A Fractional Microlocal Convolution–Projection Operator on Distribution Spaces and Its Geometric Regularization Properties

This paper presents a new class of operators, denoted by MΓα,β, that act on distributions in D′(Ω) for Ω ⊆Rn.These operators combine fractional differentiation of arbitrary positive order α, convolution against a microlo-cally defined kernel Kβ tuned by a parameter β >0, and a geometric projection associated with a smoothlyembedded submanifold Γ ⊂Ω. The resulting construction yields operators with remarkable smoothing andlocalization properties, continuity in distribution spaces, and well-defined Fourier characterizations. Weestablish existence and uniqueness results for solutions of an integral equation involving MΓα,β, and we analyzetheir spectral properties. A detailed numerical experiment applied to a Gaussian test function demonstratessignificant enhancement of high-frequency components, anisotropy aligned with Γ, and selective directionalfeature extraction. These findings thus provides contributions to advanced analytic and computational toolsthat blend harmonic analysis, microlocal analysis, and fractional calculus into a single elegant framework.