New convexity and fixed point properties in Hardy and Lebesgue-Bochner spaces (Q1328331)

It is shown that for the Hardy class of functions \(H^ 1\) with domain the ball or polydisc in \(\mathbb{C}^ N\), a certain type of uniform convexity property (the uniform Kadec-Klee-Huff property) holds with respect to the topology of pointwise convergence on the interior, which coincides with both the topology of uniform convergence on compacta and the \(\text{weak}^*\) topology on bounded subsets of \(H^ 1\). Also, it is shown that if a Banach space \(X\) has a uniform Kadec-Klee-Huff property, then the Lebesgue-Bochner space \(L_ p(\mu,X)\), \(1\leq p< \infty\), must have a related uniform Kadec-Klee-Huff property. Consequently, by known results, the above spaces have normal structure properties and fixed point properties for non-expansive mappings.