Some properties of the generalized causal and anticausal Riesz potentials Original Research Article

In this note, we study the causal (anticausal) generalized Riesz potential of order α: RCαf (RAαf) of the function f ∈ S (cf. (1.8) and (1.9), respectively). The distributional functions RCαf (RAαf) are causal (anticausal) analogues of the α-dimensional potentials in the ultrahyperbolic space defined by Nozaki (cf. [1, p. 85]). Therefore, we define the generalized causal (anticausal) Riesz derivative of order α of a function α by the formula View the MathML source is a nonnegative integer, l > α > 0 and α ≠ 1,2,3,…, where dn,l(α) and (Tlαf)C(x) are given by (2.4) and (2.1), respectively. Theorem 2 expresses that DCαRAβf = MUC−α+β + NUA−α+β, where View the MathML source, View the MathML source; Theorem 3 says that RCαRA−2kf = RCα−2kf, α ≠ n + 2r, r = 0,1,…. Similarly, we have Theorem 4: RAαRC−2kf = RCα−2kf, α ≠ n + 2r, r = 0,1… Theorem 5 expresses that (cf. (3.5)) RCα(RAβf) + RAα(RCβf) = K1RCα+βf + K2RAα+βf, f ∈ S. Finally, Theorem 6 the following formula is valid: DCα(DAαf) + DAα(DCβf) = C1DCα+βf + C2DAα+βf, where C1 and C2 appear in (3.13).