiskd. (5) 13 (2012), no. 3, 159--169

    2012

  88. [96]

    Mandouvalos and M

    N. Mandouvalos and M. Marias. Spectrum of the Laplacian and Riesz transform on locally symmetric spaces. Bull. Sci. Math. 133 (2009), 134--144

    2009

  89. [97]

    H. P. McKean. An upper bound to the spectrum of on a manifold of negative curvature. J. Differential Geom. 4 (1970), no. 3, 359--366

    1970

  90. [98]

    Martinez Carracedo and M

    C. Martinez Carracedo and M. Sanz Alix. The theory of fractional powers of operators. North-Holland Mathematics Studies, 187. North-Holland Publishing Co., Amsterdam, 2001

    2001

  91. [99]

    R. E. Megginson. An introduction to Banach space theory. Graduate Texts in Mathematics, 183. Springer-Verlag, New York, 1998

    1998

  92. [100]

    E. Milman. On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (2009), no. 1, 1--43

    2009

  93. [101]

    M\"uller

    W. M\"uller. Manifolds with Cusps of Rank One, Spectral Theory and L^2 -Index Theorem. Lecture Notes in Mathematics, vol. 1244. Springer, Berlin, 1987

    1987

  94. [102]

    van Neerven

    J. van Neerven. The L^p -Poincar\'e inequality for analytic Ornstein-Uhlenbeck semigroups. Operator semigroups meet complex analysis, harmonic analysis and mathematical physics, 353--368. Oper. Theory Adv. Appl., 250. Birkh\"auser/Springer, Cham, 2015

    2015

  95. [103]

    Q. A. Ng\^o and V. H. Nguyen. Sharp constant for Poincar\'e-type inequalities in the hyperbolic space. Acta Math. Vietnam. 44 (2019), 781--795

    2019

  96. [104]

    Pauly and J

    D. Pauly and J. Valdman. Poincar\'e--Friedrichs type constants for operators involving grad, curl, and div: Theory and numerical experiments. Comput. Math. Appl. 79 (2020), no. 11, 3027--3067

    2020

  97. [105]

    L. E. Payne and H. F. Weinberger. An optimal Poincar\'e inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960), no. 1, 286--292

    1960

  98. [106]

    M. A. Pinsky. On the spectrum of Cartan-Hadamard manifolds. Pacific J. Math. 94 (1981), no. 1, 223--230

    1981

  99. [107]

    Poincar\'e

    H. Poincar\'e. Sur les Equations aux D\'eriv\'ees Partielles de la Physique Math\'ematique. Amer. J. Math. 12, no. 3. (1890), 211--294

  100. [108]

    D. W. Robinson. Elliptic operators and Lie groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991

    1991

  101. [109]

    Rudolph and M

    G. Rudolph and M. Schmidt. Differential geometry and mathematical physics. Part II. Theoret. Math. Phys., Springer, Dordrecht, 2017

    2017

  102. [110]

    Russ and Y

    E. Russ and Y. Sire. Nonlocal Poincar\'e inequalities on Lie groups with polynomial volume growth and Riemannian manifolds. Studia Math. 203 (2011), no. 2, 105--127

    2011

  103. [111]

    Schm \"u dgen

    K. Schm \"u dgen. Unbounded self-adjoint operators on Hilbert space. Graduate Texts in Mathematics, 265. Springer, Dordrecht, 2012

    2012

  104. [112]

    Schoen and S.-T

    R. Schoen and S.-T. Yau. Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, vol. 1. International Press, Cambridge, MA, 1994

    1994

  105. [113]

    B. Simon. Harmonic Analysis: A Comprehensive Course in Analysis, Part 3. American Mathematical Society, Providence, RI, 2015

    2015

  106. [114]

    E. M. Stein. Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No. 63 Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970

    1970

  107. [115]

    R. S. Strichartz. Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52 (1983), no. 1, 48--79

    1983

  108. [116]

    K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math. 196 (2006), no. 1, 65--131

    2006

  109. [117]

    K.-T. Sturm. On the geometry of metric measure spaces. II. Acta Math. 196 (2006), no. 1, 133--177

    2006

  110. [118]

    Talagrand

    M. Talagrand. Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis' graph connectivity theorem. Geom. Funct. Anal. 3 (1993), no. 3, 295--314

    1993

  111. [119]

    M. E. Taylor. L^p -estimates on functions of the Laplace operator. Duke Math. J. 58 (1989), no. 3, 773--793

    1989

  112. [120]

    Tewodrose

    D. Tewodrose. Some functional inequalities and spectral properties of metric measure spaces with curvature bounded below. PhD thesis, Universit\'e Paris sciences et lettres and Scuola Normale Superiore di Pisa, 2018

    2018

  113. [121]

    Varopoulos, L

    N. Varopoulos, L. Saloff-Coste and T. Coulhon. Analysis and geometry on groups. Cambridge Tracts in Mathematics, 100. Cambridge University Press, Cambridge, 1992

    1992

  114. [122]

    F.-Y. Wang. Functional Inequalities, Markov Semigroups and Spectral Theory. Mathematics Monograph Series, vol. 4. Elsevier, Amsterdam, 2005

    2005

  115. [123]

    A. Weber. Heat kernel estimates and L^p -spectral theory of locally symmetric spaces. PhD thesis, Universit\"at Karlsruhe, 2007

    2007

  116. [124]

    A. Weber. Heat kernel bounds, Poincar\'e series, and L^2 spectrum for locally symmetric spaces. Bull. Aust. Math. Soc. 78 (2008), no. 1, 73--86

    2008

  117. [125]

    L. Weis. The stability of positive semigroups on L_p spaces. Proc. Amer. Math. Soc. 123 (1995), no. 10, 3089--3094

    1995

  118. [126]

    J. A. Wolf. Essential self-adjointness for the Dirac operator and its square. Indiana Univ. Math. J. 22 (1972/73), 611--640

    1972

  119. [127]

    J. A. Wolf. Spaces of Constant Curvature, sixth ed. AMS Chelsea Publishing, American Mathematical Society, Providence, RI, 2011

    2011

  120. [128]

    Xiong, Q

    X. Xiong, Q. Xu, and Z. Yin. Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori. Mem. Amer. Math. Soc. 252 (2018), no. 1203

    2018

  121. [129]

    Q. Zeng. Poincar\'e type inequalities for group measure spaces and related transportation cost inequalities. J. Funct. Anal. 266 (2014), no. 5, 3236--3264

    2014

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