## INTERPOLATION BETWEEN LOGARITHMIC SOBOLEV AND POINCARÉ INEQUALITIES

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Citations: | 17 - 7 self |

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@MISC{Arnold_interpolationbetween,

author = {Anton Arnold and Jean-philippe Bartier and Jean Dolbeault},

title = {INTERPOLATION BETWEEN LOGARITHMIC SOBOLEV AND POINCARÉ INEQUALITIES},

year = {}

}

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### Abstract

Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1.

### Citations

562 |
Methods of modern mathematical physics. IV. Analysis of operators
- Reed, Simon
- 1978
(Show Context)
Citation Context ...C∞0 (R d), be essentially self-adjoint on L2(dν) and implying a LSI. Then it has a pure point spectrum without accumulation points, see Th. 2.1 of [W2]. Since λk ր∞, (H) holds, see Theorem XIII.64 in =-=[RS]-=-. 2. With the following transformation, N can be rewritten as a Schrödinger operator: f e−V/2 =: g so that ∫ Rd |f |2 dν = ∫ Rd |g|2 dx and ∫ Rd |∇f |2 dν =∫ Rd (|∇g|2 + V1 |g| 2) dx with V1 := 1 4 |... |

276 |
Logarithmic Sobolev inequalities
- GROSS
- 1976
(Show Context)
Citation Context ...of generalized Poincaré inequalities (GPI) for the Gaussian measure that yield a sharp interpolation between the classical Poincaré inequality and the logarithmic Sobolev inequality (LSI) of L. Gross =-=[G]-=-. For any 1 ≤ p < 2 these GPIs read (1.1) 1 2 − p [ ∫ R d f 2 (∫ dµ − Rd |f| p ) ] 2/p ∫ dµ ≤ Rd |∇f| 2 dµ ∀f ∈ H 1 (dµ) , where µ(x) denotes the normal centered Gaussian distribution on R d : µ(x) :=... |

135 |
Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses. Société Mathématique de
- Ané, Blachère, et al.
- 2000
(Show Context)
Citation Context ... Figure 2.1: It shows the p-dependent constant C(p)/CLS for several values of α. 2. If the logarithmic Sobolev constant takes its minimal value CLS = CP (i.e., α = 1), we have C(p) = CLS, for all p ∈ =-=[1, 2]-=- which is the optimal constant (consider f = 1 + εg with ε → 0 in (2.1)). However, even for the Gaussian measure, Inequality (2.1) does not admit a non-trivial minimal function for any 1 < p < 2 (prov... |

113 | On logarithmic Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations
- Arnold, Markowich, et al.
- 2001
(Show Context)
Citation Context ... := −∆+x·∇. Generalizations of (1.1) to other probability measures and the quest for “sharpest” constants in such inequalities have attracted lots of interest in the last years ([AD, BCR, LO, W]). In =-=[AMTU]-=- GPIs have been derived for strictly log-concave distribution functions ν(x): (1.2) [ ∫ 1 2 − p Rd f 2 (∫ dν − Rd |f| p ) ] 2/p dν ≤ 1 ∫ κ Rd |∇f| 2 dν ∀f ∈ H 1 (dν) , where κ is the uniform convexity... |

106 | GÖTZE Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
- BOBKOV, F
- 1999
(Show Context)
Citation Context ... was kindly suggested to us by Michel Ledoux. While the Poincaré constant is here bounded for ε ∈ [0, 1], the logarithmic Sobolev constant blows up like O(1/ε), which can be estimated with Th. 1.1 of =-=[BG]-=- (also see [BR] for a simplified approach and §3 of [L] for a review of applications in geometry). On the other hand, Lemma 2 of [LO] gives a simple sufficient condition in one dimension such that α =... |

47 |
Hypercontractivité de semi-groupes de diffusion
- Bakry, Émery
- 1984
(Show Context)
Citation Context ... 1 2− p [∫ Rd f2 dν − (∫ Rd |f |p dν )2/p] ≤ 1 κ ∫ Rd |∇f |2 dν ∀f ∈ H1(dν) ,(1.2) where κ is the uniform convexity bound of − log ν(x), i.e. Hess(− log ν(x)) ≥ κ ∀x ∈ R d. This Bakry-Emery condition =-=[BE]-=- also implies a LSI with constant CLS = 1 κ , i.e., 1 2 ∫ Rd f2 log ( f2∫ Rd f2 dν ) dν ≤ CLS ∫ Rd |∇f |2 dν ∀ f ∈ H1(dν) ,(1.3) which is a special case (p→ 2 limit) of (1.2). Lata la and Oleszkiewicz... |

45 |
A generalized Poincare inequality for Gaussian measures
- Beckner
- 1989
(Show Context)
Citation Context ...e between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1. Introduction In 1989 W. Beckner =-=[B]-=- derived a family of generalized Poincaré inequalities (GPI) for the Gaussian measure that yield a sharp interpolation between the classical Poincaré inequality and the logarithmic Sobolev inequality ... |

42 |
Between Sobolev and Poincaré, in Geometric aspects of functional analysis
- la, Oleszkiewicz
- 2000
(Show Context)
Citation Context ...tribution functions ν(x): (1.2) [ ∫ 1 2 − p Rd f 2 (∫ dν − Rd |f| p ) ] 2/p dν ≤ 1 ∫ κ Rd |∇f| 2 dν ∀f ∈ H 1 (dν) , where κ is the uniform convexity bound of − log ν(x). Lata̷la and Oleszkiewicz (see =-=[LO]-=-) derived such GPIs under the weaker assumption that ν(x) satisfies a LSI with constant 0 < C < ∞, i.e., (1.3) ∫ R d f 2 ( f log 2 ∫ Rd f 2 ) dν ∫ dν ≤ 2 C Rd |∇f| 2 dν ∀ f ∈ H 1 (dν) . Under the assu... |

