The generalized Stieltjes–Poisson transform over Lebesgue spaces and distributions of compact support

This paper investigates the generalized Stieltjes–Poisson transform, deriving a real inversion formula and exploring its asymptotic behaviours over Lebesgue spaces. It further establishes a Parseval-type relations for this transform over Lebesgue spaces. We also obtain exchange formulae for the generalized Stieltjes–Poisson transform via Mellin-type convolutions over weighted Lebesgue spaces. Additionally, the study extends the transform to derive an inversion formula and analyze asymptotic behaviours over distributions of compact support.