BMO on shapes and sharp constants

TL;DR

This paper generalizes BMO spaces to arbitrary shape bases in $\

Contribution

It introduces a broad definition of BMO on domains with shape-based oscillation, analyzing properties and sharp inequalities, including a product decomposition for structured shapes.

Findings

Established basic properties of generalized BMO

Reviewed known results for classical BMO and strong BMO

Proved a product decomposition for BMO with product-structured shapes

Abstract

We consider a very general definition of BMO on a domain in $\mathbb{R}^n$, where the mean oscillation is taken with respect to a basis of shapes, i.e. a collection of open sets covering the domain. We examine the basic properties and various inequalities that can be proved for such functions, with special emphasis on sharp constants. For the standard bases of shapes consisting of balls or cubes (classic BMO), or rectangles (strong BMO), we review known results, such as the boundedness of rearrangements and its consequences. Finally, we prove a product decomposition for BMO when the shapes exhibit some product structure, as in the case of strong BMO.