A Brascamp-Lieb-Luttinger–type inequality and applications to symmetric stable processes

Abstract

Abstract. We derive an inequality for multiple integrals from which we conclude various generalized isoperimetric inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius. Our multiple integral inequality is a replacement for the classical inequality of H. J. Brascamp, E. H. Lieb and J. M. Luttinger, where instead of fixing the volume of the domain one fixes its inradius.

$L^p$--bounds for the Beurling--Ahlfors transform

“…Nazarov and Volberg [17] lower the bound to 2(p * − 1) using an analytic approach with Bellman functions that ultimately also depends on the martingale inequalities of Burkholder. A different proof of this bound is obtained in [2] using essentially the same proof as the one in [3] but applied to "heat" martingales. Finally, Dragičević and Volberg [12] refine the Bellman function/martingale techniques and make a further observation that gives the following asymptotic result:…”

“…In this note, we refine the techniques of [3] and [2], utilize certain symmetries in B and prove the following theorem.…”

“…The special martingale version for the Beurling-Ahlfors transform is as follows; we refer the reader to [2] for full details. Let B t = (Z t , T − t) denote space-time Brownian motion started at (0, T ) in R 2 × (0, ∞).…”

“…The right hand side limit term is equal to ϕ p . Again, we refer the reader to [2] for the full details of this argument. This proves Theorem 1.1.…”

Old and New on the Bass Note, the Torsion Function and the Hyperbolic Metric

“…There is a minor technical difficulty with the above argument as it stands, but it seems a shame to complicate such an elegant idea with technicalities. The interested reader may discover the problem and/or its solution in [5]. The Brascamp-Lieb-Luttinger inequality may also be used to resolve St. Venant's problem.…”