  We study a generalized class of supersolutions, so-called $p$-supercaloric
functions, to the parabolic $p$-Laplace equation. This class of functions is
defined as lower semicontinuous functions that are finite in a dense set and
satisfy the parabolic comparison principle. Their properties are relatively
well understood for $p\geq 2$, but little is known in the fast diffusion case
$1<p<2$. Every bounded $p$-supercaloric function belongs to the natural Sobolev
space and is a weak supersolution to the parabolic $p$-Laplace equation for the
entire range $1<p<\infty$. Our main result shows that unbounded
$p$-supercaloric functions are divided into two mutually exclusive classes with
sharp local integrability estimates for the function and its weak gradient in
the supercritical case $\frac{2n}{n+1}<p<2$. The Barenblatt solution and the
infinite point source solution show that both alternatives occur. Barenblatt
solutions do not exist in the subcritical case $1<p\leq \frac{2n}{n+1}$ and the
theory is not yet well understood.
