Geometric properties of Banach space valued Bochner-Lebesgue spaces with variable exponent (Q2855184)

In variable exponent Lebesgue spaces, the constant exponent \(p\) in the classical Lebesgue spaces is replaced with a variable exponent \(p(y)\). These spaces where introduced already in the 1930s by Orlicz, but applications sparked renewed interest in the early 1990s. NEWLINENEWLINENEWLINEIn this paper, Banach space-valued Bochner-Lebesgue and Bochner-Sobolev spaces with variable exponents are defined.NEWLINENEWLINELet \((A,\mathcal{A},\mu)\) be a \(\sigma\)-finite complete measure space. The set of variable exponents is the set \(\mathcal{P}(A,\mu)\) of all \(\mu\)-measurable functions \(p: A \to [1,\infty]\). For \(p \in \mathcal{P}(A,\mu)\), let \(p^-\) and \(p^+\) denote the essential infimum and supremum of \(p\) over \(A\).NEWLINENEWLINEFor a Banach space \(E\) and \(p \in \mathcal{P}(A,\mu)\), the Bochner-Lebesgue space with variable exponent \(L^{p(\cdot)}(A,E)\) is the collection of all strongly \(\mu\)-measurable functions \(f:A \to E\) with norm NEWLINE\[NEWLINE \|f\|_{L^{p(\cdot)}(A,E)} = \inf \{ \lambda > 0 : \rho_p(f/\lambda) \leq 1\} < \infty, NEWLINE\]NEWLINE where \(\rho_p(f) = \int_A \|f(y)\|^{p(y)} \; d\mu(y)\).NEWLINENEWLINEThe authors show that \(L^{p(\cdot)}(A,E)\) is a Banach space. Furthermore, if \(1 < p^- \leq p^+ < \infty\), then \(L^{p(\cdot)}(A,E)\) is reflexive whenever \(E\) is, \(L^{p(\cdot)}(A,E)\) is uniformly convex whenever \(E\) is, and \(L^{p(\cdot)}(A,E)\) is uniformly smooth whenever \(E\) is. Similar results for Bochner-Sobolev spaces are also shown.