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.
Categories: math.AP