Abstract
We study the operator-valued positive dyadic operator
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 pC (σ) → L qD (ω) boundedness of the operator T λ( · σ) is characterized by the direct and the dual L ∞ testing conditions:
,
.
Here L p C (σ) and L q D (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < p ≤ q < ∞, and locally finite Borel measures σ and ω.
In the unweighted case, we show that the L p C (μ) → L p D (μ) boundedness of the operator T λ( · μ) is equivalent to the end-point direct L ∞ testing condition:
.
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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Hänninen, T.S. Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes. Isr. J. Math. 219, 71–114 (2017). https://doi.org/10.1007/s11856-017-1474-2
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DOI: https://doi.org/10.1007/s11856-017-1474-2