On continuously differentiable vector-valued functions of non-integer order

Abstract

In this paper, we study spaces of vector-valued functions with continuous Riemann–Liouville or Caputo fractional derivatives of order $\alpha >0$. Despite the absence of a classical product rule for these derivatives, we prove that these spaces form Banach algebras, mirroring the classical case $\alpha =1$. Our work offers a comprehensive comparison with classical Hölder spaces and introduces several new contributions: we provide a sharp characterization of fractional differentiability via Hölder-type regularity; we establish optimal continuous embeddings between fractional spaces of different orders; and we describe in detail the structural differences between the Riemann–Liouville and Caputo frameworks. A cornerstone of our results is a vector-valued extension of the classical Hardy–Littlewood theorem, which establishes new inclusion criteria of Hölder spaces into fractional differentiability spaces. Many of our proofs rely on fine integral representations and delicate regularity estimates that, to the best of our knowledge, are entirely new even in the scalar-valued setting.