A Two-Parameter Memory-Weighted Velocity Operator for Time and State Variables: Foundations and Fundamental Properties

We introduce and analyze a memory-weighted velocity operator V_{{\alpha},\b{eta}} within the framework of operator theory, establishing rigorous theoretical results for systems with time-varying memory. The operator employs two independent continuous exponents {\alpha}(t) and \b{eta}(t) that separately weight past state increments and elapsed time scaling. This decoupling mechanism is the main novelty of the work: it allows the two memory aspects to evolve independently, which is essential for systems where the influence of past states and the perception of time may change in qualitatively different ways. Such situations arise naturally in adaptive materials, non-stationary transport, and evolving memory processes.
Motivated by systems across multiple physical contexts-such as viscoelastic materials with stress-dependent relaxation or anomalous transport with history-dependent characteristics-the framework addresses memory aspects that may evolve differently across disciplines.
We establish the operator's foundational properties: an explicit integral representation, linearity, and continuous dependence on the memory exponents with respect to uniform convergence. Central to the analysis are weighted pointwise estimates revealing how the exponent difference \b{eta}(t)-{\alpha}(t) modulates the operator, and we establish weighted boundedness estimates for this linear operator. These estimates exhibit a natural compensation mechanism between the two memory weightings.