A note on generalized Fujii-Wilson conditions and BMO spaces

Ombrosi, Sheldy; Pérez, Carlos; Rela, Ezequiel; Rivera-Rı́os, Israel P.

doi:10.1007/s11856-020-2031-y

Cited by 6 publications

(14 citation statements)

References 22 publications

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“…for any functional Y : Q → (0, ∞), adapting the definition of A ∞ accordingly. This would recover the main theorem of [22].…”

Section: 1supporting

confidence: 70%

BMO with respect to Banach function spaces

Lerner¹,

Lorist²,

Ombrosi³

2022

Preprint

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For every cube Q ⊂ R n we let X Q be a quasi-Banach function space over Q such that χ Q XQ ≃ 1, and forWe study necessary and sufficient conditions on X such that BMO = BMO X = BMO * X . In particular, we give a full characterization of the embedding BMO ֒→ BMO X in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding BMO * X ֒→ BMO. Our main theorems recover and improve all previously known results in this area.

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“…for any functional Y : Q → (0, ∞), adapting the definition of A ∞ accordingly. This would recover the main theorem of [22].…”

Section: 1supporting

confidence: 70%

BMO with respect to Banach function spaces

Lerner¹,

Lorist²,

Ombrosi³

2022

Preprint

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“…is valid if and only if the weight v and the functional Y satisfy the Fujii-Wilson type ]). The following theorem generalizes the aforementioned result in [OPRRR20] to the setting of doubling measures. A proof of it is provided in Appendix A.…”

Section: Introductionsupporting

confidence: 62%

“…The main contribution of our work is to provide analogous results for more general norms at the left-hand side of the inequality in Theorem A. To that end, we will take further some ideas in [PR19] and will also consider some of the concepts appearing in [OPRRR20], thus including in the theory more general BMO spaces defined by different oscillations. Now, to be able to present the main result, we need to describe the key concept for our purposes, namely the notion of local average.…”

Section: Introductionmentioning

confidence: 99%

“…In this section we provide the fundamental tools and concepts used to prove the main results of the paper. Some aspects of the previous self-improving result [PR19, Theorem 1.5] and some results in [OPRRR20] will be discussed to motivate one of the new concepts which will be introduced here, namely, the generalized A ∞ type condition adapted to general quasi-normed function spaces. The basic assumptions on the quasi-norms we will consider in this work will also be introduced here.…”

Section: Introductionmentioning

confidence: 99%

“…6) is what makes possible to work with these perturbations without assuming the A ∞ (dµ) condition on the weight. Also related with this problem, and more related to the results which will be studied here is the work [OPRRR20], where the embedding of BMO(dµ) into certain weighted BMO spaces is characterized. To be precise, they consider the following weighted BMO spaces.…”

Section: Introductionmentioning

confidence: 99%

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Quantitative John-Nirenberg inequalities at different scales

Martínez-Perales¹,

Rela²,

Rivera-Rı́os³

2020

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≤ c(µ, Y )ψ(X) f BMO(dµ) for all cubes Q in R n and every function f ∈ BMO(dµ), where µ is a doubling measure in R n , Y is some positive functional defined on cubes,is a sufficiently good quasi-norm and c(µ, Y ) and ψ(X) are positive constants depending on µ and Y , and X, respectively. That abstract scheme allows us to recover the sharp estimatefor 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 [OPRRR20, Theorem 1.2] to the setting of doubling measures and obtain a new characterization of Muckenhoupt's A∞ weights. Contents1. Introduction 1 2. The A ∞ condition of a functional with respect to a quasi-norm 7 3. A new quantitative self-improving theorem for BMO functions 13 3.1. Lemmata 13 3.2. Proof of the self-improving theorem 14 4. Applications of the self-improving theorem 17 4.1. BMO-type improvement at the Orlicz spaces scale 18 4.2. BMO-type improvement at the variable Lebesgue spaces scale 24 Appendix A. Proof of Theorem 2.1 26 Acknowledgements 30 References 31

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