Estimation of multiple Laplace transforms of convex functions with an application to analytic \((C_ 0)\)-semigroups
For a nonnegative convex function f on (0,\(\infty)\) let \[ J_ n(h_ 1,...,h_ n):=(\prod^{n}_{i=1}h_ i^{- 1})\int^{\infty}_{0}...\int^{\infty}_{0} \exp (- \sum^{n}_{i=1}h_ i^{-1}\xi_ i)f(\sum^{n}_{i=1}\xi_ i)d\xi_ 1...d\xi_ n \] for \(h_ i>0\), \(i=1,2,...,n\), \(n=1,2,..\). The paper under review shows that: (1) if \(m\leq n\), then \(J_ n(t/n,...,t/n)\leq J_ n(h_ 1,...,h_ n)\leq J_ m(t/m,...,t/m)\) for all \(h_ i\in (0,t/m)\), \(i=1,...,n\), with \(\sum^{n}_{i=t}h_ i=t\); \((2)\quad J_ m(h,...,h)=(m-1)!^{-1}h^{- m}\int^{\infty}_{0}\xi^{m-1}e^{-(\xi /h)}f(\xi)d\xi.\) In particular, if \(f(\xi)=C_{\alpha}\xi^{-\alpha},\) then (1) and (2) imply that \(J_ n(h_ 1,...,h_ n)\leq (m-1)!^{-1}C_{\alpha}\Gamma (m-\alpha)(\sum^{n}_{i=1}h_ i)^{-\alpha}\) whenever \(0\leq h_ i\leq m^{-1}\sum^{n}_{j=1}h_ j\), \(i=1,...,n\). This inequality is then applied to obtain an estimation for the iteration of resolvents of the generator A of an analytic semigroup, namely, \(\| A^{m-1}A_{\theta}\prod^{n}_{i=1}(I-h_ iA)^{-1}\| \leq C_{\alpha}(\sum^{n}_{i=1}h_ i)^{-\alpha},\) and to yield a characterization of the generator of an analytic semigroup, which involves the characterization due to \textit{M. G. Crandall}, \textit{A. Pazy} and \textit{L. Tartar} [Israel J. Math. 32, 363-374 (1979; Zbl 0436.47028)].
scientific article; zbMATH DE number 4036418