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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2606.01688 |
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| _version_ | 1866911739340128256 |
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| author | Sababe, Saeed Hashemi |
| author_facet | Sababe, Saeed Hashemi |
| contents | We study generalized \(BMO\)-type spaces associated with a normalized family of local quasi-Banach function spaces \(\mathbb X=\{X_Q\}_{Q\subset\mathbb R^n}\). For such a family we consider two oscillation seminorms: the mean-based seminorm \(BMO_{\mathbb X}\) and the best-constant seminorm \(BMO_{\mathbb X}^{*}\). The main purpose of the paper is to separate the two mechanisms that govern their comparison with classical \(BMO\).
First, we introduce a lower median-testing functional \(Λ_{\mathbb X}\), which measures the nondegeneracy of the local norms on subsets occupying a fixed positive proportion of a cube. Using the John--Strömberg median oscillation characterization of \(BMO\), we prove that the condition \(Λ_{\mathbb X}(λ)>0\) for some \(0<λ<1/2\) implies the embedding \[
BMO_{\mathbb X}^{*}\cap L^1_{\mathrm{loc}}(\mathbb R^n)
\hookrightarrow BMO . \] Second, we introduce a sparse testing seminorm \(T_{\mathbb X}\), which measures the compatibility of the local norms with sparse sums of characteristic functions. Using a sparse domination principle for \(BMO\) oscillation, we prove that \(T_X(η_0)<\infty\), where \(η_0\) is the sparsity parameter arising from the local sparse domination formula, implies \[
BMO\hookrightarrow BMO_{\mathbb X} . \] We also provide a sufficient small-set criterion for this sparse testing condition in terms of an upper testing functional \(Ψ_{\mathbb X}\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01688 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp median testing and sparse criteria for generalized \(BMO\) spaces Sababe, Saeed Hashemi Functional Analysis Operator Algebras 42B20, 42B25, 42B35, 46E30 We study generalized \(BMO\)-type spaces associated with a normalized family of local quasi-Banach function spaces \(\mathbb X=\{X_Q\}_{Q\subset\mathbb R^n}\). For such a family we consider two oscillation seminorms: the mean-based seminorm \(BMO_{\mathbb X}\) and the best-constant seminorm \(BMO_{\mathbb X}^{*}\). The main purpose of the paper is to separate the two mechanisms that govern their comparison with classical \(BMO\). First, we introduce a lower median-testing functional \(Λ_{\mathbb X}\), which measures the nondegeneracy of the local norms on subsets occupying a fixed positive proportion of a cube. Using the John--Strömberg median oscillation characterization of \(BMO\), we prove that the condition \(Λ_{\mathbb X}(λ)>0\) for some \(0<λ<1/2\) implies the embedding \[ BMO_{\mathbb X}^{*}\cap L^1_{\mathrm{loc}}(\mathbb R^n) \hookrightarrow BMO . \] Second, we introduce a sparse testing seminorm \(T_{\mathbb X}\), which measures the compatibility of the local norms with sparse sums of characteristic functions. Using a sparse domination principle for \(BMO\) oscillation, we prove that \(T_X(η_0)<\infty\), where \(η_0\) is the sparsity parameter arising from the local sparse domination formula, implies \[ BMO\hookrightarrow BMO_{\mathbb X} . \] We also provide a sufficient small-set criterion for this sparse testing condition in terms of an upper testing functional \(Ψ_{\mathbb X}\). |
| title | Sharp median testing and sparse criteria for generalized \(BMO\) spaces |
| topic | Functional Analysis Operator Algebras 42B20, 42B25, 42B35, 46E30 |
| url | https://arxiv.org/abs/2606.01688 |