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arxiv: 2604.16791 · v1 · submitted 2026-04-18 · 🧮 math.FA · math.AP· math.PR

Log-Sobolev and Beckner inequalities and stability of Poincar\'e inequality with weighted Gaussian measures

Nguyen Lam , Guozhen Lu , Andrey Russanov This is my paper

Pith reviewed 2026-05-10 07:25 UTC · model grok-4.3

classification 🧮 math.FA math.APmath.PR
keywords Beckner inequalityPoincaré inequalityLog-Sobolev inequalityweighted Gaussian measuresMarkov semigroupGamma-calculusstability estimatesHeisenberg uncertainty principle
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The pith

Markov semigroups establish generalized Beckner inequalities for weighted Gaussian measures

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

The paper uses Markov semigroup methods together with the Gamma-calculus to prove a generalized Beckner inequality for weighted Gaussian measures. A reader would care because these inequalities bound how much a function can vary or concentrate under the measure, which controls key properties in analysis and probability. The Poincaré inequality follows at once as a consequence. Duality then produces stability estimates that quantify how close the inequality comes to equality. The same framework also yields a scale-dependent Poincaré inequality, stability results for the Heisenberg uncertainty principle, and a logarithmic Sobolev inequality that recovers the Euclidean version under homogeneous log-concave weights.

Core claim

We employ a Markov semigroup approach combined with the Γ-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincaré inequality in the same setting. Subsequently, by means of a duality argument, we investigate gradient and L² stability estimates of the Poincaré inequality. Furthermore, we formulate a scale-dependent version of the Poincaré inequality for homogeneous Gaussian-type measures and apply it to analyze the stability of the Heisenberg Uncertainty Principle with homogeneous weights. Finally, we establish a Logarithmic Sobolev inequality for weighted Gaussian measures and utilize it t

What carries the argument

The Markov semigroup generated by the weighted Gaussian measure together with its Γ-calculus, which supplies the identities needed to derive the inequalities and to run the duality argument for stability.

Load-bearing premise

The weighted Gaussian measures must admit a Markov semigroup whose generator satisfies the Gamma-calculus identities that let the duality argument produce the stability estimates.

What would settle it

A specific weighted Gaussian measure for which the generalized Beckner inequality fails to hold with the constant predicted by the semigroup approach would disprove the central claim.

read the original abstract

We employ a Markov semigroup approach combined with the $\Gamma$-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincar\'e inequality in the same setting. Subsequently, by means of a duality argument, we investigate gradient and $L^2$ stability estimates of the Poincar\'e inequality. Furthermore, we formulate a scale-dependent version of the Poincar\'e inequality for homogeneous Gaussian-type measures and apply it to analyze the stability of the Heisenberg Uncertainty Principle with homogeneous weights. Finally, we establish a Logarithmic Sobolev inequality for weighted Gaussian measures and utilize it to derive the Euclidean Logarithmic Sobolev inequality with homogeneous log-concave weights.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to derive a generalized Beckner inequality for weighted Gaussian measures using a Markov semigroup approach combined with Γ-calculus. As a direct consequence, it obtains the corresponding Poincaré inequality. It then uses a duality argument to derive gradient and L² stability estimates for the Poincaré inequality. The paper further introduces a scale-dependent version of the Poincaré inequality for homogeneous Gaussian-type measures and applies it to analyze stability of the Heisenberg Uncertainty Principle with homogeneous weights. Finally, it establishes a Logarithmic Sobolev inequality for weighted Gaussian measures and derives the Euclidean Logarithmic Sobolev inequality with homogeneous log-concave weights.

