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 $BM{O}_{\mathbb{X}}$ and the best-constant seminorm $BM{O}_{\mathbb{X}}^{\ast }$. 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 ${\mathrm{\Lambda }}_{\mathbb{X}}$, 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 ${\mathrm{\Lambda }}_{\mathbb{X}}(\lambda )>0$ for some $0<\lambda <1\mathrm{/}2$ implies the embedding $$BM{O}_{\mathbb{X}}^{\ast }\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)\hookrightarrow BMO\mathrm{.}$$ 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 $T_X({\eta }_0)<\mathrm{\infty }$, where ${\eta }_0$ is the sparsity parameter arising from the local sparse domination formula, implies $$BMO\hookrightarrow BM{O}_{\mathbb{X}}\mathrm{.}$$ We also provide a sufficient small-set criterion for this sparse testing condition in terms of an upper testing functional ${\mathrm{\Psi }}_{\mathbb{X}}$.