Banach spaces for the Schwartz distributions

Abstract

This paper is a survey of a new family of Banach spaces ${KS}^2$ and $SD^2$ that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the $L^p$ spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions $\mathcal{D}$ as a continuous dense embedding. Thus, by the Hahn-Banach theorem, $\mathcal{D}' \subset \mathcal{B}'$. A new family that extend the space of functions of bounded mean oscillation $BMO[\mathbb{R}^n]$, to include the HK-integrable functions are also introduced. We provide a few applications. We use ${KS}^2$ to provide a simple solution to the generator (with unbounded coefficients) problem for Markov processes. We also use $SD^2$ to provide the best possible a priori bound for the nonlinear term of the Navier-Stokes equation.