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arXiv · 2603.26565

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

Valentia Fragkiadaki·Mishko Mitkovski·Cody B. Stockdale
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Abstract

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.

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Valentia Fragkiadaki, Mishko Mitkovski, Cody B. Stockdale. 2026-03-27. Local dyadic fractional Sobolev spaces: paraproducts, commutators, and the algebra property. https://arxiv.org/abs/2603.26565

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