Oscillatory Singular Special Function Transform

We introduce a new class of integral transforms, called the Oscillatory Singular Special Function Transform (OSSFT), whose kernels combine algebraic singularities, nonlinear oscillatory phases, and special-function components of Mittag–Leffler type. Fundamental properties of the OSSFT are established, including boundedness on weighted Lebesgue spaces, stability, compactness, and smoothing effects. Under suitable symmetry and decay conditions, a Plancherel-type theorem and Heisenberg-type uncertainty inequalities are proved. The compactness of the associated operators further yields spectral discreteness and Sobolev regularity of eigenfunctions.