Hardy–Hilbert Inequalities in Weighted Morrey Spaces: Commutators and Applications to Fractional PDEs on Bounded Domains for Financial Analysis

In this paper, we develop Hardy–Hilbert-type integral estimates in weighted Morrey spaces on bounded domains and study the associated commutators with BMO symbols. The analysis is positioned relative to existing weighted Morrey estimates for fractional and rough operators, and the hypotheses needed for the fractional-integral reduction are stated explicitly. We then derive a priori estimates for space–time fractional partial differential equations and examine a fractional-type Black–Scholes problem as a model application. The financial component is used to illustrate the role of localized weighted regularity: The Morrey parameter records how the size of a perturbation is controlled across spatial scales, while the weight permits nonuniform emphasis of different log-price regions; neither parameter is identified with a market primitive such as memory or volatility by definition. Finally, we formulate a localized discrete Morrey error functional for the numerical approximation, so that the computational discussion is consistent with the functional setting of the analysis.