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

Abstract

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 Galé–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)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 {etL}t≥0 generated by L.