This is an announcement for the paper "The $L^p$ primitive integral" by Erik Talvila.

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 $L^p$.

Archive classification: math.CA math.FA

Mathematics Subject Classification: 46E30, 46F10, 46G12 (Primary) 42A38, 42A85, 46B42, 46C05 (Secondary)

Submitted from: Erik.Talvila@ufv.ca

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1208.3694

or

http://arXiv.org/abs/1208.3694