Poincaré like inequalities for semigroups, cosine and sine operator functions (Q1014777)

The author considers inequalities of the form \[ \|\Delta_n(\cdot) y\|_{L^\nu(0, a; E)} \leq C(\nu, q) \|S(\cdot)A^n y\|_{L^q(0, a; E)}, \] where \(S(t)\) is a strongly continuous semigroup in the Banach space \(E\), with \(A\) the infinitesimal generator of \(S(t)\), \(y \in D(A^n)\), and \(\Delta_n(t)\) denotes the tail of the Taylor expansion of \(S(t)\), \[ \Delta_n(t) = S(t) y - \sum_{k=0}^{n-1} {t^k \over k!} A^k y = {1 \over (n - 1)!} \int_0^t (t - u)^{n-1}S(u) A^n y \, du . \] The constant \(C(\nu, q)\) is characterized explicitly. There are corresponding results for cosine and sine operator functions. The author applies the semigroup estimates to the heat equation and the cosine-sine estimates to the wave equation, in both cases in one space variable.