The John–Nirenberg inequality in ball Banach function spaces and application to characterization of BMO

Abstract

Our goal is to obtain the John-Nirenberg inequality for ball Banach function spaces X, provided that the Hardy-Littlewood maximal operator M is bounded on the associate space X by using the extrapolation. As an application we characterize BMO, the bounded mean oscillation, via the norm of X.

A Survey on Function Spaces of John–Nirenberg Type

“…Furthermore, the John-Nirenberg inequality on X was also obtained in [48] (Theorem 3.1), which shows that there exists some positive constant C, such that for any ball B ⊂ R n and any τ ∈ [0, ∞),…”

“…Recently, Izuki et al [48] obtained both the John-Nirenberg inequality and the equivalent characterization of BMO (R n ) on the ball Banach function space which contains Morrey spaces, (weighted, mixed-norm, variable) Lebesgue spaces, and Orlicz-slice spaces as special cases (see [48], Definition 2.8, and also [49], for the related definitions). Precisely, let X be a ball Banach function space satisfying the additional assumption that the Hardy-Littlewood maximal operator M is bounded on X (the associated space of X; see [48], Definition 2.9, for its definition), and for any b ∈ L 1 loc (R n ),…”

“…. Moreover, in [48] (Theorem 1.2), Izuki et al showed that under the above assumption of X, b ∈ BMO (R n ) if and only if b ∈ L 1 loc (R n ) and b BMO X < ∞; meanwhile,…”

A Survey on Function Spaces of John--Nirenberg Type

“…Recently, Izuki et al [53] obtained both the John-Nirenberg inequality and the equivalent characterization of BMO (R n ) on the ball Banach function space which contains Morrey spaces, (weighted, mixed-norm, variable) Lebesgue spaces, and Orlicz-slice spaces as special cases; see [53,Definition 2.8] and also [94] for the related definitions. Precisely, let X be a ball Banach function space satisfying the additional assumption that the Hardy-Littlewood maximal operator M is bounded on X ′ (the associate space of X; see [53, Definition 2.9] for its definition), and, for…”

Generalization in Ball Banach Function Spaces of Brezis--Van Schaftingen--Yung Formulae with Applications to Fractional Sobolev and Gagliardo--Nirenberg Inequalities

“…They are less restrictive than the classical Banach function spaces introduced in the book [7, Chapter 1]. For more studies on ball quasi-Banach function spaces, we refer the reader to [65,64,68,77,78,17] for the Hardy space associated with ball quasi-Banach function spaces, to [79,33,75] for the boundedness of operators on ball quasi-Banach function spaces, and to [41,42,76,37,72] for the applications of ball quasi-Banach function spaces.…”