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We consider boundedness of a certain positive dyadic operatorT  σ   :   L  p   (  σ  ;   ℓ  2   )  →   L  p   (  ω  )  ,that arose during our attempts to develop a two‐weight theory for the Hilbert transform in    L  p   . Boundedness of    T  σ    is characterized when    p  ∊  [  2  ,  ∞  )    in terms of certain testing conditions. This requires a new Carleson‐type embedding theorem that is also proved.

A TWO‐WEIGHT INEQUALITY BETWEEN L p ( ℓ 2 ) AND L p