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 $\{{\lambda }_Q:C\to D{\}}_{Q\in \mathcal{D}}$ are positive operators from a Banach lattice $C$ to a Banach lattice $D$. We assume that the Banach lattices $C$ and $D^{\ast }$ 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 $L^{\infty }$ 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 $L_C^p(\sigma )$ and $L_D^q(\omega )$ denote the Lebesgue--Bochner spaces associated with exponents $1<p\le q<\infty$, 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 $L^{\infty }$ 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.

32 pages. The main changes are: a) Banach lattice-valued functions are considered. It is assumed that the Banach lattices have the Hardy--Littlewood property. b) The unweighted norm inequality is characterized by an endpoint testing condition and some corollaries of this characterization are stated. c) Some questions about the borderline of the vector-valued testing conditions are posed

References in corpus (4)

• Non-probabilistic proof of the A_2 theorem, and sharp weighted bounds for the q-variation of singular integrals
• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, I
• Dyadic harmonic analysis beyond doubling measures
• Operator-valued dyadic shifts and the T(1) theorem

Cited by in corpus (2)

• Two weight norm inequalities for vector-valued operators
• Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting