Extensions of the John-Nirenberg theorem and applications
Abstract
The John–Nirenberg theorem states that functions of bounded mean oscillation are
exponentially integrable. In this article we give two extensions of this theorem. The first one
relates the dyadic maximal function to the sharp maximal function of Fefferman–Stein, while
the second one concerns local weighted mean oscillations, generalizing a result of Muckenhoupt
and Wheeden. Applications to the context of generalized Poincaré type inequalities and to the
context of the $C_p$ class of weights are given. Extensions to the case of polynomial BMO type
spaces are also given.