Sharp Volume-Based Bounds for BMO Functions on Compact Domains and Applications

Abstract

We establish sharp localized $L^r$ bounds for compactly supported functions of bounded mean oscillation (BMO). Classical estimates obtained via the John-Nirenberg inequality bound the $L^r$ norm of such a function in terms of the diameter of its support. We show that the diameter dependence can be replaced by the Lebesgue measure of the support, the relevant quantity for thin, highly eccentric domains. The proof relies on non-increasing rearrangements and the Bennett-DeVore-Sharpley theorem, and requires no vanishing-oscillation hypothesis. We present physical and operator-theoretic applications: geometric collapse in tubular neighborhoods, a localized Cwikel-Lieb-Rozenblum bound for Schr\"odinger operators, and decoupling bounds for the localized action of Calder\'on-Zygmund singular integrals.