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

This paper deals with the embedding of the Sobolev spaces of fractional order into the space of Hölder continuous functions. More precisely, we show that the function $f\in H^s(\mathbb{R})$ with $\frac{1}{2}<s<1$ is Hölder 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 $\mathit{\text{H.}}\mathit{\text{Triebel,}}\mathit{\text{Interpolation}}\mathit{\text{Theory,}}\mathit{\text{Function}}\mathit{\text{Spaces,}}\mathit{\text{Differential}}\mathit{\text{Operators,}}\mathit{\text{North-Holland}}\mathit{\text{Pub.}}\mathit{\text{Co.,}}\mathit{\text{Amsterdam,}}\mathit{\text{1978.}}$), but here we give an elementary proof for $H^s(\mathbb{R})$.