Sharp median testing and sparse criteria for generalized \(BMO\) spaces
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 \(Ψ_{\mathbb X}\).