Hölder continuity of functions in the fractional Sobolev spaces: 1-dimensional case

Yan Rybalko

Published 2023-08-11Version 1

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 $s-\frac{1}{2}$. Our result is an improvement of the Sobolev embedding theorem in the one dimensional case, which states that every such a function $f$ is continuous. The H\"older exponent $s-\frac{1}{2}$ is consistent with the Morrey's inequality, which yields that $f\in H^1(\mathbb{R})$ is H\"older continuous with the exponent $\frac{1}{2}$.