Two weight inequality for vector-valued positive dyadic operators by parallel stopping cubes

We study the vector-valued positive dyadic operator $$T_\lambda(f\sigma):=\sum_{Q\in\mathcal{D}} \lambda_Q \int_Q f \mathrm{d}\sigma 1_Q,$$ where the coefficients $\{\lambda_Q:C\to D\}_{Q\in\mathcal{D}}$ are positive operators from a Banach lattice $C$ to a Banach lattice $D$. We assume that the Banach lattices $C$ and $D^*$ each have the Hardy--Littlewood property. An example of a Banach lattice with the Hardy--Littlewood property is a Lebesgue space. In the two-weight case, we prove that the $L^p_C(\sigma)\to L^q_D(\omega)$ boundedness of the operator $T_\lambda( \cdot \sigma)$ is characterized by the direct and the dual $L^\infty$ testing conditions: $$\lVert 1_Q T_\lambda(1_Q f \sigma)\rVert_{L^q_D(\omega)}\lesssim \lVert f\rVert_{L^\infty_C(Q,\sigma)} \sigma(Q)^{1/p},$$ $$\lVert1_Q T^*_{\lambda}(1_Q g \omega)\rVert_{L^{p'}_{C^*}(\sigma)}\lesssim \lVert g\rVert_{L^\infty_{D^*}(Q,\omega)} \omega(Q)^{1/q'}.$$ Here $L^p_C(\sigma)$ and $L^q_D(\omega)$ denote the Lebesgue--Bochner spaces associated with exponents $1<p\leq q<\infty$, and locally finite Borel measures $\sigma$ and $\omega$. In the unweighted case, we show that the $L^p_C(\mu)\to L^p_D(\mu)$ boundedness of the operator $T_\lambda( \cdot \mu)$ is equivalent to the endpoint direct $L^\infty$ testing condition: $$\lVert1_Q T_\lambda(1_Q f \mu)\rVert_{L^1_D(\mu)}\lesssim \lVert f\rVert_{L^\infty_C(Q,\mu)} \mu(Q).$$ This condition is manifestly independent of the exponent $p$. By specializing this to particular cases, we recover some earlier results in a unified way.