Extension Properties and Boundary Estimates for a Fractional Heat Operator

Abstract

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.

References in corpus (3)

• An extension problem for the fractional derivative defined by Marchaud
• Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation
• Decay estimates for time-fractional and other non-local in time subdiffusion equations in ${\mathbb{R}}^d$