The $L^p$ primitive integral

TL;DR

This paper introduces a new space of integrable Schwartz distributions, $L'^p$, derived from $L^p$ functions, establishing its properties, duality, and applications like convolution and solving boundary value problems.

Contribution

It defines the space $L'^p$ as derivatives of $L^p$ functions, explores its structure, and demonstrates its utility in analysis and PDEs, extending classical $L^p$ theory.

Findings

$L'^p$ is isometrically isomorphic to $L^p$

$L'^p$ is a reflexive Banach lattice and $L$-space

Convolution with Poisson kernel solves the Dirichlet problem

Abstract

For each $1\leq p<\infty$ 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 $\lVert f\rVert'_p=\lVert F\rVert_p$. 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 $L^p$.