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

Abstract

We study the vector-valued positive dyadic operator \[T_λ(fσ):=\sum_{Q\in\mathcal{D}} λ_Q \int_Q f \mathrm{d}σ1_Q,\] where the coefficients $\{λ_Q:C\to D\}_{Q\in\mathcal{D}}$ 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 $L^p_C(σ)\to L^q_D(ω)$ boundedness of the operator $T_λ( \cdot σ)$ is characterized by the direct and the dual $L^\infty$ testing conditions: \[ \lVert 1_Q T_λ(1_Q f σ)\rVert_{L^q_D(ω)}\lesssim \lVert f\rVert_{L^\infty_C(Q,σ)} σ(Q)^{1/p},\] \[ \lVert1_Q T^*_λ(1_Q g ω)\rVert_{L^{p'}_{C^*}(σ)}\lesssim \lVert g\rVert_{L^\infty_{D^*}(Q,ω)} ω(Q)^{1/q'}.\] Here $L^p_C(σ)$ and $L^q_D(ω)$ denote the Lebesgue--Bochner spaces associated with exponents $1<p\leq q<\infty$, and locally finite Borel measures $σ$ and $ω$. In the unweighted case, we show that the $L^p_C(μ)\to L^p_D(μ)$ boundedness of the operator $T_λ( \cdot μ)$ is equivalent to the endpoint direct $L^\infty$ testing condition: \[ \lVert1_Q T_λ(1_Q f μ)\rVert_{L^1_D(μ)}\lesssim \lVert f\rVert_{L^\infty_C(Q,μ)} μ(Q).\] This condition is manifestly independent of the exponent $p$. By specializing this to particular cases, we recover some earlier results in a unified way.