Some properties of the generalized causal and anticausal Riesz potentials
In this note, we study the causal (anticausal) generalized Riesz potential of order α: Rα Cf (Rα Af) of the function f ∈ S (cf. (1.8) and (1.9), respectively). The distributional functions Rα Cf (Rα Af) are causal (anticausal) analogues of the α-dimensional potentials in the ultrahyperbolic space de...
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todo:paper_08939659_v13_n4_p129_Cerutti2023-10-03T15:41:50Z Some properties of the generalized causal and anticausal Riesz potentials Cerutti, R.A. Trione, S.E. Generalized Riesz potentials Riesz derivatives of order α In this note, we study the causal (anticausal) generalized Riesz potential of order α: Rα Cf (Rα Af) of the function f ∈ S (cf. (1.8) and (1.9), respectively). The distributional functions Rα Cf (Rα Af) 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 (Dα Cf)(x) = (1/dn,ℓ(α))(Tα ℓf)C(x), α ε ℂ, ℓ is a nonnegative integer, ℓ > α > 0 and α ≠ 1, 2, 3, . . . , where dn,ℓ(α) and (Tα ℓf)C(x) are given by (2.4) and (2.1), respectively. Theorem 2 expresses that Dα CRβ A = MU-α+β C + NU-α+β A, where Uα C,A = Φα C,A*f, Φα C,A = rα-n ±/Cn(α); Theorem 3 says that Rα C R-2k Af = Rα-2kCf, α ≠ n + 2r, r = 0, 1, . . . . Similarly, we have Theorem 4: Rα AR-2k Cf = Rα-2k Cf, a ≠ n + 2r, r = 0, 1, . . . . Theorem 5 expresses that (cf. (3.5)) Rα C(Rβ Af) + Rα A(Rβ Cf) = K1Rα+βCf + K2Rα+β Af, f ∈ S. Finally, Theorem 6 expresses that the following formula is valid: Dα C(Dα Af) + Dα A(Dβ Cf) = C1Dα+β Cf + C2Dα+β Af, where C1and C2 appear in (3.13). © 2000 Elsevier Science Ltd. All rights reserved. Fil:Cerutti, R.A. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Trione, S.E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_08939659_v13_n4_p129_Cerutti |
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Universidad de Buenos Aires |
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I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Generalized Riesz potentials Riesz derivatives of order α |
| spellingShingle |
Generalized Riesz potentials Riesz derivatives of order α Cerutti, R.A. Trione, S.E. Some properties of the generalized causal and anticausal Riesz potentials |
| topic_facet |
Generalized Riesz potentials Riesz derivatives of order α |
| description |
In this note, we study the causal (anticausal) generalized Riesz potential of order α: Rα Cf (Rα Af) of the function f ∈ S (cf. (1.8) and (1.9), respectively). The distributional functions Rα Cf (Rα Af) 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 (Dα Cf)(x) = (1/dn,ℓ(α))(Tα ℓf)C(x), α ε ℂ, ℓ is a nonnegative integer, ℓ > α > 0 and α ≠ 1, 2, 3, . . . , where dn,ℓ(α) and (Tα ℓf)C(x) are given by (2.4) and (2.1), respectively. Theorem 2 expresses that Dα CRβ A = MU-α+β C + NU-α+β A, where Uα C,A = Φα C,A*f, Φα C,A = rα-n ±/Cn(α); Theorem 3 says that Rα C R-2k Af = Rα-2kCf, α ≠ n + 2r, r = 0, 1, . . . . Similarly, we have Theorem 4: Rα AR-2k Cf = Rα-2k Cf, a ≠ n + 2r, r = 0, 1, . . . . Theorem 5 expresses that (cf. (3.5)) Rα C(Rβ Af) + Rα A(Rβ Cf) = K1Rα+βCf + K2Rα+β Af, f ∈ S. Finally, Theorem 6 expresses that the following formula is valid: Dα C(Dα Af) + Dα A(Dβ Cf) = C1Dα+β Cf + C2Dα+β Af, where C1and C2 appear in (3.13). © 2000 Elsevier Science Ltd. All rights reserved. |
| format |
JOUR |
| author |
Cerutti, R.A. Trione, S.E. |
| author_facet |
Cerutti, R.A. Trione, S.E. |
| author_sort |
Cerutti, R.A. |
| title |
Some properties of the generalized causal and anticausal Riesz potentials |
| title_short |
Some properties of the generalized causal and anticausal Riesz potentials |
| title_full |
Some properties of the generalized causal and anticausal Riesz potentials |
| title_fullStr |
Some properties of the generalized causal and anticausal Riesz potentials |
| title_full_unstemmed |
Some properties of the generalized causal and anticausal Riesz potentials |
| title_sort |
some properties of the generalized causal and anticausal riesz potentials |
| url |
http://hdl.handle.net/20.500.12110/paper_08939659_v13_n4_p129_Cerutti |
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AT ceruttira somepropertiesofthegeneralizedcausalandanticausalrieszpotentials AT trionese somepropertiesofthegeneralizedcausalandanticausalrieszpotentials |
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1807324599985111040 |