Quantitative John–Nirenberg inequalities at different scales

Given a family Z = { ‖ · ‖ Z Q } of norms or quasi-norms with uniformly bounded triangle inequality constants, where each Q is a cube in R n , we provide an abstract estimate of the form ‖ f - f Q , μ ‖ Z Q ≤ c ( μ ) ψ ( Z ) ‖ f ‖ BMO ( d μ ) for every function f ∈ BMO ( d μ ) , where μ is a doubling measure in R n 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 - f Q , μ ‖ L p Q , d μ ( x ) μ ( Q ) ≤ c ( μ ) p ‖ f ‖ BMO ( d μ ) , p ≥ 1 for every cube Q and every f ∈ BMO ( d μ ) , 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.