Optimal function spaces for the Laplace transform
type: Item
instance of: scholarly article
title: Optimal function spaces for the Laplace transform
zbMATH Open document ID: 1400.46022
author: Eva Buriánková
author: D. E. Edmunds
author: Luboš Pick
published in: Revista Matemática Complutense
publication date: 2017-11-02T00:00:00Z
review text: The paper under review deals with the characterization of the optimal rearrangement invariant spaces (r.i) associated to the Laplace transform. If \(\mathcal L:X \to Y\) is the Laplace transform for functions acting in \([0,\infty)\) and defined between Banach function spaces \(X\) and \(Y,\) the paper studies the following questions: (1) if \(X\) is fixed, what is the smallest r.i. Banach function space \(\hat Y\) such that the range of the operator lies in \(\hat Y\)?, and (2) if \(Y\) is fixed, what is the optimal domain, that is, the biggest rearrangement invariant space \(\hat X\) to which the operator can be extended? After introducing some definitions, the authors fix the framework. First, they prove an integral inequality for the rearrangement of the Laplace transform that is used for proving the theorems that give the answers to the abovementioned questions. Theorem 3.4 proves that, under the assumption that \(\min \{1,1/t\} \in X'\), the Köthe dual of the r.i. space \(X\), the function norm \[ \rho(g):= \Big\| \int_0^{1/t} g^*(s) \, ds \Big\|_X \] satisfies that the associate (dual) norm \(\rho'\) determines the optimal (smallest) r.i. space \(Y.\) If the function \(\min \{1,1/t\}\) does not belong to \(X'\), then there is no r.i. space \(Y\) such that \(\mathcal L:X \to Y\). The second question is answered in Theorem 3.5, where a ``dual'' result can be found. In this case, under a similar assumption on the \(\min\) function, the optimal (biggest) r.i. domain space is determined by the functional \[ \rho(f):= \Big\| \int_0^{1/t} f^*(s) \, ds \Big\|_Y, \] and a similar conclusion as in Theorem 3.4 holds if \(\min \{1,1/t\} \) does not belong to \(Y\). Concrete computations applying these results are also given for the case of Lorentz spaces, which helps to fix the abstract results obtained before for a class of well-known spaces. Technically, the authors use a way of approaching the problem of optimization from a different point of view than some recent res…
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 26D20
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 6802587
zbMATH Keywords: Laplace transform
zbMATH Keywords: rearrangement-invariant spaces
zbMATH Keywords: optimal domain
zbMATH Keywords: optimal range
zbMATH Keywords: Lorentz spaces
MaRDI profile type: Publication
full work available at URL: s13163-017-0234-5
OpenAlex ID: W2732710459
Wikidata QID: Q59311780
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DOI: 10.1007/S13163-017-0234-5
description: scientific article; zbMATH DE number 6802587