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

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 $H^{s}(\mathbb{R})$.