Abstract
The Jacobi polynomials induce a translation operator on function spaces on the interval [β 1, 1]. For any homogeneous Banach space B w.r.t. this translation, we can study the according little and big Lipschitz spaces, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathop {\rm lip}\nolimits _B(\lambda )$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathop {\rm Lip}\nolimits _B(\lambda ),$\end{document} respectively. The big Lipschitz spaces are not homogeneous themselves.
Therefore we introduce semihomogeneous Banach spaces w.r.t. Jacobi translation, of which the big Lipschitz spaces are particular examples. We study the relation between semihomogeneous Banach spaces and their homogeneous counterparts. We give a characterisation of Lipschitz spaces in terms of intermediate spaces. Our main result is that, for an arbitrary homogeneous Banach space B, the bidual of the little Lipschitz space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathop {\rm lip}\nolimits _B(\lambda )$\end{document} is the corresponding big one, namely \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathop {\rm Lip}\nolimits _B(\lambda ).$\end{document}