Significance. If the central derivations are rigorous and the necessary regularity conditions on the weights are clearly stated and verified, the results would extend classical functional inequalities (Beckner, Poincaré, Log-Sobolev) to the weighted Gaussian setting and provide new stability estimates with an application to the Heisenberg Uncertainty Principle. Such extensions are of interest in analysis, probability, and related fields, as they build on standard semigroup and duality techniques. The manuscript does not appear to introduce machine-checked proofs or fully parameter-free derivations, but the chain from Beckner to Poincaré to stability is a coherent contribution if the technical hypotheses hold.

major comments (1)
  1. [Section 2 (Preliminaries and setting)] The setting for the weighted Gaussian measure μ = w · γ (γ the standard Gaussian) and its associated diffusion generator L is introduced without explicit regularity assumptions on the weight w (e.g., C² smoothness, growth conditions at infinity, or lower bounds on derivatives). These conditions are load-bearing for the Γ-calculus identities and curvature lower bounds used to obtain the generalized Beckner inequality via the semigroup interpolation argument; without them, essential self-adjointness of L and the required algebraic relations for Γ and Γ₂ may fail, undermining the subsequent Poincaré inequality and duality stability estimates.
minor comments (2)
  1. [Abstract and §4] The term 'homogeneous Gaussian-type measures' is used in the abstract and later sections without a precise definition or reference to prior literature; adding a short definition or citation would improve readability.
  2. [Throughout] Notation for the carré-du-champ operators Γ and Γ₂, as well as the weighted measure, should be introduced once in a dedicated notation subsection and used consistently thereafter to avoid minor ambiguities in the duality argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the regularity assumptions. We address the point below and will incorporate the necessary clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: The setting for the weighted Gaussian measure μ = w · γ (γ the standard Gaussian) and its associated diffusion generator L is introduced without explicit regularity assumptions on the weight w (e.g., C² smoothness, growth conditions at infinity, or lower bounds on derivatives). These conditions are load-bearing for the Γ-calculus identities and curvature lower bounds used to obtain the generalized Beckner inequality via the semigroup interpolation argument; without them, essential self-adjointness of L and the required algebraic relations for Γ and Γ₂ may fail, undermining the subsequent Poincaré inequality and duality stability estimates.

    Authors: We agree that explicit regularity conditions on the weight w are required for the Γ-calculus framework and the semigroup arguments to hold rigorously. In the revised version we will add a dedicated paragraph in Section 2 stating the precise assumptions: w ∈ C²(ℝⁿ) with |∇w| and |D²w| satisfying suitable polynomial growth bounds at infinity, together with a uniform lower bound on the Bakry-Émery curvature term that guarantees essential self-adjointness of L on L²(μ). These hypotheses are standard in the weighted Gaussian setting and will be verified to be compatible with the homogeneous weights used in the later sections on the Heisenberg uncertainty principle. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies the standard Markov semigroup method together with Γ-calculus to weighted Gaussian measures to obtain a generalized Beckner inequality, from which the Poincaré inequality follows directly; stability estimates are then obtained via a duality argument. These steps rely on well-established functional-analytic techniques whose validity is independent of the present paper's fitted values or self-referential definitions. No load-bearing self-citation, ansatz smuggled via prior work, or renaming of known results as new derivations is indicated in the abstract or claimed chain. The derivation remains self-contained against external benchmarks of Bakry-Émery theory and semigroup interpolation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the applicability of Markov semigroup theory and Gamma-calculus to the chosen class of weighted measures; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Weighted Gaussian measures admit a Markov semigroup whose generator satisfies the Gamma-calculus identities needed for the Beckner derivation.
    Invoked at the start of the approach to establish the generalized inequality.
  • domain assumption The duality argument applies directly to produce gradient and L2 stability estimates without further restrictions on the weights.
    Used to obtain the stability results after the Poincaré inequality.

pith-pipeline@v0.9.0 · 5426 in / 1107 out tokens · 47910 ms · 2026-05-10T07:25:48.223438+00:00 · methodology

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Works this paper leans on

63 extracted references · 63 canonical work pages

  1. [1]

