Extension Properties and Boundary Estimates for a Fractional Heat Operator

TL;DR

This paper characterizes fractional powers of the heat operator as boundary maps of heat extensions, deriving properties and estimates for related parabolic equations using local extension arguments.

Contribution

It extends known results for the square root of the heat operator to general fractional powers, providing new boundary estimates and properties.

Findings

Characterization of fractional heat operators as Dirichlet-to-Neumann maps.

Boundary estimates for parabolic integro-differential equations.

Use of local extension arguments to derive properties.

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.