  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 $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite
measure space. Consider two filtrations of $\Sigma$ by successive refinement of
two atomic $\sigma$-algebras $\Sigma_\mathrm{a}, \Sigma_\mathrm{b}$ 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 $(\Sigma_\mathrm{a}, \Sigma_\mathrm{b})$
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 Calder\'on-Zygmund
operators on $(\Omega,\mu, d)$ under additional conditions on
$(\Sigma_\mathrm{a}, \Sigma_\mathrm{b})$. 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 $e^{\pm |x|^\alpha}$ for
any $\alpha > 0$ or $(1 + |x|^{\beta})^{-1}$ for any $\beta \gtrsim n^{3/2}$. A
(limited) comparison with Tolsa's RBMO is also possible. On the other hand, a
more intrinsic formulation yields a Calder\'on-Zygmund theory adapted to
regular filtrations over $(\Sigma_\mathrm{a}, \Sigma_\mathrm{b})$ 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.

Classical Analysis and ODEs;Operator Algebras