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, 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 Galé-Miana-Stinga when $0<s<1$. More precisely, we prove existence and uniqueness of solutions $U(y)$, $y\ge 0$, 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 $\{e^{tL}{\}}_{t\ge 0}$ generated by $L$.