    Bakry and M

    D. Bakry and M. ´Emery, Diffusions hypercontractives, inS´ eminaire de probabilit´ es, XIX, 1983/84, Lecture Notes in Math.1123, Springer, Berlin, 1985, 177–206

    work page 1983

  2. [2]

    Bakry, I

    D. Bakry, I. Gentil, and M. Ledoux,Analysis and Geometry of Markov Diffusion Operators, Grundlehren Math. Wiss.348, Springer, Cham, 2014

    work page 2014

  3. [3]

    Z. M. Balogh, S. Don, and A. Krist´ aly, Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications,Trans. Amer. Math. Soc.377(2024), no. 7, 5129–5163

    work page 2024

  4. [4]

    Bartsch, T

    T. Bartsch, T. Weth, and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator,Calc. Var. Partial Differential Equations18 (2003), 253–268

    work page 2003

  5. [5]

    Beckner, Inequalities in Fourier analysis onR n,Proc

    W. Beckner, Inequalities in Fourier analysis onR n,Proc. Nat. Acad. Sci. USA72(1975), 638–641

    work page 1975

  6. [6]

    Beckner, A generalized Poincar´ e inequality for Gaussian measures,Proc

    W. Beckner, A generalized Poincar´ e inequality for Gaussian measures,Proc. Amer. Math. Soc.105 (1989), 517–522

    work page 1989

  7. [7]

    Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality,Ann

    W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality,Ann. of Math.(2)138(1993), no. 1, 213–242

    work page 1993

  8. [8]

    Bianchi and H

    G. Bianchi and H. Egnell, A note on the Sobolev inequality,J. Funct. Anal.100(1991), 18–24

    work page 1991

  9. [9]

    Bonforte, J

    M. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov, Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity, and the entropy method,Mem. Amer. Math. Soc.308(2025), no. 1554, viii+166 pp

    work page 2025

  10. [10]

    M. Bonnefont, Poincar´ e inequality with explicit constant in dimensiond≥1, Global Sensitivity Analy- sis and Poincar´ e Inequalities (Toulouse, 2022) Note,https://www.math.u-bordeaux.fr/~mibonnef/ Poincare__Toulouse.pdf

    work page 2022

  11. [11]

    Brezis, Is there failure of the inverse function theorem?, inMorse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations, Proc

    H. Brezis, Is there failure of the inverse function theorem?, inMorse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations, Proc. Workshop, Chinese Acad. Sci., Beijing, 1999, 23–33, New Stud. Adv. Math.1, Int. Press, Somerville, MA, 2003

    work page 1999

  12. [12]

    Brezis and E

    H. Brezis and E. H. Lieb, Sobolev inequalities with remainder terms,J. Funct. Anal.62(1985), 73–86

    work page 1985

  13. [13]

    Cabr´ e and X

    X. Cabr´ e and X. Ros-Oton, Regularity of stable solutions up to dimension 7 in domains of double revolution,Comm. Partial Differential Equations38(2013), 135–154

    work page 2013

  14. [14]

    Cabr´ e and X

    X. Cabr´ e and X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights,J. Differ- ential Equations255(2013), no. 11, 4312–4336

    work page 2013

  15. [15]

    Cabr´ e, X

    X. Cabr´ e, X. Ros-Oton, and J. Serra, Sharp isoperimetric inequalities via the ABP method,J. Eur. Math. Soc.18(2016), 2971–2998

    work page 2016

  16. [16]

    Carlen and A

    E. Carlen and A. Figalli, Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation,Duke Math. J.162(2013), no. 3, 579–625

    work page 2013

  17. [17]

    Cazacu, J

    C. Cazacu, J. Flynn, N. Lam, and G. Lu, Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities,J. Math. Pures Appl.(9)182(2024), 253–284

    work page 2024

  18. [18]

    S. Chen, R. Frank, and T. Weth, Remainder terms in the fractional Sobolev inequality,Indiana Univ. Math. J.62(2013), no. 4, 1381–1397

    work page 2013

  19. [19]

