A functional equation of Bohnemblust and positive homogeneous functionals
Bohnenblust had a characterization of \(L^p\)-space among partially ordered normed linear spaces in the paper: \textit{H. F. Bohnenblust} [An axiomatic characterization of \(L_p\)-spaces, Duke Math. J. 6, 627-640 (1940; Zbl 0024.04101)]. By almost the same ideas, the author obtains the following: Theorem. Let \(E\) be a partially ordered linear space which is a lattice and let \({\mathcal L}\subset E\) be a cone. Let \(S\) be a subset of \({\mathcal L}\) such that if \(x,y\in S\) are orthogonal then \(x+ y\in S\). Let \(F:{\mathcal L}\to [0,\infty)\) be a \(m\) homogeneous monotone functional such that for all triples \(s_1\), \(s_2\), \(s_3\) of nonnegative real numbers there are elements \(x_1\), \(x_2\), \(x_3\) in \(S\) mutually orthogonal with \(F(x_j)= s_j\), \(j= 1,2,3\). Suppose that there is a two variables function \(h\) such that \[ F(x+ y)= h(F(x),F(y)\quad\text{for all orthogonal }x,y\in S. \] Then there exists \(p\), \(0< p\leq\infty\) such that \[ F(x+y)^p= F(x)^p+ F(y)^p\quad\text{for all orthogonal }x,y\in S. \] He obtains also the following: Theorem. Let \((\Omega,{\mathcal A},\mu)\) be a measure space with at least two disjoint sets of finite and positive measure. Suppose that \(\psi\) is a strictly increasing function and \(m\) is a nondecreasing and onto function. Then \(F\) is a \(m\)-homogeneous positive functional if and only if there exist real numbers \(p,q>0\) such that \[ m(s)= s^q,\quad \phi(s)= \phi(1)s^{pq},\quad \psi(s)= \psi(1) s^{1/p}, \] where he considers functions \(m,\psi,\phi: [0,\infty)\to [0,\infty)\) with \(\phi(0)= \psi(0)= 0\) and \(\phi\) nonnull and \(F\) is the functional defined over \({\mathcal L}\) by \[ F(f)= \psi\Biggl(\int_\Omega \phi(f) d\mu\Biggr). \] He also considers the case when the Hölder inequality is an equality.
scientific article; zbMATH DE number 1496349