40 |
The free Markoff field
- Nelson
- 1973
(Show Context)
Citation Context ...e get ∫ R d |fk| 2 ∫ dµ − Rd ∣ ∣e −t N fk ∣ 2 k0 t 1 − e−2 dµ ≤ k0 ∫ R d |∇fk| 2 dµ , which proves the result by summation. □ The second preliminary result is Nelson’s hypercontractive estimates, see =-=[N]-=-. To make this note selfcontained we include a sketch of the proof given in [G]. Lemma 2.2. For any f ∈ L p (dµ), p ∈ (1, 2), it holds ∥ ∥e −tN f∥ L2 (dµ) ≤ ‖f‖Lp (dµ) ∀ t ≥ − 1 log(p − 1) . 2 Proof. ... |

39 | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
- Barthe, Cattiaux, et al.
(Show Context)
Citation Context ...is result to more general measures dν. Generalizations of (1.1) to other probability measures and the quest for “sharpest” constants in such inequalities have attracted much interest in recent years (=-=[2, 5, 6, 11, 15]-=-). In [3] GPIs have been derived for strictly log-concave distribution functions ν(x): 1 2− p [∫ Rd f2 dν − (∫ Rd |f |p dν )2/p] ≤ 1 κ ∫ Rd |∇f |2 dν ∀f ∈ H1(dν), (1.2) ∗Received: July 18, 2007; accep... |

37 | The geometry of Markov diffusion generators
- Ledoux
- 2000
(Show Context)
Citation Context ...Poincaré constant is here bounded for ε ∈ [0, 1], the logarithmic Sobolev constant blows up like O(1/ε), which can be estimated with Th. 1.1 of [BG] (also see [BR] for a simplified approach and §3 of =-=[L]-=- for a review of applications in geometry). On the other hand, Lemma 2 of [LO] gives a simple sufficient condition in one dimension such that α = 1, when k0 = 1. If the logarithmic Sobolev constant ta... |

35 | Functional inequalities for empty essential spectrum
- Wang
(Show Context)
Citation Context ...ple conditions for (H) to hold: 1. Let N, defined on C∞0 (R d), be essentially self-adjoint on L2(dν) and implying a LSI. Then it has a pure point spectrum without accumulation points, see Th. 2.1 of =-=[W2]-=-. Since λk ր∞, (H) holds, see Theorem XIII.64 in [RS]. 2. With the following transformation, N can be rewritten as a Schrödinger operator: f e−V/2 =: g so that ∫ Rd |f |2 dν = ∫ Rd |g|2 dx and ∫ Rd |... |

31 |
Sobolev inequalities for probability measures on the real line
- Barthe, Roberto
(Show Context)
Citation Context ...gested to us by Michel Ledoux. While the Poincaré constant is here bounded for ε ∈ [0, 1], the logarithmic Sobolev constant blows up like O(1/ε), which can be estimated with Th. 1.1 of [BG] (also see =-=[BR]-=- for a simplified approach and §3 of [L] for a review of applications in geometry). On the other hand, Lemma 2 of [LO] gives a simple sufficient condition in one dimension such that α = 1, when k0 = 1... |

13 | Refined convex Sobolev inequalities
- Arnold, Dolbeault
- 2004
(Show Context)
Citation Context ...ce this LSI implies a Poincaré inequality (with constant C), the second constant in the above min just follows from Hölder’s inequality (∫ Rd f dν ) 2 (∫ ≤ Rd |f| p dν ) 2/p 2 = ‖f‖Lp (dν) (cf. §3 in =-=[AD]-=-). Our second result, Theorem 3.1, improves upon the p-dependent constant on the r.h.s. of (1.4). 1As a third result we shall derive “refined convex Sobolev inequalities” under the assumption that ν(... |

9 |
A generalization of Poincaré and log-Sobolev inequalities
- Wang
- 2005
(Show Context)
Citation Context ...is result to more general measures dν. Generalizations of (1.1) to other probability measures and the quest for “sharpest” constants in such inequalities have attracted much interest in recent years (=-=[2, 5, 6, 11, 15]-=-). In [3] GPIs have been derived for strictly log-concave distribution functions ν(x): 1 2− p [∫ Rd f2 dν − (∫ Rd |f |p dν )2/p] ≤ 1 κ ∫ Rd |∇f |2 dν ∀f ∈ H1(dν), (1.2) ∗Received: July 18, 2007; accep... |

3 |
Entropies, convexity, and functional inequalities
- Chafaı̈
(Show Context)
Citation Context ...h formally corresponds to α→∞: for fixed p we have from (2.2) lim α→∞ C(p) = CP (2− p) , which corresponds to the second constant in the min of inequality (1.4) (cf. also Theorem 4 in [2] and §2.2 in =-=[9]-=-). Remark 2.6. (On case (b).) Even in the Gaussian case ν = µ0, Theorem 2.4(b) improves on Beckner’s result for any k0 ≥ 2. 976 CONVEX SOBOLEV INEQUALITIES Example 2.7. 1. CLS = CP clearly holds for t... |

2 | A Generalization of Poincare - Wang |

1 | A Generalization of Poincaré and Log-Sobolev - Wang |