Large BMO spaces vs interpolation

Abstract

In this paper we introduce a class of BMO spaces which interpolate with $L_p$ and are sufficiently large to serve as endpoints for new singular integral operators. More precisely, let $(Ω, Σ, μ)$ be a $σ$-finite measure space. Consider two filtrations of $Σ$ by successive refinement of two atomic $σ$-algebras $Σ_\mathrm{a}, Σ_\mathrm{b}$ having trivial intersection. Construct the corresponding truncated martingale BMO spaces. Then, the intersection seminorm only leaves out constants and we provide a quite flexible condition on $(Σ_\mathrm{a}, Σ_\mathrm{b})$ so that the resulting space interpolates with $L_p$ in the expected way. In the presence of a metric $d$, we obtain endpoint estimates for Calderón-Zygmund operators on $(Ω,μ, d)$ under additional conditions on $(Σ_\mathrm{a}, Σ_\mathrm{b})$. These are weak forms of the \lq isoperimetric\rq${}$ and the \lq locally doubling\rq${}$ properties of Carbonaro/Mauceri/Meda which admit less concentration at the boundary. Examples of particular interest include densities of the form $e^{\pm |x|^α}$ for any $α> 0$ or $(1 + |x|^β)^{-1}$ for any $β\gtrsim n^{3/2}$. A (limited) comparison with Tolsa's RBMO is also possible. On the other hand, a more intrinsic formulation yields a Calderón-Zygmund theory adapted to regular filtrations over $(Σ_\mathrm{a}, Σ_\mathrm{b})$ without using a metric. This generalizes well-known estimates for perfect dyadic and Haar shift operators. In contrast to previous approaches, ours extends to matrix-valued functions (via recent results from noncommutative martingale theory) for which only limited results are known and no satisfactory nondoubling theory exists so far.