We consider boundedness of a certain positive dyadic operator

$$\begin{eqnarray}T^{\unicode[STIX]{x1D70E}}:L^{p}(\unicode[STIX]{x1D70E};\ell ^{2})\rightarrow L^{p}(\unicode[STIX]{x1D714}),\end{eqnarray}$$

$L^{p}$

$T^{\unicode[STIX]{x1D70E}}$

$p\in [2,\infty )$

We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.

The boundedness of Calderón–Zygmund operators is proved in the scale of the mixed Lebesgue spaces. As a consequence, the boundedness of the bilinear null forms $Q_{i j} (u,v) \,{=}\,\p_i u\p_j v \,{-}\, \p_j u\p_i v$, $Q_0(u,v)\,{=}\,u_t v_t \,{-}\,\nabla_x u\,{\cdot}\, \nabla_x v$ on various space–time mixed Sobolev–Lebesgue spaces is shown.