Towards a geometric interpretation of generalized fractional integrals - Erdelyi-Kober type integrals on $R^N$ as an example

A family of generalized Erdelyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on $R^N$. Based on this geometric view, several extensions are discussed.