The one-dimensional heat equation in the Alexiewicz norm

A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock-Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let Theta(t) (x) = exp(-x(2)/(4t))/root 4 pi t be the heat kernel. With initial data f that is the distributional derivative of a continuous function, it is shown that u(t)(x) := u(x, t) := f * Theta(t)(x) is a classical solution of the heat equation u(11) = u(2). The estimate parallel to f * Theta(t)parallel to(infinity) <= parallel to f parallel to/root pi t holds. The Alexiewicz norm is parallel to f parallel to := sup(I) vertical bar integral(I) f vertical bar, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, parallel to u(t) - f parallel to -> 0 as t -> 0(+). The solution of the heat equation is unique under the assumptions that parallel to u(t)parallel to is bounded and u(t) -> f in the Alexiewicz norm for some integrable f. The heat equation is also considered with initial data that is the n th derivative of a continuous function and in weighted spaces such that integral(infinity)(-infinity)f (x) exp(-ax(2)) dx exists for some a > 0. Similar results are obtained.