## Brascamp–Lieb–Luttinger inequalities for convex domains of finite inradius (2002)

Venue: | Duke Math. J |

Citations: | 9 - 0 self |

### BibTeX

@ARTICLE{Méndez-hernández02brascamp–lieb–luttingerinequalities,

author = {Pedro J. Méndez-hernández},

title = {Brascamp–Lieb–Luttinger inequalities for convex domains of finite inradius},

journal = {Duke Math. J},

year = {2002},

pages = {93--131}

}

### OpenURL

### Abstract

We prove a multiple integral inequality for convex domains in R n of finite inradius. This inequality is a version of the classical inequality of H. Brascamp, E. Lieb, and J. Luttinger, but here, instead of fixing the volume of the domain, one fixes its inradius rD and the ball is replaced by (−rD, rD) × R n−1. We also obtain a sharper version of our multiple integral inequality, which generalizes the results in [6], for two-dimensional bounded convex domains where we replace infinite strips by rectangles. It is well known by now that the Brascamp-Lieb-Luttinger inequality provides a powerful and elegant method for obtaining and extending many of the classical geometric and physical isoperimetric inequalities of G. Pólya and G. Szegö. In a similar fashion, the new multiple integral inequalities in this paper yield various new isoperimetric-type inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius which refine in various ways the results in [2], [3], [4], [5], and [20]. These include extensions to heat kernels, heat content, and torsional rigidity. Finally, our results also apply to the processes studied in [10] whose generators are relativistic Schrödinger operators. 1.

### Citations

674 |
Lévy processes and infinitely divisible distributions (english ed.)., volume 68 of Cambridge studies in advanced mathematics
- Sato
- 2005
(Show Context)
Citation Context ...esses studied in [10] whose generators are the so-called “relativistic” Schrödinger operators, and any processes of the form BAt , where Bt is a Brownian motion and At is a subordinator (see [13] and =-=[22]-=-). The paper is organized as follows. We prove Theorem 1.3 in §2. In §3 we give some preliminary results on polyhedral domains in R n which are fundamental in the proof of Theorem 1.2. The proof of Th... |

98 |
Isoperimetric inequalities and applications
- Bandle
- 1980
(Show Context)
Citation Context ...elativistic Schrödinger operators. 1. Introduction Let D ⊂ Rn be a domain of finite volume, and denote by D∗ the ball in Rn centered at the origin with the same volume as D. As described in C. Bandle =-=[1]-=- and Pólya and Szegö [18], there is a large class of analytical quantities related to the Dirichlet Laplacian in D which are maximized or minimized by the corresponding quantities for D∗ . These inequ... |

38 |
Intrinsic ultracontractivity and conditional gauge for symmetric stable processes
- Chen, Song
- 1997
(Show Context)
Citation Context ...processes has been extensively studied for several years. For some of the recent developments and basic properties of pα D,V (t, z, w) and Gα D,V (z, w), we refer the reader to Z.-Q. Chen and R. Song =-=[11]-=-, [12]. If V = 0, we just write pα D (t, z, w) for the heat kernel and Gα D (z, w) for the Green function. Let us denote the space of C∞-functions with compact support in D by C∞ 0 (D). By the Feynman... |

36 |
A general rearrangement inequality for multiple integrals
- Brascamp, Lieb, et al.
- 1974
(Show Context)
Citation Context ...s of multiple integrals, to prove (1) as well as some of the other generalized isoperimetric inequalities of Pólya and Szegö [18]. The following inequality, proved by Brascamp, Lieb, and Luttinger in =-=[9]-=-, is a refinement of the original inequality of Luttinger. THEOREM 1.1 ([9]) Let p1, . . . , p2m be nonnegative functions in Rn , and let p∗ 1 , . . . , p∗ 2m be their symmetric decreasing rearrangeme... |

22 | Relativistic Schrödinger operators: asymptotic behavior of the eigenvalues
- Carmona, Masters, et al.
- 1990
(Show Context)
Citation Context ...n various ways the results in [2], [3], [4], [5], and [20]. These include extensions to heat kernels, heat content, and torsional rigidity. Finally, our results also apply to the processes studied in =-=[10]-=- whose generators are relativistic Schrödinger operators. 1. Introduction Let D ⊂ Rn be a domain of finite volume, and denote by D∗ the ball in Rn centered at the origin with the same volume as D. As ... |

