Extension Properties and Boundary Estimates for a Fractional Heat Operator

The square root of the heat operator $\sqrt{{\mathrm{\partial }}_t-\mathrm{\Delta }}$, 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, $({\mathrm{\partial }}_t-\mathrm{\Delta }{)}^s$, $s\in (0,1)$. Using the characterizations we derive properties and boundary estimates for parabolic integro-differential equations from purely local arguments in the extension problem.