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}$}$arepositiveoperatorsfromaBanachlattice$C$ to a Banach lattice $D$. We assume that the Banach lattices $C$ and $D^{\ast }$ 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 $L_C^p(\sigma )\to L_D^q(\omega )$ boundedness of the operator $T_{\lambda }(\cdot \sigma )$ is characterized by the direct and the dual $L^{\mathrm{\infty }}$ testing conditions: $$\parallel 1_Q{T}_{\lambda }(1_{Q}f\sigma ){\parallel }_{L_D^q(\omega )}\lesssim \parallel f{\parallel }_{L_C^{\mathrm{\infty }}(Q,\sigma )}\sigma (Q{)}^{1\mathrm{/}p},$$ $$\parallel 1_Q{T}_{\lambda }^{\ast }(1_{Q}g\omega ){\parallel }_{L_{C^{\ast }}^{p^{\mathrm{\prime }}}(\sigma )}\lesssim \parallel g{\parallel }_{L_{D^{\ast }}^{\mathrm{\infty }}(Q,\omega )}\omega (Q{)}^{1\mathrm{/}{q}^{\mathrm{\prime }}}\mathrm{.}$$ Here $L_C^p(\sigma )$ and $L_D^q(\omega )$ denote the Lebesgue–Bochner spaces associated with exponents $1<p\le q<\mathrm{\infty }$, and locally finite Borel measures $\sigma$ and $\omega$. In the unweighted case, we show that the $L_C^p(\mu )\to L_D^p(\mu )$ boundedness of the operator $T_{\lambda }(\cdot \mu )$ is equivalent to the endpoint direct $L^{\mathrm{\infty }}$ testing condition: $$\parallel 1_Q{T}_{\lambda }(1_{Q}f\mu ){\parallel }_{L_D^1(\mu )}\lesssim \parallel f{\parallel }_{L_C^{\mathrm{\infty }}(Q,\mu )}\mu (Q)\mathrm{.}$$ This condition is manifestly independent of the exponent $p$. By specializing this to particular cases, we recover some earlier results in a unified way.