  For each $1\leq p<\infty$ a Banach space of integrable Schwartz distributions
is defined by taking the distributional derivative of all functions in
$L^p({\mathbb R})$. Such distributions can be integrated when multiplied by a
function that is the integral of a function in $L^q({\mathbb R})$, where $q$ is
the conjugate exponent of $p$. The heat equation on the real line is solved in
this space of distributions. The initial data is taken to be the distributional
derivative of an $L^p({\mathbb R})$ function. The solutions are shown to be
smooth functions. Initial conditions are taken on in norm. Sharp estimates of
solutions are obtained and a uniqueness theorem is proved.
