Dyadic fractional Sobolev spaces: Embeddings and algebra property

This paper studies a dyadic version of fractional Sobolev spaces in ${\mathbb{R}}^n$ for $n\ge 1$. It provides new proofs of the corresponding fractional Sobolev embedding as well as the algebra property of the spaces, which rely solely on dyadic techniques and in particular bypass the Fourier transform. Specific counterexamples are constructed to verify the failure of the algebra property in low-regularity ranges.