H\"older continuity of functions in the fractional Sobolev spaces: 1-dimensional case

TL;DR

This paper provides an elementary proof that functions in fractional Sobolev spaces on the real line are H"older continuous with a specific exponent, illustrating a particular embedding theorem in a simple setting.

Contribution

It offers a straightforward proof of the H"older continuity of fractional Sobolev functions in one dimension, simplifying the understanding of this embedding.

Findings

Functions in H^s(R) with 1/2 < s < 1 are H"older continuous with exponent s - 1/2

Elementary proof of a special case of a known embedding theorem

Clarifies the regularity properties of fractional Sobolev functions in one dimension

Abstract

This paper deals with the embedding of the Sobolev spaces of fractional order into the space of H\"older continuous functions. More precisely, we show that the function $f\in H^s(\mathbb{R})$ with $\frac{1}{2}<s<1$ is H\"older continuous with the exponent $s-\frac{1}{2}$. This is a particular case of the much stronger embedding theorems (see Section 2.8.1 in \textit{H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Pub. Co., Amsterdam, 1978.}), but here we give an elementary proof for $H^{s}(\mathbb{R})$.