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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Autores principales:	Cerutti, Ruben Alejandro, Trione, Susana Elena
Publicado:	2000
Materias:	Generalized Riesz potentials Riesz derivatives of order α
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08939659_v13_n4_p129_Cerutti http://hdl.handle.net/20.500.12110/paper_08939659_v13_n4_p129_Cerutti
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

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spelling	paper:paper_08939659_v13_n4_p129_Cerutti2023-06-08T15:47:36Z Some properties of the generalized causal and anticausal Riesz potentials Cerutti, Ruben Alejandro Trione, Susana Elena 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. 2000 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08939659_v13_n4_p129_Cerutti http://hdl.handle.net/20.500.12110/paper_08939659_v13_n4_p129_Cerutti
institution	Universidad de Buenos Aires
institution_str	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, Ruben Alejandro Trione, Susana Elena 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.
author	Cerutti, Ruben Alejandro Trione, Susana Elena
author_facet	Cerutti, Ruben Alejandro Trione, Susana Elena
author_sort	Cerutti, Ruben Alejandro
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
publishDate	2000
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08939659_v13_n4_p129_Cerutti http://hdl.handle.net/20.500.12110/paper_08939659_v13_n4_p129_Cerutti
work_keys_str_mv	AT ceruttirubenalejandro somepropertiesofthegeneralizedcausalandanticausalrieszpotentials AT trionesusanaelena somepropertiesofthegeneralizedcausalandanticausalrieszpotentials
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Some properties of the generalized causal and anticausal Riesz potentials

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