15 |
Generalized isoperimetric inequalities
- Luttinger
- 1973
(Show Context)
Citation Context ...ics Subject Classification. Primary 31C60. Author’s work supported in part by Purdue Research Foundation grant number 690-1395-3149. 9394 PEDRO J. MÉNDEZ-HERNÁNDEZ spectively, then λD ∗ ≤ λD. (1) In =-=[15]-=-, [16], and [17], Luttinger provided a new method, based on the FeynmanKac representation of the heat kernel in terms of multiple integrals, to prove (1) as well as some of the other generalized isope... |

12 | Sharp inequalities for heat kernels of Schrödinger operators and applications to spectral gaps
- Bañuelos, Méndez-Hernández
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Citation Context ...ns of fixed volume, provides a powerful and versatile tool for studying extremal inequalities for Dirichlet heat kernels in various other settings. This method was also used by Bañuelos and Méndez in =-=[7]-=- to study extremal inequalities for ratios of heat kernels and spectral gaps in convex domains of fixed diameter. Not only do Theorems 1.2 and 1.3 yield isoperimetric-type inequalities for the Dirichl... |

12 |
Isoperimetric inequalities
- Polya, Szegö
- 1951
(Show Context)
Citation Context ...operators. 1. Introduction Let D ⊂ Rn be a domain of finite volume, and denote by D∗ the ball in Rn centered at the origin with the same volume as D. As described in C. Bandle [1] and Pólya and Szegö =-=[18]-=-, there is a large class of analytical quantities related to the Dirichlet Laplacian in D which are maximized or minimized by the corresponding quantities for D∗ . These inequalities are often called ... |

11 | Brownian motion and the fundamental frequency of a drum - Bañuelos, Carroll - 1994 |

10 | A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes
- Bañuelos, Latala, et al.
- 2001
(Show Context)
Citation Context ...] result to symmetric stable processes. It is well known that π 2 λI (D),2 = 4r 2 D However, for α ∈ (0, 2), very little seems to be known about the explicit value of λI (D),α. We refer the reader to =-=[6]-=- and [19] for some estimates on these eigenvalues. We now use Theorem 1.3 to obtain a new set of isoperimetric-type inequalities that include sharper versions of Theorem 5.1 and its consequences for c... |

5 |
A Lower Bound for the Fundamental Frequency of a Convex Region
- Protter
- 1981
(Show Context)
Citation Context ... various new isoperimetric-type inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius which refine in various ways the results in [2], [3], [4], [5], and =-=[20]-=-. These include extensions to heat kernels, heat content, and torsional rigidity. Finally, our results also apply to the processes studied in [10] whose generators are relativistic Schrödinger operato... |

5 |
J.-F.: Mean curvature and the heat equation
- Berg, Gall
- 1994
(Show Context)
Citation Context ...e t is given by the integral ∫ ∫ Qt(D) = pD(t, z, w) dz dw, D D and the torsional rigidity of D is the integral of the heat content in time from zero to ∞. (For more on these quantities, see [18] and =-=[24]-=-.) We show that Theorem 1.3BRASCAMP-LIEB-LUTTINGER INEQUALITIES 99 implies that for all t > 0 the heat content of D is less than the heat content of C(D). That is, for all t > 0, ( ) Qt(D) ≤ Qt C(D) ... |

4 | Time changes hidden in Brownian subordination
- Geman, Madan, et al.
- 2000
(Show Context)
Citation Context ...stic processes studied in [10] whose generators are the so-called “relativistic” Schrödinger operators, and any processes of the form BAt , where Bt is a Brownian motion and At is a subordinator (see =-=[13]-=- and [22]). The paper is organized as follows. We prove Theorem 1.3 in §2. In §3 we give some preliminary results on polyhedral domains in R n which are fundamental in the proof of Theorem 1.2. The pr... |

3 |
Some Theorems on Symmetric Stable Processes
- Blumenthal, Geetor
- 1960
(Show Context)
Citation Context ...d. That is, if α = 2, then Xt = B2t and so If 0 < α < 2, then p 2 t (z, w) = 1 exp (4πt) n/2 [ −|z − w| 2 4t ] . Xt = B2σt , (55) where σt is a stable subordinator of index α/2 independent of Bt (see =-=[8]-=-). Therefore, p α t (z, w) = ∫ ∞ 0 p 2 u (z, w)gα/2(t, u) du,124 PEDRO J. MÉNDEZ-HERNÁNDEZ where gα/2(t, u) is the transition density of σt. Thus, for every positive t, pα t (z, w) = f α t (|z − w|) ... |

