The Sneddon ℛ-Transform and Its Inverse over Lebesgue Spaces

We study the Sneddon R-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel xcos(xt)+hsin(xt), the R-transform acts as a unifying operator for sine- and cosine-type integral transforms. Boundedness, continuity, and weighted Lp-estimates are established in an appropriate Banach space framework, together with Parseval–Goldstein type identities. Initial and final value theorems are derived for generalized functions in Zemanian-type spaces, yielding precise asymptotic behaviour at the origin and at infinity. A finite-interval theory is also developed, leading to polynomial growth estimates and final value theorems for the finite R-transform.