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

We prove sharp characterizations of higher order fractional powers (-L)(s), where s>0 is noninteger, of generators L of uniformly bounded C0-semigroups on Banach spaces via extension problems, which in particular include results of Caffarelli-Silvestre, Stinga-Torrea and Gal & eacute;-Miana-Stinga when 0 < s < 1. More precisely, we prove existence and uniqueness of solutions U(y), y >= 0, to initial value problems for both higher order and second order extension problems and characterizations of (-L)(s)u, 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 >= 0) generated by L. (c) 2024 Elsevier Inc. All rights reserved.