The $L^p$ primitive integral

For each $1\le p<\mathrm{\infty }$ 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$1/p+1/q=1$\mathrm{.}Anormis$\lVert f\rVert’_p=\lVert F\rVert_p$. 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 H"older 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 $L^p$.