    L. Chen, G. Lu, and H. Tang, Sharp stability of log-Sobolev and Moser-Onofri inequalities on the sphere,J. Funct. Anal.285(2023), no. 5, Paper No. 110022, 24 pp

    work page 2023

  20. [20]

    L. Chen, G. Lu, and H. Tang, Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds,Adv. Math.450(2024), Paper No. 109778, 28 pp

    work page 2024

  21. [21]

    L. Chen, G. Lu, and H. Tang, Optimal asymptotic lower bound for stability of fractional Sobolev inequality and the global stability of log-Sobolev inequality on the sphere,Adv. Math.479(2025), part B, Paper No. 110438, 32 pp

    work page 2025

  22. [22]

    L. Chen, G. Lu, and H. Tang, Optimal stability of Hardy-Littlewood-Sobolev and Sobolev inequalities of arbitrary orders with dimension-dependent constants,Math. Ann.394(2026), no. 4, 77

    work page 2026

  23. [23]

    L. Chen, G. Lu, H. Tang, and B. Wang, Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants,J. Math. Pures Appl.(9)206(2026), Paper No. 103832, 29 pp

    work page 2026

  24. [24]

    Cianchi, N

    A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli, The sharp Sobolev inequality in quantitative form,J. Eur. Math. Soc.(JEMS)11(2009), no. 5, 1105–1139

    work page 2009

  25. [25]

    Conrado and D

    F. Conrado and D. Zhou, An eigenvalue estimate for self-shrinkers in a Ricci shrinker,Adv. Nonlinear Stud.25(2025), no. 4, 1032–1046

    work page 2025

  26. [26]

    N. A. Dao, N. T. Duy, N. Lam, and N. V. Phong, Hardy-Sobolev inequalities with Dunkl weights,Acta Math. Vietnam.48(2023), no. 1, 133–149. BECKNER, POINCAR ´E AND LOG SOBOLEV INEQUALITIES 27

    work page 2023

  27. [27]

    A. X. Do, N. Lam, and G. Lu, Sharp stability of the Heisenberg uncertainty principle: second-order and curl-free field cases,J. Funct. Anal.290(2026), no. 7, Paper No. 111321, 33 pp

    work page 2026

  28. [28]

    Dolbeault, M

    J. Dolbeault, M. J. Esteban, A. Figalli, R. Frank, and M. Loss, Sharp stability for Sobolev and log- Sobolev inequalities, with optimal dimensional dependence,Camb. J. Math.13(2025), no. 2, 359–430

    work page 2025

  29. [29]

    A. T. Duong and V. H. Nguyen, On the stability estimate for the sharp second order uncertainty principle,Calc. Var. Partial Differential Equations64(2025), no. 4, Paper No. 129, 24 pp

    work page 2025

  30. [30]

    N. T. Duy, N. Lam, and G. Lu,p-Bessel pairs, Hardy’s identities and inequalities and Hardy-Sobolev inequalities with monomial weights,J. Geom. Anal.32(2022), no. 4, Paper No. 109, 36 pp

    work page 2022

  31. [31]

    N. T. Duy, N. V. Phong, and P. T. T. Hien, Hardy inequalities with Bessel pair for Dunkl operator, Adv. Nonlinear Stud.25(2025), no. 4, 1127–1141

    work page 2025

  32. [32]

    Fathi, A short proof of quantitative stability for the Heisenberg-Pauli-Weyl inequality,Nonlinear Anal.210(2021), Paper No

    M. Fathi, A short proof of quantitative stability for the Heisenberg-Pauli-Weyl inequality,Nonlinear Anal.210(2021), Paper No. 112403, 3 pp

    work page 2021

  33. [33]

    Fathi, E

    M. Fathi, E. Indrei, and M. Ledoux, Quantitative logarithmic Sobolev inequalities and stability esti- mates,Discrete Contin. Dyn. Syst.36(2016), no. 12, 6835–6853

    work page 2016

  34. [34]

