A full theory of Besov and Triebel-Lizorkin spaces with operator-valued Ap weights is constructed in Banach spaces using a weak reverse Hölder inequality implied by the weight condition.
Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark
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Real-variable theory of function spaces with operator-valued $A_p$ weights in Banach spaces
A full theory of Besov and Triebel-Lizorkin spaces with operator-valued Ap weights is constructed in Banach spaces using a weak reverse Hölder inequality implied by the weight condition.