2 | An isoperimetric-type inequality for integrals of Green’s functions - BAÑUELOS, HOUSWORTH - 1995 |

2 |
la fréquence fondamentale d’une membrane vibrante: évaluations par défaut et principe de maximum
- HERSH, Sur
- 1960
(Show Context)
Citation Context ...alities for convex domains in R n where, instead of fixing the volume of D, one fixes its inradius rD. Recall that the inradius is the supremum of the radius of all the balls contained in D. J. Hersh =-=[14]-=- (n = 2) and M. Protter [20] (n ≥ 3) proved that if D ⊂ R n is a convex domain of inradius rD, then π 2 4r 2 D = λI (D) = λS(D) ≤ λD, (2) where I (D) = (−rD, rD) and S(D) is the infinite slab (infinit... |

2 |
Maximum Principles and Their
- SPERB
- 1981
(Show Context)
Citation Context ..., w) dw ≤ pI (D)(t, 0, w) dw (4) D for all z ∈ D and all t > 0. Here and for the rest of the paper, we use 0 to denote the origin in Rn . Upon integrating (4) in time, we obtain R. Sperb’s inequality =-=[23]-=- on integrals of Green’s functions. It is important to note here that inequalities (2) and (4) are false if the convexity assumption of the domain is removed. This can be seen by taking a slit disk. W... |

1 |
Inradius and integral means for Green’s Functions and conformal mappings
- BAÑUELOS, CARROLL, et al.
- 1998
(Show Context)
Citation Context ...in this paper yield various new isoperimetric-type inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius which refine in various ways the results in [2], =-=[3]-=-, [4], [5], and [20]. These include extensions to heat kernels, heat content, and torsional rigidity. Finally, our results also apply to the processes studied in [10] whose generators are relativistic... |

1 |
Isoperimetric-type bounds for solutions of the heat equation
- BAÑUELOS, KRÖGER
- 1997
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Citation Context ...per yield various new isoperimetric-type inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius which refine in various ways the results in [2], [3], [4], =-=[5]-=-, and [20]. These include extensions to heat kernels, heat content, and torsional rigidity. Finally, our results also apply to the processes studied in [10] whose generators are relativistic Schröding... |

1 |
on Green functions and Poisson kernels for symmetric stable processes
- Estimates
- 1998
(Show Context)
Citation Context ...ses has been extensively studied for several years. For some of the recent developments and basic properties of pα D,V (t, z, w) and Gα D,V (z, w), we refer the reader to Z.-Q. Chen and R. Song [11], =-=[12]-=-. If V = 0, we just write pα D (t, z, w) for the heat kernel and Gα D (z, w) for the Green function. Let us denote the space of C∞-functions with compact support in D by C∞ 0 (D). By the Feynman-Kac f... |

1 |
isoperimetric inequalities
- Generalized
- 1973
(Show Context)
Citation Context ...bject Classification. Primary 31C60. Author’s work supported in part by Purdue Research Foundation grant number 690-1395-3149. 9394 PEDRO J. MÉNDEZ-HERNÁNDEZ spectively, then λD ∗ ≤ λD. (1) In [15], =-=[16]-=-, and [17], Luttinger provided a new method, based on the FeynmanKac representation of the heat kernel in terms of multiple integrals, to prove (1) as well as some of the other generalized isoperimetr... |

1 |
A two-sided bound on the lowest eigenvalue of an operator that characterizes stable processes, Theory Probab
- POZIN, SAKHNOVICH
- 1992
(Show Context)
Citation Context ... to symmetric stable processes. It is well known that π 2 λI (D),2 = 4r 2 D However, for α ∈ (0, 2), very little seems to be known about the explicit value of λI (D),α. We refer the reader to [6] and =-=[19]-=- for some estimates on these eigenvalues. We now use Theorem 1.3 to obtain a new set of isoperimetric-type inequalities that include sharper versions of Theorem 5.1 and its consequences for convex pla... |