On Continuously Differentiable Vector-Valued Functions of Non-Integer Order

TL;DR

The paper addresses the problem of defining and analyzing spaces of continuously fractional differentiable functions of order $α>0$ using the Riemann-Liouville and Caputo derivatives and relating them to Hölder spaces. It develops a rigorous framework by defining the spaces $RL^α([t_0,t_1];X)$ and $C^α([t_0,t_1];X)$, establishing their basic properties, and detailing how they embed into Hölder-type spaces, including equality only at integer orders and strict inclusions otherwise. It also characterizes the interplay between RL and Caputo derivatives, proves Hardy–Littlewood-type inclusions, and demonstrates the Banach-algebra structure of these spaces (and their vector-valued versions) under multiplication, with appropriate product rules. Collectively, these results provide a solid analytical foundation for fractional differentiability, enabling further study and applications in fractional differential equations and related areas.

Abstract

The function spaces of continuously differentiable functions are extensively studied and appear in various mathematical settings. In this context, we investigate the spaces of continuously fractional differentiable functions of order $α>0$, considering both the Riemann-Liouville and Caputo fractional derivatives. We explore several fundamental properties of these spaces and, inspired by a result of Hardy and Littlewood, we compare them with the space of Hölder continuous functions. Our main objective is to establish a rigorous theoretical framework to support the study and further advancement of this subject.

Theorems & Definitions (43)

• Definition 1
• Remark 2
• Proposition 3
• proof
• Theorem 4
• proof
• Definition 5
• Remark 6
• Proposition 7
• proof