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 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∞ weightsJ. C. M.-P. is supported by the Basque Government through the BERC 2018-2021 program and by Spanish Ministry of Science, Innovation and Universities through BCAM Severo Ochoa accreditation SEV-2017-0718. He is also supported by MINECO through the MTM2017-82160-C2-1-P project funded by (AEI/FEDER, UE), acronym “HAQMEC”, through ”la Caixa” Grant, and through the MATHROCKS project, funded by European Commission with Grant Agreement Number 777778 (H2020- MSCA-RISE-2017). He is also grateful to the people of the Universidad de Buenos Aires and the Universidad Nacional del Sur for their hospitality during his visit to Argentina in 2019. E.R. is partially supported by Grants UBACyT 20020170200057BA and PIP (CONICET) 11220110101018. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No 777822. I.P.R.-R. is partially supported by Grants PICT 2018-02501 and PICT 2019-00018 (Agencia I+D+i) and by Grant UMA18-FEDERJA-002 (FEDER). Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Funding for open access charge: Universidad de Málaga / CBU