Fractional integral transforms associated with generalized Mittag-Leffler functions: Abelian theorems, Tauberian conditions, and applications to fractional differential equations

A unified class of fractional integral transforms whose kernels involve the three-parameter Mittag–Leffler (Prabhakar) function is studied in a distributional setting. The operators encompass, as special cases, the Riemann–Liouville, Erdélyi–Kober and Weyl fractional integrals. Continuity on Zemanian-type test-function spaces is proved via a Mellin–Barnes representation of the convolution, with the two standing hypotheses on the kernel (absolute integrability and Mellin-transform decay) established from the parameters rather than assumed. Abelian initial- and final-value theorems are proved in full; in particular we show that the Prabhakar kernel has total mass identically zero whenever it is integrable, which sharpens the final-value theorem considerably. A Tauberian converse is derived under Karamata’s classical theorem, with the exact asymptotic exponent and constant computed and checked against the Riemann–Liouville limit. The theory is applied to distributed-order relaxation equations; for a genuinely distributed Beta-density order weight, the long-time relaxation is shown to be logarithmic (ultraslow), consistent with the literature on continuously distributed fractional order. Numerical experiments, with full reproducibility details, corroborate the analytic results.