Extension properties and boundary estimates for a fractional heat operator

The square root of the heat operator ∂t−Δ, can be realized as the Dirichlet to Neumann map of the heat extension of data on Rn+1Rn+1 to R+n+2. In this note we obtain similar characterizations for general fractional powers of the heat operator, (∂t−Δ)s(∂t−Δ)s, s∈(0,1)s∈(0,1). Using the characterizations we derive properties and boundary estimates for parabolic integro-differential equations from purely local arguments in the extension problem.