    Federbush, A partially alternate derivation of a result of Nelson,J

    P. Federbush, A partially alternate derivation of a result of Nelson,J. Math. Phys.10(1969), 50–52

    work page 1969

  35. [35]

    Figalli and D

    A. Figalli and D. Jerison, Quantitative stability for sumsets inR n,J. Eur. Math. Soc.(JEMS)17 (2015), no. 5, 1079–1106

    work page 2015

  36. [36]

    Figalli and D

    A. Figalli and D. Jerison, Quantitative stability for the Brunn-Minkowski inequality,Adv. Math.314 (2017), 1–47

    work page 2017

  37. [37]

    Figalli and R

    A. Figalli and R. Neumayer, Gradient stability for the Sobolev inequality: the casep≥2,J. Eur. Math. Soc.(JEMS)21(2019), no. 2, 319–354

    work page 2019

  38. [38]

    Figalli and Y

    A. Figalli and Y. Zhang, Sharp gradient stability for the Sobolev inequality,Duke Math. J.171(2022), no. 12, 2407–2459

    work page 2022

  39. [39]

    Gentil, The general optimalL p-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equa- tions,J

    I. Gentil, The general optimalL p-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equa- tions,J. Funct. Anal.202(2003), no. 2, 591–599

    work page 2003

  40. [40]

    Gross, Logarithmic Sobolev inequalities,Amer

    L. Gross, Logarithmic Sobolev inequalities,Amer. J. Math.97(1975), 1061–1083

    work page 1975

  41. [41]

    Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford Dirichlet form,Duke Math

    L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford Dirichlet form,Duke Math. J.42(1975), 383–396

    work page 1975

  42. [42]

    Gross, Hypercontractivity, logarithmic Sobolev inequalities, and applications: a survey of surveys, in Diffusion, Quantum Theory, and Radically Elementary Mathematics, Math

    L. Gross, Hypercontractivity, logarithmic Sobolev inequalities, and applications: a survey of surveys, in Diffusion, Quantum Theory, and Radically Elementary Mathematics, Math. Notes47, Princeton Univ. Press, Princeton, NJ, 2006, 45–73

    work page 2006

  43. [43]

    Gross, Invariance of intrinsic hypercontractivity under perturbation of Schr¨ odinger operators,Adv

    L. Gross, Invariance of intrinsic hypercontractivity under perturbation of Schr¨ odinger operators,Adv. Nonlinear Stud.25(2025), no. 4, 951–1024

    work page 2025

  44. [44]

    Gwynne, Functional Inequalities for Gaussian and Log-Concave Probability Measures, Bachelor’s thesis, Northwestern Univ., Evanston, IL, 2013

    E. Gwynne, Functional Inequalities for Gaussian and Log-Concave Probability Measures, Bachelor’s thesis, Northwestern Univ., Evanston, IL, 2013

    work page 2013

  45. [45]

    K¨ onig, On the sharp constant in the Bianchi-Egnell stability inequality,Bull

    T. K¨ onig, On the sharp constant in the Bianchi-Egnell stability inequality,Bull. Lond. Math. Soc.55 (2023), no. 4, 2070–2075

    work page 2023

  46. [46]

    K¨ onig, Stability for the Sobolev inequality: existence of a minimizer, arXiv:2211.14185, to appear inJ

    T. K¨ onig, Stability for the Sobolev inequality: existence of a minimizer, arXiv:2211.14185, to appear inJ. Eur. Math. Soc

    work page arXiv

  47. [47]

    Lam, Sharp Trudinger-Moser inequalities with monomial weights,NoDEA Nonlinear Differential Equations Appl.24(2017), no

    N. Lam, Sharp Trudinger-Moser inequalities with monomial weights,NoDEA Nonlinear Differential Equations Appl.24(2017), no. 4, Art. 39, 21 pp

    work page 2017

  48. [48]

