The heat equation with the $L^p$ primitive integral

For each $1\le p<\mathrm{\infty }$ 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.