Boundedness of Erdélyi--Kober Integrals and Mellin Fractional Integrals on Weighted Lebesgue Spaces

What the math gives to ML

The paper defines a fractional integral whose useful transferable structure is a causal convolution in logarithmic scale, rather than ordinary additive coordinates. After setting x=e^s and t=xe^{-u}, the Erdelyi-Kober operator becomes a one-sided log-scale convolution with a kernel whose fractional order controls near-scale behavior and whose power parameter controls long-range decay. This suggests an interpretable, scale-equivariant mixer for feature pyramids or multiscale tokens, using a normalized analytic kernel instead of dense cross-scale attention.