Bounded Mean Oscillation: an $\mathbb{R}$-Function with Multi-$\mathbb{K}_6$ Cubes. Dual of the Hardy Space $H^1$ and Banach Extent

TL;DR

This paper explores harmonic functions of bounded mean oscillation, establishing fundamental properties, duality with Hardy space, and exponential integrability, providing new insights into their mathematical structure.

Contribution

It offers a renewed formalism for BMO functions, proves a main theorem, and discusses duality with Hardy space $H^1$, advancing theoretical understanding.

Findings

Main theorem relating to BMO functions and their properties

Exponential integrability of BMO functions

Duality between Hardy space $H^1$ and BMO

Abstract

This paper investigates the concept of harmonic functions of bounded mean oscillation, starting from John-Nirenberg's pioneering studies, under a renewed formalism, suitable for bringing out some fundamental properties inherent in it. In more detail: after a quick introduction, the second Section presents the main theorem, plus complete proof, relating to this function; in the third Section there is a suggestion on the exponential integrability (theorem and sketch of proof), while the fourth Section deals with the duality of Hardy Space $H^1$ and bounded mean oscillation, with some ideas for a demonstration. The writing closes with a graphic appendix.