Brascamp-Lieb-Luttinger inequalities for convex domains of finite inradius

摘要

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 r(D) and the ball is replaced by (-r(D), r(D)) x Rn-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. Polya and G. Szego. 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 off red 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 Schrodinger operators. [References: 24]