The Fourier transform in Lebesgue spaces

Abstract

For each $f\in L^p(\mathbb{R})$ ($1\le p<\infty$) 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.

Error corrected in Theorem 4.2(g). Thank you to André Kowacs for pointing this out