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

Venue: | Duke Math. J |

Citations: | 10 - 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

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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... |

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1 |
on Green functions and Poisson kernels for symmetric stable processes
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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... |

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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... |