Estimation of multiple Laplace transforms of convex functions with an application to analytic \((C_ 0)\)-semigroups
type: Item
instance of: scholarly article
title: Estimation of multiple Laplace transforms of convex functions with an application to analytic \((C_ 0)\)-semigroups
zbMATH Open document ID: 0636.44001
author: Gen Nakamura
author: Shinnosuke Oharu
published in: Proceedings of the Japan Academy. Series A
publication date: 1986-01-01T00:00:00Z
review text: 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)].
reviewed by: Sen-Yen Shaw
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 44A30
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 26A51
zbMATH DE Number: 4036418
zbMATH Keywords: multiple Laplace transforms
zbMATH Keywords: convex functions
zbMATH Keywords: iteration of resolvents
zbMATH Keywords: Banach space
zbMATH Keywords: analytic semigroup
MaRDI profile type: Publication
full work available at URL: pjaa.62.253
OpenAlex ID: W2069052155
cites work: Remarks on generators of analytic semigroups
cites work: Semigroups of linear operators and applications to partial differential equations
DOI: 10.3792/PJAA.62.253
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description: scientific article; zbMATH DE number 4036418