Israel Journal of Mathematics

, Volume 219, Issue 1, pp 71–114 | Cite as

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

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Abstract

We study the operator-valued positive dyadic operator
$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$
where the coefficients {λ Q : CD} QD 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 (σ) → L D q (ω) boundedness of the operator T λ( · σ) is characterized by the direct and the dual L testing conditions:
$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$
,
$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$
.

Here L C p (σ) and L D q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < pq < ∞, and locally finite Borel measures σ and ω.

In the unweighted case, we show that the L C p (μ) → L D p (μ) boundedness of the operator T λ( · μ) is equivalent to the end-point direct L testing condition:
$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$
.

This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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