Quantitative John–Nirenberg inequalities at different scales

Abstract

AbstractGiven a family $${\mathcal {Z}}=\{\Vert \cdot \Vert _{Z_Q}\}$$ Z = { ‖ · ‖ Z Q } of norms or quasi-norms with uniformly bounded triangle inequality constants, where each Q is a cube in $${\mathbb {R}}^n$$ R n , we provide an abstract estimate of the form $$\begin{aligned} \Vert f-f_{Q,\mu }\Vert _{Z_Q}\le c(\mu )\psi ({\mathcal {Z}})\Vert f\Vert _{\mathrm {BMO}(\mathrm {d}\mu )} \end{aligned}$$ ‖ f - f Q , μ ‖ Z Q ≤ c ( μ ) ψ ( Z ) ‖ f ‖ BMO ( d μ ) for every function $$f\in \mathrm {BMO}(\mathrm {d}\mu )$$ f ∈ BMO ( d μ ) , where $$\mu $$ μ is a doubling measure in $${\mathbb {R}}^n$$ R n and $$c(\mu )$$ c ( μ ) and $$\psi ({\mathcal {Z}})$$ ψ ( Z ) are positive constants depending on $$\mu $$ μ and $${\mathcal {Z}}$$ Z , respectively. That abstract scheme allows us to recover the sharp estimate $$\begin{aligned} \Vert f-f_{Q,\mu }\Vert _{L^p \left( Q,\frac{\mathrm {d}\mu (x)}{\mu (Q)}\right) }\le c(\mu )p\Vert f\Vert _{\mathrm {BMO}(\mathrm {d}\mu )}, \qquad p\ge 1 \end{aligned}$$ ‖ 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 $$f\in \mathrm {BMO}(\mathrm {d}\mu )$$ f ∈ BMO ( d μ ) , which is known to be equivalent to the John–Nirenberg inequality, and also enables us to obtain quantitative counterparts when $$L^p$$ L p is replaced by suitable strong and weak Orlicz spaces and $$L^{p(\cdot )}$$ 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 $$A_\infty $$ A ∞ weights.