Weakly absolutely continuous functions without weak, but fractional weak derivatives

Abstract

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.

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Notes

  1. 1.

    The typewriter sequence

    $$\begin{aligned} \psi _n(t)=\chi _{\left[ \frac{n-2^k}{2^k},\frac{n-2^k+1}{2^k}\right] }\text{, }\quad k\ge 0 \text { being the unique integer with }2^k\le n<2^{k+1}, \end{aligned}$$

    has this property. Actually, \(\{ \psi _n\}\) is a sequence of indicator functions of intervals of decreasing length, marching across the unit interval [0,1] over and over again.

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Correspondence to Hussein A. H. Salem.

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Salem, H.A.H. Weakly absolutely continuous functions without weak, but fractional weak derivatives. J. Pseudo-Differ. Oper. Appl. 10, 941–954 (2019). https://doi.org/10.1007/s11868-019-00274-6

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Keywords

  • Fractional calculus
  • Orlicz spaces
  • Pettis integrals

Mathematics Subject Classification

  • 26A33
  • 34G20