  The square root of the heat operator $\sqrt{\partial_t-\Delta}$, can be
realized as the Dirichlet to Neumann map of the heat extension of data on
$\mathbb R^{n+1}$ to $\mathbb R^{n+2}_+$. In this note we obtain similar
characterizations for general fractional powers of the heat operator,
$(\partial_t-\Delta)^s$, $s\in (0,1)$. Using the characterizations we derive
properties and boundary estimates for parabolic integro-differential equations
from purely local arguments in the extension problem.

Analysis of PDEs