Weakly absolutely continuous functions without weak, but fractional weak derivatives - Salem, Hussein A. H. (Journal of pseudo-differential operators and applications, 2019)
Let E be an infinite-dimensional Banach space and I be a compact interval of the real line. The aim of this paper is two-fold: On the one hand, we construct an example of a weakly absolutely continuous function taking its values in E that is nowhere weakly differentiable on I , but has weakly continuous fractional weak derivatives of some critical orders less than one. This also holds for (nearly) all orders less than one if E failing cotype. We believe that this results are of independent interest and discuss it in a rather general setting. On the other hand, we establish some examples of weakly continuous functions taking its values in Gauge space fail to be pseudo differentiable on I , but have fractional-pseudo derivatives of “all” order less than one. An application will be given.