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
