Quantitative John–Nirenberg inequalities at different scales

Abstract

Given a family Z = { · Z Q } of norms or quasi-norms with uniformly bounded triangle inequality constants, where each Q is a cube in Rn, we provide an abstract estimate of the form f − fQ,μZ Q ≤ c(μ)ψ(Z) f BMO(dμ) for every function f ∈ BMO(dμ), where μ is a doubling measure in Rn and c(μ) and ψ(Z) are positive constants depending on μ and Z, respectively. That abstract scheme allows us to recover the sharp estimate f − fQ,μL p

Q, dμ(x) μ(Q) ≤ c(μ)p f BMO(dμ), p ≥ 1 for every cube Q and every f ∈ BMO(dμ), which is known to be equivalent to the John–Nirenberg inequality, and also enables us to obtain quantitative counterparts when L p is replaced by suitable strong and weak Orlicz spaces and L p(·) spaces. Besides the aforementioned results we also generalize [(Ombrosi in Isr J Math 238:571-591, 2020), Theorem 1.2] to the setting of doubling measures and obtain a new characterization of Muckenhoupt’s A∞ weights