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.Comment: 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 pose

On Two-Weight Norm Inequalities for Positive Dyadic Operators

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Equivalence of sparse and Carleson coefficients for general sets

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

We study the operator-valued positive dyadic operator T-lambda (f sigma) := Sigma(Q is an element of D) lambda(Q) integral(Q) f d sigma 1(Q,) where the coefficients {lambda(Q) : C -> D}(Q is an element of 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-C(p) (sigma) -> L-D(q)(omega) boundedness of the operator T-lambda(. sigma) is characterized by the direct and the dual L-infinity testing conditions: parallel to 1(Q)T(lambda) (1(Q)f sigma)parallel to(LDq) ((omega)) less than or similar to parallel to f parallel to(LC infinity) ((Q,sigma)) sigma(Q)(1/p), parallel to 1(Q)T(lambda)* (1(Q)g omega)parallel to(LC*p') ((sigma)) less than or similar to parallel to g parallel to(LD*infinity) ((Q,omega)) omega(Q)(1/q'), Here L-C(p) (sigma) and L-D(q) (omega) denote the Lebesgue-Bochner spaces associated with exponents 1 L-D(p) (mu) boundedness of the operator T-lambda(. mu) is equivalent to the end-point direct L-infinity testing condition: parallel to 1(Q)T(lambda) (1(Q)f mu)parallel to(LD1) (mu) less than or similar to parallel to f parallel to(LC infinity) ((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.Peer reviewe