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 :
C →
D}
Q∈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 (
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 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