Causal (anticausal) Bessel derivative and the ultrahyperbolic Bessel operator

Abstract

Let B^α_C and B^α_A be ultrahyperbolic Bessel operator causal (anticausal) of the order α defined by B^α_C f = G_α ( P + i0, m, n) ∗ f , B^α f = G_α ( P −i0, m, n) ∗ f and let D^α_C and D^α_A be generalized causal (anticausal) Bessel derivative of order α defined by D^α_C f = G_{−α} ( P − i0, m, n) ∗ f , D^α_A f =G_{−α} ( P + i0, m, n) ∗ f . In this note we give a sense to several relations of type:
                              B^α_C (B^β_ A f ) + B^α_A (B^β_C f ),

                               D^ α_C (D^β_ A f ) + D^α_A (D^β_C f ),
                              
                                              .  .  .