$p$-adic Bochner–Riesz operators and their functional dynamics

In this article, we introduced and investigated two new families of $p$-adic pseudo-differential operators arising from Bochner–Riesz-type constructions in the non-Archimedean setting. The first family is given by $p$-adic Bochner–Riesz operators, while the second consists of operators whose symbols are naturally derived from and closely related to the Bochner–Riesz framework. These classes of operators provide new examples of nonlocal operators in $p$-adic analysis and offer a broader perspective on the role of radial symbols in the study of evolution processes over ultrametric spaces. Our main objective was to analyze the evolutionary dynamics generated by these operators through the construction and study of the associated convolution semigroups. In particular, we derived and investigated the corresponding evolution equations and examined the fundamental kernels that govern their solutions. Special attention was devoted to understanding how the different symbolic structures of the two operator families affect the analytical and qualitative properties of the resulting kernels, including their regularity, propagation behavior, and semigroup characteristics. The results revealed significant differences between the dynamics generated by the two classes of operators, highlighting the influence of the underlying symbols on the associated evolution processes. Furthermore, our approach establishes new connections between harmonic analysis on $p$-adic fields, pseudo-differential operator theory, and non-Archimedean evolution equations. These findings contribute to the development of $p$-adic analysis and provide a foundation for future applications in non-Archimedean mathematical physics, stochastic processes, and operator theory.