Seminar: Bellman Functions for VMO

Abstract: We consider a quantitative version of VMO on an interval I, equipped with a Campanato−type norm corresponding to a modulus ξ. We fix a non−negative real−valued function f on ℝ and, under certain mild assumptions on f and ξ, construct the exact Bellman functions for the upper and lower bounds on the integral ∫_I f(φ) for φ∈VMO in terms of ξ, averages of φ and φ² on I, and the length of I. This disposes of the traditional Bellman−function requirement of scale invariance and is the first Bellman application directly addressing such "non−autonomous" estimates. As applications, we present sharp estimates for several choices of ξ and f, as well as a universal weak−type inequality for those ξ that give VMO with unbounded elements. This is joint work with Pavel Zatitskii.

Faculty Host: Brett Wick