ON THE COMPOSITION FORMULAS OF THE SOLUTIONS OF THE ULTRAHYPERBOLIC AND KLEIN-GORDON OPERATORS

Susana Trione
Reviata de la Uni6n Matematic:a Argentina   unpublished
Let t = (tl,t2,"" t a) be a point of fl!l. We shall write t~+ ... +t!-t!+l-" .-t!+ .. = 2y1 u, p. + 11 = n. We put, by definition, Ra(u) = ;n(O); here 0 is a complex parameter, n the dimension of the space and the constant ](n(o) is defined by (1,2). Ro(u) is the Marcel Riesz' ultrahyperbolic kernel. The distributional kernel Ro(u) share many properties with the Riemann-Liouville kernel of which they are n-dimensional ultrahyperbolic analogues. In this pa.per we prove the following composition
more » ... lowing composition formula: Ro * Rp(u) = Ro+p(u), 0 and P E C (d. form. (II,6». We remark that this formula has been proved by Nogin (d. [1]), by a completely different method. In paragraph III we put, by definition, (m-2u)~ 2 t Wo(u,m) = !!yl~ Ja-n(m u) ; 71'2 r(i)'-here 0 is .a complex parameter, m a real nonnegative number and n the dimension ot th~ space. Wo(u, m), which is an ordinary function if Re a ~ n, is an entire distributional function of o. Wo(u, m) is the ultrahyperbolic solution of the Klein-Gordon operator. We shall evaluate the composition formula Wo * Wp(u,m) = Wo+p(u,m), o,p E C. (d. form. (IV ,6». The particular case p. = 1 of this formula was proved in [2] by a completely different manner. I. The Marcel Riesz' ultrahyperbolic kernel. Let t = (t1. t2, ... ,tn) be a point of Rn. We shall write t~ + ... + t!-t!+l- ...-t!+ .. = u, p. + 11 = n. By r + we designate the interior of the forward cone: r + = {t E R n /t1 > 0, u > OJ. We consider the following family of functions introduced by Nozaki (d. [3], p. 72):
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