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Distributions, their primitives and integrals with applications to differential equations

In this paper we will study integrability of distributions whose primitives are left regulated functions and locally or globally integrable in the Henstock--Kurzweil, Lebesgue or Riemann sense. Corresponding spaces of distributions and their primitives are defined and their properties are studied. Basic properties of primitive integrals are derived and applications to systems of first order nonlinear distributional differential equations and to an $m$th order distributional differential equation are presented. The domain of solutions can be unbounded, as shown by concrete examples.