Local dyadic fractional Sobolev spaces: paraproducts, commutators, and the algebra property

We characterize the boundedness and compactness of dyadic paraproducts on local dyadic fractional Sobolev spaces, $H^s$. We apply this result to establish the algebra property for $H^s$ when $s\in (\frac{1}{2},1)$ and to deduce the boundedness and compactness of commutators with the Haar shift on $H^s$. Our conditions are stated in terms of new dyadic fractional ${\text{BMO}}^s$ and ${\text{CMO}}^s$ conditions involving the dyadic fractional Sobolev capacity, and our proof uses a new dyadic fractional version of the Carleson embedding theorem.