The Fourier transform in Lebesgue spaces

For each $f\in L^p(\mathbb{R})$ ($1\le p<\mathrm{\infty }$) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each $p$ a norm is defined so that the space Fourier transforms is isometrically isomorphic to $L^p(\mathbb{R})$. There is an exchange theorem and inversion in norm.