Sharp extension problem characterizations for higher fractional power operators in Banach spaces

TL;DR

This paper provides sharp characterizations of higher fractional powers of generators of semigroups in Banach spaces through extension problems, establishing existence, uniqueness, and boundary conditions, and introduces new subordination formulas.

Contribution

It extends extension problem characterizations to higher order fractional powers, resolving initial condition issues and deriving explicit subordination formulas.

Findings

Existence and uniqueness of solutions to extension problems for fractional powers.

Boundary derivative characterizations of fractional powers in Banach spaces.

New explicit subordination formulas for solutions in terms of semigroup operators.

Abstract

We prove sharp characterizations of higher order fractional powers $(-L)^s$, where $s>0$ is noninteger, ofgenerators $L$ of uniformly bounded $C_0$-semigroups on Banach spaces via extension problems, which in particular include results of Caffarelli-Silvestre, Stinga-Torrea and Gal\'e-Miana-Stinga when $0<s<1$. More precisely, we prove existence and uniqueness of solutions $U(y)$, $y\geq0$, to initial value problems for both higher order and second order extension problems and characterizations of $(-L)^su$, $s>0$, in terms of boundary derivatives of $U$ at $y=0$, under the sharp hypothesis that $u$ is in the domain of $(-L)^s$. Our results resolve the question of setting up the correct initial conditions that guarantee well-posedness of both extension problems. Furthermore, we discover new explicit subordination formulas for the solution $U$ in terms of the semigroup $\{e^{tL}\}_{t\geq0}$ generated by $L$.