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

TL;DR

This paper characterizes the boundedness and compactness of dyadic paraproducts on local dyadic fractional Sobolev spaces, establishing algebra properties and commutator boundedness using new dyadic fractional conditions and a Carleson embedding theorem.

Contribution

It introduces new dyadic fractional BMO and CMO conditions and a dyadic fractional Carleson embedding theorem to analyze operators on Sobolev spaces.

Findings

Characterization of boundedness and compactness of dyadic paraproducts on $H^s$

Establishment of algebra property for $H^s$ when $s in (1/2,1)$

Boundedness and compactness results for commutators with Haar shift

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.