The $L^p$ primitive integral

Abstract

For each $1\le p<\infty$ a space of integrable Schwartz distributions, $L^'^{\,p}$, is defined by taking the distributional derivative of all functions in $L^p$. Here, $L^p$ is with respect to Lebesgue measure on the real line. If $f\in L^'^{\,p}$ such that $f$ is the distributional derivative of $F\in L^p$ then the integral is defined as ${\int }_{-\infty }^{\infty }fG=-{\int }_{-\infty }^{\infty }F(x)g(x)dx$, where $g\in L^q$, $G(x)={\int }_0^{x}g(t)dt$ and $1/p+1/q=1$. A norm is $\Vert f{\Vert }_p^{\prime }=\Vert F{\Vert }_p$. The spaces $L^'^{\,p}$ and $L^p$ are isometrically isomorphic. Distributions in $L^'^{\,p}$ share many properties with functions in $L^p$. Hence, $L^'^{\,p}$ is reflexive, its dual space is identified with $L^q$, there is a type of Hölder inequality, continuity in norm, convergence theorems, Gateaux derivative. It is a Banach lattice and abstract $L$-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well-defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes $L^'^{\,1}$ into a Banach algebra isometrically isomorphic to the convolution algebra on $L^1$. Spaces of higher order derivatives of $L^p$ functions are defined. These are also Banach spaces isometrically isomorphic to $L^p$.