Optimal function spaces for the Laplace transform
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 results that are also cited in the article and are based on the integration of vector measures (on \(\sigma\)-algebras). The authors use function norms, classical integration inequalities regarding decreasing rearrangements, and interpolation instead. In a sense, this is more adequate, since the spaces are considered to be composed by functions acting in a proper \(\sigma\)-finite space (\([0,\infty)\) with Lebesgue measure). Finally, this reviewer learned from the bibliography of the work that he is no longer co-author of the book [\textit{S. Okada} et al., Optimal domain and integral extension of operators, acting in function spaces. Basel: Birkhäuser (2008; Zbl 1145.47027)], which was published along with Professors S.\,Okada and W.\,Ricker in 2008.... Bad luck!
scientific article; zbMATH DE number 6802587
scientific article