Research Article Fractional Operators in p-adic Variable Exponent Lebesgue Spaces and Application to p-adic Derivative Leonardo Fabio Chacón-Cortés 1 and Humberto Rafeiro 2 1 Departamento de Matemáticas, Facultad de Ciencias, Pontificia Universidad Javeriana, Cra. 7 No. 43-82, Bogotá, Colombia 2 Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, UAE Correspondence should be addressed to Humberto Rafeiro; hrafeiro@gmail.com Received 3 June 2021; Accepted 7 August 2021; Published 18 September 2021 Academic Editor: Douadi Drihem Copyright © 2021 Leonardo Fabio Chacón-Cortés and Humberto Rafeiro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p-adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the p-adic variable exponent Lebesgue spaces. 1. Introduction The field of p-adic numbers are an interesting and useful tool to study phenomena in physics, biology, and medicine, among other sciences; see, e.g., [1–4] and references therein. For this reason, the study of operators that allows us to describe such phenomena is essential. Even more so when in the p-adic setting it is not possible to define the derivative in the classical sense. Variable exponent Lebesgue spaces generalize the notion of q-integrability in the classical Lebesgue spaces, allowing the exponent to be a measurable function. These spaces were introduced in 1931 by Orlicz [5] but lay essentially dormant for more than 50 years. They received a thrust in the paper [6] and are now an active area of research having many known applications, e.g., in the modeling of thermorheologi- cal fluids [7] as well as electrorheological fluids [8–11], in differential equations with nonstandard growth [12, 13], and in the study of image processing [14–20]. For a thor- ough history, theory, and applications of variable exponent Lebesgue spaces, see [6, 21–24]. In this article, we are interested in the boundedness of the fractional integral and maximal fractional operator on the p-adic Lebesgue spaces with a variable exponent. The corresponding result for classical p-adic Lebesgue space is known (cf. [25]). These operators play an important role in such areas such as Sobolev spaces, potential theory, PDEs, and integral geometry, to name a few. This work is divided as follows. Section 2 contains a quick description of the preliminary on the topic of the p-adic analysis and variable exponent Lebesgue spaces on the p-adic numbers, necessary for the development of this work. In Section 3, the boundedness of the fractional maximal operator M α fx ðÞ = sup γ∈ℤ 1 p γ n−α ð Þ ð B n γ x ðÞ fy ðÞ j jdy, x ∈ ℚ n p , ð1Þ is studied in the framework of variable exponent p-adic Lebesgue spaces. The boundedness of the special case M 0 , the so-called Hardy-Littlewood maximal operator, was obtained in [26] under appropriate conditions on the expo- nent function. We prove, using a suitable pointwise estimate, the boundedness of the fractional maximal operator from L qð·Þ ðℚ n p Þ to L q # ð·Þ ðℚ n p Þ, where q # is the Sobolev limiting exponent; see (31) for the corresponding definition. The boundedness of the fractional integral operator I α fx ðÞ = ð ℚ n p fy ðÞ x − y k k n−α p dy, x ∈ ℚ n p , ð2Þ is obtained from the boundedness of the fractional maximal operator and Welland’s pointwise inequality tailored for the Hindawi Journal of Function Spaces Volume 2021, Article ID 3096701, 9 pages https://doi.org/10.1155/2021/3096701