Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper, we investigate the continuity of a class of infinite-order differential operators acting on spaces of entire hyperholomorphic functions. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different from each other, in both cases, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite-order differential operators acting on these two classes of functions. This is particularly remarkable since the exponential function is not in the kernel of the Dirac operator, but it plays an important role in the theory of entire monogenic functions with growth conditions.

无限次微分算子出现在数学和物理学的不同领域，在过去的十年中，它们被证明对于作为Schrödinger方程的初始基准的超振动演化的研究至关重要。在量子力学中产生的算符的启发下，本文研究了一类在整个超亚纯函数空间上作用的无穷微分算符的连续性。我们将考虑两类超亚纯函数，它们都是一个复杂变量的全纯函数的自然扩展。我们证明，即使这两种超亚纯函数的概念彼此完全不同，但在两种情况下，具有指数界的整个超亚纯函数在作用于这两类函数的无穷微分算子的连续性中起着至关重要的作用。这是特别引人注目的，因为指数函数不在Dirac算子的核中，而是在具有生长条件的整个单基因函数的理论中起着重要作用。