Fractional Sobolev embeddings and algebra property: A dyadic view

TL;DR

This paper offers a dyadic, Fourier-free approach to fractional Sobolev embeddings and algebra properties, providing explicit counterexamples and broadening understanding in low-regularity contexts.

Contribution

It introduces a dyadic decomposition method to study fractional Sobolev spaces, avoiding Fourier analysis and enabling applications in non-Fourier settings.

Findings

Dyadic approach reproduces classical embeddings

Counterexamples show algebra property failure at low regularity

Method applicable beyond Fourier-based frameworks

Abstract

This paper revisits classical fractional Sobolev embedding theorems and the algebra property of the fractional Sobolev space $H^s(\mathbb{R})$ by means of Haar functions and dyadic decompositions. The aim is to provide an alternative, hands-on approach without Fourier transform that may be transferred to settings where the latter is not available. Explicit counterexamples are constructed to show the failure of the algebra property in the low-regularity regime.