    Lam, Sharp weighted isoperimetric and Caffarelli-Kohn-Nirenberg inequalities,Adv

    N. Lam, Sharp weighted isoperimetric and Caffarelli-Kohn-Nirenberg inequalities,Adv. Calc. Var.14 (2021), no. 2, 153–169

    work page 2021

  49. [49]

    N. Lam, G. Lu, and A. Russanov, Stability of Gaussian Poincar´ e inequalities and Heisenberg Uncer- tainty Principle with monomial weights,Math. Z.312(2026), no. 2, Paper No. 42

    work page 2026

  50. [50]

    N. Lam, G. Lu, and L. Zhang, Factorizations and Hardy’s type identities and inequalities on upper half spaces,Calc. Var. Partial Differential Equations58(2019), no. 6, Paper No. 183, 31 pp

    work page 2019

  51. [51]

    Ledoux, The geometry of Markov diffusion generators,Ann

    M. Ledoux, The geometry of Markov diffusion generators,Ann. Fac. Sci. Toulouse Math.(6)9(2000), no. 2, 305–366

    work page 2000

  52. [52]

    Ledoux,The Concentration of Measure Phenomenon, Math

    M. Ledoux,The Concentration of Measure Phenomenon, Math. Surveys Monogr.89, Amer. Math. Soc., Providence, RI, 2001

    work page 2001

  53. [53]

    Lu and J

    G. Lu and J. Wei, On a Sobolev inequality with remainder terms,Proc. Amer. Math. Soc.128(1999), 75–84

    work page 1999

  54. [54]

    McCurdy and R

    S. McCurdy and R. Venkatraman, Quantitative stability for the Heisenberg-Pauli-Weyl inequality, Nonlinear Anal.202(2021), Paper No. 112147, 13 pp

    work page 2021

  55. [55]

    Nelson, The free Markoff field,J

    E. Nelson, The free Markoff field,J. Functional Analysis12(1973), 211–227

    work page 1973

  56. [56]

    V. H. Nguyen, Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on half-spaces via mass transport and consequences,Proc. Lond. Math. Soc.(3)111(2015), no. 1, 127–148. 28 NGUYEN LAM, GUOZHEN LU, AND ANDREY RUSSANOV

    work page 2015

  57. [57]

    A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon,Inf. Control2(1959), 101–112

    work page 1959

  58. [58]

    Wang,Functional Inequalities, Markov Semigroups and Spectral Theory, Science Press, Beijing, 2005

    F.-Y. Wang,Functional Inequalities, Markov Semigroups and Spectral Theory, Science Press, Beijing, 2005

    work page 2005

  59. [59]

    Wang, Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants, Ann

    F.-Y. Wang, Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants, Ann. Probab.27(1999), no. 2, 653–663

    work page 1999

  60. [60]

    Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds,Probab

    F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds,Probab. Theory Related Fields109(1997), no. 3, 417–424

    work page 1997

  61. [61]

    Wang, Log-Sobolev inequalities: different roles of Ric and Hess,Ann

    F.-Y. Wang, Log-Sobolev inequalities: different roles of Ric and Hess,Ann. Probab.37(2009), no. 4, 1587–1604

    work page 2009

  62. [62]

    Wang,L p Hardy’s identities and inequalities for Dunkl operators,Adv

    J. Wang,L p Hardy’s identities and inequalities for Dunkl operators,Adv. Nonlinear Stud.22(2022), no. 1, 416–435

    work page 2022

  63. [63]

    F. B. Weissler, Logarithmic Sobolev inequalities for the heat-diffusion semigroup,Trans. Amer. Math. Soc.237(1978), 255–269. Nguyen Lam: School of Science and the Environment, Grenfell Campus, Memorial Uni- versity of Newfoundland, Corner Brook, NL A2H5G4, Canada Email address:nlam@mun.ca Guozhen Lu: Department of Mathematics, University of Connecticut, S...

    work page 1978