Dyadic fractional Sobolev spaces: Embeddings and algebra property

TL;DR

This paper introduces dyadic fractional Sobolev spaces, providing new dyadic-based proofs for embeddings and algebra properties, and constructs counterexamples for low-regularity failure.

Contribution

It offers novel dyadic techniques for fractional Sobolev spaces, bypassing Fourier analysis, and clarifies algebra property limitations in low-regularity regimes.

Findings

New dyadic proofs of Sobolev embeddings

Verification of algebra property in certain ranges

Counterexamples showing failure at low regularity

Abstract

This paper studies a dyadic version of fractional Sobolev spaces in $\mathbb{R}^n$ for $n\geq 1$. 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.