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Let w be in the Muckenhoupt $A_{\infty }$ weight class. We show that the Riesz transforms are bounded on the weighted Carleson measure space $CM{O}_w^p$, the dual of the weighted Hardy space $H_w^p$, 0 < p ≤ 1.

We first show that a linear operator which is bounded on $L{²}_w$ with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space $h{¹}_w$ if and only if this operator is uniformly bounded on all $h{¹}_w$-atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to $h{¹}_w$.