Smooth renormings of the Lebesgue-Bochner function space \(L^1(\mu ,X)\) (Q2891268)

An important field of research in the area of vector-valued integration theory is the analysis of the possible or impossible transfer of properties of the norm on a Banach space \(X\) to the norm on the Lebesgue-Bochner spaces \(L^p(\mu,X)\). Typically, the rotundity and locally uniform rotundity results are positive for \(p\geq 1\), and smoothness usually works well for \(p>1\). For \(p=1\), there is no way to expect transferring smoothness -- the \(\|\cdot\|_1\) norm on \(L^1(\mu)\) is not smooth, and \(L_1(\mu)\) embeds in \(L_1(\mu,X)\). It is natural then to ask for the existence of equivalent renormings of \(L_1(\mu,X)\) preserving smoothness properties of the norm on \(X\).NEWLINENEWLINEIn the present paper, positive answers to this question are given for the case of Gâteaux (resp., uniformly Gâteaux) smoothness on \(L^{1}(\mu, X)\) (Theorem 2.1), and for Fréchet (resp., uniformly Fréchet) smoothness on every reflexive subspace of \(L^1(\mu,X)\) (Theorem 3.1) -- observe that a complete analogue to Theorem 2.1 makes no sense in this case.NEWLINENEWLINETo be precise, the first result shows that, if \((X,\|\cdot\|)\) is Gâteaux (uniformly Gâteaux) smooth, then \(L^1(\mu,X)\) has an equivalent Gâteaux (resp., uniformly Gâteaux) smooth norm, and the second that, if \((X,\|\cdot\|)\) is Fréchet (uniformly Fréchet) smooth, then \(L^1(\mu,X)\) has an equivalent norm whose restriction to every reflexive subspace is Fréchet (resp., uniformly Fréchet) smooth.NEWLINENEWLINEThe defined equivalent norms on \(L^1(\mu,X)\) are of Orlicz-Bochner type. The authors mention that the parenthetic part in both theorems follows indirectly from [\textit{M. Fabian} et al., Isr. J. Math. 124, 243--252 (2001; Zbl 1027.46012)] for the first result, and from [\textit{T. Figiel} and \textit{G. Pisier}, C. R. Acad. Sci., Paris, Sér. A 279, 611--614 (1974; Zbl 0326.46007)] and [\textit{M. Fabian} et al., Rocky Mt. J. Math. 39, No. 6, 1885--1893 (2009; Zbl 1190.46011)] for the second.