Extension problem and fractional operators: semigroups and wave equations

TL;DR

This paper generalizes the extension problem for fractional operators, connecting it with semigroup theory and wave equations, and provides integral representations for these fractional powers.

Contribution

It extends existing results to a broader class of operators, including generators of semigroups and operators with imaginary symbols, using heat and wave equation solutions.

Findings

Extended the extension problem to generators of semigroups and operators with imaginary symbols.

Provided integral representations involving heat and wave equations.

Broadened the applicability of fractional operator characterizations.

Abstract

We extend results of Caffarelli--Silvestre and Stinga--Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded $C_0$ semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation.